{"id":1630,"date":"2024-10-20T00:18:46","date_gmt":"2024-10-19T15:18:46","guid":{"rendered":"https:\/\/xn--k10aa.com\/?p=1630"},"modified":"2024-12-21T18:41:48","modified_gmt":"2024-12-21T09:41:48","slug":"plt04","status":"publish","type":"post","link":"https:\/\/remoooo.com\/en\/plt04\/","title":{"rendered":"Wave Optics Rendering: Full Wave Reference Simulation - Study Notes - 4"},"content":{"rendered":"\n<p>\u4eca\u5929\u7ee7\u7eed\u6765\u7c97\u7565\u770b\u770bSig23\u8fd9\u7bc7\u8ba1\u7b97\u8868\u9762\u53cd\u5c04\u7684\u5168\u6ce2\u53c2\u8003\u6a21\u62df\u5668paper\u3002<\/p>\n\n\n\n<p>\u5173\u952e\u8bcd\uff1a\u56fe\u5f62\u5b66\u5165\u95e8\u3001\u6ce2\u52a8\u5149\u5b66\u6e32\u67d3\u3001BEM\u3001AIM\u3001BRDF<br>Wave Optics Render, Full Wave Reference Simulator<\/p>\n\n\n\n<p class=\"has-text-align-center\">\u539f\u6587\uff1a<a href=\"https:\/\/zhuanlan.zhihu.com\/p\/1471147574\">https:\/\/zhuanlan.zhihu.com\/p\/1471147574<\/a><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u524d\u8a00<\/h2>\n\n\n\n<p>\u7a0d\u5fae\u603b\u7ed3\u4e00\u4e0b\u76ee\u524d\u7684\u7406\u89e3\u3002<\/p>\n\n\n\n<p>\u57fa\u4e8e\u6ce2\u52a8\u5149\u5b66\u7684\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u5bf9\u56fe\u5f62\u5b66\u7814\u7a76\u8005\u800c\u8a00\u65e0\u7591\u662f\u5de8\u5927\u7684\u6311\u6218\u3002\u5b83\u4e0d\u4ec5\u8981\u5728\u8ba1\u7b97\u4e0a\u5904\u7406\u7535\u78c1\u6ce2\uff0c\u8fd8\u6d89\u53ca\u4e00\u4e9b\u91cf\u5b50\u529b\u5b66\u7684\u7406\u8bba\uff0c\u60f3\u8981\u6253\u901a\u7269\u7406\u5b66\u548c\u56fe\u5f62\u5b66\u4e4b\u95f4\u7684\u9694\u9602\u5c5e\u5b9e\u4e0d\u6613\u3002\u5728\u57fa\u4e8e\u6ce2\u52a8\u5149\u5b66\u7684\u56fe\u5f62\u5b66\u4e2d\uff0c\u5149\u5728\u4ecb\u8d28\u4e2d\u7684\u4f20\u64ad\u4e0d\u518d\u53ea\u662f\u76f4\u7ebf\uff0c\u800c\u662f\u4f1a\u56e0\u4e3a\u6ce2\u957f\u4e0d\u540c\uff0c\u8868\u73b0\u51fa\u81ea\u65cb\u3001\u504f\u6298\u548c\u884d\u5c04\u7b49\u591a\u79cd\u7279\u6027\u3002\u4ece\u80a5\u7682\u6ce1\u3001\u504f\u632f\u955c\u3001\u6cb9\u819c\u5149\u6cfd\u6216\u7cd6\u5206\u4ecb\u8d28\u5bfc\u81f4\u7684\u65cb\u5149\u518d\u5230\u65e0\u7ebf\u7535\u8f90\u5c04\u5982\u4f55\u5728\u57ce\u5e02\u5efa\u7b51\u4e4b\u95f4\u4f20\u64ad\uff0c\u90fd\u79bb\u4e0d\u5f00\u6ce2\u52a8\u5149\u5b66\u7406\u8bba\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-471-1024x597.png\" alt=\"\" class=\"wp-image-1631 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"597\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-471-1024x597.png\" alt=\"\" class=\"wp-image-1631 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-471-1024x597.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-471-300x175.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-471-768x448.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-471-1536x895.png 1536w, https:\/\/remoooo.com\/wp-content\/uploads\/image-471-2048x1193.png 2048w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u5728\u77e5\u4e4e\u4e0a\u4e5f\u6709\u5f88\u591a\u5927\u4f6c\u5206\u4eab\u4e86\u975e\u5e38\u8be6\u7ec6\u7684\u6ce2\u52a8\u5149\u5b66\u6e32\u67d3\u7406\u8bba\u57fa\u7840\u6559\u5b66\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/zhuanlan.zhihu.com\/p\/565692297\">\u6ce2\u52a8\u5149\u5b66\u6e32\u67d3\u65b9\u7a0b &#8211; Heskey0\u7684\u6587\u7ae0 &#8211; \u77e5\u4e4e<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/zhuanlan.zhihu.com\/p\/589291609\">\u9ad8\u4e2d\u751f\u90fd\u80fd\u770b\u61c2\u7684\u6ce2\u52a8\u5149\u5b66\u6e32\u67d3\uff08PLT\uff09 &#8211; Aegle\u7684\u6587\u7ae0 &#8211; \u77e5\u4e4e<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/zhuanlan.zhihu.com\/p\/660806855\">\u6ce2\u52a8\u5149\u5b66\u6e32\u67d3\uff08\u5165\u95e81\uff09\uff1a\u7b97\u6cd5\u8fc7\u7a0b\u6982\u8ff0 &#8211; \u7d22\u7f57\u7279\u7684\u6587\u7ae0 &#8211; \u77e5\u4e4e<\/a><\/li>\n<\/ul>\n\n\n\n<p>\u4f46\u662f\u5948\u4f55\u9605\u8bfb\u95e8\u69db\u90fd\u8f83\u9ad8\uff0c\u4e14\u8ddd\u79bb\u5b9e\u9645\u5e94\u7528\u592a\u8fdc\u3002\u6ce2\u52a8\u5149\u5b66\u6e32\u67d3\u6d89\u53ca\u7684\u6570\u5b66\u7269\u7406\u77e5\u8bc6\u592a\u591a\uff0c\u5c31\u7b97\u662f\u4e00\u884c\u4e00\u884c\u7ec6\u770b\u95eb\u8001\u5e08\u548c\u4ed6\u535a\u58eb\u751f\u7684paper\uff0c\u4e5f\u662f\u5bf8\u6b65\u96be\u884c\u3002<\/p>\n\n\n\n<p>\u56fe\u5f62\u5b66\u7684\u5723\u676f\u662f\u5149\u7ebf\u8ffd\u8e2a\uff0c\u800c\u5168\u5c40\u6ce2\u52a8\u5149\u5b66\u6e32\u67d3\u5219\u662f\u56fe\u5f62\u5b66\u5723\u676f\u4e2d\u7684\u5723\u676f\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-472-1024x577.png\" alt=\"\" class=\"wp-image-1632 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"577\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-472-1024x577.png\" alt=\"\" class=\"wp-image-1632 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-472-1024x577.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-472-300x169.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-472-768x433.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-472-1536x866.png 1536w, https:\/\/remoooo.com\/wp-content\/uploads\/image-472.png 1660w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u5728Sig21\u4e0a\uff0cShlomi\u63d0\u51fa\u4e86\u5c06\u8def\u5f84\u8ffd\u8e2a\u4e0e\u7269\u7406\u5149\u5b66\u76f8\u7ed3\u5408\u5b9e\u73b0\u7535\u78c1\u8f90\u5c04\u4f20\u64ad\u7684\u771f\u5b9e\u6a21\u62df\u3002\u76ee\u524d\u5df2\u7ecf\u6709\u4e0d\u5c11\u65b9\u6cd5\u8ba1\u7b97\u7535\u78c1\u6ce2\u7684\u4f20\u8f93\uff0c\u4ece\u9ad8\u7cbe\u5ea6\u4f46\u8ba1\u7b97\u5bc6\u96c6\u7684\u6ce2\u52a8\u6c42\u89e3\u5668\u5230\u5feb\u901f\u4f46\u4e0d\u591f\u7cbe\u786e\u7684\u51e0\u4f55\u5149\u5b66\u65b9\u6cd5\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u5c55\u793a\u4e86\u5f53\u524d\u7535\u78c1\u6ce2\u4f20\u8f93\u7684\u8ba1\u7b97\u65b9\u6cd5\u3002\u5de6\u8fb9\u6700\u51c6\u786e\uff0c\u53f3\u8fb9\u6700\u5feb\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-473-1024x307.png\" alt=\"\" class=\"wp-image-1633 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"307\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-473-1024x307.png\" alt=\"\" class=\"wp-image-1633 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-473-1024x307.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-473-300x90.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-473-768x230.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-473-1536x461.png 1536w, https:\/\/remoooo.com\/wp-content\/uploads\/image-473-2048x614.png 2048w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u6ce2\u52a8\u6c42\u89e3\u5668\uff08Wave Solvers\uff09\u4e13\u6ce8\u4e8e\u6c42\u89e3\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u7684\u7cbe\u786e\u89e3\uff0c\u4f46\u5bf9\u4e8e\u5927\u578b\u573a\u666f\u6765\u8bf4\u5e76\u4e0d\u5b9e\u7528\uff0c\u4e00\u822c\u7528FDTD\u3001BEM\u6216FEM\u6765\u505a\u3002<\/p>\n\n\n\n<p>PO\u57fa\u4e8e\u9ad8\u9891\u8fd1\u4f3c\u7684\u7535\u78c1\u6ce2\u8ba1\u7b97\uff0c\u4f46\u662f\u5bf9\u4e8e\u53ef\u89c1\u5149\u8fd9\u4e2a\u9891\u7387\u6765\u8bf4\u5176\u5b9e\u52c9\u5f3a\u591f\u7528\u4e86\u3002PS\uff1a[Xia 2023]\u7684\u9ed1\u72d7\u6bdb\u5c31\u5c5e\u4e8ePhysical Optics\u65b9\u6cd5\u3002\u800c\u672c\u6587\u7684\u5168\u6ce2\u53c2\u8003\u5e94\u8be5\u8fd8\u662f\u5c5e\u4e8eWave Solvers\u7684\u65b9\u5f0f\uff0c\u4f46\u662f\u5728BEM\u7684\u57fa\u7840\u4e0a\u4f7f\u7528\u4e86PO\u7684\u4e00\u4e9b\u601d\u60f3\uff08\u7b49\u6548\u7535\u6d41\u7b49\uff09\u3002<\/p>\n\n\n\n<p>\u9664\u4e86PO\uff0c\u8fd8\u6709\u4e00\u79cd\u65b9\u6cd5\u4ecb\u4e8eWave Solvers\u548cGeometrical Optics\u4e4b\u95f4\uff0c\u79f0\u4e3aHybrid GO-PO\uff0c\u6211\u4e2a\u4eba\u89c9\u5f97\u5e94\u8be5\u53eb\u505a\u51e0\u4f55\u5149\u5b66-\u7269\u7406\u5149\u5b66\u6df7\u5408\u65b9\u6cd5\u3002\u7edf\u4e00\u884d\u5c04\u7406\u8bba\uff08Uniform Theory of Diffraction, UTD\uff09\u5c06\u884d\u5c04\u6548\u5e94\u7eb3\u5165\u51e0\u4f55\u5149\u5b66\u6765\u8ba1\u7b97\u9ad8\u9891\u6761\u4ef6\u4e0b\u7684\u7535\u78c1\u6ce2\u4f20\u8f93\u3002\u4e2a\u4eba\u7406\u89e3\uff0cUTD\u901a\u8fc7\u8ba1\u7b97\u7ed5\u5c04\u7cfb\u6570\u6765\u8865\u507f\u51e0\u4f55\u5149\u5b66\u5c04\u7ebf\u8fb9\u754c\u6761\u4ef6\u7684\u4e0d\u8db3\uff0c\u4e5f\u5c31\u662f\u8bf4\u51e0\u4f55\u5149\u5b66\u7684\u5c04\u7ebf\u4e5f\u53ef\u4ee5\u8f6c\u5f2f\u4e86\u3002\u8fd9\u79cd\u64cd\u4f5c\u5728\u96f7\u8fbe\u63a2\u6d4b\u5929\u7ebf\u8bbe\u8ba1\u9886\u57df\u975e\u5e38\u5b9e\u7528\u3002\u9664\u4e86UTD\uff0cHybrid GO-PO\u8fd8\u6d89\u53ca\u4e00\u79cd\u53eb\u5c04\u7ebf\u53d1\u5c04\u548c\u53cd\u5f39\u65b9\u6cd5\uff08Shooting and Bouncing Rays, 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decoding=\"async\" width=\"1024\" height=\"858\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-475-1024x858.png\" alt=\"\" class=\"wp-image-1635 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-475-1024x858.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-475-300x251.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-475-768x643.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-475-1536x1286.png 1536w, https:\/\/remoooo.com\/wp-content\/uploads\/image-475.png 1796w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u89c6\u9891\u901a\u8fc7\u53f3\u65cb\u624b\u6027\u4ecb\u8d28\u4f1a\u4f7f\u5149\u201c\u65cb\u8f6c\u201d\u8fd9\u4e2a\u7279\u6b8a\u6ce2\u52a8\u5149\u5b66\u73b0\u8c61\uff0c\u79d1\u666e\u4e86\u4e00\u7cfb\u5217\u6ce2\u52a8\u5149\u5b66\u7684\u6709\u8da3\u73b0\u8c61\u4ee5\u53ca\u80cc\u540e\u7684\u539f\u7406\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-476-1024x440.png\" alt=\"\" class=\"wp-image-1636 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"440\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-476-1024x440.png\" alt=\"\" class=\"wp-image-1636 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-476-1024x440.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-476-300x129.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-476-768x330.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-476-1536x660.png 1536w, https:\/\/remoooo.com\/wp-content\/uploads\/image-476-2048x879.png 2048w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-477-1024x640.png\" alt=\"\" class=\"wp-image-1637 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"640\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-477-1024x640.png\" alt=\"\" class=\"wp-image-1637 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-477-1024x640.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-477-300x187.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-477-768x480.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-477-1536x960.png 1536w, https:\/\/remoooo.com\/wp-content\/uploads\/image-477.png 1738w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u975e\u5e38\u5f15\u4eba\u6df1\u601d\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-478-1024x501.png\" alt=\"\" class=\"wp-image-1639 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"501\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-478-1024x501.png\" alt=\"\" class=\"wp-image-1639 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-478-1024x501.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-478-300x147.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-478-768x376.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-478-1536x752.png 1536w, https:\/\/remoooo.com\/wp-content\/uploads\/image-478-2048x1003.png 2048w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-480-1024x499.png\" alt=\"\" class=\"wp-image-1641 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"499\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-480-1024x499.png\" alt=\"\" class=\"wp-image-1641 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-480-1024x499.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-480-300x146.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-480-768x374.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-480-1536x749.png 1536w, https:\/\/remoooo.com\/wp-content\/uploads\/image-480-2048x998.png 2048w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u6700\u540e\u751a\u81f3\u8fd8\u8bb2\u5230\u4e86\u5982\u4f55\u5229\u7528\u7269\u8d28\u6ce2\u6784\u5efa\u5168\u606f\u5f71\u50cf\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=QCX62YJCmGk\">This tests your understanding of light | Optics puzzles 1<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=aXRTczANuIs\">How wiggling charges give rise to light | Optics puzzles 2<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=KTzGBJPuJwM&amp;t=695s\">But why would light &#8220;slow down&#8221;? | Optics puzzles 3<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=Cz4Q4QOuoo8&amp;t=3s\">How well do you understand refraction? | Optics puzzles 4<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=EmKQsSDlaa4\">How are holograms possible? | Optics puzzles 5<\/a><\/li>\n<\/ul>\n\n\n\n<p>\u5728\u7ecf\u5178\u7535\u78c1\u573a\u7406\u8bba\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528<strong>\u5e73\u9762\u6ce2<\/strong>\u5c55\u5f00\u7535\u78c1\u573a\uff0c\u6bcf\u4e2a\u6a21\u5f0f\u7684\u5149\u5b50\u5728\u7a7a\u95f4\u4e0a\u662f<strong>\u975e\u5b9a\u57df<\/strong>\u7684\uff0c\u5177\u6709\u65e0\u9650\u7684\u7a7a\u95f4\u5ef6\u5c55\u3002\u8fd9\u79cd\u5c55\u5f00\u65b9\u5f0f\u4e0b\uff0c<strong>\u4e00\u4e2a\u5149\u5b50\u5904\u4e8e\u4e00\u4e2a\u9891\u7387\u548c\u6ce2\u77e2\u660e\u786e\u7684\u6a21\u5f0f\u4e0a<\/strong>\uff0c\u56e0\u6b64\u5b83\u5728\u7a7a\u95f4\u4e2d\u201c\u5145\u6ee1\u201d\u4e86\u6574\u4e2a\u5e73\u9762\u6ce2\u7684\u533a\u57df\u3002\u5bf9\u4e8e\u81ea\u7531\u7a7a\u95f4\u4e2d\u7684\u7535\u78c1\u573a\u6a21\u5f0f\u975e\u5e38\u5e38\u89c1\uff0c\u4f46\u8fd9\u79cd\u5e73\u9762\u6ce2\u5e76\u4e0d\u9002\u5408\u63cf\u8ff0\u5c40\u57df\u6027\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" 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\/><\/noscript><\/figure>\n\n\n\n<p>\u5728\u7ecf\u5178\u7535\u78c1\u7406\u8bba\u4e2d\uff0c\u4e00\u4e2a\u6ce2\u5305\u53ef\u4ee5\u5bf9\u5e94\u4e8e\u4e00\u4e2a\u80fd\u91cf\u96c6\u4e2d\u7684\u533a\u57df\u3002<\/p>\n\n\n\n<p>\u4f46\u662f\u6ce8\u610f\u5149\u5b50\u4e5f\u4e0d\u80fd\u5355\u7eaf\u7406\u89e3\u4e3a\u4e00\u79cd\u6ce2\u5305\uff0c\u800c\u662f\u7406\u89e3\u4e3a\u4e00\u79cd\u6982\u7387\u6ce2\u3002\u91cf\u5b50\u529b\u5b66\u4e2d\u7684<strong>\u6982\u7387\u6ce2\u51fd\u6570<\/strong>\u63cf\u8ff0\u4e86\u5149\u5b50\u7684\u5b58\u5728\u51e0\u7387\u3002\u6ce2\u5305\u8d8a\u96c6\u4e2d\uff0c\u7c92\u5b50\u6027\u5c31\u8d8a\u660e\u663e\u3002\u8fd9\u5c31\u662f\u91cf\u5b50\u7535\u52a8\u529b\u5b66\u89e3\u91ca\u7684\u6ce2\u7c92\u4e8c\u8c61\u6027\u3002\u4e00\u4e2a\u5149\u5b50\u5e76\u4e0d\u4e00\u5b9a\u53ea\u80fd\u5728\u4e00\u4e2a\u6ce2\u5305\u4e2d\uff0c\u5b83\u4e5f\u53ef\u4ee5\u63cf\u8ff0\u4e3a\u591a\u4e2a\u6ce2\u5305\u7684\u53e0\u52a0\u3002\u56e0\u4e3a\u5149\u5b50\u72b6\u6001\u672c\u8d28\u4e0a\u662f\u91cf\u5b50\u573a\u7684\u6fc0\u53d1\uff0c<strong>\u5141\u8bb8\u5728\u4e0d\u540c\u4f4d\u7f6e\u7684\u6ce2\u5305\u4e0a\u8fdb\u884c\u53e0\u52a0<\/strong>\u3002<\/p>\n\n\n\n<p>\u4e5f\u5c31\u662f\u8bf4\uff0c<strong>\u4e00\u4e2a\u5149\u5b50\u53ef\u4ee5\u201c\u8de8\u8d8a\u201d\u591a\u4e2a\u6ce2\u5305<\/strong>\uff0c\u5373\u5b83\u7684\u6ce2\u51fd\u6570\u53ef\u4ee5\u5728\u7a7a\u95f4\u4e0a\u4ee5\u591a\u4e2a\u6ce2\u5305\u7684\u5f62\u5f0f\u5b58\u5728\uff0c\u800c\u4e0d\u5c40\u9650\u4e8e\u67d0\u4e2a\u7279\u5b9a\u4f4d\u7f6e\u3002\u5f53\u4e00\u4e2a\u6ce2\u5305\u6a21\u5f0f\u4e0a\u6709\u4e00\u4e2a\u5149\u5b50\u65f6\uff0c\u8fd9\u4e2a\u6ce2\u5305\u53ef\u4ee5\u770b\u4f5c\u662f<strong>\u6700\u4f4e\u6fc0\u53d1\u6001<\/strong>\uff1b\u800c\u5982\u679c\u6ce2\u5305\u4e0a\u6709\u591a\u4e2a\u5149\u5b50\uff08\u9ad8\u9636\u6fc0\u53d1\u6001\uff09\uff0c\u5219\u4f1a\u4f53\u73b0\u51fa\u66f4\u9ad8\u7684\u80fd\u91cf\u3002<\/p>\n\n\n\n<p>\u672c\u6587\u4e2d\u63d0\u5230\u7684\u4e00\u79cd\u7535\u78c1\u6ce2\u675f<strong>\u9ad8\u65af\u5149\u675f<\/strong>\u662f\u4e00\u79cd\u7279\u5b9a\u7684\u7535\u78c1\u6ce2\u89e3\uff0c\u53ef\u4ee5\u89c6\u4e3a\u4e00\u79cd\u6ce2\u5305\u3002<strong>\u9ad8\u65af\u5149\u675f<\/strong>\u662f\u4e00\u4e2a\u5177\u6709\u7a33\u5b9a\u632f\u5e45\u548c\u76f8\u4f4d\u7ed3\u6784\u7684<strong>\u5355\u4e00\u6ce2\u5305<\/strong>\uff0c\u5b83\u5728\u6a2a\u622a\u9762\u4e0a\u8868\u73b0\u4e3a\u9ad8\u65af\u5206\u5e03\u3002\u4e0d\u540c\u4e8e\u6211\u4eec\u5e73\u5e38\u5728\u7ecf\u5178\u7535\u78c1\u7406\u8bba\u4e2d\u5e38\u7528\u7684\u5e73\u9762\u6ce2\u89e3\uff0c\u56e0\u4e3a\u9ad8\u65af\u5149\u675f\u7684\u6ce2\u524d\u662f\u66f2\u7387\u53d8\u5316\u7684\uff0c\u4e0d\u662f\u65e0\u9650\u5ef6\u4f38\u7684\u5e73\u9762\u3002<\/p>\n\n\n\n<p><strong>\u6ce2\u7684\u7cfb\u7efc\uff08ensemble of