Progress in Computational Physics (PiCP)

Mathematical and Numerical Techniques for Open Periodic Waveguides

Author(s): Johannes Tausch

Pp: 50-72 (23)

DOI: 10.2174/978160805150211001010050

* (Excluding Mailing and Handling)

Abstract

The propagation of electromagnetic waves in dielectric slab waveguides with periodic corrugations is described by the spectrum of the Helmholtz operator on an infinite strip with quasiperiodic boundary conditions. This chapter reviews the basic properties of this spectrum, which typically consists of guided modes, radiation modes and leaky modes. A great deal of attention will be devoted to planar waveguides which share some of the important features of the periodic case. To compute the eigenmodes and the associated propagation constants numerically, one usually truncates the domain that contains the grating and imposes certain radiation conditions on the artificial boundary. An alternative to this approach is to decompose the infinite strip into a rectangle, which contains the grating, and two semi-infinite domains. The guided and leaky modes can be computed by matching the Dirichlet-to-Neumann operator on the interfaces of these three domains. The discretized eigenvalue problem is nonlinear because of the appearance of the propagation constant in the artificial boundary condition. We will discuss how such problems can be solved by numerical continuation. In this approach, one starts with an approximating planar waveguide and then follows the solutions by a continuous transition to the multilayer periodic structure. The chapter is concluded with a brief description of how the perfectly matched layer can be used to compute the guided modes of a waveguide.