waves\uff09<\/strong>\u6307\u7684\u662f\u591a\u4e2a\u6ce2\u7684\u96c6\u5408\uff0c\u8fd9\u4e9b\u6ce2\u53ef\u80fd\u5177\u6709\u4e0d\u540c\u7684\u9891\u7387\u3001\u76f8\u4f4d\u6216\u4f20\u64ad\u65b9\u5411\u3002\u5982\u679c\u8003\u8651\u4e00\u4e2a\u7cfb\u7edf\u4e2d\u591a\u4e2a\u72ec\u7acb\u6ce2\u5305\uff08\u4f8b\u5982\u591a\u4e2a\u8109\u51b2\u6fc0\u5149\u675f\uff09\u76f8\u4e92\u53e0\u52a0\uff0c\u53ef\u4ee5\u5c06\u8fd9\u4e9b\u6ce2\u5305\u7406\u89e3\u4e3a\u4e00\u4e2a<strong>\u6ce2\u7684\u7cfb\u7efc<\/strong>\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5982\u679c\u6709\u591a\u4e2a\u4e0d\u76f8\u5173\u7684\u6ce2\u5305\uff0c\u5c24\u5176\u662f\u968f\u673a\u76f8\u4f4d\u7684\u6ce2\u5305\u53e0\u52a0\u5728\u4e00\u8d77\uff0c\u5b83\u4eec\u5728\u7edf\u8ba1\u4e0a\u53ef\u6784\u6210\u6ce2\u7684\u7cfb\u7efc\u3002<\/p>\n\n\n\n<p><p>\u5f53\u6211\u4eec\u628a\u7535\u78c1\u573a\u5c55\u5f00\u4e3a\u4e00\u7cfb\u5217\u6ce2\u5305\uff0c\u90a3\u4e48\u5c31\u53ef\u4ee5\u5c06\u6bcf\u4e2a\u6ce2\u5305\u89c6\u4e3a\u4e00\u4e2a\u968f\u673a\u4e8b\u4ef6\uff0c\u4ed6\u4eec\u7684\u5230\u8fbe\u65f6\u95f4\u3001\u76f8\u4f4d\u90fd\u662f\u968f\u673a\u53d8\u91cf\u3002\u5bf9\u4e8e\u591a\u4e2a\u6ce2\u5305\u7684\u96c6\u5408\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u4e0d\u540c\u65f6\u95f4\u3001\u4e0d\u540c\u4f4d\u7f6e\u89c2\u5bdf\u5230\u8fd9\u4e9b\u6ce2\u5305\u7684\u7279\u5f81\u3002\u5728\u7edf\u8ba1\u610f\u4e49\u4e0a\uff0c\u4f7f\u7528<strong>\u7cfb\u7efc\u5e73\u5747<\/strong>\u6765\u5206\u6790\u5149\u7684\u80fd\u91cf\u5206\u5e03\u548c\u6ce2\u52a8\u884c\u4e3a\u3002<\/p>\n<p><br>$$<br>\\langle U(\\vec{r}, t) \\rangle = \\lim_{T \\to \\infty} \\frac{1}{2T} \\int_{-T}^{T} U(\\vec{r}, t) \\, dt<br>$$<\/p><\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u7531\u4e8e\u4e0d\u662f\u7269\u7406\u58ec\uff0c\u5fae\u5143\u5199\u5728\u79ef\u5206\u5f0f\u524d\u6211\u662f\u4e0d\u8ba4\u53ef\u7684\uff08<\/p>\n<\/blockquote>\n\n\n\n<p>\u4e0a\u9762\u8bf4\u5230\uff0c\u5149\u53ef\u4ee5\u89c6\u4e3a\u591a\u4e2a\u6ce2\u5305\u7684\u96c6\u5408\u3002\u800c\u4e0d\u540c\u6ce2\u5305\u4e4b\u95f4\u53ef\u80fd\u4f1a\u5b58\u5728\u76f8\u4f4d\u5dee\u548c\u65f6\u95f4\u5dee\uff0c\u56e0\u6b64\u9700\u8981\u5f15\u5165<strong>\u4e92\u76f8\u5173\u51fd\u6570<\/strong>\u548c<strong>\u4e92\u76f8\u5e72\u51fd\u6570<\/strong>\u6765\u63cf\u8ff0\u4e24\u4e2a\u4e0d\u540c\u4f4d\u7f6e\u7684\u6ce2\u5305\u4e4b\u95f4\u7684\u76f8\u5e72\u6027\u548c\u76f8\u5bf9\u76f8\u4f4d\uff0c\u63d0\u4f9b\u4e86\u5bf9\u8fd9\u4e9b\u6ce2\u5305\u5728<strong>\u65f6\u7a7a\u4e0a\u76f8\u4f3c\u6027<\/strong>\u7684\u91cf\u5316\u63cf\u8ff0\u3002\u5982\u679c\u8bf4<strong>\u7cfb\u7efc\u5e73\u5747<\/strong>\u7528\u4e8e\u63cf\u8ff0\u4e00\u4e2a\u4fe1\u53f7\u5728\u7edf\u8ba1\u610f\u4e49\u4e0b\u7684\u5e73\u5747\u884c\u4e3a\uff0c\u90a3\u4e48<strong>\u4e92\u76f8\u5173\u51fd\u6570<\/strong>\u662f\u5728\u65f6\u95f4\u5e73\u5747\u7684\u89d2\u5ea6\u4e0a\uff0c\u63cf\u8ff0\u4e86<strong>\u4e0d\u540c\u4f4d\u7f6e\u548c\u4e0d\u540c\u65f6\u95f4\u4e4b\u95f4\u7684\u76f8\u5e72\u5173\u7cfb<\/strong>\u548c<strong>\u6ce2\u52a8\u5173\u8054\u6027<\/strong>\u3002<\/p>\n\n\n\n<p>\u5177\u4f53\u800c\u8a00\uff0c\u4e92\u76f8\u5173\u51fd\u6570\u63cf\u8ff0\u4e86\u4f4d\u7f6e $\\vec{r}_1$ \u548c $\\vec{r}_2$ \u4e4b\u95f4\u7684\u6ce2\u52a8\u76f8\u5e72\u6027\u3002\u5982\u679c\u4e24\u4e2a\u6ce2\u76f8\u5e72\uff0c\u90a3\u4e48\u8fd9\u4e2aGamma\u503c\u5c31\u4f1a\u6bd4\u8f83\u5927\uff1a<\/p>\n\n\n\n<p><br>$$<br>\\Gamma(\\vec{r}_1, \\vec{r}_2, \\tau) = \\langle U(\\vec{r}_1, t) U^*(\\vec{r}_2, t + \\tau) \\rangle<br>$$<\/p>\n\n\n\n<p><br>\u4ea4\u53c9\u8c31\u5bc6\u5ea6\uff08CSD\uff09\u77e9\u9635 $W(\\vec{r}_1, \\vec{r}_2, \\omega)$ \u662f\u4e92\u76f8\u5173\u51fd\u6570\u7684<strong>\u5085\u91cc\u53f6\u53d8\u6362<\/strong>\uff0c\u8868\u793a\u5728\u9891\u7387\u57df\u4e2d<strong>\u4e24\u4e2a\u4f4d\u7f6e\u4e4b\u95f4\u7684\u76f8\u5e72\u6027<\/strong>\u3002<\/p>\n\n\n\n<p>\u8fd8\u8bb0\u5f97\u6211\u4eec\u4e4b\u524d\u5728Xia2023\u90a3\u7bc7\u9ed1\u72d7\u6bdb\u4e2d\u8ba8\u8bba\u7684\u81ea\u76f8\u5173\u51fd\u6570\u5417\uff1f\u4e92\u76f8\u5173\u51fd\u6570\u9700\u8981\u4e24\u4e2a\u4fe1\u53f7\u6765\u63cf\u8ff0\uff0c\u800c\u81ea\u76f8\u5173\u51fd\u6570\u53ea\u6709\u4e00\u4e2a\u4fe1\u53f7\u3002\u53e6\u4e00\u4e2a\u89d2\u5ea6\u6765\u7406\u89e3\uff0c\u81ea\u76f8\u5173\u51fd\u6570\uff08ACF\uff09\u662f\u4e92\u76f8\u5173\u51fd\u6570\u7684\u4e00\u79cd\u7279\u6b8a\u60c5\u51b5\u3002\u81ea\u76f8\u5173\u51fd\u6570\u63cf\u8ff0\u7684\u662f<strong>\u4fe1\u53f7\u4e0e\u81ea\u8eab\u5728\u4e0d\u540c\u65f6\u523b\u6216\u4e0d\u540c\u4f4d\u7f6e\u7684\u76f8\u5173\u6027<\/strong>\uff0c\u800c\u4e92\u76f8\u5173\u51fd\u6570\u5219\u662f<strong>\u4e24\u4e2a\u4e0d\u540c\u4fe1\u53f7\u6216\u540c\u4e00\u4fe1\u53f7\u5728\u4e0d\u540c\u4f4d\u7f6e\u7684\u76f8\u5173\u6027<\/strong>\u3002\u4e92\u76f8\u5173\u51fd\u6570\u4e0e\u4ea4\u53c9\u8c31\u5bc6\u5ea6\u4e3a\u4e00\u5bf9\u5085\u91cc\u53f6\u53d8\u6362\u5bf9\uff0c\u81ea\u76f8\u5173\u51fd\u6570\u4e0e\u529f\u7387\u8c31\u5bc6\u5ea6\u4e3a\u4e00\u5bf9\u5085\u91cc\u53f6\u53d8\u6362\u5bf9\u3002<\/p>\n\n\n\n<p>\u4ea4\u53c9\u8c31\u5bc6\u5ea6\uff08CSD\uff09\u548c\u4e92\u76f8\u5e72\u51fd\u6570\u63cf\u8ff0\u5728\u4e0d\u540c\u4f4d\u7f6e\u95f4\u6ce2\u52a8\u7684\u76f8\u5e72\u6027\u3002\u800c<strong>\u8f90\u5c04\u4e92\u8c31\u5bc6\u5ea6\uff08Radiance Cross-Spectral Density, RCSD\uff09<\/strong>\u662fCSD\u7684\u63a8\u5e7f\uff0c\u63cf\u8ff0\u7684\u662f\u5149\u8f90\u5c04\uff08\u5373\u80fd\u91cf\u5bc6\u5ea6\uff09\u5728\u4e0d\u540c\u4f4d\u7f6e\u548c\u4e0d\u540c\u65b9\u5411\u4e4b\u95f4\u7684\u76f8\u5173\u6027\u3002\u53ef\u4ee5\u7406\u89e3\u4e3a\uff0cRCSD\u5728\u8f90\u5c04\u5ea6\u91cf\u7684\u57fa\u7840\u4e0a\u63d0\u4f9b\u4e86\u7c7b\u4f3c\u4e8eCSD\u7684\u76f8\u5e72\u6027\u63cf\u8ff0\u3002\u516c\u5f0f\u4e2d\uff0c $L(\\vec{r}_1, \\vec{r}_2, \\omega)$ \u8868\u793a\u5728\u9891\u7387 $\\omega$ \u4e0b\uff0c\u4f4d\u7f6e $\\vec{r}_1$ \u548c $\\vec{r}_2$ \u4e4b\u95f4\u7684\u8f90\u5c04\u5f3a\u5ea6\u76f8\u5e72\u6027\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-484-1024x582.png\" alt=\"\" class=\"wp-image-1645 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"582\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-484-1024x582.png\" alt=\"\" class=\"wp-image-1645 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-484-1024x582.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-484-300x171.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-484-768x437.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-484.png 1358w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p><strong>\u8f90\u5c04\u4e92\u8c31\u5bc6\u5ea6\u4f20\u8f93\u65b9\u7a0b\uff08SDTE\uff09<\/strong>\u4e0e\u7ecf\u5178\u7684<strong>\u5149\u4f20\u8f93\u65b9\u7a0b\uff08Light Transport Equation, LTE\uff09<\/strong>\u76f8\u4f3c\uff0c\u4f46\u662f\u66f4\u52a0\u9002\u5408\u6ce2\u52a8\u5149\u5b66\u3002LTE\u63cf\u8ff0\u7684\u662f\u4ece\u70b9\u5230\u70b9\u7684\u5149\u8f90\u5c04\u5ea6\u4f20\u8f93\uff0c\u800cSDTE\u5219\u4f7f\u7528RCSD\u51fd\u6570\u6765\u63cf\u8ff0\u5149\u8f90\u5c04\u7684\u4f20\u64ad\uff0c\u76f8\u5f53\u4e8e\u5c06\u4f20\u8f93\u89c6\u4e3a\u533a\u57df\u95f4\u7684\u76f8\u5e72\u4f20\u8f93\u3002<\/p>\n\n\n\n<p>SDTE\u4e2d\u7684RCSD\u901a\u8fc7\u533a\u57df\u95f4\u7684\u79ef\u5206\u5f62\u5f0f\u8868\u8fbe\u8f90\u5c04\u7684\u4f20\u64ad\uff0c\u53ef\u4ee5\u7406\u89e3\u4e3a\u7528RCSD\u77e9\u9635\u548c\u884d\u5c04\u7b97\u5b50\u4ee3\u66ff\u4e86\u4f20\u7edf\u7684\u53cd\u5c04\u548c\u6563\u5c04\u3002\u5e76\u4e14\u6ce8\u610f\uff0cSDTE\u57fa\u4e8e<strong>RCSD\u51fd\u6570<\/strong>\uff0c\u800c\u4e0d\u662f\u5177\u4f53\u7684\u5149\u5f3a\u503c\uff0c\u8fd9\u4e00\u70b9\u548cLTE\u662f\u6709\u8f83\u5927\u533a\u522b\u7684\u3002<\/p>\n\n\n\n<p>\u5728\u8fdb\u4e00\u6b65\u8ba8\u8bba\u5229\u7528<strong>\u8fb9\u754c\u5143\u65b9\u6cd5\uff08Boundary Element Method, BEM\uff09<\/strong>\u548c<strong>\u81ea\u9002\u5e94\u79ef\u5206\u6cd5\uff08Adaptive Integral Method, AIM\uff09<\/strong>\u52a0\u901f\u7684\u5168\u6ce2\u53c2\u8003\u6a21\u62df\u5668\u4e4b\u524d\uff0c\u9996\u5148\u7b80\u5355\u56de\u987e\u4e0b\u524d\u6587\u7684\u5185\u5bb9\u3002\u524d\u6587\u4ecb\u7ecd\u4e86PO\u65b9\u6cd5\u548cSDTE\/RCSD\u7406\u8bba\u3002\u8fd9\u4e9b\u65b9\u6cd5\u7528\u4e8e\u4e0d\u540c\u7684\u6563\u5c04\u8ba1\u7b97\u9700\u6c42\uff0c\u4f46\u5b83\u4eec\u7684\u57fa\u672c\u7406\u8bba\u548c\u9002\u7528\u8303\u56f4\u4e0d\u540c\u3002\u672c\u6587\u5c06\u8ba8\u8bba\u4e00\u79cd\u901a\u8fc7BEM\u548cAIM\u7ed3\u5408\u63d0\u4f9b\u9ad8\u7cbe\u5ea6\u7684\u9762\u6563\u5c04\u6a21\u62df\u7684\u65b9\u6cd5\u3002<\/p>\n\n\n\n<p>\u7528\u539f\u4f5c\u8005\u7684\u8bdd\u6765\u603b\u7ed3\uff0cWave optics\u5728\u57fa\u4e8e\u7269\u7406\u6e32\u67d3\u91cc\u9762\u5c5e\u4e8e\u5f88\u65b0\u7684\u5206\u652f\u3002\u867d\u7136\u6ce2\u52a8\u5149\u5b66\u73b0\u8c61\u5728\u751f\u6d3b\u4e2d\u968f\u5904\u53ef\u89c1\uff0c\u4f46\u662f\u5bf9\u753b\u9762\u7684\u5f71\u54cd\u5e76\u4e0d\u662f\u5f88\u5927\u3002\u8fd9\u4e2a\u65b9\u5411\u5176\u5b9e\u8fd8\u6709\u5f88\u591a\u53ef\u4ee5\u505a\u7684\u5730\u65b9\u3002<\/p>\n\n\n\n<p>\u5176\u4ed6\u76f8\u5173\u53c2\u8003\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/zhuanlan.zhihu.com\/p\/696565568\">[Tannor 2007] 3.2 Gauss\u6ce2\u5305\u7684\u4e00\u822c\u6027\u8d28 &#8211; \u7b20\u9053\u6893\u7684\u6587\u7ae0 &#8211; \u77e5\u4e4e<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.zhihu.com\/question\/534511391\/answer\/2501271591\">\u9ad8\u65af\u5149\u675f\uff0c\u5e73\u9762\u6ce2\uff0c\u7403\u9762\u6ce2\u4e09\u8005\u95f4\u6709\u4ec0\u4e48\u5173\u7cfb? &#8211; \u5b9a\u5236\u6ee4\u5149\u7247\u7684\u56de\u7b54 &#8211; \u77e5\u4e4e<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.zhihu.com\/question\/650601936\/answer\/3451120677\">\u6ce2\u52a8\u5149\u5b66\u4e2d\u7684\u4e00\u4e2a\u6ce2\u5305\u548c\u91cf\u5b50\u5149\u5b66\u4e2d\u7684\u4e00\u4e2a\u5149\u5b50\u6709\u4ec0\u4e48\u5173\u7cfb\u5417\uff1f &#8211; Godfly\u7684\u56de\u7b54 &#8211; \u77e5\u4e4e<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/zhuanlan.zhihu.com\/p\/434739417\">\u4e0d\u786e\u5b9a\u6027\u5173\u7cfb\u5373\u5085\u91cc\u53f6\u5e26\u5bbd\u5b9a\u7406\u4e4b\u8bc1\u660e &#8211; Michael Lieman\u7684\u6587\u7ae0 &#8211; \u77e5\u4e4e<\/a><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">A Full-Wave Reference Simulator for Computing Surface Reflectance<\/h1>\n\n\n\n<p>Paper\u4e3b\u9875\uff1a<\/p>\n\n\n\n<p><a href=\"https:\/\/blaire9989.github.io\/assets\/1_BEMsim3D\/project.html\">https:\/\/blaire9989.github.io\/assets\/1_BEMsim3D\/project.html<\/a><\/p>\n\n\n\n<p>ACM\u4e3b\u9875\uff1a<\/p>\n\n\n\n<p><a href=\"https:\/\/dl.acm.org\/doi\/10.1145\/3592414\">https:\/\/dl.acm.org\/doi\/10.1145\/3592414<\/a><\/p>\n\n\n\n<p>ACM Citations\uff1a<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>Yunchen Yu, Mengqi Xia, Bruce Walter, Eric Michielssen, and Steve Marschner. 2023. A Full-Wave Reference Simulator for Computing Surface Reflectance. ACM Trans. Graph. 42, 4, Article 109 (August 2023), 17 pages. https:\/\/doi.org\/10.1145\/3592414<\/p>\n<\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\">\u6f14\u8bb2\u62a5\u544a<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-487-1024x570.png\" alt=\"\" class=\"wp-image-1648 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"570\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-487-1024x570.png\" alt=\"\" class=\"wp-image-1648 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-487-1024x570.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-487-300x167.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-487-768x427.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-487.png 1440w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u4e0e\u66f4\u5e38\u7528\u7684\u5149\u7ebf\u8ffd\u8e2a\u6280\u672f\u76f8\u6bd4\uff0c\u6ce2\u52a8\u5149\u6a21\u62df\u66f4\u8d39\u65f6\u95f4\u4f46\u4e5f\u66f4\u7cbe\u786e\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-488-1024x546.png\" alt=\"\" class=\"wp-image-1649 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"546\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-488-1024x546.png\" alt=\"\" class=\"wp-image-1649 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-488-1024x546.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-488-300x160.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-488-768x410.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-488.png 1440w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u6bd4\u65b9\u8bf4\u91d1\u5c5e\u8868\u9762\u7684\u5fae\u89c2\u5212\u75d5\u3001\u6bdb\u53d1\u7ea4\u7ef4\u7ed3\u6784\uff0c\u4f20\u7edf\u7684\u5149\u5b66\u6a21\u578b\u6e32\u67d3\u7684\uff0c\u65e0\u6cd5\u6e32\u67d3\u51fa\u6211\u4eec\u73b0\u5b9e\u751f\u6d3b\u4e2d\u89c2\u5bdf\u5230\u7684\u4e94\u5f69\u6591\u6593\u7684\u8272\u5f69\u6548\u679c\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-489-1024x553.png\" alt=\"\" class=\"wp-image-1650 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"553\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-489-1024x553.png\" alt=\"\" class=\"wp-image-1650 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-489-1024x553.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-489-300x162.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-489-768x414.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-489.png 1440w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u57fa\u4e8e\u6ce2\u52a8\u5149\u5b66\u7684\u6e32\u67d3\u662f\u4e00\u4e2a\u96be\u9898\uff0c\u56e0\u4e3a\u89e3\u51b3\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u7ec4\u9700\u8981\u5927\u91cf\u590d\u6742\u7684\u8ba1\u7b97\u3002\u73b0\u6709\u7684\u57fa\u4e8e\u6ce2\u7684\u5916\u89c2\u6a21\u578b\u901a\u5e38\u91c7\u7528\u4e00\u4e9b\u8fd1\u4f3c\u65b9\u6cd5\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-490-1024x572.png\" alt=\"\" class=\"wp-image-1651 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"572\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-490-1024x572.png\" alt=\"\" class=\"wp-image-1651 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-490-1024x572.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-490-300x168.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-490-768x429.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-490.png 1440w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u4e00\u79cd\u8fd1\u4f3c\u65b9\u6cd5\u662f\u6807\u91cf\u573a\u8fd1\u4f3c\uff08scalar field approximation\uff09\u3002<\/p>\n\n\n\n<p>\u5728\u6ce2\u6563\u5c04\u95ee\u9898\uff08wave scattering problem\uff09\u4e2d\uff0c\u7535\u573a\u548c\u78c1\u573a\u662f\u4e0d\u540c\u7684\u77e2\u91cf\u573a\u91cf\uff0c\u5177\u6709\u4e0d\u540c\u6781\u5316\u65b9\u5411\uff08polarizations\uff09\u7684\u5149\u7531\u6307\u5411\u4e0d\u540c\u65b9\u5411\u7684\u573a\u91cf\u7ec4\u6210\u3002<\/p>\n\n\n\n<p>\u6709\u4e9b\u8fd1\u4f3c\u6a21\u578b\u5c06\u8fd9\u4e24\u4e2a\u77e2\u91cf\u573a\u91cf\u66ff\u6362\u4e3a\u5355\u4e00\u6807\u91cf\u51fd\u6570\uff08single scalar function\uff09\uff0c\u56e0\u6b64\u53ef\u4ee5\u8ba1\u7b97\u5149\u80fd\u7684\u5f3a\u5ea6\uff0c\u4f46\u653e\u5f03\u4e86\u5bf9\u6781\u5316\u7684\u5efa\u6a21\uff08modeling polarization\uff09\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-491-1024x564.png\" alt=\"\" class=\"wp-image-1652 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"564\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-491-1024x564.png\" alt=\"\" class=\"wp-image-1652 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-491-1024x564.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-491-300x165.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-491-768x423.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-491.png 1440w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u53e6\u4e00\u79cd\u8fd1\u4f3c\u662f\u4e00\u9636\u8fd1\u4f3c\uff08first-order approximation\uff09\u3002\u5047\u8bbe\u5149\u7ebf\u4ec5\u4e0e\u6a21\u578b\u7ed3\u6784\u7684\u6bcf\u4e2a\u90e8\u5206\u53d1\u751f\u4e00\u6b21\u53cd\u5c04\uff0c\u5ffd\u7565\u4e86\u591a\u6b21\u53cd\u5c04\u3002\u7136\u800c\uff0c\u8bb8\u591a\u60c5\u51b5\u4e0b\u8fd9\u4e9b\u8fd1\u4f3c\u90fd\u4e0d\u9002\u7528\u3002<\/p>\n\n\n\n<p>\u4f8b\u5982 Yu \u7b49\u4eba\u4e0e Dr. Lawrence \u56e2\u961f\u7684\u5408\u4f5c\uff0cPenn State University \u5236\u4f5c\u4e86\u5e26\u6709\u5706\u67f1\u5f62\u6a2a\u622a\u9762\u7684\u8868\u9762\uff0c\u8fd9\u4e9b\u8868\u9762\u4f1a\u5f15\u8d77\u591a\u6b21\u5149\u53cd\u5c04\u5e76\u4ea7\u751f\u7ed3\u6784\u8272\u5f69\uff0c\u4f7f\u7528\u8fd1\u4f3c\u6a21\u578b\u65e0\u6cd5\u5f88\u597d\u5730\u7406\u89e3\u6216\u9884\u6d4b\u8fd9\u4e9b\u73b0\u8c61\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-492-1024x530.png\" alt=\"\" class=\"wp-image-1653 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"530\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-492-1024x530.png\" alt=\"\" class=\"wp-image-1653 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-492-1024x530.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-492-300x155.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-492-768x397.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-492.png 1440w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u4f5c\u8005\u5e0c\u671b\u901a\u8fc7\u8ba1\u7b97\u53cc\u5411\u53cd\u5c04\u5206\u5e03\u51fd\u6570\uff08BRDF\uff09\u6765\u5c3d\u53ef\u80fd\u7cbe\u786e\u5730\u8868\u5f81\u8868\u9762\u6563\u5c04\u3002<\/p>\n\n\n\n<p>\u73b0\u6709\u6a21\u578b\u90fd\u91c7\u7528\u5404\u79cd\u8fd1\u4f3c\u65b9\u6cd5\uff0c\u6bd4\u5982\u57fa\u4e8e\u5149\u7ebf\u3001\u6807\u91cf\u6216\u4e00\u6b21\u8fd1\u4f3c\u7684\u6a21\u578b\uff0c\u5728\u6ca1\u6709\u53c2\u8003\u8d28\u91cf\uff08reference quality\uff09\u7684BRDF\u7684\u60c5\u51b5\u4e0b\uff0c\u5f88\u96be\u770b\u51fa\u6bcf\u79cd\u53cd\u5c04\u6a21\u578b\u7f3a\u5c11\u4e86\u4ec0\u4e48\u6216\u9002\u7528\u4e8e\u4ec0\u4e48\u573a\u666f\u3002<\/p>\n\n\n\n<p>\u4f5c\u8005\u7684\u89e3\u51b3\u65b9\u6848\u662f\u6784\u5efa\u4e00\u4e2a\u4e09\u7ef4\u56db\u6ce2\u6a21\u62df\u5668\uff083D 4-way simulation\uff09\uff0c\u7528\u4e8e\u8ba1\u7b97\u5177\u6709\u660e\u786e\u5fae\u89c2\u51e0\u4f55\u7ed3\u6784\u7684\u8868\u9762\u7684BRDF\u3002<\/p>\n\n\n\n<p>\u58f0\u79f0\u4e0d\u4f7f\u7528\u4efb\u4f55\u8fd1\u4f3c\u65b9\u6cd5\uff0c\u4e3a\u8868\u9762\u6837\u672c\u8ba1\u7b97\u51fa\u53c2\u8003\u8d28\u91cf\u7684BRDF\u3002<\/p>\n\n\n\n<p>\u901f\u5ea6\u65b9\u9762dddd\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-493-1024x546.png\" alt=\"\" class=\"wp-image-1654 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"546\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-493-1024x546.png\" alt=\"\" class=\"wp-image-1654 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-493-1024x546.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-493-300x160.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-493-768x410.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-493.png 1440w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u4f5c\u8005\u63a5\u4e0b\u6765\u4ecb\u7ecd\u4ed6\u4eec\u7684\u6a21\u62df\u5982\u4f55\u5de5\u4f5c\u4ee5\u53ca\u5b83\u4e0eBRDF\u7684\u5173\u7cfb\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-494-1024x552.png\" alt=\"\" class=\"wp-image-1655 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"552\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-494-1024x552.png\" alt=\"\" class=\"wp-image-1655 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-494-1024x552.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-494-300x162.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-494-768x414.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-494.png 1440w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u9996\u5148\uff0c\u5728\u5de6\u8fb9\u8fd9\u5e45\u56fe\uff0cInput\u4e00\u4e2a\u8868\u9762\u6837\u672c\uff08\u5b9a\u4e49\u4e3a\u9ad8\u5ea6\u573a\uff09\u4ee5\u53ca\u4e00\u4e2a\u5165\u5c04\u65b9\u5411\uff08\u5728\u6295\u5f71\u534a\u7403\u4e0a\u8868\u793a\uff09\u3002\u5b9a\u4e49\u4e86\u4e00\u4e2a\u5411\u8868\u9762\u4f20\u64ad\u7684\u5165\u5c04\u573a\uff0c\u5e76\u4ece\u8868\u9762\u8ba1\u7b97\u51fa\u4e00\u4e2a\u6563\u5c04\u573a\u3002<\/p>\n\n\n\n<p>\u4e2d\u95f4\u8fd9\u5e45\u56fe\uff0c\u5149\u675f\u662f\u5165\u5c04\u573a\uff0c\u80cc\u666f\u4e2d\u7684\u6563\u5c04\u573a\u4e5f\u663e\u793a\u5728\u6b64\u56fe\u4e2d\u3002<\/p>\n\n\n\n<p>\u8f93\u51fa\u662f\u5bf9\u5e94\u7ed9\u5b9a\u5165\u5c04\u65b9\u5411\u7684BRDF\u6a21\u5f0f\uff0c\u534a\u7403\u6bcf\u4e2a\u70b9\u4ee3\u8868\u4e00\u4e2a\u51fa\u5c04\u65b9\u5411\uff0c\u989c\u8272\u4ee3\u8868\u76f8\u5e94\u65b9\u5411\u4e0a\u53cd\u5c04\u5149\u7684\u989c\u8272\u3002BRDF\u4ee5RGB\u989c\u8272\u8868\u793a\uff0c\u8fd9\u4e9b\u989c\u8272\u662f\u4ece\u5149\u8c31\u6570\u636e\u8f6c\u6362\u800c\u6765\u3002<\/p>\n\n\n\n<p>\u5bf9\u4e8e\u5f88\u591a\u7c97\u7cd9\u5ea6\u4e0d\u9ad8\u7684\u8868\u9762\uff0c\u53cd\u5c04\u6a21\u5f0f\u56f4\u7ed5\u955c\u9762\u65b9\u5411\u5bf9\u79f0\u5e76\u968f\u5165\u5c04\u5149\u65b9\u5411\u7684\u79fb\u52a8\u800c\u53d8\u5316\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-495-1024x554.png\" alt=\"\" class=\"wp-image-1656 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"554\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-495-1024x554.png\" alt=\"\" class=\"wp-image-1656 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-495-1024x554.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-495-300x162.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-495-768x416.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-495.png 1400w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u63a5\u4e0b\u6765\u8bb2\u7684\u662f\u5982\u4f55\u4f7f\u7528\u8fb9\u754c\u5143\u6cd5\u6765\u89e3\u51b3\u4ec5\u5728\u8868\u9762\u4e0a\u7684\u4fe1\u53f7\u6563\u5c04\u95ee\u9898\uff0c\u4ece\u800c\u964d\u4f4e\u4e86\u95ee\u9898\u7684\u7ef4\u5ea6\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-496-1024x543.png\" alt=\"\" class=\"wp-image-1657 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"543\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-496-1024x543.png\" alt=\"\" class=\"wp-image-1657 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-496-1024x543.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-496-300x159.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-496-768x407.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-496.png 1396w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u8868\u9762\u4fe1\u53f7\u662f\u4ece\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u89e3\u51fa\u7684\u8868\u9762\u7535\u6d41\uff0c\u5728\u79bb\u6563\u5316\u540e\uff0c\u5c06\u95ee\u9898\u6784\u5efa\u4e3a\u4e00\u4e2a\u7ebf\u6027\u7cfb\u7edf\uff0c\u5e76\u6c42\u89e3\u51fa\u8868\u9762\u7535\u6d41\u548c\u6563\u5c04\u573a\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-497-1024x547.png\" alt=\"\" class=\"wp-image-1658 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"547\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-497-1024x547.png\" alt=\"\" class=\"wp-image-1658 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-497-1024x547.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-497-300x160.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-497-768x410.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-497.png 1394w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u4e3a\u4e86\u4f7f\u8ba1\u7b97\u53ef\u884c\uff0c\u4f5c\u8005\u5c06\u7ebf\u6027\u7cfb\u7edf\u5bf9\u79f0\u5316\uff0c\u5e76\u4f7f\u7528\u4e00\u4e2a\u9002\u5408\u5bf9\u79f0\u77e9\u9635\u7684\u6700\u5c0f\u6b8b\u5dee\u6c42\u89e3\u5668\u3002<\/p>\n\n\n\n<p>\u6b64\u5916\uff0c\u4f7f\u7528\u81ea\u9002\u5e94\u79ef\u5206\u65b9\u6cd5\u52a0\u901f\u77e9\u9635\u5411\u91cf\u4e58\u6cd5\uff0c\u8fd9\u662f\u4e00\u79cd\u57fa\u4e8e\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\u7684\u52a0\u901f\u65b9\u6cd5\uff0c\u6700\u521d\u7528\u4e8e\u96f7\u8fbe\u8ba1\u7b97\u3002<\/p>\n\n\n\n<p>\u4ee3\u7801\u5927\u90e8\u5206\u4f7f\u7528\u4e86Cuda C++\u5305\u8fdb\u884c\u52a0\u901f\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-498-1024x530.png\" alt=\"\" class=\"wp-image-1659 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"530\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-498-1024x530.png\" alt=\"\" class=\"wp-image-1659 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-498-1024x530.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-498-300x155.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-498-768x398.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-498.png 1422w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u63a5\u4e0b\u6765\uff0c\u5c55\u793a\u4e86\u4e00\u4e9b\u7ed3\u679c\uff0c\u8bf4\u660e\u5176\u8ba1\u7b97\u7684BRDF\u4e0e\u4e4b\u524d\u65b9\u6cd5\u5f97\u51fa\u7684BRDF\u7684\u6bd4\u8f83\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-499-1024x552.png\" alt=\"\" class=\"wp-image-1660 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"552\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-499-1024x552.png\" alt=\"\" class=\"wp-image-1660 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-499-1024x552.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-499-300x162.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-499-768x414.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-499.png 1396w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>[Yan 2018] \u4f7f\u7528\u6807\u91cf\u573a\u8fd1\u4f3c\u7684BRDF\u6a21\u578b\uff0c\u8fd9\u4e9b\u6a21\u578b\u53ea\u8003\u8651\u4e00\u6b21\u6298\u5c04\u3002<\/p>\n\n\n\n<p>[Xia 2023] \u8fd9\u7bc7\u4f7f\u7528\u77e2\u91cf\u573a\u91cf\u4f46\u4e5f\u53ea\u8003\u8651\u4e00\u6b21\u6298\u5c04\u3002<\/p>\n\n\n\n<p>\u6700\u7cbe\u786e\u7684\u65b9\u6cd5\u8fd8\u5f97\u662f\u54b1\u4eec\u7684\u3002\u4e0d\u4ec5\u4f7f\u7528\u77e2\u91cf\u573a\u91cf\uff08vector field quantities\uff09\uff0c\u800c\u4e14\u8003\u8651\u4e86\u6240\u6709\u6b21\u5e8f\u7684\u53cd\u5c04\uff08reflections of all orders\uff09\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-500-1024x561.png\" alt=\"\" class=\"wp-image-1661 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"561\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-500-1024x561.png\" alt=\"\" class=\"wp-image-1661 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-500-1024x561.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-500-300x164.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-500-768x421.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-500.png 1412w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u4e0a\u56fe\u6bcf\u4e2a\u5165\u5c04\u65b9\u5411\u5bf9\u5e94\u4e94\u79cdBRDF\uff0c\u5206\u522b\u4ee3\u8868\u4e0d\u540c\u7684\u8ba1\u7b97\u65b9\u6cd5\u3002<\/p>\n\n\n\n<p>\u8868\u9762\u76f8\u5bf9\u5e73\u6ed1\u7684\u6750\u6599\uff0c\u8868\u9762\u4e0a\u8986\u76d6\u4e86\u4e00\u4e9b\u5404\u5411\u540c\u6027\u7684\u51f8\u8d77\uff08a bunch of isotropic bombs\uff09\u3002<\/p>\n\n\n\n<p>\u7b2c\u4e00\u884c\u663e\u793a\u7684\u662f\u6cd5\u7ebf\u5165\u5c04\uff08normal instance\uff09\uff0c\u53cd\u5c04\u6a21\u5f0f\u57fa\u672c\u4e0a\u5c45\u4e2d\uff08reflection pattern is pretty much centered\uff09\u3002<\/p>\n\n\n\n<p>\u7b2c\u4e8c\u884c\u7531\u4e8e\u5165\u5c04\u5149\u65b9\u5411\u4ece\u67d0\u4e2a\u503e\u659c\u65b9\u5411\u6765\uff0c\u6a21\u5f0f\u5411\u5de6\u504f\u79fb\u3002<\/p>\n\n\n\n<p>\u7531\u4e8e\u8868\u9762\u4e0d\u592a\u7c97\u7cd9\uff0c\u4e94\u79cd\u65b9\u6cd5\u7684\u7ed3\u679c\u975e\u5e38\u76f8\u4f3c\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-501-1024x565.png\" alt=\"\" class=\"wp-image-1662 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"565\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-501-1024x565.png\" alt=\"\" class=\"wp-image-1662 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-501-1024x565.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-501-300x166.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-501-768x424.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-501.png 1410w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u53e6\u4e00\u79cd\u6750\u8d28\u5462\uff0c\u6709\u4e00\u4e9b\u68f1\u68f1\u89d2\u89d2\uff08corner cubes\uff09\u3002\u6bcf\u4e2acorner cubes\u7684\u4e09\u4e2a\u9762\u4f1a\u8ba9\u5149\u591a\u6b21\u53cd\u5c04\uff0c\u4f7f\u53cd\u5c04\u5149\u6cbf\u5165\u5c04\u65b9\u5411\u8fd4\u56de\u3002\u53eb\u505a\u9006\u5411\u53cd\u5c04\uff08retroreflection\uff09\u3002<\/p>\n\n\n\n<p>\u54b1\u4eec\u7684\u6a21\u62df\u5668\u4e5f\u53ef\u4ee5\u6a21\u62df\u8fd9\u79cd\u73b0\u8c61\u3002\u5de6\u8fb9\u56db\u79cd\u65b9\u6cd5\u90fd\u8d25\u4e0b\u9635\u6765\u3002<\/p>\n\n\n\n<p>\u539f\u56e0\u5728\u4e8e\u5982\u679c\u53ea\u8003\u8651\u4e00\u6b21\u53cd\u5c04\uff0c\u5f53\u5149\u7ebf\u51fb\u4e2dcorner cubes\u7684\u4e00\u4e2a\u9762\u540e\uff0c\u4f1a\u88ab\u9884\u6d4b\u4e3a\u5411\u4e0b\u8fdb\u5165\u4e0b\u534a\u7403\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-502-1024x571.png\" alt=\"\" class=\"wp-image-1663 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"571\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-502-1024x571.png\" alt=\"\" class=\"wp-image-1663 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-502-1024x571.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-502-300x167.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-502-768x428.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-502.png 1406w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u6700\u540e\u7684\u4f8b\u5b50\u662f\u4e00\u4e2a\u8868\u9762\u8986\u76d6\u4e86\u4e00\u4e9b\u7403\u5f62\u5751\u3002<\/p>\n\n\n\n<p>\u7531\u4e8e\u5751\u8fb9\u7f18\u7684\u9ad8\u5761\u5ea6\uff08high slopes of the surface at the edges of the pits\uff09\uff0c\u5bfc\u81f4\u591a\u6b21\u53cd\u5c04\u7684\u51fa\u73b0\u3002<\/p>\n\n\n\n<p>\u4e0d\u540c\u7684\u65b9\u6cd5\u770b\u5230\u660e\u663e\u5dee\u5f02\u3002<\/p>\n\n\n\n<p>\u53ef\u4ee5\u770b\u5230\u9884\u6d4b\u7684\u989d\u5916\u53cd\u5c04\u5cf0\uff08extra lobe predicting\uff09\u3002\uff08\u4e2d\u95f4\u504f\u53f3\u90a3\u4e2a\u90e8\u5206\uff09<\/p>\n\n\n\n<p>\u53e6\u5916\u6574\u4f53\u66f4\u4eae\u4e86\u3002<\/p>\n\n\n\n<p>\u8fd9\u4e9b\u5dee\u5f02\u6e90\u4e8e\u4e0d\u540c\u6b21\u5e8f\u53cd\u5c04\u95f4\u7684\u5e72\u6d89\u6548\u679c\uff08interference between reflections of different orders\uff09\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-503-1024x547.png\" alt=\"\" class=\"wp-image-1664 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"547\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-503-1024x547.png\" alt=\"\" class=\"wp-image-1664 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-503-1024x547.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-503-300x160.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-503-768x410.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-503.png 1386w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u6700\u540e\uff0c\u7b80\u5355\u4ecb\u7ecd\u9ad8\u6548\u8ba1\u7b97\u975e\u5e38\u591a\u7684\u5bc6\u96c6\u91c7\u6837\u65b9\u5411\u7684BRDF\u7684\u6280\u672f\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-504-1024x548.png\" alt=\"\" class=\"wp-image-1665 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"548\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-504-1024x548.png\" alt=\"\" class=\"wp-image-1665 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-504-1024x548.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-504-300x161.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-504-768x411.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-504.png 1394w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u5982\u679c\u9700\u8981\u6a21\u62df\u7684\u8868\u9762\u5f88\u5927\uff0c\u901f\u5ea6\u5c31\u4f1a\u5f88\u6162\uff0c\u5e76\u9700\u8981\u5927\u91cf\u7684GPU\u5185\u5b58\u3002<\/p>\n\n\n\n<p>\u4f46\u662f\u8ba1\u7b97\u662f\u7ebf\u6027\u7684\uff0c\u53ef\u4ee5\u5c06\u5927\u9762\u79ef\u8868\u9762\u5206\u89e3\u4e3a\u591a\u4e2a\u8f83\u5c0f\u7684\u5b50\u533a\u57df\u3002<\/p>\n\n\n\n<p>\u6bcf\u4e2a\u5b50\u533a\u57df\u4e0a\u6295\u5c04\u5165\u5c04\u573a\uff0c\u9996\u5148\u6267\u884c\u8f83\u5c0f\u7684\u5b50\u533a\u57df\u6a21\u62df\uff0c\u7136\u540e\u5c06\u6563\u5c04\u573a\u6574\u5408\uff0c\u5f97\u51fa\u6574\u4e2a\u5927\u9762\u79ef\u8868\u9762\u7684BRDF\u3002<\/p>\n\n\n\n<p>\u5bf9\u4e0d\u540c\u5b50\u533a\u57df\u7684\u6563\u5c04\u573a\u5e94\u7528\u4e0d\u540c\u7684\u590d\u6570\u503c\u7f29\u653e\u56e0\u5b50\uff08different complex value scale factors\uff09\uff0c\u5c31\u53ef\u4ee5\u7efc\u5408\u51fa\u5bf9\u5e94\u4e0d\u540c\u5165\u5c04\u65b9\u5411\u7684\u5927\u9762\u79ef\u8868\u9762\u7684BRDF\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-505-1024x575.png\" alt=\"\" class=\"wp-image-1666 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"575\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-505-1024x575.png\" alt=\"\" class=\"wp-image-1666 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-505-1024x575.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-505-300x169.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-505-768x432.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-505.png 1406w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u8fd9\u662f\u56e0\u4e3a\u5bf9\u6bcf\u4e2a\u5b50\u533a\u57df\u7684\u5c40\u90e8\u5165\u5c04\u573a\uff0c\u5e94\u7528\u9002\u5f53\u7684\u76f8\u79fb\u4f1a\u5728\u8868\u9762\u4e0a\u4ea7\u751f\u5177\u6709\u4e0d\u540c\u51c0\u65b9\u5411\u7684\u603b\u5165\u5c04\u573a\u3002\u5728\u8fd9\u5f20\u56fe\u4e2d\uff0c\u5165\u5c04\u573a\u5782\u76f4\u4f20\u64ad\u3002\u5982\u679c\u5c06\u4e94\u4e2a\u7126\u70b9\u5728\u4e0d\u540c\u7a7a\u95f4\u4f4d\u7f6e\u7684\u76f8\u540c\u573a\u53e0\u52a0\uff0c\u5f97\u5230\u4e00\u4e2a\u7a7a\u95f4\u4e0a\u66f4\u5bbd\u7684\u573a\uff0c\u4ecd\u6cbf\u5782\u76f4\u65b9\u5411\u4f20\u64ad\u3002\u5982\u679c\u7ebf\u6027\u7ec4\u5408\u8fd9\u4e9b\u573a\uff0c\u5e76\u5bf9\u6bcf\u4e2a\u5b50\u533a\u57df\u7684\u573a\u5e94\u7528\u9002\u5f53\u7684\u590d\u6570\u503c\u7f29\u653e\u56e0\u5b50\uff0c\u6574\u4f53\u573a\u53ef\u4ee5\u4ee5\u7565\u5fae\u503e\u659c\u7684\u65b9\u5411\u4f20\u64ad\u3002<\/p>\n\n\n\n<p>\u8fd9\u91cc\u89e3\u91ca\u4e00\u4e0b\u4e3a\u5565\u590d\u6570\u7f29\u653e\u56e0\u5b50\uff08different complex value scale factors\uff09\u53ef\u4ee5\u4ea7\u751f\u4e0d\u540c\u7684\u5165\u5c04\u65b9\u5411\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u8fd9\u4e2a\u56e0\u5b50\u53ef\u4ee5\u8c03\u6574\u6ce2\u7684\u5e45\u5ea6\u548c\u76f8\u4f4d\u3002\u5c31\u6bd4\u5982\u5f80\u6c34\u9762\u4e22\u77f3\u5934\uff0c\u4e24\u9897\u77f3\u5934\u540c\u65f6\u4e22\uff0c\u6c34\u9762\u7684\u6ce2\u5c31\u4f1a\u66f4\u5f3a\u3002\u5982\u679c\u4e00\u9897\u7a0d\u5fae\u665a\u4e00\u4e9b\u4e22\uff0c\u6ce2\u7eb9\u5c31\u53ef\u80fd\u4f1a\u76f8\u4e92\u62b5\u6d88\uff08\u76f8\u6d88\u5e72\u6d89\uff09\u3002\u8fd9\u4e2a\u56e0\u5b50\u5c31\u662f\u63a7\u5236\u77f3\u5934\u6295\u63b7\u7684\u65f6\u95f4\u3002\u901a\u8fc7\u8c03\u6574\u76f8\u4f4d\u63a7\u5236\u6ce2\u7684\u53e0\u52a0\u65b9\u5f0f\uff0c\u8fdb\u800c\u201c\u5f15\u5bfc\u201d\u6ce2\u5411\u4e0d\u540c\u7684\u65b9\u5411\u4f20\u64ad\u3002\u8be6\u7ec6\u7684\u53bb\u641c\u7d22\u300cBeamforming\u300d\uff0c\u96f7\u8fbe\u3001\u65e0\u7ebf\u901a\u4fe1\u548c\u58f0\u7eb3\u7b49\u9886\u57df\u5e94\u7528\u5f88\u5e7f\u6cdb\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u8fd9\u4e09\u5f20\u56fe\u4ee3\u8868\u5149\u6ce2\u7684\u6ce2\u524d\u3002\u5373\u6ce2\u5cf0\u3002\u53ef\u4ee5\u7406\u89e3\u4e3a\u5149\u5728\u4f20\u64ad\u4e2d\u7684\u6ce2\u5f62\u622a\u9762\u3002<\/p>\n\n\n\n<p>\u4e0a\u65b9\u7684\u56fe\uff0c\u5782\u76f4\u5730\u5c04\u5411\u8868\u9762\uff0c\u5165\u5c04\u573a\u5206\u5e03\u96c6\u4e2d\u5728\u4e2d\u5fc3\u3002\u5149\u573a\u96c6\u4e2d\u5728\u4e2d\u5fc3\uff0c\u6cbf\u5782\u76f4\u65b9\u5411\u4f20\u64ad\uff08\u5373\u4e2d\u95f4\u7684\u9ec4\u7ebf\u65b9\u5411\uff09\u3002<\/p>\n\n\n\n<p>\u5de6\u4e0b\u65b9\uff0c\u591a\u4e2a\u76f8\u540c\u5165\u5c04\u573a\u7684\u53e0\u52a0\u6548\u679c\uff0c\u4f46\u53e0\u52a0\u65f6\u76f8\u4f4d\u4fdd\u6301\u4e00\u81f4\uff08\u5373\u6ca1\u6709\u76f8\u4f4d\u5dee\uff09\u3002\u591a\u4e2a\u5165\u5c04\u573a\u76f8\u52a0\uff0c\u4f7f\u5f97\u6574\u4e2a\u573a\u5728\u7a7a\u95f4\u4e0a\u5206\u5e03\u66f4\u5bbd\u4e86\uff0c\u4f46\u4f20\u64ad\u65b9\u5411\u8fd8\u662f\u4fdd\u6301\u5782\u76f4\u3002<\/p>\n\n\n\n<p>\u53f3\u4e0b\u65b9\uff0c\u591a\u4e2a\u5165\u5c04\u573a\u7684\u53e0\u52a0\u6548\u679c\uff0c\u4f46\u662f\u5728\u53e0\u52a0\u65f6\u52a0\u5165\u4e86\u76f8\u4f4d\u5dee\u3002\u76f8\u5f53\u4e8e\u201c\u504f\u79fb\u201d\u4e86\u5165\u5c04\u573a\u7684\u65b9\u5411\u3002\u5448\u73b0\u51fa\u4e00\u4e2a\u503e\u659c\u65b9\u5411\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-506-1024x578.png\" alt=\"\" class=\"wp-image-1667 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"578\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-506-1024x578.png\" alt=\"\" class=\"wp-image-1667 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-506-1024x578.