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      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • (select j.user_access_id,j.volume_id,j.user_id,j.from_ip,j.to_ip,j.type,j.track,"Trial Access" as access_base_text, CASE j.type WHEN 111 THEN true ELSE false END as terms_popup, "S" as content_download_type, null as restricted_user_access_key, null as restricted_user_access_type, CASE j.track WHEN 0 THEN false ELSE true END as tracksAccessByVolume,"chapter" as access_level, j.chapter_id from `user_access_chapter_trail` as `j` inner join `user_access_info` as `info` on `j`.`user_access_id` = `info`.`user_access_id` where ((j.from_ip <= 316642695 and j.to_ip >=316642695 )) and ((j.from_date <= 1739972291 and j.to_date >= 1739972291 and j.perpetual= 0) or (j.from_date is null and j.to_date is null and j.perpetual= 1)) and j.chapter_id = 2723) union (select j.user_access_id,j.volume_id,j.user_id,j.from_ip,j.to_ip,j.type,j.track,"Subscribed" as access_base_text, false as terms_popup, "S" as content_download_type, null as restricted_user_access_key, null as restricted_user_access_type, CASE j.track WHEN 0 THEN false ELSE true END as tracksAccessByVolume,"chapter" as access_level, j.chapter_id from `user_access_chapter_corporate` as `j` inner join `user_access_info` as `info` on `j`.`user_access_id` = `info`.`user_access_id` where ((j.from_ip <= 316642695 and j.to_ip >=316642695 )) and ((j.from_date <= 1739972291 and j.to_date >= 1739972291 and j.perpetual= 0) or (j.from_date is null and j.to_date is null and j.perpetual= 1)) and j.chapter_id = 2723) union (select j.user_access_id,j.volume_id,j.user_id,j.from_ip,j.to_ip,j.type,j.track,"Token Based Access" as access_base_text, false as terms_popup, "R" as content_download_type, j.uc_id as restricted_user_access_key, "v" as restricted_user_access_type, CASE j.track WHEN 0 THEN false ELSE true END as tracksAccessByVolume,"chapter" as access_level, j.chapter_id from `user_access_chapter_token` as `j` inner join `user_access_info` as `info` on `j`.`user_access_id` = `info`.`user_access_id` where ((j.from_ip <= 316642695 and j.to_ip >=316642695 )) and ((j.from_date <= 1739972291 and j.to_date >= 1739972291 and j.perpetual= 0) or (j.from_date is null and j.to_date is null and j.perpetual= 1)) and j.chapter_id = 2723)
      680μsalphaeurekaselec_live_10_06_2022UserAccess.php#1266
      Backtrace
      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1549
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • select b.banner_link, b.banner_name, b.banner_script, b.banner_remarks, b.banner_file_name, b.banner_placement, bd.content_type, bd.content_id from `tbl_banner` as `b` inner join `tbl_banner_detail` as `bd` on `bd`.`banner_id` = `b`.`banner_id` inner join `users` as `u` on `u`.`id` = `b`.`updated_by` where `b`.`banner_ebook` = 1 and `bd`.`content_type` = 'B' and `bd`.`content_id` = 975 and `b`.`banner_from_date` <= 1739972291 and `b`.`banner_to_date` >= 1739972291 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      1.09msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 975
      • 3: 1739972291
      • 4: 1739972291
      • 5: A
      Backtrace
      • 13. app/Models/Banner.php:95
      • 14. app/Http/helpers.php:404
      • 17. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 18. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    • select count(*) as aggregate from `uc_cart_products` where cart_id='df2dd514c3af83e16dc463921ab8c695'
      280μsalphaeurekaselec_live_10_06_2022Cart.php#271
      Backtrace
      • 15. app/Models/Cart.php:271
      • 16. view::layouts._header:260
      • 18. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 20. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Keywords
    1Keywords.php#?
    App\Models\Ebook
    1Ebook.php#?
        _token
        kyX7nDzbRxIlnXaLdnX7spf9d7jrIectiHjQjfeG
        uc_cart_id
        df2dd514c3af83e16dc463921ab8c695
        _previous
        array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/2723" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /chapter/2723
        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
        array:17 [ "x-https" => array:1 [ 0 => "1" ] "host" => array:1 [ 0 => "alpha.eurekaselect.com" ] "priority" => array:1 [ 0 => "u=0, i" ] "cookie" => array:1 [ 0 => "wayfindertarget=eyJpdiI6Ikp3dXpmNUVKbVdQdkNYV2RaT0tTZ3c9PSIsInZhbHVlIjoieUtRenVCRThmK1Via2RPYWVZNmRHZHNOOExQdDkxK05kcytVZi9Gb0NjRT0iLCJtYWMiOiJlNmEyZWM5YmU2N2U2ZmFiNWJmYTAyZWFiY2VlZmVhMTJiNGFlZjkyNDkyYTQ5MjRjMTc4YmQ0ZWQ0MjI2YjgwIiwidGFnIjoiIn0%3D; XSRF-TOKEN=eyJpdiI6ImoxOUNEeTY5d0xGNEJaUmI1LzllbWc9PSIsInZhbHVlIjoiSGk3ZERvKytqbVFsdmo0MWx0ak9ZcDNha3UweUpXcCtTM1dSU3Q5NlRGNngzMk42MlF6Q0JHUkN5UXFCK1NPazB4QzhUMk80Y3c5aDl6WGlYcTNZdlUyRUpNN2Fwc0NJeWxrZk1vcjREbjM4WEMyb2tUZEpCL0w4K29xREtFd3QiLCJtYWMiOiI4ZmQ1YmI0NGNjODc1ZmY4MDYxYjFkZGMzOWE4YzBjMzllNWEyOWQ3MDg5M2YzOGYzNmE4N2JjNDgyOTdmNTg4IiwidGFnIjoiIn0%3D; alphaeurekaselectnet_session=eyJpdiI6ImFwR2hBUFk0OTFqZWo4aE5rNDYvN3c9PSIsInZhbHVlIjoiZERqY3AwbTdrL1VJU2xZU0p2NzBtYmRQV1Y4NTQrYWgrclE3QmJGZUFKTTBISkJ5enp4NUZWK0VhTU9hSzc3d2NGSG1BTnFZK2Nia0VOVnZPdkRTY3dYbFE4KzJrQ1lsL0R2S1MyNDlUTS9zbjdIQkVFNjB3RTFaQ0svNEhLQ1AiLCJtYWMiOiJhYjMyM2MzMTU0NGI2NjFlYjQwNTZiODZmNWVhOGI0NGVkMTUwNTg3ZWQ0NDRlZDYxYzI1ZjU4OWJhMzllOWQ5IiwidGFnIjoiIn0%3Dwayfindertarget=eyJpdiI6Ikp3dXpmNUVKbVdQdkNYV2RaT0tTZ3c9PSIsInZhbHVlIjoieUtRenVCRThmK1Via2RPYWVZNmRHZHNOOExQdDkxK05kcytVZi9Gb0NjRT0iLCJtYWMiOiJlNmEyZWM5YmU2N2U2" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] ]
        request_cookies
        0 of 0
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        response_headers
        0 of 0
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        session_attributes
        0 of 0
        array:5 [ "_token" => "kyX7nDzbRxIlnXaLdnX7spf9d7jrIectiHjQjfeG" "uc_cart_id" => "df2dd514c3af83e16dc463921ab8c695" "_previous" => array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/2723" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]
        ClearShow all
        Date ↕MethodURLData
        #12025-02-19 13:38:11GET/chapter/2723112193