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-506-300x169.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-506-768x433.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-506.png 1400w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u6f14\u793a\u4e2d\uff0c\u4e24\u4e2a\u89c6\u9891\u5c55\u793a\u4e86BRDF\u6a21\u5f0f\u7684\u79fb\u52a8\u3002\uff08\u6211\u61d2\u5f97\u622aGIF\u4e86\uff09<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-507-1024x506.png\" alt=\"\" class=\"wp-image-1668 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"506\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-507-1024x506.png\" alt=\"\" class=\"wp-image-1668 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-507-1024x506.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-507-300x148.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-507-768x380.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-507.png 1440w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u6700\u540e\u662f\u5927\u4f6c\u6bd4\u5fc3\u5408\u5f71\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u8bba\u6587<\/h2>\n\n\n\n<p>\u6211\u7684<a href=\"https:\/\/zhuanlan.zhihu.com\/p\/776529221\">\u7b2c\u4e00\u7bc7\u6587\u7ae0<\/a>\u5df2\u7ecf\u7c97\u7565\u4ecb\u7ecd\u4e86\u8fd9\u9879\u5de5\u4f5c\u7684\u5185\u5bb9\u548c\u6210\u679c\uff0c\u63a5\u4e0b\u6765\u76f4\u63a5\u8fdb\u5165\u7406\u8bba\u63a8\u5bfc(Section3-5)\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3 FULL-WAVE SIMULATION<\/h3>\n\n\n\n<p>\u6574\u4f53\u65b9\u6cd5\u4ece\u4e00\u4e2a\u8868\u9762\u6a21\u578b\u5f00\u59cb\uff0c\u8868\u9762\u7531\u9ad8\u5ea6\u573a\u53ca\u5176\u6750\u6599\u5c5e\u6027\uff08\u5982\u590d\u6298\u5c04\u7387\uff09\u63cf\u8ff0\uff0c\u5e76\u6307\u5b9a\u4e00\u4e2a\u76ee\u6807\u70b9\u3002<\/p>\n\n\n\n<p>\u4e3a\u4e86\u8ba1\u7b97\u7ed9\u5b9a\u5165\u5c04\u65b9\u5411\u7684BRDF\uff0c\u5b9a\u4e49\u4e86\u4e00\u4e2a\u5165\u5c04\u573a\uff0c\u8be5\u573a\u4ece\u7279\u5b9a\u65b9\u5411\u4f20\u64ad\u81f3\u76ee\u6807\u70b9\u3002<\/p>\n\n\n\n<p>\u8fd9\u4e2a\u5165\u5c04\u573a\u4f5c\u4e3a\u8f93\u5165\uff0c\u901a\u8fc7\u8868\u9762\u6563\u5c04\u6a21\u62df\u8fdb\u884c\u5904\u7406\uff0c\u4ece\u800c\u6c42\u89e3\u51fa\u76f8\u5e94\u7684\u6563\u5c04\u7535\u78c1\u573a\u3002<\/p>\n\n\n\n<p>\u5728\u672c\u8282\uff08FULL-WAVE SIMULATION\uff09\u4e2d\uff0c\u5c06\u91cd\u70b9\u4ecb\u7ecdBEM\u5728\u5e94\u7528\u573a\u666f\u4e2d\u7684\u539f\u7406\u3002\u4e0b\u4e00\u8282\u4f1a\u8bb2\u89e3\u5982\u4f55\u9ad8\u6548\u5b9e\u73b0BEM\u7b97\u6cd5\uff0c\u6700\u540e\u4e00\u8282\u4f1a\u4ecb\u7ecd\u5982\u4f55\u7ed3\u5408\u591a\u4e2a\u6a21\u62df\u7ed3\u679c\uff0c\u4ee5\u5408\u6210\u5728\u5165\u5c04\u548c\u6563\u5c04\u65b9\u5411\u4e0a\u5bc6\u96c6\u91c7\u6837\u7684BRDF\u3002<\/p>\n\n\n\n<p>\u5f00\u5c40\u5148\u6765\u4e00\u5f20\u7b26\u53f7\u8868\u5413\u4e00\u5413\u4f60\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-508.png\" alt=\"\" class=\"wp-image-1669 lazyload\"\/><noscript><img decoding=\"async\" width=\"954\" height=\"834\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-508.png\" alt=\"\" class=\"wp-image-1669 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-508.png 954w, https:\/\/remoooo.com\/wp-content\/uploads\/image-508-300x262.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-508-768x671.png 768w\" sizes=\"(max-width: 954px) 100vw, 954px\" \/><\/noscript><\/figure>\n\n\n\n<h4 class=\"wp-block-heading\">3.1 Boundary Element Method: The Basics<\/h4>\n\n\n\n<p>\u8fb9\u754c\u5143\u6cd5\uff08BEM\uff09\u4e3b\u8981\u89e3\u51b3\u5355\u4e00\u9891\u7387\u7684\u6563\u5c04\u95ee\u9898\uff0c\u5373\u7279\u5b9a\u9891\u7387\u7684\u7535\u78c1\u6ce2\uff08\u5305\u62ec\u7535\u573a\u548c\u78c1\u573a\uff09\u5982\u4f55\u5728\u4e0d\u540c\u4ecb\u8d28\u7684\u8fb9\u754c\u4e0a\u53cd\u5c04\u548c\u6563\u5c04\u3002\u8fd9\u91cc\u7684\u8fb9\u754c\u5c06\u7a7a\u95f4\u5206\u6210\u4e86\u4e24\u4e2a\u5747\u5300\u533a\u57df\uff0c\u4e24\u4e2a\u533a\u57df\u7684\u6750\u6599\u7279\u6027\uff08\u5165\u5c04\u573a\u6240\u5904\u7684\u4ecb\u8d28\u53c2\u6570\uff09\u7528( $\\epsilon_1, \\mu_1$ )\u548c( $\\epsilon_2, \\mu_2$ )\u8868\u793a\u3002\u5176\u4e2d\uff0c $\\epsilon$ \u4ee3\u8868\u4ecb\u7535\u5e38\u6570\uff08\u4ecb\u7535\u7387\uff09\uff0c$\\mu$ \u4ee3\u8868\u78c1\u5bfc\u7387\uff08\u78c1\u5bfc\u7cfb\u6570\uff09\u3002<\/p>\n\n\n\n<p>\u5728\u8fd9\u79cd\u65b9\u6cd5\u4e2d\uff0c\u6211\u4eec\u5904\u7406\u7684\u662f\u590d\u6570\u503c\u573a\u91cf\uff0c\u8fd9\u4e9b\u573a\u91cf\u65e2\u5305\u542b\u632f\u5e45\u4fe1\u606f\uff0c\u4e5f\u5305\u542b\u76f8\u4f4d\u4fe1\u606f\uff08\u5373\u6ce2\u7684\u4f20\u64ad\u72b6\u6001\uff09\u3002\u4e3a\u4e86\u7b80\u5316\u516c\u5f0f\uff0c\u6211\u4eec\u5047\u8bbe\u6240\u6709\u6ce2\u90fd\u662f\u201c\u65f6\u95f4\u8c10\u548c\u201d\u7684\u2014\u2014\u4e5f\u5c31\u662f\u6ce2\u968f\u65f6\u95f4\u6309\u7167\u7279\u5b9a\u7684\u5468\u671f\u53d8\u5316\u3002\u5728\u6574\u4e2a\u6587\u4e2d\uff0c $e^{j\\omega t}$ \u9879\u88ab\u7701\u7565\uff0c\u4ee5\u7b80\u5316\u8868\u8ff0\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">3.1.1 Maxwell\u2019s Equations and Surface Currents<\/h5>\n\n\n\n<p>\u9996\u5148\uff0c\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u63cf\u8ff0\u4e86\u7535\u573a\uff08E\uff09\u548c\u78c1\u573a\uff08H\uff09\u662f\u5982\u4f55\u76f8\u4e92\u5f71\u54cd\u7684\uff0c\u51b3\u5b9a\u4e86\u5149\u6ce2\u5982\u4f55\u5728\u4e0d\u540c\u6750\u6599\u4e4b\u95f4\u4f20\u64ad\u548c\u6563\u5c04\u3002\u4e3a\u4e86\u5316\u7b80\uff0c\u8fd9\u91cc\u7528\u201c\u65f6\u95f4\u8c10\u548c\u201d\u7684\u5f62\u5f0f\uff1a<\/p>\n\n\n\n<p>$$ \\begin{align} \\nabla \\times \\mathbf{E} &amp;= -\\mathbf{M} &#8211; j \\omega \\mu \\mathbf{H} \\\\ \\nabla \\times \\mathbf{H} &amp;= \\mathbf{J} + j \\omega \\epsilon \\mathbf{E} \\tag{1} \\end{align} $$<\/p>\n\n\n\n<p><br>\u7b49\u53f7\u5de6\u8fb9\u63cf\u8ff0\u7535\u573a\u548c\u78c1\u573a\u5728\u7a7a\u95f4\u4e2d\u7684\u201c\u65cb\u8f6c\u201d\u7a0b\u5ea6\u3002 $M$ \u548c $J$ \u662f<strong>\u8868\u9762\u7535\u6d41\u5bc6\u5ea6<\/strong>\uff08\u5047\u60f3\u7684\u7535\u6d41\uff09\uff0c\u5206\u522b\u8868\u793a\u78c1\u6d41\u548c\u7535\u6d41\u7684\u5bc6\u5ea6\uff08electric and magnetic current densities\uff09\u3002\u8fd9\u4e2a\u516c\u5f0f\u53ef\u4ee5\u7406\u89e3\u4e3a\uff0c\u7535\u573a\u5728\u8fb9\u754c\u9644\u8fd1\u201c\u65cb\u8f6c\u201d\u65f6\uff0c\u4ea7\u751f\u78c1\u6d41\u548c\u78c1\u573a\u7684\u53d8\u5316\uff1b\u78c1\u573a\u7684\u65cb\u8f6c\u4e5f\u4f1a\u4ea7\u751f\u7535\u573a\u548c\u7535\u6d41\u7684\u53d8\u5316\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-509.png\" alt=\"\" class=\"wp-image-1670 lazyload\"\/><noscript><img decoding=\"async\" width=\"964\" height=\"616\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-509.png\" alt=\"\" class=\"wp-image-1670 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-509.png 964w, https:\/\/remoooo.com\/wp-content\/uploads\/image-509-300x192.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-509-768x491.png 768w\" sizes=\"(max-width: 964px) 100vw, 964px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u8fb9\u754c\u5143\u6cd5\u7684\u6838\u5fc3\u601d\u60f3\u662f\uff1a<strong>\u5728\u8fb9\u754c\u4e0a\u5f15\u5165\u8868\u9762\u7535\u6d41<\/strong>\uff0c\u7528\u8fd9\u4e9b\u7535\u6d41\u6765\u95f4\u63a5\u63cf\u8ff0\u573a\u7684\u5206\u5e03\uff0c\u800c\u4e0d\u9700\u8981\u8ba1\u7b97\u6bcf\u4e2a\u533a\u57df\u4e2d\u7684\u6240\u6709\u70b9\u3002\u4e09\u7ef4\u95ee\u9898\u5316\u7b80\u4e3a\u8fb9\u754c\u4e0a\u7684\u4e8c\u7ef4\u95ee\u9898\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">3.1.2 Source-Field Relationships<\/h5>\n\n\n\n<p>\u8868\u9762\u4e0a\u7684\u5047\u60f3\u7535\u6d41\uff08\u7535\u78c1\u6ce2\u7684\u201c\u6e90\u201d\uff09\u662f\u5982\u4f55\u4ea7\u751f\u6563\u5c04\u7684\u7535\u78c1\u573a\uff08\u201c\u573a\u201d\uff09\u7684\u5462\uff1f<\/p>\n\n\n\n<p>\u5982Fig2.\u6240\u793a\uff0c\u5728\u533a\u57df $R_1$ \uff0c\u603b\u573a\uff08\u5165\u5c04\u573a\u548c\u6563\u5c04\u573a\uff09\u5206\u522b\u8868\u793a\u4e3a $E_1$ \u548c $H_1$ \uff1b<\/p>\n\n\n\n<p>$$ \\begin{align} &amp;\\mathbf{E}_1 = \\mathbf{E}_i + \\mathbf{E}_s \\\\ &amp;\\mathbf{H}_1 = \\mathbf{H}_i + \\mathbf{H}_s \\tag{2} \\end{align} $$<\/p>\n\n\n\n<p><br>\u5728\u533a\u57df $R_2$ \uff0c\u603b\u573a\u8868\u793a\u4e3a $E_2$ \u548c $H_2$ \u3002\u4e0a\u65b9\u7684\u6563\u5c04\u573a\u7531\u4e0a\u65b9\u7684\u7535\u78c1\u6d41\u4ea7\u751f\uff1b\u4e0b\u9762\u7684\u6563\u5c04\u573a\u7531\u4e0b\u65b9\u7684\u7535\u78c1\u6d41\u4ea7\u751f\u3002<\/p>\n\n\n\n<p>\u5728\u5747\u5300\u4ecb\u8d28\u4e2d\uff0c\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u7ec4\u53ef\u4ee5\u5199\u6210\u79ef\u5206\u7684\u5f62\u5f0f\uff0c\u63cf\u8ff0\u7535\u573a\u548c\u78c1\u573a\u7684\u751f\u6210\u65b9\u5f0f\u3002<\/p>\n\n\n\n<p>\\begin{align} \\mathbf{E}(\\mathbf{r}) &amp;= -j \\omega \\mu (\\mathcal{L} \\mathbf{J})(\\mathbf{r}) &#8211; (\\mathcal{K} \\mathbf{M})(\\mathbf{r}) \\\\ \\mathbf{H}(\\mathbf{r}) &amp;= -j \\omega \\epsilon (\\mathcal{L} \\mathbf{M})(\\mathbf{r}) + (\\mathcal{K} \\mathbf{J})(\\mathbf{r}) \\tag{3} \\end{align}<\/p>\n\n\n\n<p><br>\u7b49\u53f7\u5de6\u8fb9\u5206\u522b\u8868\u793a\u7535\u78c1\u573a\u5728 $r$ \u5904\u7684\u7535\u78c1\u573a\u5f3a\u5ea6\uff0c\u5373\u63cf\u8ff0\u7684\u662f\u573a\u7684\u201c\u4f5c\u7528\u6548\u679c\u201d\u3002 $\\mathcal{L}$ \u548c $\\mathcal{K}$ \u662f\u79ef\u5206\u7b97\u5b50\uff0c\u8868\u793a\u573a\u5982\u4f55\u4ece\u8868\u9762\u7535\u6d41\u548c\u78c1\u5316\u5f3a\u5ea6\u4ea7\u751f\u3002\u8fd9\u4e24\u4e2a\u7b97\u5b50\u5b9a\u4e49\u4e3a\uff1a<\/p>\n\n\n\n<p>$$ \\begin{aligned} &amp; (\\mathcal{L} \\mathbf{X})(\\mathbf{r})=\\left[1+\\frac{1}{k^2} \\nabla \\nabla \\cdot\\right] \\int_V G\\left(\\mathbf{r}, \\mathbf{r}^{\\prime}\\right) \\mathbf{X}\\left(\\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} \\\\ &amp; (\\mathcal{K} \\mathbf{X})(\\mathbf{r})=\\nabla \\times \\int_V G\\left(\\mathbf{r}, \\mathbf{r}^{\\prime}\\right) \\mathbf{X}\\left(\\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} \\end{aligned} \\tag{4} $$<\/p>\n\n\n\n<p><br>$G(r, r{\\prime})$ \u662f\u7528\u4e8e\u6807\u91cf\u4ea5\u59c6\u970d\u5179\u65b9\u7a0b\u7684\u4e09\u7ef4\u683c\u6797\u51fd\u6570\uff0c\u5b9a\u4e49\u4e3a\uff1a<\/p>\n\n\n\n<p>$$<br>G(\\mathbf{r}, \\mathbf{r}^{\\prime}) = \\frac{e^{-jkr}}{4 \\pi r} \\quad \\text{where } r = |\\mathbf{r} &#8211; \\mathbf{r}^{\\prime}|<br>\\tag{5}<br>$$<\/p>\n\n\n\n<p><br>\u8fd9\u4e2a\u51fd\u6570\u628a\u6563\u5c04\u8868\u9762\u7684\u6e90\u573a\u8f6c\u6362\u4e3a\u6563\u5c04\u533a\u57df\u5185\u7535\u78c1\u573a\u7684\u5206\u5e03\u3002<\/p>\n\n\n\n<p>\u672c\u8bba\u6587\u8fd9\u91cc\u548c[Xia 2023]\u4e2d\u7684\u516c\u5f0f(11)\u5176\u5b9e\u662f\u4e00\u6837\u7684\uff0c\u4f46\u672c\u8bba\u6587\u5c06\u683c\u6797\u51fd\u6570\u9690\u542b\u5728\u7b97\u5b50\u4e2d\uff0c\u4e14\u79ef\u5206\u57df\u66f4\u4e3a\u5e7f\u6cdb\u3002\u672c\u8d28\u4e0a\u90fd\u662f\u63cf\u8ff0\u5982\u4f55\u4ece\u7535\u6d41\u5bc6\u5ea6 $\\mathbf{J}$ \u548c\u78c1\u6d41\u5bc6\u5ea6 $\\mathbf{M}$ \u751f\u6210\u6563\u5c04\u7535\u573a $E(r)$\u3002<\/p>\n\n\n\n<p>\u6c42\u89e3\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u65f6\uff0c\u901a\u8fc7\u683c\u6797\u51fd\u6570\u6765\u6574\u5408\u5404\u5904\u7684\u6e90\uff08\u5982\u7535\u6d41\u3001\u7535\u8377\uff09\u5bf9\u7a7a\u95f4\u4e2d\u7535\u78c1\u573a\u7684\u5f71\u54cd\u3002\u5047\u8bbe\u7535\u78c1\u573a\u4ee5 $e^{j\\omega t}$ \u7684\u5f62\u5f0f\u968f\u65f6\u95f4\u53d8\u5316\uff08\u5355\u9891\uff09\uff0c\u5c31\u53ef\u4ee5\u5f97\u5230\u7c7b\u4f3c\u4e8e\u4ea5\u59c6\u970d\u5179\u65b9\u7a0b\u7684\u5f62\u5f0f\uff1a$(\\nabla^2 + k^2) \\mathbf{E} = -j \\omega \\mu \\mathbf{J}$ \uff0c\u5b9e\u9645\u4e0a\u8fd9\u662f\u4e00\u79cd\u201c\u9891\u57df\u201d\u5f62\u5f0f\u7684\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u3002\u5f15\u5165\u683c\u6797\u51fd\u6570\u5efa\u7acb\u4e00\u79cd\u6e90-\u573a\u5173\u7cfb\uff0c\u5373\u5c06\u7535\u6d41 $\\mathbf{J}$ \u548c\u7535\u8377 $\\rho$ \u4f5c\u4e3a\u201c\u6e90\u201d\u6765\u8ba1\u7b97\u7535\u78c1\u573a $\\mathbf{E}$ \u548c $\\mathbf{H}$ \u7684\u5206\u5e03\u3002\u90a3\u4e48\u683c\u6797\u51fd\u6570\u662f\u6ee1\u8db3\u5982\u4e0b\u65b9\u7a0b\u7684\uff1a$(\\nabla^2 + k^2) G(\\mathbf{r}, \\mathbf{r}{\\prime}) = -\\delta(\\mathbf{r} &#8211; \\mathbf{r}{\\prime})$ \uff0c$\\delta$ \u662f\u72c4\u62c9\u514b\u03b4\u51fd\u6570\uff0c\u8868\u793a\u201c\u70b9\u6e90\u201d\u5728\u7a7a\u95f4\u4e2d\u4ea7\u751f\u7684\u6807\u51c6\u6ce2\u5f62\uff0c\u63cf\u8ff0\u7684\u662f\u5728\u7a7a\u95f4\u4e2d\u7531\u4e00\u4e2a\u201c\u70b9\u6e90\u201d\u6fc0\u53d1\u7684\u6ce2\u573a\u3002\u901a\u8fc7\u683c\u6797\u51fd\u6570\uff0c\u53ef\u4ee5\u8868\u8fbe\u4efb\u610f\u7535\u6d41\u5206\u5e03 $\\mathbf{J}$ \u5728\u7a7a\u95f4\u4e2d\u5bf9\u573a\u70b9 $\\mathbf{r}$ \u4ea7\u751f\u7684\u5f71\u54cd\u4e86\uff01\u63a5\u7740\uff0c\u6bcf\u4e2a\u6e90\u70b9\u4e0a\u7684\u201c\u7535\u6d41\u201d\u6216\u201c\u7535\u8377\u201d\u90fd\u901a\u8fc7\u683c\u6797\u51fd\u6570\u6269\u6563\u5230\u6574\u4e2a\u7a7a\u95f4\uff0c\u5bf9\u6bcf\u4e00\u4e2a\u573a\u70b9\u4ea7\u751f\u7d2f\u79ef\u5f71\u54cd\u3002$\\mathbf{E}(\\mathbf{r}) = \\int_V G(\\mathbf{r}, \\mathbf{r}{\\prime}) \\mathbf{J}(\\mathbf{r}{\\prime}) d \\mathbf{r}{\\prime}$ \u8fd9\u516c\u5f0f\u5c31\u662f\u7535\u573a\u8868\u793a\u4e3a\u6e90\u7535\u6d41\u7684\u79ef\u5206\u53e0\u52a0\u3002\u6700\u540e\uff0c\u7ed3\u5408\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u7ec4\u7684\u79ef\u5206\u5f62\u5f0f\u548c\u683c\u6797\u51fd\u6570\u7684\u601d\u60f3\u3002\u6bd4\u5982\u7535\u573a\u7684\u79ef\u5206\u5f62\u5f0f\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a$\\mathbf{E}(\\mathbf{r}) = -j \\omega \\mu \\int_V G(\\mathbf{r}, \\mathbf{r}{\\prime}) \\mathbf{J}(\\mathbf{r}{\\prime}) d \\mathbf{r}{\\prime} &#8211; \\nabla \\times \\int_V G(\\mathbf{r}, \\mathbf{r}{\\prime}) \\mathbf{M}(\\mathbf{r}{\\prime}) d \\mathbf{r}{\\prime}$ \uff0c\u901a\u8fc7\u5377\u79ef\u53ef\u4ee5\u83b7\u5f97\u6bcf\u4e2a\u6e90\u70b9\u4e0a\u7684\u5fae\u5c0f\u7535\u6d41\u5206\u5e03\u5bf9\u573a\u70b9\u7684\u7d2f\u79ef\u5f71\u54cd\uff0c\u8fd9\u662f\u901a\u8fc7\u683c\u6797\u51fd\u6570\u7684\u79ef\u5206\u5728\u7a7a\u95f4\u4e2d\u4f20\u64ad\u548c\u53e0\u52a0\u7684\u7ed3\u679c\u3002\u683c\u6797\u51fd\u6570\u4f5c\u4e3a\u5377\u79ef\u6838\uff0c\u5c06\u7a7a\u95f4\u4e2d\u5404\u70b9\u7684\u7535\u6d41\u5206\u5e03\u901a\u8fc7\u79ef\u5206\u4f20\u64ad\u5230\u76ee\u6807\u573a\u70b9\uff0c\u5b9e\u73b0\u4e86\u7a7a\u95f4\u4e2d\u5404\u6e90\u70b9\u7535\u6d41\u5bf9\u6574\u4e2a\u573a\u7684\u7d2f\u79ef\u5f71\u54cd\u3002<\/p>\n\n\n\n<p>\u4ecb\u8d28\u533a\u57df $R_1$ \u548c $R_2$ \u5185\u7684\u7535\u573a\u5c31\u5206\u522b\u8868\u793a\u4e3a\u516c\u5f0f(6)(7)\uff0c\u5305\u542b\u4e86\u4e24\u4e2a\u7b97\u5b50\u3002<\/p>\n\n\n\n<p>\u5728\u533a\u57df $R_2$ \u5185\uff1a<\/p>\n\n\n\n<p>$$ \\begin{align} &amp; \\mathbf{E}_1(\\mathbf{r}) = -j \\omega \\mu_1 (\\mathcal{L}_1 \\mathbf{J}_1)(\\mathbf{r}) &#8211; (\\mathcal{K}_1 \\mathbf{M}_1)(\\mathbf{r}) \\\\ &amp; \\mathbf{H}_1(\\mathbf{r}) = -j \\omega \\epsilon_1 (\\mathcal{L}_1 \\mathbf{M}_1)(\\mathbf{r}) + (\\mathcal{K}_1 \\mathbf{J}_1)(\\mathbf{r}) \\end{align} \\tag{6} $$<\/p>\n\n\n\n<p><br>\u5728\u533a\u57df $R_2$ \u5185\uff1a<\/p>\n\n\n\n<p>\\begin{align} \\mathbf{E}_2(\\mathbf{r}) &amp;= -j \\omega \\mu_2 (\\mathcal{L}_2 \\mathbf{J}_2)(\\mathbf{r}) &#8211; (\\mathcal{K}_2 \\mathbf{M}_2)(\\mathbf{r}) \\\\ \\mathbf{H}_2(\\mathbf{r}) &amp;= -j \\omega \\epsilon_2 (\\mathcal{L}_2 \\mathbf{M}_2)(\\mathbf{r}) + (\\mathcal{K}_2 \\mathbf{J}_2)(\\mathbf{r}) \\tag{7} \\end{align}<\/p>\n\n\n\n<p><br>\u8fd9\u6837\u5c31\u5f97\u5230\u4e86\u5728\u4e0d\u540c\u4ecb\u8d28\u533a\u57df\u4e2d\u7535\u573a\u548c\u78c1\u573a\u7684\u8868\u73b0\u5f62\u5f0f\u3002<\/p>\n\n\n\n<p>\u603b\u7ed3\u4e00\u4e0b\uff0c\u8fd9\u4e00\u5c0f\u8282\u901a\u8fc7\u5047\u60f3\u7684\u8868\u9762\u7535\u6d41\u4ea7\u751f\u6563\u5c04\u7535\u78c1\u573a\uff0c\u5c06\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u7ec4\u8f6c\u5316\u4e3a\u79ef\u5206\u8868\u8fbe\uff0c\u683c\u6797\u51fd\u6570\u5c06\u7535\u6d41\u548c\u7535\u8377\u5206\u5e03\u8f6c\u5316\u4e3a\u7535\u78c1\u573a\u7684\u79ef\u5206\u53e0\u52a0\uff0c\u5c55\u793a\u4e86\u6e90-\u573a\u5173\u7cfb\u7684\u5177\u4f53\u5b9e\u73b0\u65b9\u5f0f\u3002\u6700\u540e\u7ed9\u51fa\u4e86\u533a\u57df $R_1$ \u548c $R_2$ \u5185\u7535\u573a\u548c\u78c1\u573a\u7684\u8868\u8fbe\u5f0f\uff0c\u5c55\u793a\u4e86\u4e0d\u540c\u4ecb\u8d28\u53c2\u6570\u5bf9\u573a\u7684\u5f71\u54cd\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">3.1.3 Boundary Conditions<\/h5>\n\n\n\n<p>\u5f53\u7535\u78c1\u6ce2\u5728\u4e24\u79cd\u4e0d\u540c\u4ecb\u8d28\u7684\u8fb9\u754c\u4f20\u64ad\u65f6\uff0c\u4f1a\u53d1\u751f\u53cd\u5c04\u548c\u6298\u5c04\u3002\u6b64\u65f6\u6ce2\u7684\u80fd\u91cf\u4e0d\u53ef\u80fd\u51ed\u7a7a\u6d88\u5931\uff0c\u800c\u662f\u5728\u754c\u9762\u4e0a\u5e73\u6ed1\u8fc7\u6e21\u7684\u3002\u5982\u679c\u7535\u573a\u6216\u78c1\u573a\u5728\u8fb9\u754c\u4e0a\u4e0d\u8fde\u7eed\uff0c\u5c31\u4f1a\u51fa\u73b0\u4e0d\u7b26\u5408\u5b9e\u9645\u7684\u80fd\u91cf\u8dc3\u53d8\uff08\u5373\u80fd\u91cf\u7a81\u7136\u6d88\u5931\u6216\u589e\u52a0\uff09\uff0c\u8fd9\u8fdd\u53cd\u4e86\u80fd\u91cf\u5b88\u6052\u3002<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u5177\u4f53\u53ef\u4ee5\u641c\u7d22\uff1a\u300cInterface conditions for electromagnetic fields\u300d\u3002<\/p>\n<\/blockquote>\n\n\n\n<p>\u56e0\u6b64\uff0c\u5fc5\u987b\u5728\u4ecb\u8d28\u8fb9\u754c\u4e0a\u6ee1\u8db3\u4e00\u5b9a\u7684\u8fb9\u754c\u6761\u4ef6\uff0c\u4ee5\u786e\u4fdd\u7535\u78c1\u573a\u7684\u8fde\u7eed\u6027\u548c\u80fd\u91cf\u5b88\u6052\u3002\u5177\u4f53\u800c\u8a00\uff0c\u5f53\u7535\u78c1\u6ce2\u5728\u4e24\u79cd\u4e0d\u540c\u4ecb\u8d28\u7684\u754c\u9762\u4e0a\u4f20\u64ad\u65f6\uff0c\u7535\u573a\u548c\u78c1\u573a\u7684\u5207\u5411\u5206\u91cf\uff08tangential component\uff09\u9700\u8981\u5728\u8fb9\u754c\u4e0a\u4fdd\u6301\u8fde\u7eed\u6027\uff1a<\/p>\n\n\n\n<p><br>$$<br>\\begin{aligned}<br>&amp; \\mathbf{n} \\times (\\mathbf{E}_1 &#8211; \\mathbf{E}_2) = 0 \\\\<br>&amp; \\mathbf{n} \\times (\\mathbf{H}_1 &#8211; \\mathbf{H}_2) = 0<br>\\end{aligned}<br>\\tag{8}<br>$$<\/p>\n\n\n\n<p><br>\u5e76\u4e14\u8fb9\u754c\u4e0a\u7684\u51c0\u7535\u78c1\u6d41\u5bc6\u5ea6\u4e3a\u96f6\uff0c\u5373\u8fb9\u754c\u4e24\u4fa7\u7535\u78c1\u6d41\u5bc6\u5ea6\u65b9\u5411\u76f8\u53cd\u5927\u5c0f\u76f8\u7b49\u3002<\/p>\n\n\n\n<p><br>$$<br>\\begin{aligned}<br>&amp; \\mathbf{J} = \\mathbf{J}_1 = -\\mathbf{J}_2 \\\\<br>&amp; \\mathbf{M} = \\mathbf{M}_1 = -\\mathbf{M}_2<br>\\end{aligned}<br>\\tag{9}<br>$$<\/p>\n\n\n\n<p><br>\u8fd9\u4e24\u4e2a\u6761\u4ef6\u540c\u65f6\u6ee1\u8db3\uff0c\u624d\u4e0d\u4f1a\u7834\u574f\u7269\u7406\u5b88\u6052\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">3.1.4 Integral Equations<\/h5>\n\n\n\n<p>\u7ed3\u5408\u4e0a\u9762\u516c\u5f0f (6)\u3001(7)\u3001(8) \u548c (9)\uff0c\u5f97\u5230\u5173\u4e8e\u7535\u573a\u548c\u78c1\u573a\u7684\u79ef\u5206\u65b9\u7a0b\uff0c\u79f0\u4e4b\u4e3a PMCHWT\uff08Poggio-Miller-Chang-Harrington-Wu-Tsai\uff09\u65b9\u7a0b\uff1a<\/p>\n\n\n\n<p>$$<br>\\begin{aligned}<br>&amp; {\\left[j \\omega \\mu_1\\left(\\mathcal{L}1 \\mathbf{J}\\right)(\\mathbf{r})+j \\omega \\mu_2\\left(\\mathcal{L}_2 \\mathbf{J}\\right)(\\mathbf{r})+\\left(\\mathcal{K}_1 \\mathbf{M}\\right)(\\mathbf{r})+\\right.}\\left.\\left(\\mathcal{K}_2 \\mathbf{M}\\right)(\\mathbf{r})\\right]{\\tan } \\\\<br>&amp;=\\left[\\mathbf{E}i(\\mathbf{r})\\right]{\\tan } \\\\<br>&amp; {\\left[\\left(\\mathcal{K}1 \\mathbf{J}\\right)(\\mathbf{r})+\\left(\\mathcal{K}_2 \\mathbf{J}\\right)(\\mathbf{r})-j \\omega \\varepsilon_1\\left(\\mathcal{L}_1 \\mathbf{M}\\right)(\\mathbf{r})-j \\omega \\varepsilon_2\\left(\\mathcal{L}_2 \\mathbf{M}\\right)(\\mathbf{r})\\right]{\\tan } } \\\\<br>&amp;=-\\left[\\mathbf{H}i(\\mathbf{r})\\right]{\\tan }<br>\\end{aligned}<br>\\tag{10}<br>$$<\/p>\n\n\n\n<p><br>\u8fd9\u4e24\u4e2a\u65b9\u7a0b\u5206\u522b\u8fd8\u6709\u4e2a\u540d\u5b57\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>EFIE<\/strong>\uff08Electric Field Integral Equation\uff09\uff0c\u7535\u573a\u79ef\u5206\u65b9\u7a0b\u3002 $j \\omega \\mu_1 (\\mathcal{L}_1 \\mathbf{J})(\\mathbf{r})$ \u548c $j \\omega \\mu_2 (\\mathcal{L}_2 \\mathbf{J})(\\mathbf{r})$ \u8868\u793a\u5728\u4ecb\u8d28 1 \u548c \u4ecb\u8d28 2 \u4e2d\uff0c\u7535\u6d41\u5bc6\u5ea6 $\\mathbf{J}$ \u5bf9\u7535\u573a\u7684\u8d21\u732e\u3002 $\\mathcal{K}_1 \\mathbf{M}$ \u548c $\\mathcal{K}_2 \\mathbf{M}$ \u8868\u793a\u5728\u4ecb\u8d28 1 \u548c\u4ecb\u8d28 2 \u4e2d\uff0c\u78c1\u6d41\u5bc6\u5ea6 $\\mathbf{M}$ \u5bf9\u7535\u573a\u7684\u8d21\u732e\u3002<\/li>\n\n\n\n<li><strong>MFIE<\/strong>\uff08Magnetic Field Integral Equation\uff09\uff0c\u78c1\u573a\u79ef\u5206\u65b9\u7a0b\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u603b\u7684\u6765\u8bf4\uff0c\u8fd9\u662f\u4e00\u79cd\u8fb9\u754c\u79ef\u5206\u65b9\u7a0b\uff0c\u4e13\u95e8\u7528\u6765\u6c42\u4ecb\u7535\u7269\u4f53\u5f15\u8d77\u7684\u7535\u78c1\u6563\u5c04\u95ee\u9898\u3002\u6709\u4e86\u8fd9\u4e2aPMCHWT\u65b9\u7a0b\uff0c\u5c31\u53ef\u4ee5\u51c6\u786e\u8ba1\u7b97\u7535\u78c1\u573a\u7684\u5206\u5e03\u4e86\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">3.1.5 Solving for Current Densities<\/h5>\n\n\n\n<p>\u8fd9\u4e00\u8282\uff0c\u9700\u8981\u901a\u8fc7\u4e0a\u9762\u7684PMCHWT\u65b9\u7a0b\u8ba1\u7b97\u7269\u4f53\u8868\u9762\u4e0a\u7684\u201c\u7535\u6d41\u201d\u548c\u201c\u78c1\u6d41\u201d\u5206\u5e03\uff0c\u8fd9\u4e9b\u5206\u5e03\u51b3\u5b9a\u4e86\u7535\u78c1\u6ce2\u78b0\u5230\u7269\u4f53\u540e\u4f1a\u600e\u4e48\u201c\u53cd\u5c04\u201d\u6216\u8005\u201c\u6298\u5c04\u201d\u3002<\/p>\n\n\n\n<p>\u6c42\u89e3\u8868\u9762\u7535\u6d41\u5bc6\u5ea6 $\\mathbf{J}$ \u548c\u78c1\u6d41\u5bc6\u5ea6 $\\mathbf{M}$ \u662f\u901a\u8fc7\u5c06\u8fb9\u754c\u5143\u79bb\u6563\u5316\u3002\u5bf9\u4e8e\u79bb\u6563\u5355\u5143\u5b9a\u4e49\u4e00\u4e2a\u57fa\u51fd\u6570 $f_m(\\mathbf{r})$ \uff0c\u7528\u57fa\u51fd\u6570\u5c55\u5f00\u6cd5\u8868\u793a\u7535\u6d41\u5bc6\u5ea6\u548c\u78c1\u6d41\u5bc6\u5ea6\u5206\u5e03\u3002<\/p>\n\n\n\n<p><br>$$<br>\\mathbf{J}(\\mathbf{r}) = \\sum_{m=1}^{N} I_{J_m} f_m(\\mathbf{r}); \\quad \\mathbf{M}(\\mathbf{r}) = \\sum_{n=1}^{N} I_{M_n} f_n(\\mathbf{r})<br>\\tag{11}<br>$$<\/p>\n\n\n\n<p><br>N \u662f\u57fa\u51fd\u6570\u7684\u603b\u6570\uff1b$ I_{J_m}$ \u548c $I_{M_n}$ \u662f\u5bf9\u5e94\u57fa\u51fd\u6570\u7684\u672a\u77e5\u7cfb\u6570\uff0c\u4ee3\u8868\u4e86\u6bcf\u4e2a\u5355\u5143\u4e0a\u7684\u7535\u6d41\u548c\u78c1\u6d41\u5f3a\u5ea6\u3002<\/p>\n\n\n\n<p>\u901a\u8fc7\u8fd9\u79cd\u57fa\u51fd\u6570\u5c55\u5f00\uff0c\u8fde\u7eed\u7684\u8868\u9762\u7535\u6d41\u5bc6\u5ea6\u548c\u78c1\u6d41\u5bc6\u5ea6\u88ab\u5206\u89e3\u6210\u4e00\u7cfb\u5217\u57fa\u51fd\u6570\u7684\u7ebf\u6027\u7ec4\u5408\u3002<\/p>\n\n\n\n<p>\u4e3a\u4e86\u6c42\u89e3\u5bf9\u5e94\u57fa\u51fd\u6570\u7684\u672a\u77e5\u7cfb\u6570\uff0c\u5c06\u7535\u573a\u79ef\u5206\u65b9\u7a0b (EFIE) \u548c\u78c1\u573a\u79ef\u5206\u65b9\u7a0b (MFIE) \u8f6c\u5316\u4e3a\u4e00\u4e2a\u7ebf\u6027\u65b9\u7a0b\u7ec4\u3002\u5177\u4f53\u4f7f\u7528\u4f3d\u8fbd\u91d1\u6cd5\uff08Galerkin Method\uff09\u5b8c\u6210\u3002\u8fd9\u4e2a\u65b9\u6cd5\u57fa\u672c\u601d\u60f3\u662f\uff0c\u5c06\u79ef\u5206\u65b9\u7a0b\u4f5c\u7528\u5728\u6bcf\u4e2a\u57fa\u51fd\u6570\u4e0a\uff0c\u52a0\u6743\u5e73\u5747\u4f7f\u5f97\u79ef\u5206\u65b9\u7a0b\u5728\u6bcf\u4e2a\u57fa\u51fd\u6570\u7684\u6295\u5f71\u65b9\u5411\u4e0a\u90fd\u6210\u7acb\u3002\u8be5\u65b9\u6cd5\u7b80\u5355\u7684\u8bf4\u5c31\u662f\u79bb\u6563\u5316\u3001\u627e\u57fa\u5e95\u3001\u7b97\u7cfb\u6570\u3002\u4e00\u4e2a\u9ad8\u7ef4\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u53ef\u4ee5\u7528\u7ebf\u6027\u4ee3\u6570\u65b9\u6cd5\u7b80\u5316\u3002<\/p>\n\n\n\n<p>\u8fd9\u6837\u53ef\u4ee5\u5c06\u539f\u672c\u8fde\u7eed\u5f62\u5f0f\u7684PMCHWT\u65b9\u7a0b\u7684EFIE\u90e8\u5206\u8f6c\u6362\u4e3a\u6709\u9650\u4e2a\u7ebf\u6027\u65b9\u7a0b\uff0c\u628a\u95ee\u9898\u8f6c\u5316\u4e3a\u89e3\u5982\u4e0b\u77e9\u9635\u65b9\u7a0b\u3002<\/p>\n\n\n\n<p><br>$$<br>\\begin{bmatrix} A_{EJ} &amp; A_{EM} \\ A_{HJ} &amp; A_{HM} \\end{bmatrix} \\begin{bmatrix} I_J \\ I_M \\end{bmatrix} = \\begin{bmatrix} V_E \\ V_H \\end{bmatrix}<br>\\tag{12}<br>$$<\/p>\n\n\n\n<p><br>\u5176\u4e2d<\/p>\n\n\n\n<p><br>$$<br>A_{\\mathrm{EJ}}^{m n} =\\int_S \\mathbf{f}_m(\\mathbf{r}) \\cdot\\left[j \\omega \\mu_1\\left(\\mathcal{L}_1 \\mathbf{f}_n\\right)(\\mathbf{r})+j \\omega \\mu_2\\left(\\mathcal{L}_2 \\mathbf{f}_n\\right)(\\mathbf{r})\\right] d \\mathbf{r} \\tag{13}<br>$$<\/p>\n\n\n\n<p>$$<br>A_{\\mathrm{EM}}^{m n} =\\int_S \\mathbf{f}_m(\\mathbf{r}) \\cdot\\left[\\left(\\mathcal{K}_1 \\mathbf{f}_n\\right)(\\mathbf{r})+\\left(\\mathcal{K}_2 \\mathbf{f}_n\\right)(\\mathbf{r})\\right] d \\mathbf{r} \\tag{14}<br>$$<\/p>\n\n\n\n<p>$$<br>A_{\\mathrm{HJ}}^{m n} =\\int_S \\mathbf{f}_m(\\mathbf{r}) \\cdot\\left[\\left(\\mathcal{K}_1 \\mathbf{f}_n\\right)(\\mathbf{r})+\\left(\\mathcal{K}_2 \\mathbf{f}_n\\right)(\\mathbf{r})\\right] d \\mathbf{r} \\tag{15}<br>$$<\/p>\n\n\n\n<p>$$<br>A_{\\mathrm{HM}}^{m n} =-\\int_S \\mathbf{f}_m(\\mathbf{r}) \\cdot\\left[j \\omega \\varepsilon_1\\left(\\mathcal{L}_1 \\mathbf{f}_n\\right)(\\mathbf{r})+j \\omega \\varepsilon_2\\left(\\mathcal{L}_2 \\mathbf{f}_n\\right)(\\mathbf{r})\\right] d \\mathbf{r} \\tag{16}<br>$$<\/p>\n\n\n\n<p>\u548c<\/p>\n\n\n\n<p><br>$$<br>V_{\\mathrm{E}}^m =\\int_S \\mathbf{f}_m(\\mathbf{r}) \\cdot \\mathbf{E}_i(\\mathbf{r}) d \\mathbf{r}\\tag{17}<br>$$<\/p>\n\n\n\n<p>$$<br>V_{\\mathrm{H}}^m =-\\int_S \\mathbf{f}_m(\\mathbf{r}) \\cdot \\mathbf{H}_i(\\mathbf{r}) d \\mathbf{r}\\tag{18}<br>$$<\/p>\n\n\n\n<p>\u516c\u5f0f(12)\u4e2d\uff0c\u9700\u8981\u6c42\u51fa $I_J$ \u548c $I_M$ \u3002<\/p>\n\n\n\n<p>\u4e0d\u4e25\u8c28\u5730\u8bb2\uff0c\u516c\u5f0f(13)-(16)\u5206\u522b\u8868\u793a\u6bcf\u4e00\u5c0f\u5757\u7684\u7535\u6d41\u5bc6\u5ea6\u5bf9\u4ea7\u751f\u7535\u573a\u7684\u8d21\u732e\u3001\u6bcf\u4e2a\u5c0f\u5757\u7684\u78c1\u6d41\u5bc6\u5ea6\u5bf9\u4ea7\u751f\u7535\u573a\u7684\u8d21\u732e\u3001\u6bcf\u4e2a\u5c0f\u5757\u7684\u7535\u6d41\u5bc6\u5ea6\u5bf9\u4ea7\u751f\u78c1\u573a\u7684\u8d21\u732e\u548c\u6bcf\u4e2a\u5c0f\u5757\u7684\u78c1\u6d41\u5bc6\u5ea6\u5bf9\u4ea7\u751f\u78c1\u573a\u7684\u8d21\u732e\u3002\u516c\u5f0f(17)(18)\u5206\u522b\u8868\u793a\u5916\u754c\u5165\u5c04\u7535\u573a\u5bf9\u8fd9\u4e2a\u5c0f\u5757\u7535\u6d41\u7684\u201c\u63a8\u52a8\u529b\u201d\u548c\u5916\u754c\u5165\u5c04\u78c1\u573a\u5bf9\u8fd9\u4e2a\u5c0f\u5757\u78c1\u6d41\u7684\u201c\u63a8\u52a8\u529b\u201d\u3002\u5f3a\u8c03\u4e00\u4e0b\u77e9\u9635\u91cc\u9762\u7684\u5143\u7d20\uff0c\u6bd4\u5982 $A_{EJ}^{mn}$ \uff0c\u8fd9\u5176\u5b9e\u662f\u4e00\u4e2a\u4e8c\u91cd\u79ef\u5206\u3002\u7531\u4e8e\u5bf9\u6e90\u70b9 $\\mathbf{r}{\\prime}$ \u7684\u79ef\u5206\u5df2\u7ecf\u5728 $\\mathcal{L}_1$ \u548c $\\mathcal{L}_2$ \u4e2d\u5b8c\u6210\u4e86\uff0c\u56e0\u6b64\u5bfc\u81f4\u539fpaper\u770b\u8d77\u6765\u662f\u4e2a\u4e00\u91cd\u79ef\u5206\u3002<\/p>\n\n\n\n<p>\u867d\u7136\u539f\u8bba\u6587\u6ca1\u6709\u5199\uff0c\u4f46\u662f\u5efa\u8bae\u806a\u660e\u7684\u8bfb\u8005\u81ea\u5df1\u63a8\u5bfc\u4e00\u4e0b\u3002\u5c1d\u8bd5\u6839\u636e\u4e0a\u6587\u63d0\u5230\u7684\u516c\u5f0f(4)\u628a\u516c\u5f0f(13)\u5c55\u5f00\u3002\u6211\u8fd9\u91cc\u5c1d\u8bd5\u63a8\u5bfc\u4e86\u4e00\u4e0b\uff0c\u6709\u9519\u8bef\u8bf7\u6307\u6b63\u3002\u9996\u5148\u628a\u4e24\u4e2a\u7b97\u5b50\u4ee3\u5165\uff0c\u6ce8\u610f\u8fd9\u91cc\u68af\u5ea6\u7b97\u5b50\u7684\u4f4d\u7f6e\uff1a<\/p>\n\n\n\n<p><br>$$<br>\\begin{aligned}<br>A_{\\mathrm{EJ}}^{mn} &amp;= j \\omega \\mu_1 \\int_S \\mathbf{f}_m(\\mathbf{r}) \\cdot \\left\\{ \\int_V G_1(\\mathbf{r}, \\mathbf{r}{\\prime}) \\mathbf{f}_n(\\mathbf{r}{\\prime}) d\\mathbf{r}{\\prime} + \\frac{1}{k_1^2} \\nabla \\left[ \\nabla \\cdot \\int_V G_1(\\mathbf{r}, \\mathbf{r}{\\prime}) \\mathbf{f}_n(\\mathbf{r}{\\prime}) d\\mathbf{r}{\\prime} \\right] \\right\\} d\\mathbf{r} \\\\<br>&amp;\\quad + j \\omega \\mu_2 \\int_S \\mathbf{f}_m(\\mathbf{r}) \\cdot \\left\\{ \\int_V G_2(\\mathbf{r}, \\mathbf{r}{\\prime}) \\mathbf{f}_n(\\mathbf{r}{\\prime}) d\\mathbf{r}{\\prime} + \\frac{1}{k_2^2} \\nabla \\left[ \\nabla \\cdot \\int_V G_2(\\mathbf{r}, \\mathbf{r}{\\prime}) \\mathbf{f}_n(\\mathbf{r}{\\prime}) d\\mathbf{r}{\\prime} \\right] \\right\\} d\\mathbf{r}<br>\\end{aligned}<br>$$<\/p>\n\n\n\n<p>\u5148\u8003\u8651\u5176\u4e2d\u4e00\u4e2a\u68af\u5ea6\u9879\uff1a<\/p>\n\n\n\n<p>$$<br>\\int_S \\mathbf{f}_m(\\mathbf{r}) \\cdot \\nabla \\left[ \\nabla \\cdot \\int_V G(\\mathbf{r}, \\mathbf{r}{\\prime}) \\mathbf{f}_n(\\mathbf{r}{\\prime}) d\\mathbf{r}{\\prime} \\right] d\\mathbf{r}<br>$$<\/p>\n\n\n\n<p>\u7528\u5411\u91cf\u5206\u5e03\u79ef\u5206\u5c55\u5f00\uff1a<\/p>\n\n\n\n<p>$$<br>\\int_S \\mathbf{f}m \\cdot \\nabla B \\, d r = -\\int_S B (\\nabla \\cdot \\mathbf{f}_m) \\, dr + \\int{\\partial S} B (\\mathbf{A} \\cdot \\mathbf{n}) \\, dr<br>$$<\/p>\n\n\n\n<p>\u5176\u4e2d\uff0c\u6563\u5ea6\u9879 $B$ \uff1a<\/p>\n\n\n\n<p>$$<br>B = \\nabla \\cdot \\int_V G(\\mathbf{r}, \\mathbf{r}{\\prime}) \\mathbf{f}_n(\\mathbf{r}{\\prime}) d\\mathbf{r}{\\prime}<br>$$<\/p>\n\n\n\n<p>\u4f46\u662f\u5f88\u62b1\u6b49\uff0c\u8fd9\u91cc\u662f\u7269\u7406\u3002\u5728\u8fb9\u754c\u6761\u4ef6\u4e0b\uff0c\u8fb9\u754c\u9879\u76f4\u63a5\u7b80\u5316\u4e3a\u96f6\uff0c\u5f97\u5230\uff1a<\/p>\n\n\n\n<p>$$<br>\\int_S \\mathbf{f}_m \\cdot \\nabla B \\, dS = -\\int_S B (\\nabla \\cdot \\mathbf{f}_m) \\, dS<br>$$<\/p>\n\n\n\n<p>\u5bf9\u4e8e\u6563\u5ea6\u9879 $B$ \uff0c\u53ef\u4ee5\u76f4\u63a5\u5c55\u5f00\u6563\u5ea6\uff1a<\/p>\n\n\n\n<p>$$<br>B = \\nabla \\cdot \\int_V G(\\mathbf{r}, \\mathbf{r}{\\prime}) \\mathbf{f}_n(\\mathbf{r}{\\prime}) d\\mathbf{r}{\\prime} = \\int_V (\\nabla \\cdot G(\\mathbf{r}, \\mathbf{r}{\\prime}) \\mathbf{f}_n(\\mathbf{r}{\\prime})) d\\mathbf{r}{\\prime}<br>$$<\/p>\n\n\n\n<p>\u4e00\u4e2a\u662f\u6807\u91cf\u51fd\u6570\uff0c\u4e00\u4e2a\u662f\u77e9\u9635\u51fd\u6570\uff0c\u56e0\u6b64\u6839\u636e\u6563\u5ea6\u7684\u4e58\u79ef\u6cd5\u5219\uff1a<\/p>\n\n\n\n<p>$$<br>\\nabla \\cdot (G \\mathbf{f}_n) = (\\nabla G) \\cdot \\mathbf{f}_n + G (\\nabla \\cdot \\mathbf{f}_n)<br>$$<\/p>\n\n\n\n<p>\u4f46\u662f\u5f88\u62b1\u6b49\uff0c\u8fd9\u91cc\u662f\u7269\u7406\u3002\u7531\u4e8e\u57fa\u51fd\u6570\u6ee1\u8db3\u65e0\u6563\u5ea6\u6761\u4ef6\uff0c\u56e0\u6b64\u8fd9\u91cc\u76f4\u63a5\u7b80\u5316\uff1a<\/p>\n\n\n\n<p>$$<br>\\nabla \\cdot (G \\mathbf{f}_n) = (\\nabla G) \\cdot \\mathbf{f}_n<br>$$<\/p>\n\n\n\n<p>\u540c\u65f6\u6211\u4eec\u6ce8\u610f\u5230\u683c\u6797\u51fd\u6570\u7684\u5bf9\u79f0\u6027 $\\nabla G(\\mathbf{r}, \\mathbf{r}{\\prime}) = -\\nabla{\\prime} G(\\mathbf{r}, \\mathbf{r}{\\prime})$ \uff0c\u4e8e\u662f\u6709\uff1a<\/p>\n\n\n\n<p>$$<br>B = \\nabla \\cdot \\int_V G(\\mathbf{r}, \\mathbf{r}{\\prime}) \\mathbf{f}_n(\\mathbf{r}{\\prime}) d\\mathbf{r}{\\prime} = -\\int_V \\nabla{\\prime} G(\\mathbf{r}, \\mathbf{r}{\\prime}) \\cdot \\mathbf{f}_n(\\mathbf{r}{\\prime}) d\\mathbf{r}{\\prime}<br>$$<\/p>\n\n\n\n<p>\u5bf9\u8fd9\u4e00\u9879\u4e5f\u8fdb\u884c\u5206\u5e03\u79ef\u5206\uff1a<\/p>\n\n\n\n<p>$$<br>\\int_V \\nabla{\\prime} G(\\mathbf{r}, \\mathbf{r}{\\prime}) \\cdot \\mathbf{f}_n(\\mathbf{r}{\\prime}) d\\mathbf{r}{\\prime} = \\int_V \\nabla{\\prime} \\cdot (G \\mathbf{f}_n) d\\mathbf{r}{\\prime} &#8211; \\int_V G (\\nabla{\\prime} \\cdot \\mathbf{f}_n) d\\mathbf{r}{\\prime}<br>$$<\/p>\n\n\n\n<p>\u4f46\u662f\u5f88\u62b1\u6b49\uff0c\u8fd9\u91cc\u662f\u7269\u7406\u3002\u8fb9\u754c\u9879\u518d\u6b21\u5316\u7b80\uff1a<\/p>\n\n\n\n<p>$$<br>\\int_V \\nabla{\\prime} G(\\mathbf{r}, \\mathbf{r}{\\prime}) \\cdot \\mathbf{f}_n(\\mathbf{r}{\\prime}) d\\mathbf{r}{\\prime} = -\\int_V G(\\mathbf{r}, \\mathbf{r}{\\prime}) (\\nabla{\\prime} \\cdot \\mathbf{f}_n(\\mathbf{r}{\\prime})) d\\mathbf{r}{\\prime}<br>$$<\/p>\n\n\n\n<p>\u6700\u7ec8\u5f97\u5230\uff1a<\/p>\n\n\n\n<p>$$<br>B = \\nabla \\cdot \\int_V G(\\mathbf{r}, \\mathbf{r}{\\prime}) \\mathbf{f}_n(\\mathbf{r}{\\prime}) d\\mathbf{r}{\\prime} = \\int_V G(\\mathbf{r}, \\mathbf{r}{\\prime}) (\\nabla{\\prime} \\cdot \\mathbf{f}_n(\\mathbf{r}{\\prime})) d\\mathbf{r}{\\prime}<br>$$<\/p>\n\n\n\n<p>\u6ce2\u6570 $k_i$ \u4e8e\u4ecb\u8d28\u53c2\u6570\u7684\u5173\u7cfb\uff1a<\/p>\n\n\n\n<p>$$<br>k_i^2 = \\omega^2 \\mu_i \\varepsilon_i \\quad \\Rightarrow \\quad \\frac{1}{k_i^2} = \\frac{1}{\\omega^2 \\mu_i \\varepsilon_i}<br>$$<\/p>\n\n\n\n<p>\u53e6\u4e00\u4e2a\u68af\u5ea6\u9879\u4e5f\u662f\u540c\u7406\uff0c\u6700\u540e\u5f97\u5230\u6700\u7ec8\u7684\u8868\u8fbe\u5f0f $A_{\\mathrm{EJ}}^{mn}$ \uff1a<\/p>\n\n\n\n<p>$$<br>\\begin{aligned} A_{\\mathrm{EJ}}^{mn} &amp;= j \\omega \\mu_1 \\int_S \\int_{V_1} \\mathbf{f}m(\\mathbf{r}) \\cdot \\mathbf{f}n(\\mathbf{r}{\\prime}) G_1(\\mathbf{r}, \\mathbf{r}{\\prime}) d\\mathbf{r}{\\prime} d\\mathbf{r} \\\\ &amp;\\quad &#8211; \\frac{j}{\\omega \\varepsilon_1} \\int_S \\int{V_1} (\\nabla \\cdot \\mathbf{f}m(\\mathbf{r})) G_1(\\mathbf{r}, \\mathbf{r}{\\prime}) (\\nabla{\\prime} \\cdot \\mathbf{f}n(\\mathbf{r}{\\prime})) d\\mathbf{r}{\\prime} d\\mathbf{r} \\\\ &amp;\\quad + j \\omega \\mu_2 \\int_S \\int{V_2} \\mathbf{f}m(\\mathbf{r}) \\cdot \\mathbf{f}n(\\mathbf{r}{\\prime}) G_2(\\mathbf{r}, \\mathbf{r}{\\prime}) d\\mathbf{r}{\\prime} d\\mathbf{r} \\\\ &amp;\\quad &#8211; \\frac{j}{\\omega \\varepsilon_2} \\int_S \\int{V_2} (\\nabla \\cdot \\mathbf{f}_m(\\mathbf{r})) G_2(\\mathbf{r}, \\mathbf{r}{\\prime}) (\\nabla{\\prime} \\cdot \\mathbf{f}_n(\\mathbf{r}{\\prime})) d\\mathbf{r}{\\prime} d\\mathbf{r} \\end{aligned}<br>$$<\/p>\n\n\n\n<p>\u540c\u6837\u7684\u64cd\u4f5c\uff0c\u5f97\u5230\u5269\u4e0b\u7684\u77e9\u9635\u5143\u7d20\uff0c\u8fd9\u91cc\u76f4\u63a5\u6284\u8bba\u6587\u9644\u52a0\u6750\u6599\u7684\u5185\u5bb9\u4e86\uff1a<\/p>\n\n\n\n<p>$$<br>\\begin{aligned} A_{\\mathrm{EM}}^{m n}= &amp; A_{\\mathrm{HJ}}^{m n}=\\int_{\\mathbf{f}_m} \\int_{\\mathbf{f}_n} \\mathbf{f}_m(\\mathbf{r}) \\cdot\\left[\\nabla G_1\\left(\\mathbf{r}, \\mathbf{r}^{\\prime}\\right) \\times \\mathbf{f}_n\\left(\\mathbf{r}^{\\prime}\\right)\\right] d \\mathbf{r}^{\\prime} d \\mathbf{r} \\\\ &amp; +\\int_{\\mathbf{f}_m} \\int_{\\mathbf{f}_n} \\mathbf{f}_m(\\mathbf{r}) \\cdot\\left[\\nabla G_2\\left(\\mathbf{r}, \\mathbf{r}^{\\prime}\\right) \\times \\mathbf{f}_n\\left(\\mathbf{r}^{\\prime}\\right)\\right] d \\mathbf{r}^{\\prime} d \\mathbf{r} \\\\ A_{\\mathrm{HM}}^{m n} &amp; =-j \\omega \\varepsilon_1 \\int_{\\mathbf{f}_m} \\int_{\\mathbf{f}_n} \\mathbf{f}_m(\\mathbf{r}) \\cdot \\mathbf{f}_n\\left(\\mathbf{r}^{\\prime}\\right) G_1\\left(\\mathbf{r}, \\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} d \\mathbf{r} \\\\ &amp; +\\frac{j}{\\omega \\mu_1} \\int_{\\mathbf{f}_m} \\int_{\\mathbf{f}_n} \\nabla \\cdot \\mathbf{f}_m(\\mathbf{r}) G_1\\left(\\mathbf{r}, \\mathbf{r}^{\\prime}\\right) \\nabla^{\\prime} \\cdot \\mathbf{f}_n\\left(\\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} d \\mathbf{r} \\\\ &amp; -j \\omega \\varepsilon_2 \\int_{\\mathbf{f}_m} \\int_{\\mathbf{f}_n} \\mathbf{f}_m(\\mathbf{r}) \\cdot \\mathbf{f}_n\\left(\\mathbf{r}^{\\prime}\\right) G_2\\left(\\mathbf{r}, \\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} d \\mathbf{r} \\\\ &amp; +\\frac{j}{\\omega \\mu_2} \\int_{\\mathbf{f}_m} \\int_{\\mathbf{f}_n} \\nabla \\cdot \\mathbf{f}_m(\\mathbf{r}) G_2\\left(\\mathbf{r}, \\mathbf{r}^{\\prime}\\right) \\nabla^{\\prime} \\cdot \\mathbf{f}_n\\left(\\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} d \\mathbf{r}\\end{aligned}<br>$$<\/p>\n\n\n\n<p>\u5f97\u5230\u77e9\u9635\u7684\u6bcf\u4e00\u4e2a\u5143\u7d20\u4e4b\u540e\uff0c\u5f15\u5165\u5e73\u79fb\u4e0d\u53d8\u51fd\u6570\uff08Shift-invariant Functions\uff09\u3002\u5e2e\u52a9\u5f97\u5230\u5728\u4e0d\u540c\u5750\u6807\u7cfb\u4e0b\u8868\u793a\u683c\u6797\u51fd\u6570\u53ca\u5176\u68af\u5ea6\u3002<\/p>\n\n\n\n<p>$$<br>\\begin{aligned}<br>&amp; g_{1, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right)=G_i\\left(\\mathbf{r}, \\mathbf{r}^{\\prime}\\right)=\\frac{e^{-j k_i r}}{4 \\pi r} \\\\<br>&amp; g_{2, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right)=\\hat{\\mathbf{x}} \\cdot \\nabla G_i\\left(\\mathbf{r}, \\mathbf{r}^{\\prime}\\right)=-\\left(x-x^{\\prime}\\right)\\left(\\frac{1+j k_i r}{4 \\pi r^3}\\right) e^{-j k_i r} \\\\<br>&amp; g_{3, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right)=\\hat{\\mathbf{y}} \\cdot \\nabla G_i\\left(\\mathbf{r}, \\mathbf{r}^{\\prime}\\right)=-\\left(y-y^{\\prime}\\right)\\left(\\frac{1+j k_i r}{4 \\pi r^3}\\right) e^{-j k_i r} \\\\<br>&amp; g_{4, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right)=\\hat{\\mathbf{z}} \\cdot \\nabla G_i\\left(\\mathbf{r}, \\mathbf{r}^{\\prime}\\right)=-\\left(z-z^{\\prime}\\right)\\left(\\frac{1+j k_i r}{4 \\pi r^3}\\right) e^{-j k_i r} \\quad \\text { where } r=\\left|\\mathbf{r}-\\mathbf{r}^{\\prime}\\right|<br>\\end{aligned}<br>$$<\/p>\n\n\n\n<p>\u7136\u540e\u5c06\u77e9\u9635\u5143\u7d20\u5c55\u5f00\u4e3a\u57fa\u51fd\u6570\u7684\u4e0d\u540c\u5206\u91cf\u7684\u7ec4\u5408\uff0c\u5206\u91cf\u4f5c\u7528\u5728\u5e73\u79fb\u4e0d\u53d8\u51fd\u6570\u4e0a\u3002\u6700\u7ec8\u77e9\u9635\u7684\u6240\u6709\u5143\u7d20\uff0c\u90fd\u6709\u5982\u4e0b\u5f62\u5f0f\u3002\u8fd9\u6837\u7684\u5f62\u5f0f\u53ef\u4ee5\u52a0\u901f\u8fb9\u754c\u5143\u77e9\u9635\u7684\u6784\u9020\u548c\u6c42\u89e3\u3002<\/p>\n\n\n\n<p>$$<br>\\int_{\\mathbf{f}m} \\int{\\mathbf{f}_n} \\psi_m(\\mathbf{r}) g\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right) \\xi_n\\left(\\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} d \\mathbf{r}<br>$$<\/p>\n\n\n\n<p>\u6700\u7ec8\u5f97\u5230\uff1a<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>Here, $x, y, z, x^{\\prime}, y^{\\prime}, z^{\\prime}$ are the Cartesian components of $\\mathbf{r}, \\mathbf{r}^{\\prime}$. Now we have for $i=1,2$ :<\/p>\n\n\n\n<p>$$ \\begin{aligned} A_{\\mathrm{EJ}, i}^{m n} &amp;= j \\omega \\mu_i \\int_{\\mathbf{f}_m} \\int_{\\mathbf{f}_n} \\mathbf{f}_{m x}(\\mathbf{r}) \\, g_{1, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right) \\, \\mathbf{f}_{n x}\\left(\\mathbf{r}^{\\prime}\\right) \\, d \\mathbf{r}^{\\prime} \\, d \\mathbf{r} \\\\ &amp; \\quad + j \\omega \\mu_i \\int_{\\mathbf{f}_m} \\int_{\\mathbf{f}_n} \\mathbf{f}_{m y}(\\mathbf{r}) \\, g_{1, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right) \\, \\mathbf{f}_{n y}\\left(\\mathbf{r}^{\\prime}\\right) \\, d \\mathbf{r}^{\\prime} \\, d \\mathbf{r} \\\\ &amp; \\quad + j \\omega \\mu_i \\int_{\\mathbf{f}_m} \\int_{\\mathbf{f}_n} \\mathbf{f}_{m z}(\\mathbf{r}) \\, g_{1, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right) \\, \\mathbf{f}_{n z}\\left(\\mathbf{r}^{\\prime}\\right) \\, d \\mathbf{r}^{\\prime} \\, d \\mathbf{r} \\\\ &amp; \\quad &#8211; \\frac{j}{\\omega \\varepsilon_i} \\int_{\\mathbf{f}_m} \\int_{\\mathbf{f}_n} \\nabla \\cdot \\mathbf{f}_m(\\mathbf{r}) \\, g_{1, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right) \\, \\nabla^{\\prime} \\cdot \\mathbf{f}_n\\left(\\mathbf{r}^{\\prime}\\right) \\, d \\mathbf{r}^{\\prime} \\, d \\mathbf{r} \\end{aligned} \\tag{S.18} $$<\/p>\n\n\n\n<p>where $\\mathbf{f}<em>{m x}, \\mathbf{f}<\/em>{m y}, \\mathbf{f}_{m z}$ are the $x, y, z$ components of the vector basis function $\\mathbf{f}_m$ . Similarly, we have:<\/p>\n\n\n\n<p>$$ \\begin{aligned} &amp; A_{\\mathrm{EM}, i}^{m n}=A_{\\mathrm{HJ}, i}^{m n}=\\int_{\\mathbf{f}m} \\int{\\mathbf{f}n} \\mathbf{f}{m z}(\\mathbf{r}) g_{2, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right) \\mathbf{f}{n y}\\left(\\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} d \\mathbf{r} \\\\ &amp; -\\int{\\mathbf{f}m} \\int{\\mathbf{f}n} \\mathbf{f}{m y}(\\mathbf{r}) g_{2, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right) \\mathbf{f}{n z}\\left(\\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} d \\mathbf{r} \\\\ &amp; +\\int{\\mathbf{f}m} \\int{\\mathbf{f}n} \\mathbf{f}{m x}(\\mathbf{r}) g_{3, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right) \\mathbf{f}{n z}\\left(\\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} d \\mathbf{r} \\\\ &amp; -\\int{\\mathbf{f}m} \\int{\\mathbf{f}n} \\mathbf{f}{m z}(\\mathbf{r}) g_{3, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right) \\mathbf{f}{n x}\\left(\\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} d \\mathbf{r} \\\\ &amp; +\\int{\\mathbf{f}m} \\int{\\mathbf{f}n} \\mathbf{f}{m y}(\\mathbf{r}) g_{4, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right) \\mathbf{f}{n x}\\left(\\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} d \\mathbf{r} \\\\ &amp; -\\int{\\mathbf{f}m} \\int{\\mathbf{f}n} \\mathbf{f}{m x}(\\mathbf{r}) g_{4, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right) \\mathbf{f}_{n y}\\left(\\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} d \\mathbf{r} \\end{aligned} \\tag{S.19} $$<\/p>\n\n\n\n<p>Lastly, we also have:<\/p>\n\n\n\n<p>$$ \\begin{aligned} A_{\\mathrm{HM}, i}^{m n} &amp; =-j \\omega \\varepsilon_i \\int_{\\mathbf{f}m} \\int{\\mathbf{f}n} \\mathbf{f}{m x}(\\mathbf{r}) g_{1, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right) \\mathbf{f}{n x}\\left(\\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} d \\mathbf{r} \\\\ &amp; -j \\omega \\varepsilon_i \\int{\\mathbf{f}m} \\int{\\mathbf{f}n} \\mathbf{f}{m y}(\\mathbf{r}) g_{1, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right) \\mathbf{f}{n y}\\left(\\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} d \\mathbf{r} \\\\ &amp; -j \\omega \\varepsilon_i \\int{\\mathbf{f}m} \\int{\\mathbf{f}n} \\mathbf{f}{m z}(\\mathbf{r}) g_{1, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right) \\mathbf{f}{n z}\\left(\\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} d \\mathbf{r} \\\\ &amp; +\\frac{j}{\\omega \\mu_i} \\int{\\mathbf{f}m} \\int{\\mathbf{f}n} \\nabla \\cdot \\mathbf{f}_m(\\mathbf{r}) g{1, i}\\left(\\mathbf{r}-\\mathbf{r}^{\\prime}\\right) \\nabla^{\\prime} \\cdot \\mathbf{f}_n\\left(\\mathbf{r}^{\\prime}\\right) d \\mathbf{r}^{\\prime} d \\mathbf{r} \\end{aligned} \\tag{S.20} $$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>\u8fd9\u90e8\u5206\u4ee3\u7801\u5728<a href=\"https:\/\/github.com\/blaire9989\/BEMsim3D\/blob\/main\/MVProd0.cpp\">MVProd\u7c7b<\/a>\u4e2d\uff0c\u770b\u4e0d\u61c2\u4e5f\u6ca1\u95ee\u9898\uff0c\u56e0\u4e3a\u4e0a\u9762\u90fd\u662f\u6211\u778e\u5199\u778e\u6284\u7684\uff0c\u5df2\u7ecf\u8d85\u51fa\u56fe\u5f62\u5b66\u7684\u7814\u7a76\u8303\u7574\u4e86\u3002<\/p>\n\n\n\n<p>\u53e6\u5916\u4f5c\u8005\u8fd8\u8ba8\u8bba\u4e86EFIE\u548cMFIE\u7684\u5bf9\u79f0\u6027\u3002\u6b63\u662f\u8fd9\u79cd\u5bf9\u79f0\u6027\u4f7f\u5f97\u8ba1\u7b97\u6548\u7387\u548c\u7a7a\u95f4\u5229\u7528\u7387\u66f4\u4f4e\u3002\u6ce8\u610f\u5230\uff1a<\/p>\n\n\n\n<p><br>$$<br>A_{EJ} = A_{EJ}^T, \\quad A_{HM} = A_{HM}^T \\tag{19}<br>$$<\/p>\n\n\n\n<p>$$<br>A_{EM} = A_{EM}^T, \\quad A_{HJ} = A_{HJ}^T, \\quad A_{EM} = A_{HJ}\\tag{20}<br>$$<\/p>\n\n\n\n<p>\u7531\u4e8e\u77e9\u9635\u662f\u5bf9\u79f0\u7684\uff0c\u6211\u4eec\u4e0d\u9700\u8981\u8ba1\u7b97\u6240\u6709\u7684\u77e9\u9635\u5143\u7d20\uff0c\u4e5f\u4e0d\u7528\u5b58\u50a8\u6240\u6709\u77e9\u9635\u5143\u7d20\u3002<\/p>\n\n\n\n<p>\u5728\u89e3\u51fa\u8868\u9762\u7535\u6d41\u5bc6\u5ea6\u540e\uff0c\u53ef\u4ee5\u518d\u4f7f\u7528\u516c\u5f0f(6)\u6765\u8ba1\u7b97\u4ece\u6563\u5c04\u8868\u9762\u5411\u5916\u4f20\u64ad\u7684\u6563\u5c04\u573a\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">3.2 Rough Surface Scattering: The Specifics<\/h4>\n\n\n\n<p>\u5728\u6a21\u62df\u7535\u78c1\u6ce2\u4e0e\u7c97\u7cd9\u8868\u9762\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u65f6\uff0c\u8868\u9762\u7684\u4e0d\u89c4\u5219\u51e0\u4f55\u7ed3\u6784\u4f1a\u5bf9\u6563\u5c04\u7279\u6027\u9020\u6210\u663e\u8457\u5f71\u54cd\u3002\u4f5c\u8005\u5bf9\u7c97\u7cd9\u8868\u9762\u79bb\u6563\u5316\uff0c\u5c06\u8fde\u7eed\u7684\u8868\u9762\u5212\u5206\u4e3a\u591a\u4e2a\u5355\u5143\uff0c\u6bcf\u4e2a\u5355\u5143\u90fd\u53ef\u4ee5\u5e94\u7528PMCHWT\u65b9\u7a0b\u505a\u6570\u503c\u8ba1\u7b97\u7684\u65b9\u6cd5\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">3.2.1 Rough Surface Samples<\/h5>\n\n\n\n<p>\u5c06\u7c97\u7cd9\u8868\u9762\u8868\u793a\u4e3a\u4e00\u4e2a\u4e8c\u7ef4\u7684<strong>\u9ad8\u5ea6\u573a\uff08height field\uff09<\/strong>\uff0c\u7136\u540e\u5c06\u8be5\u9ad8\u5ea6\u573a\u8fdb\u884c\u79bb\u6563\u5316\u5904\u7406\uff0c\u5212\u5206\u4e3a\u591a\u4e2a\u77e9\u5f62\u5355\u5143\u3002<\/p>\n\n\n\n<p>\u6bcf\u6b21\u6a21\u62df\u53ea\u8003\u8651\u5c3a\u5bf8\u4e3a $$L_x \\times L_y$$ \u7684\u8868\u9762\u6837\u672c\u3002\u5e76\u4e14\u9009\u62e9\u4e00\u4e2a\u6b65\u8fdb $d$ \u6765\u5b9a\u4e49\u79bb\u6563\u5316\u7684\u7f51\u683c\u3002<\/p>\n\n\n\n<p><br>$$<br>x_s = s \\cdot d, \\quad s = 0, 1, \\ldots, N_x<br>\\<br>y_t = t \\cdot d, \\quad t = 0, 1, \\ldots, N_y<br>\\tag{21}<br>$$<\/p>\n\n\n\n<p><br>\u5176\u4e2d\uff0c$ N_x = L_x \/ d$ \u548c $N_y = L_y \/ d$ \uff0c\u5206\u522b\u8868\u793a\u5728 $x$ \u548c $y$ \u65b9\u5411\u4e0a\u88ab\u5212\u5206\u6210\u7684\u5355\u5143\u6570\u3002\u5728\u6bcf\u4e2a\u79bb\u6563\u70b9 $(x_s, y_t)$ \u90fd\u6709\u4e00\u4e2a\u9ad8\u5ea6\u573a $h(x_s, y_t)$ \u51fd\u6570\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-510-1024x505.png\" alt=\"\" class=\"wp-image-1671 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"505\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-510-1024x505.png\" alt=\"\" class=\"wp-image-1671 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-510-1024x505.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-510-300x148.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-510-768x379.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-510.png 1266w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u4f5c\u8005\u603b\u7ed3\u4e86\uff0c\u7c97\u7cd9\u8868\u9762\u7684\u9ad8\u5ea6\u53d8\u5316\u5c3a\u5ea6\u975e\u5e38\u5c0f\uff0c\u901a\u5e38\u53ea\u6709\u51e0\u5fae\u7c73\u3002\u4e5f\u5c31\u662f\u548c\u53ef\u89c1\u5149\u7535\u78c1\u6ce2\u6ce2\u957f\u76f8\u5f53\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">3.2.2 Basis Elements and Functions<\/h5>\n\n\n\n<p>\u6bcf\u4e2a\u57fa\u5143\u6709\u56db\u4e2a\u89d2\uff0c\u6bcf\u4e2a\u89d2\u6709\u4e0d\u540c\u7684\u9ad8\u5ea6\uff0c\u5e76\u4e14\u6bcf\u4e2a\u89d2\u5bf9\u7535\u6d41\u78c1\u6d41\u8d21\u732e\u90fd\u4e0d\u540c\u3002\u56e0\u6b64\u5728\u6bcf\u4e2a\u5c0f\u65b9\u5757\u4e0a\uff0c\u90fd\u5b9a\u4e49\u4e86\u56db\u4e2a\u57fa\u51fd\u6570\uff0c\u8fd1\u4f3c\u8868\u793a\u6bcf\u4e2a\u5c0f\u65b9\u5757\u4e0a\u7684\u60c5\u51b5\u3002<\/p>\n\n\n\n<p>\u5728\u5927\u591a\u6570\u6a21\u62df\u4e2d\uff0c\u6b65\u957f $d$ \u9009\u53d6\u4e3a\u6ce2\u957f\u7684 $\\lambda \/ 16$ \u5de6\u53f3\uff0c\u4ee5\u786e\u4fdd\u7cbe\u5ea6\u3002<\/p>\n\n\n\n<p>\u6bcf\u4e2a\u57fa\u5143\u7531\u4e24\u4e2a\u53c2\u6570 u \u548c v \u53c2\u6570\u5316\uff0c\u8303\u56f4\u5747\u5728 [-1, 1]\u3002<\/p>\n\n\n\n<p>\u57fa\u5143\u7684\u5f62\u72b6\u901a\u8fc7\u4e00\u4e2a\u53cc\u7ebf\u6027\u51fd\u6570 $\\mathbf{r}(u, v)$ \u6765\u8868\u793a\uff0c\u5176\u4e2d $(s, t)$ \u8868\u793a\u5f53\u524d\u57fa\u5143\u7684\u7d22\u5f15\uff0c\u4e14\u57fa\u5143\u7531\u56db\u4e2a\u9876\u70b9\u7684\u5750\u6807\u51b3\u5b9a\uff1a<\/p>\n\n\n\n$$\n\\begin{aligned} \\mathbf{r}(u, v) = \n&#038;\\frac{(1 &#8211; u)(1 &#8211; v)}{4} \\mathbf{p}_{s-1, t-1} + \\\\ \n&#038;\\frac{(1 &#8211; u)(1 + v)}{4} \\mathbf{p}_{s-1, t} + \\\\ \n&#038;\\frac{(1 + u)(1 &#8211; v)}{4} \\mathbf{p}_{s, t-1} + \\\\\n&#038;\\frac{(1 + u)(1 + v)}{4} \\mathbf{p}_{s, t} \n\\end{aligned} \\tag{22}\n$$\n\n\n\n<p>\u5176\u4e2d\uff0c $\\mathbf{p}{s, t} = (x_s, y_t, z{s, t})$ \u662f\u57fa\u5143\u7684\u56db\u4e2a\u9876\u70b9\u5750\u6807\u3002<\/p>\n\n\n\n<p>\u5728\u6bcf\u4e2a\u77e9\u5f62\u57fa\u5143\u4e0a\u5b9a\u4e49\u56db\u4e2a\u57fa\u51fd\u6570 $f_1(u, v), f_2(u, v), f_3(u, v), f_4(u, v)$ \uff0c\u5b83\u4eec\u7684\u5f62\u5f0f\u4e3a\uff1a<\/p>\n\n\n\n<p><br>$$<br>\\begin{aligned}<br>&amp; f_1(u, v) = \\frac{(1 &#8211; u)}{J(u, v)} \\frac{\\partial \\mathbf{r}(u, v)}{\\partial u}, \\quad f_2(u, v) = \\frac{(1 + u)}{J(u, v)} \\frac{\\partial \\mathbf{r}(u, v)}{\\partial u} \\\\<br>&amp; f_3(u, v) = \\frac{(1 &#8211; v)}{J(u, v)} \\frac{\\partial \\mathbf{r}(u, v)}{\\partial v}, \\quad f_4(u, v) = \\frac{(1 + v)}{J(u, v)} \\frac{\\partial \\mathbf{r}(u, v)}{\\partial v}<br>\\end{aligned}<br>\\tag{23}<br>$$<\/p>\n\n\n\n<p><br>\u8fd9\u91cc\u7684 Jacobian $J(u, v)$ \u8868\u793a\u5982\u4e0b\uff1a<\/p>\n\n\n\n<p><br>$$<br>J(u, v) = \\left| \\frac{\\partial \\mathbf{r}(u, v)}{\\partial u} \\times \\frac{\\partial \\mathbf{r}(u, v)}{\\partial v} \\right|<br>\\tag{24}<br>$$<\/p>\n\n\n\n<p><br>Jacobian \u7684\u5f15\u5165\u7528\u4e8e\u8f6c\u6362\u5750\u6807\u7cfb\u5e76\u786e\u4fdd\u57fa\u51fd\u6570\u5728\u4e0d\u540c\u7684 $u, v$ \u65b9\u5411\u4e0a\u5177\u6709\u5408\u9002\u7684\u6bd4\u4f8b\u5173\u7cfb\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-511-1024x733.png\" alt=\"\" class=\"wp-image-1672 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"733\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-511-1024x733.png\" alt=\"\" class=\"wp-image-1672 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-511-1024x733.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-511-300x215.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-511-768x549.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-511.png 1244w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<h5 class=\"wp-block-heading\">3.2.3 Gaussian Beam Incidence<\/h5>\n\n\n\n<p>\u7531\u4e8e\u4e0d\u592a\u4e86\u89e3\u5149\u5b66\uff0c\u4ee5\u4e0b\u4e3a\u4e2a\u4eba\u7406\u89e3\u3002\u9ad8\u65af\u5149\u675f\u662f\u6fc0\u5149\u53d1\u5c04\u51fa\u53bb\u7684\u65f6\u5019\uff0c\u5728\u884c\u6ce2\u573a\u4e2d\u95f4\u90e8\u5206\u51fa\u73b0\u5f80\u5185\u90e8\u51f9\u9677\u7684\u4e00\u79cd\u73b0\u8c61\uff0c\u6362\u4e00\u53e5\u8bdd\u8bf4\u9ad8\u65af\u5149\u675f\u662f\u63cf\u8ff0\u5149\u5728\u6a2a\u622a\u9762\u4e0a\u7684\u80fd\u91cf\u5206\u5e03\u3002\u800c\u5e73\u9762\u6ce2\u548c\u7403\u9762\u6ce2\u7684\u91cd\u70b9\u662f\u63cf\u8ff0\u80fd\u91cf\u4f20\u64ad\u7684\u65b9\u5411\u3002\u5728\u4f20\u64ad\u8fc7\u7a0b\u4e2d\uff0c\u9ad8\u65af\u5149\u675f\u7684\u6ce2\u524d\u5f62\u72b6\u8fd1\u4f3c\u4e3a\u7403\u9762\u6ce2\u3002<\/p>\n\n\n\n<p>Gouy \u76f8\u4f4d\uff08Gouy phase\uff09\u662f\u9ad8\u65af\u5149\u675f\u4f20\u64ad\u4e2d\u7684\u4e00\u79cd\u76f8\u4f4d\u5ef6\u8fdf\u6548\u5e94\uff0c\u5149\u675f\u901a\u8fc7\u7126\u70b9\u540e\u76f8\u4f4d\u4f1a\u989d\u5916\u589e\u52a0\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u9ad8\u65af\u5149\u675f\u6ee1\u8db3\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u7ec4\u5728\u508d\u8f74\u6761\u4ef6\u4e0b\u7684\u4e00\u4e2a\u89e3\uff0c\u53ef\u4ee5\u8fd1\u4f3c\u4e3a\u975e\u5747\u5300\u7684\u7403\u9762\u6ce2\u3002\u611f\u89c9\u76ee\u524d\u4e5f\u4e0d\u7528\u592a\u6df1\u5165\u7814\u7a76\u3002<\/p>\n\n\n\n<p>\u6709\u5173\u9ad8\u65af\u5149\u675f\u7684\u8d44\u6599\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>https:\/\/en.wikipedia.org\/wiki\/Gaussian_beam<\/li>\n\n\n\n<li><a href=\"https:\/\/zhuanlan.zhihu.com\/p\/563563288\">\u6fc0\u5149\u539f\u7406(Principle of Laser)\u7b14\u8bb0 &#8211; Senner\u7684\u6587\u7ae0 &#8211; \u77e5\u4e4e<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/mp.weixin.qq.com\/s\/ajih-cfdUl2znolc05f2Iw\">\u9ad8\u65af\u5149\u675f\uff1a\u57fa\u672c\u516c\u5f0f\u548c\u5e94\u7528 &#8211; \u4e2d\u56fd\u5149\u5b66<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/zhuanlan.zhihu.com\/p\/149476202\">\u9ad8\u65af\u5149\u675f\uff1a\u57fa\u672c\u516c\u5f0f\u548c\u5e94\u7528 &#8211; Andy\u7684\u6587\u7ae0 &#8211; \u77e5\u4e4e<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/zhuanlan.zhihu.com\/p\/697532459\">\u5149\u5b50\u7684\u8f68\u9053\u89d2\u52a8\u91cf &#8211; Godfly\u7684\u6587\u7ae0 &#8211; \u77e5\u4e4e<\/a><\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image aligncenter\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-6.gif\" alt=\"\" class=\"wp-image-1685 lazyload\"\/><noscript><img decoding=\"async\" width=\"548\" height=\"124\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-6.gif\" alt=\"\" class=\"wp-image-1685 lazyload\"\/><\/noscript><\/figure>\n\n\n\n<p>\u56de\u5230\u539f\u8bba\u6587\uff0c\u9ad8\u65af\u5149\u675f\u597d\u5904\u5728\u4e8e\u53ef\u4ee5\u63a7\u5236\u5165\u5c04\u573a\u7684\u5927\u5c0f\uff0c\u8fdb\u800c\u80fd\u591f\u63a7\u5236\u4e00\u4e2a\u7a0d\u5fae\u6bd4\u7167\u5c04\u533a\u57df\u5927\u7684\u533a\u57df\u7684\u8868\u9762\u611f\u5e94\u7535\u6d41\u5bc6\u5ea6\u662f\u975e\u96f6\u6570\u503c\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-513.png\" alt=\"\" class=\"wp-image-1674 lazyload\"\/><noscript><img decoding=\"async\" width=\"880\" height=\"328\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-513.png\" alt=\"\" class=\"wp-image-1674 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-513.png 880w, https:\/\/remoooo.com\/wp-content\/uploads\/image-513-300x112.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-513-768x286.png 768w\" sizes=\"(max-width: 880px) 100vw, 880px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u9ad8\u65af\u5149\u675f\u662f\u4e00\u79cd\u7535\u78c1\u6ce2\uff0c\u5176\u632f\u5e45\u5728\u5782\u76f4\u4e8e\u4f20\u64ad\u65b9\u5411\u7684\u5e73\u9762\u4e0a\u5448\u4e8c\u7ef4\u9ad8\u65af\u5206\u5e03[Paschotta 2008]\uff0c\u5b83\u7684\u80fd\u91cf\u4e3b\u8981\u96c6\u4e2d\u5728\u5149\u675f\u4e2d\u5fc3\u9644\u8fd1\u3002\u770b\u4e0a\u65b9\u56fe(a)\uff0c\u9ad8\u65af\u5149\u675f\u53ef\u4ee5\u7528\u7126\u5e73\u9762 $P$ \u3001\u4e2d\u5fc3\u70b9 $o$ \u3001\u548c\u5149\u675f\u8170\u5f84\uff08beam waist\uff09 $w$ \u6765\u63cf\u8ff0\u3002\u573a\u5f3a\u968f\u4f4d\u7f6e\u7684\u8870\u51cf\u5173\u7cfb\u4e3a $e^{-r^2 \/ w^2}$ \uff0c\u5f53\u8ddd\u79bb\u4e2d\u5fc3\u8d85\u8fc7 $2.5w$ \u65f6\uff0c\u573a\u5f3a\u8870\u51cf\u81f3\u6781\u5c0f\uff0c\u53ef\u8ba4\u4e3a\u51e0\u4e4e\u4e3a\u96f6\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-514.png\" alt=\"\" class=\"wp-image-1675 lazyload\"\/><noscript><img decoding=\"async\" width=\"656\" height=\"370\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-514.png\" alt=\"\" class=\"wp-image-1675 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-514.png 656w, https:\/\/remoooo.com\/wp-content\/uploads\/image-514-300x169.png 300w\" sizes=\"(max-width: 656px) 100vw, 656px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u867d\u7136\u4f46\u662f\uff0c\u9ad8\u65af\u5149\u675f\u4e5f\u5177\u6709\u4e00\u5b9a\u7684\u53d1\u6563\u6027\u3002\u53d1\u6563\u89d2 $\\theta$ \u8fd1\u4f3c\u4e0e\u6ce2\u957f $\\lambda$ \u6210\u6b63\u6bd4\uff0c\u4e0e\u5149\u675f\u8170\u5f84 $w$ \u6210\u53cd\u6bd4\u3002\u516c\u5f0f\uff1a<\/p>\n\n\n\n<p><br>$$<br>\\theta = \\frac{\\lambda}{\\pi \\eta w} \\tag{25}<br>$$<\/p>\n\n\n\n<p><br>\u5f53\u5149\u675f\u659c\u7740\u5c04\u5165\u8868\u9762\u65f6\uff0c\u9ad8\u65af\u5149\u675f\u5728\u7126\u5e73\u9762\u4e0a\u6709\u4e00\u4e2a\u692d\u5706\u5f62\u7684\u6a2a\u622a\u9762\u3002\u5982\u4e0b\u56fe\u6240\u793a\u3002\u5728\u4e24\u4e2a\u5782\u76f4\u65b9\u5411\u4e0a\u6709\u4e0d\u540c\u7684\u8170\u6591\u5c3a\u5bf8\uff08beam waist\uff09\uff1a\u4e00\u4e2a\u5e73\u884c\u4e8e\u5165\u5c04\u65b9\u5411\u548c\u8868\u9762\u6cd5\u7ebf\u7684\u5e73\u9762\uff0c\u53e6\u4e00\u4e2a\u4e0e\u4e4b\u5782\u76f4\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-515.png\" alt=\"\" class=\"wp-image-1676 lazyload\"\/><noscript><img decoding=\"async\" width=\"822\" height=\"342\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-515.png\" alt=\"\" class=\"wp-image-1676 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-515.png 822w, https:\/\/remoooo.com\/wp-content\/uploads\/image-515-300x125.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-515-768x320.png 768w\" sizes=\"(max-width: 822px) 100vw, 822px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u4e3a\u4e86\u4fdd\u8bc1\u4e0d\u540c\u65b9\u5411\u4e0a\u7684\u9ad8\u65af\u5149\u675f\u5728\u8868\u9762\u4e0a\u7684\u7167\u5c04\u533a\u57df\u76f8\u540c\uff0c\u8fd9\u91cc\u5f15\u5165\u4e86\u4e24\u4e2a\u6a2a\u5411\u65b9\u5411\u7684\u675f\u8170\u5bbd\u5ea6\uff1a<\/p>\n\n\n\n<p><br>$$<br>w_1 = w, \\quad w_2 = w \\cos \\theta_i\\tag{26}<br>$$<\/p>\n\n\n\n<p><br>\u5373\u4f7f\u5728\u4e0d\u540c\u5165\u5c04\u89d2\u5ea6\u4e0b\uff0c\u8868\u9762\u4e0a\u7684\u7167\u5c04\u9762\u79ef\u4fdd\u6301\u4e00\u81f4\u3002\u8fd9\u5bf9\u63a8\u5bfcBRDF\u800c\u8a00\u975e\u5e38\u91cd\u8981\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-516.png\" alt=\"\" class=\"wp-image-1677 lazyload\"\/><noscript><img decoding=\"async\" width=\"944\" height=\"548\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-516.png\" alt=\"\" class=\"wp-image-1677 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-516.png 944w, https:\/\/remoooo.com\/wp-content\/uploads\/image-516-300x174.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-516-768x446.png 768w\" sizes=\"(max-width: 944px) 100vw, 944px\" \/><\/noscript><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">4. IMPLEMENTATION AND ACCELERATION<\/h3>\n\n\n\n<p>\u4f46\u662f\u5982\u679c\u6309\u4e0a\u9762\u7684\u65b9\u6cd5\u786c\u7b97\uff0c\u662f\u4e0d\u53ef\u80fd\u4f1a\u6709\u7ed3\u679c\u7684\u3002\u56e0\u6b64\u9700\u8981\u4e00\u4e9b\u52a0\u901f\u624b\u6bb5\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">4.1 The Adaptive Integral Method<\/h4>\n\n\n\n<p>\u60f3\u8981\u76f4\u63a5\u8ba1\u7b97\u4e0a\u9762\u516c\u5f0f(12)\u7684\u65b9\u7a0b\u7ec4\uff0c\u8ba1\u7b97\u91cf\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\u3002<\/p>\n\n\n\n<p>$$ \\begin{bmatrix} A_{EJ} &amp; A_{EM} \\ A_{HJ} &amp; A_{HM} \\end{bmatrix} \\begin{bmatrix} I_J \\ I_M \\end{bmatrix} = \\begin{bmatrix} V_E \\ V_H \\end{bmatrix} \\tag{12} $$<\/p>\n\n\n\n<p><br>\u6309\u7167\u539f\u672c\u7684\u601d\u8def\uff0c\u7528 $N$ \u4e2a\u57fa\u51fd\u6570\u6765\u8868\u793a\u7535\u6d41\u78c1\u6d41\u5bc6\u5ea6\uff0c\u77e9\u9635\u7684\u89c4\u6a21\u5c31\u662f $2N \\times 2N$ \u3002\u5982\u679c\u76f4\u63a5\u6c42\u89e3\u77e9\u9635\uff08LU\u5206\u89e3\uff0cCholesky\u5206\u89e3\u7b49\uff09\uff0c\u603b\u590d\u6742\u5ea6\u53ef\u80fd\u5728 $\\mathcal{O}(N^3)$ \u3002\u5c31\u7b97\u7528\u5171\u8f6d\u68af\u5ea6\u6cd5\u603b\u590d\u6742\u5ea6\u4e5f\u5728 $\\mathcal{O}(N^2)$ \u3002\u5728\u4e00\u4e9b\u5c0f\u89c4\u6a21\u7684\u4eff\u771f\u4e2d\uff0c\u57fa\u51fd\u6570\u7684\u89c4\u6a21\u5927\u6982\u5728 $960*960$ \uff0c\u5b58\u50a8\u9700\u6c42\u5927\u6982\u5c31\u5728 29.4GB \u3002\u7528\u4e86Adaptive Integral Method, AIM\uff0c\u63098\u5b57\u8282\u6765\u7b97\u603b\u5b58\u50a8\u9700\u6c42\u7ea6\u5728 76.8 MB\u3002\u7a76\u7adf\u662f\u4ec0\u4e48\u65b9\u6cd5\u8fd9\u4e48\u795e\u5947\u5462\uff1f\u5c0f\u7f16\u63a5\u4e0b\u6765\u5c31\u5e26\u5927\u5bb6\u4e00\u8d77\u770b\u770b\u5427\uff01<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">4.1.1 Approximating Matrix Elements<\/h5>\n\n\n\n<p>AIM\u6700\u521d\u662f\u7531Bleszynski\u7b49\u4eba[1996]\u63d0\u51fa\u7684\u3002AIM\u7684\u6838\u5fc3\u601d\u60f3\u662f\u5c06\u6bcf\u4e2a\u57fa\u51fd\u6570\u7684\u4f5c\u7528\u8fd1\u4f3c\u4e3a\u4e00\u7ec4\u70b9\u6e90\u7684\u4f5c\u7528\uff0c\u907f\u514d\u76f4\u63a5\u8ba1\u7b97\u6bcf\u4e00\u5bf9\u57fa\u51fd\u6570\u4e4b\u95f4\u7684\u7cbe\u786e\u76f8\u4e92\u4f5c\u7528\uff0c\u540c\u65f6\u901a\u8fc7FFT\u5c06\u5404\u57fa\u51fd\u6570\u7684\u5f71\u54cd\u4f20\u64ad\u5f00\u6765\uff0c\u63d0\u5347\u8ba1\u7b97\u6548\u7387\u3002<\/p>\n\n\n\n<p>AIM\u4e2d\u77e9\u9635\u5143\u7d20\u7684\u8ba1\u7b97\u65b9\u5f0f\u662f\u5bf9\u77e9\u9635\u5143\u7d20\u7684\u67d0\u4e9b\u9879\u7684\u7ebf\u6027\u7ec4\u5408\u6765\u8fd1\u4f3c\u8ba1\u7b97\u3002\u8fd9\u5c31\u662f\u4e3a\u4ec0\u4e48\u5728\u4e0a\u9762\u516c\u5f0f(13)-(16)\u540e\u9762\u8ba9\u8bfb\u8005\u4eec\u8fdb\u4e00\u6b65\u63a8\u5bfc\uff0c\u63a8\u5bfc\u6700\u7ec8\u7684\u7ed3\u679c\u4f7f\u5176\u7b26\u5408AIM\u7684\u5f62\u5f0f\u3002<\/p>\n\n\n\n<p>$$<br>\\int_{f_m} \\int_{f_n} \\psi_m(r) g(r &#8211; r{\\prime}) \\xi_n(r{\\prime}) \\, dr{\\prime} \\, dr<br>\\tag{27}<br>$$<\/p>\n\n\n\n<p><br>AIM\u9996\u5148\u4f1a\u5728\u5305\u542b\u7535\u78c1\u573a\u548c\u573a\u6e90\u7684\u7a7a\u95f4\u5185\u521b\u5efa\u4e00\u4e2a\u5168\u5c403D\u7b1b\u5361\u5c14\u7f51\u683c\uff0c\u5982\u4e0b\u56fe(6)\u6240\u793a\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-517-1024x583.png\" alt=\"\" class=\"wp-image-1678 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"583\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-517-1024x583.png\" alt=\"\" class=\"wp-image-1678 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-517-1024x583.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-517-300x171.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-517-768x438.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-517.png 1362w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u4e3a\u4e86\u8fdb\u4e00\u6b65\u5316\u7b80\u516c\u5f0f(27)\uff0cAIM\u7b97\u6cd5\u4f1a\u8ba9\u539f\u672c\u7684\u57fa\u51fd\u6570\u8fd1\u4f3c\u4e3a\u5728\u8fd9\u4e2a\u4e09\u7ef4\u7b1b\u5361\u5c14\u5750\u6807\u4e0a\u4e00\u7ec4\u7f51\u683c\u70b9\u4e0a\u7684\u70b9\u6e90\u3002\u8bf4\u767d\u4e86\u5c31\u662f\u8fde\u7eed\u53d8\u79bb\u6563\uff0c\u5e76\u4e14\u65b9\u4fbf\u540e\u7eedFFT\u3002<\/p>\n\n\n\n<p>$$ \\psi_m(r) \\approx \\tilde{\\psi}m(r) := \\sum{p \\in S_m} \\Lambda_{mp} \\delta^3(r &#8211; p) \\\\ \\xi_n(r{\\prime}) \\approx \\tilde{\\xi}n(r{\\prime}) := \\sum{q \\in S_n} \\Lambda{\\prime}_{nq} \\delta^3(r{\\prime} &#8211; q) \\tag{28} $$<\/p>\n\n\n\n<p>\u5c06\u516c\u5f0f(27)\u4ee3\u5165\u516c\u5f0f(28)\uff0c\u6362\u53e5\u8bdd\u8bf4\uff0c\u5c31\u662f\u5c06\u53cc\u91cd\u79ef\u5206\u7684\u5f62\u5f0f\u8f6c\u5316\u4e3a\u4e86\u4e00\u4e2a\u53cc\u91cd\u6c42\u548c\u7684\u5f62\u5f0f\u3002<\/p>\n\n\n\n<p>$$<br>\\sum{p \\in S_m} \\sum_{q \\in S_n} \\Lambda_{mp} g(p &#8211; q) \\Lambda{\\prime}_{nq}<br>\\tag{29}<br>$$<\/p>\n\n\n\n<p>&nbsp;<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u65b9\u6cd5\u8be6\u7ec6\u53c2\u8003\uff1a<\/p>\n\n\n\n<p>Kai Yang and Ali E Yilmaz. 2011. Comparison of precorrected FFT\/adaptive integral method matching schemes. Microwave and Optical Technology Letters 53, 6 (2011), 1368\u20131372.<\/p>\n<\/blockquote>\n\n\n\n<h5 class=\"wp-block-heading\">4.1.2 Base and Correction Matrices<\/h5>\n\n\n\n<p>\u57fa\u4e8e\u516c\u5f0f(29)\u5b9a\u4e49\u4e00\u7ec4\u57fa\u7840\u8fd1\u4f3c\uff08base approximation\uff09\u77e9\u9635 $B_{EJ}, B_{EM}, B_{HJ}, B_{HM}$ \u4f5c\u4e3a\u8fd1\u4f3c\uff0c\u4e13\u95e8\u5904\u7406\u8ddd\u79bb\u8f83\u8fdc\u7684\u57fa\u51fd\u6570\u5bf9\u3002\u8fd9\u4e9b\u77e9\u9635\u901a\u8fc7\u5f15\u5165 $\\Lambda$ \u77e9\u9635\u548c\u5377\u79ef\u7b49\u64cd\u4f5c\u5316\u7b80\u8ba1\u7b97\u3002\u540c\u65f6\uff0c\u5bf9\u4e8e\u8ddd\u79bb\u8f83\u8fd1\uff08$d_{near}$\uff09\u7684\u57fa\u51fd\u6570\u5bf9\uff0c\u518d\u5f15\u5165\u4fee\u6b63\u77e9\u9635\uff08Correction Matrices\uff09\u6765\u51cf\u5c11\u8bef\u5dee\u3002 $C_{EJ}, C_{EM}, C_{HJ}, C_{HM}$ \u662f\u4e00\u79cd\u7a00\u758f\u77e9\u9635\uff0c\u5b9a\u4e49\u5982\u4e0b\uff1a<\/p>\n\n\n\n<p>$$ C_{\\mathrm{X}}^{m n}=\\left\\{\\begin{array}{ll} A_{\\mathrm{X}}^{m n}-B_{\\mathrm{X}}^{m n} &amp; d_{m n} \\leq d_{\\text {near }} \\\\ 0 &amp; \\text { otherwise } \\end{array} \\quad \\mathrm{X} \\in\\{\\mathrm{EJ}, \\mathrm{EM}, \\mathrm{HJ}, \\mathrm{HM}\\}\\right. \\tag{30} $$<\/p>\n\n\n\n<p>$A_X^{mn}$ \u662f\u539f\u59cb\u77e9\u9635\u7684\u7cbe\u786e\u503c\uff0c\u800c $B_X^{mn}$ \u662f\u57fa\u7840\u77e9\u9635\u7684\u8fd1\u4f3c\u503c\u3002\u901a\u8fc7\u51cf\u53bb\u57fa\u7840\u77e9\u9635\u7684\u8fd1\u4f3c\u503c\uff0c\u5f97\u5230\u4e00\u4e2a\u8f83\u51c6\u786e\u7684\u4fee\u6b63\u9879\uff0c\u7528\u4e8e\u8865\u507f\u8fd1\u8ddd\u79bb\u57fa\u51fd\u6570\u5bf9\u7684\u8bef\u5dee\u3002<\/p>\n\n\n\n<p>\u7efc\u4e0a\u6240\u8ff0\uff0cAIM\u65b9\u6cd5\u4e2d\u6bcf\u4e2a\u77e9\u9635\u7684\u6700\u7ec8\u8fd1\u4f3c\u5f62\u5f0f\u53ef\u4ee5\u5199\u6210\u5982\u4e0b\u5173\u7cfb\uff1a<\/p>\n\n\n\n<p>$$ \\begin{aligned} A_{\\mathrm{EJ}} \\approx B_{\\mathrm{EJ}}+C_{\\mathrm{EJ}} ; &amp; A_{\\mathrm{EM}} \\approx B_{\\mathrm{EM}}+C_{\\mathrm{EM}} ; \\\\ A_{\\mathrm{HJ}} \\approx B_{\\mathrm{HJ}}+C_{\\mathrm{HJ}} ; &amp; A_{\\mathrm{HM}} \\approx B_{\\mathrm{HM}}+C_{\\mathrm{HM}} \\end{aligned} $$<\/p>\n\n\n\n<p><br>\u6362\u53e5\u8bdd\u8bf4\uff0c\u539f\u59cb\u77e9\u9635\u53ef\u4ee5\u901a\u8fc7\u57fa\u7840\u77e9\u9635\u548c\u4fee\u6b63\u77e9\u9635\u7684\u7ec4\u5408\u6765\u8fd1\u4f3c\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">4.1.3 Fast Matrix-Vector Multiplication<\/h5>\n\n\n\n<p>\u5feb\u901f\u77e9\u9635-\u5411\u91cf\u4e58\u6cd5\uff08Fast Matrix-Vector Multiplication\uff09\u662fAIM\u7684\u6838\u5fc3\u3002<\/p>\n\n\n\n<p>\u7531\u4e8e\u4e0a\u9762\u5f97\u5230\u7684\u4fee\u6b63\u77e9\u9635 $C$ \u662f\u7a00\u758f\u77e9\u9635\uff0c $C$ \u53ea\u5728\u8fd1\u8ddd\u79bb\u7684\u57fa\u51fd\u6570\u4e0a\u6709\u975e\u96f6\u503c\uff0c\u56e0\u6b64\u77e9\u9635 $C$ \u7684\u4e58\u6cd5\u64cd\u4f5c\u662f\u5f88\u5feb\u7684\u3002<\/p>\n\n\n\n<p>\u5229\u7528\u57fa\u7840\u8fd1\u4f3c\u77e9\u9635 $B$ \u7684\u5377\u79ef\u7279\u6027\uff0c\u8ba1\u7b97\u4e86\u77e9\u9635 $B$ \u4e0e\u5411\u91cf\u7684\u4e58\u79ef\u3002\u8ba1\u7b97\u8fc7\u7a0b\u5206\u4e3a\u4e09\u6b65\uff1a<\/p>\n\n\n\n<p><br>$$<br>y_1 = \\Lambda_2^T x, \\quad y_2 = G y_1, \\quad y_3 = \\Lambda_1 y_2<br>\\tag{32}<br>$$<\/p>\n\n\n\n<p><br>\u7b2c\u4e00\u6b65\u628a\u5411\u91cf\u6295\u5f71\u5230\u4e00\u4e2a\u7a00\u758f\u77e9\u9635\u7f51\u683c\u4e0a\u3002<\/p>\n\n\n\n<p>\u7b2c\u4e8c\u6b65\u4e5f\u662f\u6838\u5fc3\u6b65\u9aa4\uff0c\u628a\u7f51\u683c\u70b9\u7684\u6570\u636e\u4f20\u64ad\u5230\u6574\u4e2a\u7f51\u7edc\uff0c\u4e5f\u5c31\u662f\u8ba1\u7b97\u6bcf\u4e2a\u70b9\u5bf9\u5176\u4ed6\u70b9\u7684\u5f71\u54cd\u3002\u4e24\u4e2a\u70b9\u6bd4\u8f83\u63a5\u8fd1\uff0c\u77e9\u9635 $G$ \u4e2d\u7684\u4f20\u64ad\u51fd\u6570\u5c31\u5927\u3002\u901a\u8fc7FFT\u6765\u505a\u52a0\u901f\u3002<\/p>\n\n\n\n<p><br>$$<br>y_2 = \\mathcal{F}^{-1} { \\mathcal{F}(g) \\mathcal{F}(y_1) }<br>\\tag{33}<br>$$<\/p>\n\n\n\n<p><br>\u7b2c\u4e09\u6b65\u9aa4\u5c06\u7ed3\u679c\u6620\u5c04\u56de\u539f\u6765\u7684\u57fa\u51fd\u6570\u7a7a\u95f4\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">4.2 GPU-Accelerated Iterative Solving<\/h4>\n\n\n\n<p>\u5728GPU\u4e0a\u5c06AIM\u65b9\u6cd5\u4e2d\u7684\u8ba1\u7b97\u91cd\u70b9\u8f6c\u79fb\u5230\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\uff08FFT\uff09\u548c\u7a00\u758f\u77e9\u9635\u64cd\u4f5c\u4e0a\u3002\u603b\u7ed3\u4e00\u4e0b\uff0c\u76ee\u524d\u5c06\u5927\u578b\u77e9\u9635\u5206\u4e3a\u57fa\u7840\u77e9\u9635 $B$ \u548c\u4fee\u6b63\u77e9\u9635 $C$ \uff0c\u5206\u522b\u5904\u7406\u8fdc\u8ddd\u79bb\u548c\u8fd1\u8ddd\u79bb\u7684\u57fa\u51fd\u6570\u5bf9\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>cuFFT\uff1a\u5c06\u57fa\u7840\u77e9\u9635\u7684\u4e58\u6cd5\u64cd\u4f5c\u8f6c\u6362\u4e3a\u9891\u57df\u4e2d\u7684\u5377\u79ef\u8ba1\u7b97<\/li>\n\n\n\n<li>cuSPARSE\uff1a\u7b97\u7a00\u758f\u77e9\u9635\u52a0\u901f\u4fee\u6b63\u77e9\u9635 C<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-518-1024x624.png\" alt=\"\" class=\"wp-image-1679 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"624\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-518-1024x624.png\" alt=\"\" class=\"wp-image-1679 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-518-1024x624.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-518-300x183.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-518-768x468.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-518.png 1392w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u5e76\u4e14\u4f18\u5316\u8ba1\u7b97\u7b56\u7565\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5bf9\u4e8e\u5c0f\u89c4\u6a21\u4eff\u771f\u4efb\u52a1\uff0c\u53ea\u9700\u8981\u4f7f\u75281\u4e2aGPU<\/li>\n\n\n\n<li>\u5bf9\u4e8e\u5927\u89c4\u6a21\u4eff\u771f\u4efb\u52a1\uff0c\u5c06\u4efb\u52a1\u5206\u914d\u52304\u4e2aGPU<\/li>\n<\/ul>\n\n\n\n<h5 class=\"wp-block-heading\">4.2.1 Small-Scale Simulations<\/h5>\n\n\n\n<p>\u5bf9\u4e8e\u5c0f\u89c4\u6a21\u4eff\u771f\u4efb\u52a1\uff08\u4f8b\u5982 $12 \\mu m \\times 12 \\mu m$\uff09\uff0c \u5fc5\u987b\u4e8b\u5148\u8ba1\u7b97\u5e76\u5b58\u50a8\u4f20\u64ad\u51fd\u6570\u7684\u5085\u91cc\u53f6\u53d8\u6362\u503c\uff08\u5373\u77e9\u9635 $G$ \u7684\u5085\u91cc\u53f6\u53d8\u6362\uff09\u3002\u5728\u5c0f\u89c4\u6a21\u4efb\u52a1\u4e2d\u7a00\u758f\u4fee\u6b63\u77e9\u9635 C \u5360\u7528\u7684\u663e\u5b58\u4e0d\u52305GB\uff0c\u4e00\u5f20GPU\u5c31\u53ef\u4ee5\u641e\u6382\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">4.2.2 Large-Scale Simulations<\/h5>\n\n\n\n<p>\u5bf9\u4e8e\u5927\u89c4\u6a21\u4eff\u771f\u4efb\u52a1\uff08\u4f8b\u5982 $24 \\mu m \\times 24 \\mu m$\uff09\uff0c\u7531\u4e8e\u5355\u4e2aGPU\u7684\u663e\u5b58\u4e0d\u8db3\u4ee5\u5b58\u50a8\u6240\u6709\u6570\u636e\uff0c\u4f5c\u8005\u5c06\u8ba1\u7b97\u4efb\u52a1\u5206\u914d\u52304\u4e2aGPU\u4e0a\u3002\u5728\u8fd9\u4e2a\u5c3a\u5ea6\u7684\u4eff\u771f\u4e0a\uff0c\u57fa\u51fd\u6570\u7684\u4e2a\u6570\u4f1a\u8fbe\u5230960 \u00d7 960\u4e2a\uff0c\u5b58\u50a8\u6240\u6709\u4fee\u6b63\u77e9\u9635\u7684\u975e\u96f6\u5143\u7d20\uff08\u5305\u62ec\u884c\u5217\u7d22\u5f15\u548c\u590d\u6570\u6d6e\u70b9\u6570\u503c\uff09\u5927\u7ea6\u9700\u898120GB\u663e\u5b58\u3002\u7b56\u7565\u8fd8\u662f\u548c\u5c0f\u89c4\u6a21\u4e00\u6837\uff0c\u6bcf\u4e2aGPU\u5206\u914d\u5927\u7ea65GB\u5185\u5b58\u6765\u5b58\u50a8\u4fee\u6b63\u77e9\u9635 $C$ \u3002<\/p>\n\n\n\n<p>MINRES\u6c42\u89e3\u5668\u5728\u4e3b\u673aCPU\u4e0a\u6267\u884c\uff0c\u800c\u77e9\u9635-\u5411\u91cf\u4e58\u79ef $y = Ax$ \u7684\u8ba1\u7b97\u5728GPU\u4e0a\u5b8c\u6210\u3002\u4f46\u662f\u4e0d\u9700\u8981\u62c5\u5fc3\u4f20\u8f93\u65f6\u95f4\uff0c\u8fd9\u4e2a\u5411\u91cf\u5927\u6982\u53ea\u670930MB\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">4.3 FFT-Accelerated Scattered Field Evaluation<\/h4>\n\n\n\n<p>\u7528FFT\u52a0\u901f\u8ba1\u7b97\u6563\u5c04\u573a\u3002\u5728\u8fdc\u573a\u533a\u57df\u4e0a\u8bc4\u4f30\u4ece\u8868\u9762\u6563\u5c04\u7684\u573a\u6700\u7ec8\u6c42\u51fa\u8868\u9762BRDF\u3002<\/p>\n\n\n\n<p>\u5728\u6c42\u89e3BEM\u540e\uff0c\u5f97\u5230\u4e86\u8868\u9762\u7684\u7535\u6d41\u5bc6\u5ea6 $\\mathbf{J}$ \u548c\u78c1\u6d41\u5bc6\u5ea6 $\\mathbf{M}$ \u3002\u8fd9\u4e9b\u5bc6\u5ea6\u5206\u5e03\u5b9a\u4e49\u4e86\u8868\u9762\u4e0a\u7684\u7535\u78c1\u6e90\uff0c\u53ef\u4ee5\u7528\u6765\u8ba1\u7b97\u5728\u8fdc\u573a\u533a\u57df\u4e0a\u7684\u6563\u5c04\u573a\u3002\u516c\u5f0f\u5f88\u7b80\u5355\uff0c\u968f\u7740\u8ddd\u79bb\u8870\u5f31\u540c\u65f6\u8fd8\u4f1a\u5177\u6709\u4e00\u5b9a\u7684\u76f8\u4f4d\u53d8\u5316\uff1a<\/p>\n\n\n\n<p><br>$$<br>\\mathbf{E_s}(r) \\approx \\mathbf{E}(\\hat{r}) \\frac{e^{-jkr}}{r}; \\quad \\mathbf{H_s}(r) \\approx \\mathbf{H}(\\hat{r}) \\frac{e^{-jkr}}{r}<br>\\tag{36}<br>$$<\/p>\n\n\n\n<p><br>\u516c\u5f0f\u53f3\u8fb9\u7684 $\\mathbf{E}(\\hat{r})$ \u548c $\\mathbf{H}(\\hat{r})$ \u662f\u5728\u8fdc\u573a\u4e0a\u7279\u5b9a\u65b9\u5411 $\\hat{r}$ \u7684\u632f\u5e45\u3002\u5728\u4e0d\u540c\u7684\u65b9\u5411\u4e0a\uff0c\u6563\u5c04\u573a\u7684\u5f3a\u5ea6\u53ef\u80fd\u4e0d\u540c\u3002<\/p>\n\n\n\n<p><br>$$<br>\\begin{aligned}<br>F_1(\\hat{\\mathbf{r}})=\\int_V J_x\\left(\\mathbf{r}^{\\prime}\\right) e^{j k \\mathbf{r}^{\\prime} \\cdot \\hat{\\mathbf{r}}} d \\mathbf{r}^{\\prime} ; &amp; F_2(\\hat{\\mathbf{r}})=\\int_V J_y\\left(\\mathbf{r}^{\\prime}\\right) e^{j k \\mathbf{r}^{\\prime} \\cdot \\hat{\\mathbf{r}}} d \\mathbf{r}^{\\prime} \\\\<br>F_3(\\hat{\\mathbf{r}})=\\int_V J_z\\left(\\mathbf{r}^{\\prime}\\right) e^{j k \\mathbf{r}^{\\prime} \\cdot \\hat{\\mathbf{r}}} d \\mathbf{r}^{\\prime} ; &amp; F_4(\\hat{\\mathbf{r}})=\\int_V M_x\\left(\\mathbf{r}^{\\prime}\\right) e^{j k \\mathbf{r}^{\\prime} \\cdot \\hat{\\mathbf{r}}} d \\mathbf{r}^{\\prime} \\\\<br>F_5(\\hat{\\mathbf{r}})=\\int_V M_y\\left(\\mathbf{r}^{\\prime}\\right) e^{j k \\mathbf{r}^{\\prime} \\cdot \\hat{\\mathbf{r}}} d \\mathbf{r}^{\\prime} ; &amp; F_6(\\hat{\\mathbf{r}})=\\int_V M_z\\left(\\mathbf{r}^{\\prime}\\right) e^{j k \\mathbf{r}^{\\prime} \\cdot \\hat{\\mathbf{r}}} d \\mathbf{r}^{\\prime}<br>\\end{aligned}<br>\\tag{37}<br>$$<\/p>\n\n\n\n<p><br>\u4e3a\u4e86\u907f\u514d\u76f4\u63a5\u6c42\u89e3\u8fd9\u4e9b\u79ef\u5206\uff0c\u4f5c\u8005\u5229\u7528\u5148\u524d\uff084.1.2\uff09\u5bf9 $\\mathbf{J}$ \u548c $\\mathbf{M}$ \u7684\u70b9\u6e90\u8fd1\u4f3c\uff08$\\Lambda$ \u77e9\u9635\uff09\uff0c\u5c06\u6bcf\u4e2a\u79ef\u5206\u9879 $F_i(\\hat{r})$ \u79bb\u6563\u5316\u5e76\u91cd\u5199\u4e3a\u5085\u91cc\u53f6\u53d8\u6362\u7684\u5f62\u5f0f\uff0c\u5982\u516c\u5f0f (38) \u6240\u793a\uff1a<\/p>\n\n\n\n<p><br>$$<br>F_i(\\hat{r}) = \\sum_{p \\in S} h_i(p) e^{jp \\cdot k \\hat{r}}<br>\\tag{38}<br>$$<\/p>\n\n\n\n<p><br>\u5c06\u8fde\u7eed\u7684\u573a\u5f3a\u8ba1\u7b97\u8f6c\u5316\u4e3a\u79bb\u6563\u7684\u6c42\u548c\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u7528FFT\u5feb\u901f\u8ba1\u7b97\u4e86\u3002\u5e76\u4e14\u4ece\u4e0a\u5f0f\u53ef\u4ee5\u89c2\u5bdf\u5230\uff0c $F_i(\\hat{r})$ \u5b9e\u9645\u4e0a\u662f $h_i(p)$ \u5728\u7a7a\u95f4\u9891\u7387 $-k\\hat{r}$ \u4e0a\u7684\u5085\u91cc\u53f6\u5206\u91cf\u3002<\/p>\n\n\n\n<p>The required spatial frequencies are not on the FFT grid but can be interpolated; we add zero padding prior to the FFT step, to ensure enough resolution in the frequency domain for the trilinear interpolation to be sufficiently accurate.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">5 HIGH RESOLUTION BRDF GENERATION<\/h3>\n\n\n\n<p>\u641e\u4e86\u4e00\u5927\u5806\uff0c\u7ec8\u4e8e\u56de\u5230\u719f\u6089\u7684BRDF\u8ba1\u7b97\u4e86\u3002\u8fd9\u91cc\u5173\u952e\u5728\u4e8e\u4f7f\u7528\u5c0f\u5c3a\u5ea6\u6a21\u62df\u7684\u7ebf\u6027\u53e0\u52a0\u6765\u91cd\u5efa\u5927\u5c3a\u5ea6\u5165\u5c04\u573a\u7684\u8fdc\u573a\u6563\u5c04\uff0c\u800c\u4e0d\u662f\u4e00\u53e3\u5403\u6210\u5927\u80d6\u5b50\u3002$N^2$\u4e2a\u5c0f\u5c3a\u5ea6\u7684\u9ad8\u65af\u5149\u675f\u6784\u6210\u7684\u7f51\u683c\u6765\u7ebf\u6027\u7ec4\u5408\u6210\u8fd1\u4f3c\u5927\u5c3a\u5ea6\u7684\u5165\u5c04\u573a\u3002<\/p>\n\n\n\n<p>\u8fd9\u91cc\u7528\u5230\u201c\u6ce2\u675f\u5f15\u5bfc\u201d\uff08beam steering\uff09\u6280\u672f\u3002\u8fd9\u79cd\u65b9\u6cd5\u4e0d\u7528\u5bf9\u6bcf\u4e2a\u65b9\u5411\u90fd\u8fdb\u884c\u4e00\u6b21\u6a21\u62df\uff0c\u4ece\u800c\u5927\u5e45\u964d\u4f4e\u8ba1\u7b97\u6210\u672c\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">5.1 Basic and Derived Incident Directions<\/h4>\n\n\n\n<p>\u9996\u5148\uff0c\u6cbf\u67d0\u65b9\u5411 $\\mathbf{u}$ \u4f20\u64ad\u7684 $N^2$ \u4e2a\u9ad8\u65af\u5149\u675f\u7ec4\u6210\u5728\u63a5\u6536\u5e73\u9762\u7684\u4e00\u4e2a $N \\times N$ \u70b9\u7684\u7f51\u683c\u3002\u8fd9\u4e9b\u5149\u675f\u7ec4\u5408\u540e\u80fd\u591f\u751f\u6210\u4e00\u4e2a\u5927\u7684\u603b\u573a\u3002<\/p>\n\n\n\n<p>\u7136\u540e\u7ed9\u6bcf\u4e2a\u9ad8\u65af\u5149\u675f\u5f15\u5165\u590d\u6570\u7f29\u653e\u56e0\u5b50\uff0c\u8c03\u6574\u6bcf\u4e2a\u5149\u675f\u7684\u76f8\u4f4d\uff0c\u8fdb\u800c\u8c03\u6574\u7ec4\u5408\u573a\u7684\u4f20\u64ad\u65b9\u5411\uff0c\u8fd9\u4e9b\u65b9\u5411\u79f0\u4e3adesired direction\u3002<\/p>\n\n\n\n<p><br>$$<br>a_{st} = e^{j k \\mathbf{p}{st} \\cdot \\omega_i} \\tag{39}<br>$$<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-519-1024x900.png\" alt=\"\" class=\"wp-image-1680 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"900\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-519-1024x900.png\" alt=\"\" class=\"wp-image-1680 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-519-1024x900.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-519-300x264.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-519-768x675.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-519.png 1356w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u5f53\u76ee\u6807\u5165\u5c04\u65b9\u5411 $-\\omega_i$ \u4e0e\u57fa\u672c\u65b9\u5411 $\\mathbf{u}$ \u4e4b\u95f4\u7684\u5939\u89d2\u63a5\u8fd1\u5404\u9ad8\u65af\u5149\u675f\u7684\u53d1\u6563\u89d2\uff08divergence of the small beams\uff09\u65f6\uff0caliasing artifacts begin to appear. An example is shown in Fig. 8 (d).<\/p>\n\n\n\n<p>In our framework, we decide on a primary waist w and choose a collection of basic incident directions. In general, \u8f83\u5c0f\u7684\u8170\u5bbd\u610f\u5473\u7740\u66f4\u5927\u7684\u53d1\u6563\u89d2\uff0c\u8fd9\u6837\u53ef\u4ee5\u4ece\u6bcf\u4e2a\u57fa\u672c\u65b9\u5411\u6d3e\u751f\u51fa\u66f4\u591a\u7684\u5165\u5c04\u65b9\u5411\uff0c\u4ece\u800c\u51cf\u5c11\u6240\u9700\u7684\u57fa\u672c\u65b9\u5411\u6570\u91cf\u3002\u8f83\u5927\u7684\u8170\u5bbd\u4f1a\u964d\u4f4e\u6bcf\u4e2a\u9ad8\u65af\u5149\u675f\u7684\u53d1\u6563\u89d2\uff0c\u4f7f\u5f97\u7ec4\u5408\u540e\u7684\u603b\u573a\u53d1\u6563\u66f4\u5c0f\uff0c\u4ece\u800c\u4ea7\u751f\u66f4\u7cbe\u786e\u7684\u5165\u5c04\u65b9\u5411\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-521-1024x858.png\" alt=\"\" class=\"wp-image-1682 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"858\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-521-1024x858.png\" alt=\"\" class=\"wp-image-1682 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-521-1024x858.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-521-300x251.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-521-768x644.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-521.png 1308w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u6bcf\u4e2a\u516d\u8fb9\u5f62\u7684\u4e2d\u5fc3\u5bf9\u5e94\u4e00\u4e2a\u57fa\u672c\u5165\u5c04\u65b9\u5411\u3002\u6574\u4e2a\u534a\u7403\u7684\u6240\u6709\u5165\u5c04\u65b9\u5411\u5212\u5206\u4e3a\u82e5\u5e72territories\uff0c\u6bcf\u4e2aterritories\u5f52\u5c5e\u4e8e\u4e00\u4e2a\u57fa\u672c\u5165\u5c04\u65b9\u5411\u3002\u5728\u534a\u7403\u6295\u5f71\u4e2d\uff0c\u8fd9\u79cd\u6bd4\u4f8b\u5173\u7cfb\u4e0e\u4f59\u5f26\u56e0\u5b50\u76f8\u4e92\u62b5\u6d88\uff0c\u56e0\u6b64\u53ef\u4ee5\u4f7f\u7528<strong>\u5927\u5c0f\u76f8\u7b49\u7684\u516d\u8fb9\u5f62<\/strong>\u6765\u8868\u793a\u3002<\/p>\n\n\n\n<p>\u5bf9\u4e8e\u6bcf\u4e2a\u57fa\u672c\u65b9\u5411\uff0c\u53ef\u4ee5\u901a\u8fc7\u201c\u6ce2\u675f\u5f15\u5bfc\u201d\u4ea7\u751f\u4e00\u4e9b\u504f\u79bb\u8be5\u57fa\u672c\u65b9\u5411\u7684\u6d3e\u751f\u65b9\u5411\u3002<\/p>\n\n\n\n<p>\u5f53\u5149\u7684\u5165\u5c04\u89d2\u5f88\u5c0f\u65f6\uff08\u4f8b\u5982\u63a5\u8fd1\u8868\u9762\u6cd5\u7ebf\u65b9\u5411\uff09\uff0c\u57fa\u672c\u5165\u5c04\u65b9\u5411\u9644\u8fd1\u7684\u6d3e\u751f\u65b9\u5411\u8303\u56f4\u5f88\u5c0f\uff0c\u56e0\u4e3a\u5c0f\u89d2\u5ea6\u7684\u5149\u66f4\u96c6\u4e2d\uff0c\u4e0d\u4f1a\u6709\u5f88\u5927\u7684\u6269\u6563\u3002\u53cd\u4e4b\uff0c\u6d3e\u751f\u65b9\u5411\u8303\u56f4\u4f1a\u66f4\u5927\u3002\u603b\u7ed3\u5c31\u662f\uff0c\u5165\u5c04\u89d2\u8d8a\u5927\uff08\u89d2\u5ea6\u8d8a\u63a5\u8fd1\u6c34\u5e73\uff09\uff0c\u6d3e\u751f\u65b9\u5411\u7684\u8986\u76d6\u8303\u56f4\u5c31\u8d8a\u5927\u3002\u516c\u5f0f\u4f5c\u8005\u4e5f\u63d0\u5230\u4e86\uff1a$1\/\\cos \\theta_i$\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">5.2 Individual Simulations and Synthesized Results<\/h4>\n\n\n\n<p>\u4e3a\u4e86\u8ba1\u7b97BRDF\uff0c\u6211\u4eec\u9700\u8981\u77e5\u9053\u8fd9\u4e2a\u5927\u9762\u79ef\u5165\u5c04\u5149\u7684\u6563\u5c04\u60c5\u51b5\u3002\u7136\u800c\uff0c\u76f4\u63a5\u6a21\u62df\u8fd9\u6837\u4e00\u4e2a\u5927\u9762\u79ef\u7684\u5149\u4f1a\u9700\u8981\u5f88\u9ad8\u7684\u8ba1\u7b97\u6210\u672c\u3002\u56e0\u6b64\uff0c\u6211\u4eec\uff1a<\/p>\n\n\n\n<p><strong>\u4f7f\u7528\u5c0f\u5c3a\u5ea6\u6a21\u62df\u7684\u53e0\u52a0\u6765\u6a21\u62df\u5927\u5c3a\u5ea6\u5165\u5c04\u573a\u3002<\/strong><\/p>\n\n\n\n<p>\u53ef\u4ee5\u7406\u89e3\u4e3a\uff0c\u7528\u5f88\u591a\u4e2a\u5c0f\u7684\u624b\u7535\u7b52\uff08\u9ad8\u65af\u5149\u675f\uff09\u6765\u8986\u76d6\u4e00\u4e2a\u533a\u57df\uff0c\u800c\u4e0d\u662f\u7528\u4e00\u4e2a\u5de8\u5927\u7684\u63a2\u7167\u706f\u3002<\/p>\n\n\n\n<p>\u9996\u5148\u51b3\u5b9a\u624b\u7535\u7b52\u7684\u5c3a\u5bf8\uff08\u5149\u675f\u7684\u5927\u5c0f\uff0c\u5373\u8170\u5bbd $w$\uff09\uff0c\u5e76\u5728\u8868\u9762\u4e0a\u5747\u5300\u5730\u5206\u5e03\u8fd9\u4e9b\u624b\u7535\u7b52\u3002\u5199\u6210\u516c\u5f0f\uff0c\u8fd9\u4e2a\u624b\u7535\u7b52\u7684\u6392\u5217\u5c31\u662f\u7f51\u683c\u70b9${x_s}, {y_t}$\uff0c\u4ee3\u8868\u6bcf\u4e2a\u9ad8\u65af\u5149\u675f\u7684\u4e2d\u5fc3\u4f4d\u7f6e\u3002\u7f51\u683c\u95f4\u8ddd\u4e00\u822c\u548c\u8170\u5bbd\u4e00\u81f4\uff0c\u786e\u4fdd\u5149\u7684\u5747\u5300\u8986\u76d6\uff0c\u5e76\u4fdd\u6301\u8f83\u4f4e\u7684\u53d1\u6563\u89d2\u3002<\/p>\n\n\n\n<p>\u8ba9\u6bcf\u4e2a\u9ad8\u65af\u5149\u675f\u5728\u5176\u4e2d\u5fc3\u533a\u57df\u4ea7\u751f\u76f8\u540c\u7684\u7535\u78c1\u573a\uff0c\u53ea\u662f\u5728\u4e0d\u540c\u7684\u4f4d\u7f6e\u4e0a\u91cd\u590d\u8fd9\u4e00\u6548\u679c\u3002<\/p>\n\n\n\n<p>\u63a5\u4e0b\u6765\uff0c\u60f3\u8981\u5f97\u5230\u5927\u5149\u675f\u7684\u603b\u6563\u5c04\u573a\uff0c\u8fd9\u91cc\u9700\u8981<strong>\u76f8\u4f4d\u56e0\u5b50<\/strong>\u6765\u8fdb\u884c\u201c\u8c03\u6574\u201d\u548c\u201c\u53e0\u52a0\u201d\uff0c\u8be6\u7ec6\u8bf7\u56de\u770b\u516c\u5f0f(39)\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-522-1024x528.png\" alt=\"\" class=\"wp-image-1683 lazyload\"\/><noscript><img decoding=\"async\" width=\"1024\" height=\"528\" src=\"https:\/\/\u80a5\u80a5.com\/wp-content\/uploads\/image-522-1024x528.png\" alt=\"\" class=\"wp-image-1683 lazyload\" srcset=\"https:\/\/remoooo.com\/wp-content\/uploads\/image-522-1024x528.png 1024w, https:\/\/remoooo.com\/wp-content\/uploads\/image-522-300x155.png 300w, https:\/\/remoooo.com\/wp-content\/uploads\/image-522-768x396.png 768w, https:\/\/remoooo.com\/wp-content\/uploads\/image-522.png 1222w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n\n\n\n<p>\u6700\u7ec8\uff0cCombining with Eq. 39, the scattered fields in the far field region corresponding to the pair of directions $(w_i,w_o)$ are given by:<\/p>\n\n\n\n<p>$$ \\begin{aligned} \\mathbf{E}\\left(\\omega_i, \\omega_o\\right) &amp; =\\sum_{s=1}^n \\sum_{t=1}^n e^{j k \\mathbf{p}{s t} \\cdot\\left(\\omega_i+\\omega_o\\right)} \\mathbf{E}{s t}\\left(\\omega_o\\right) \\\\ \\mathbf{H}\\left(\\omega_i, \\omega_o\\right) &amp; =\\sum_{s=1}^n \\sum_{t=1}^n e^{j k \\mathbf{p}{s t} \\cdot\\left(\\omega_i+\\omega_o\\right)} \\mathbf{H}{s t}\\left(\\omega_o\\right) \\end{aligned} \\tag{41} $$<\/p>\n\n\n\n<p>where $\\mathbf{E}, \\mathbf{H}$ refer to the far field quantities only associated with directions (without the $e^{-j k r} \/ r$ term).<\/p>\n\n\n\n<p>Lastly, we can compute the surface BRDF value as<\/p>\n\n\n\n<p>$$<br>f_r\\left(\\omega_i, \\omega_o\\right)=\\frac{\\frac{1}{2}\\left|\\mathbf{E}\\left(\\omega_i, \\omega_o\\right) \\times \\mathbf{H}\\left(\\omega_i, \\omega_o\\right)^*\\right|}{\\Phi_i \\cos \\theta_r}<br>\\tag{42}<br>$$<\/p>\n\n\n\n<p>where the incident power $\\Phi_i$ is computed by integrating the incident irradiance over the surface:<\/p>\n\n\n\n<p>$$<br>\\Phi_i=\\frac{1}{2} \\int_S\\left|\\left[\\mathbf{E}_i\\left(\\mathbf{r}^{\\prime}\\right) \\times \\mathbf{H}_i\\left(\\mathbf{r}^{\\prime}\\right)^*\\right] \\cdot \\mathbf{n}\\right| d \\mathbf{r}^{\\prime}<br>\\tag{43}<br>$$<\/p>\n\n\n\n<p>where $\\mathbf{n}$ is the surface normal at the macro scale ( $+\\mathbf{z}$ ). Note that Eq. 42 and Eq. 43 can also be applied in single simulations, where $\\Phi_i$ is computed from a single Gaussian beam.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4eca\u5929\u7ee7\u7eed\u6765\u7c97\u7565\u770b\u770bSig23\u8fd9\u7bc7\u8ba1\u7b97\u8868\u9762\u53cd\u5c04\u7684\u5168\u6ce2\u53c2\u8003\u6a21\u62df\u5668paper\u3002 \u5173\u952e\u8bcd\uff1a\u56fe\u5f62\u5b66\u5165\u95e8\u3001\u6ce2\u52a8\u5149\u5b66\u6e32\u67d3\u3001B [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":1631,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[53],"tags":[56,58],"class_list":["post-1630","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-tech","tag-cg","tag-wave-optics"],"_links":{"self":[{"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/posts\/1630","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/comments?post=1630"}],"version-history":[{"count":83,"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/posts\/1630\/revisions"}],"predecessor-version":[{"id":2118,"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/posts\/1630\/revisions\/2118"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/media\/1631"}],"wp:attachment":[{"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/media?parent=1630"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/categories?post=1630"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/remoooo.com\/en\/wp-json\/wp\/v2\/tags?post=1630"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}