Numerical evaluation of truncated Kramers-Kronig transforms

Frederick W. King

doi:10.1364/JOSAB.24.001589

Journal of the Optical Society of America B
Vol. 24,
Issue 7,
pp. 1589-1595
(2007)
•https://doi.org/10.1364/JOSAB.24.001589

Numerical evaluation of truncated Kramers–Kronig transforms

Frederick W. King

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Frederick W. King, "Numerical evaluation of truncated Kramers-Kronig transforms," J. Opt. Soc. Am. B 24, 1589-1595 (2007)

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Abstract

The truncated Kramers–Kronig transform, used widely in the analysis of optical data, is recast into a form that avoids the need to evaluate a Cauchy principal-value integral. A specialized Gaussian quadrature involving the weight function $\log_{e} x^{- 1}$ is employed. This approach yields accurate results for functions that lead to kernels with relatively rapid decay, which covers the cases most commonly encountered in optical data analysis. An application to the reststrahlen region of the $GaAs$ spectrum is made.

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Equations (38)

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$f (x)$	$\frac{1}{π} P \int_{- 1}^{1} \frac{f (s) d s}{x - s}$	Numerical Quadrature	From the Exact
	(Exact Result)	Result	Result
x	$\frac{x}{π} \log ∣ \frac{1 + x}{1 - x} ∣ - \frac{2}{π}$	$- 0.4617701961$	$- 0.4617701961$
$e^{x}$	$\frac{e^{x}}{π} {E_{1} (1 - x) - E_{1} (1 + x) - 2 Shi (1 - x)}$	$- 0.2908672551$	$- 0.2908672551$
$\sin x$	$\frac{\sin x}{π} {Ci (1 + x) - Ci (1 - x)} - \frac{\cos x}{π} {Si (1 + x) + Si (1 - x)}$	$- 0.4088775094$	$- 0.4088775094$

Abscissas $({\bar{x}}_{i})$	Weights $({\bar{w}}_{i})$
0.3242993314 4248107289 12086753 E-03	0.7155181881 1300426417 62569777 E-02
0.1860416905 5784504158 15950880 E-02	0.1373694600 1689768187 39119679 E-01
0.4696573639 0991652078 39421141 E-02	0.1869537888 1776323261 40629494 E-01
0.8833904836 9572925024 05254807 E-02	0.2263474845 1526893488 03176064 E-01
0.1426594295 0014852993 59075483 E-01	0.2582207148 4993127288 09788143 E-01
0.2098127619 9176948993 75918976 E-01	0.2840945573 6376243830 23740854 E-01
0.2896429015 1974968507 77796613 E-01	0.3049634644 8417452363 64635613 E-01
0.3819550862 7577167845 76524517 E-01	0.3215378277 2867450134 83128685 E-01
0.4865179933 4554167127 17892587 E-01	0.3343582004 8381905227 23927553 E-01
0.6030652447 3874537777 99581031 E-01	0.3438560534 1658051990 39257271 E-01
0.7312966757 2377715272 19560949 E-01	0.3503889570 1292293530 72188840 E-01
0.8708795071 6818465500 47868478 E-01	0.3542622661 8137415111 60834543 E-01
0.1021449492 7718703334 37534345	0.3557431317 9722807461 68939944 E-01
0.1182612079 0207493527 67365360	0.3550698901 9998310718 19109188 E-01
0.1353943598 7942803410 62150527	0.3524585360 1232078426 25832897 E-01
0.1534992510 2791649982 44103932	0.3481072831 2964683853 77116284 E-01
0.1725280687 4247676881 07516152	0.3421998324 7375104565 33520258 E-01
0.1924304764 8440543587 94060473	0.3349077416 9485313548 99206955 E-01
0.2131537537 9162642166 49791475	0.3263921574 4805421225 31608779 E-01
0.2346429417 4030179976 88820397	0.3168050869 5948533893 63340905 E-01
0.2568409936 8793538512 01640973	0.3062903316 6486108988 64536888 E-01
0.2796889310 5491968614 55351222	0.2949841701 0523366161 83963033 E-01
0.3031260038 4673172558 90597421	0.2830158532 0746897358 54189801 E-01
0.3270898555 7690101950 71618612	0.2705079584 7117274794 61080971 E-01
0.3515166922 1778264157 37389705	0.2575766378 9968035821 51906590 E-01
0.3763414547 7968952572 17397767	0.2443317861 4624767855 67389969 E-01
0.4014979950 9749646457 95562510	0.2308771492 5795409049 59389643 E-01
0.4269192543 8636326868 53936516	0.2173103899 0143052525 78216135 E-01
0.4525374441 1401036027 93196121	0.2037231215 8329769188 99028894 E-01
0.4782842287 2551210517 52776207	0.1902009218 1711193980 25609730 E-01
0.5040909097 4749546810 89911828	0.1768233322 1730712784 24231726 E-01
0.5298886107 9168916656 71621170	0.1636638519 6378294852 73754391 E-01
0.5556084629 7277580090 61005602	0.1507899298 6756158028 59216140 E-01
0.5811817902 5231879837 89849734	0.1382629592 9824698944 60722438 E-01
0.6065402942 1906302504 31149301	0.1261382794 4941953790 76135105 E-01
0.6316162378 1606124476 92522564	0.1144651857 7548128854 46957183 E-01
0.6563426275 2679276478 30505384	0.1032869519 0076314604 93804968 E-01
0.6806533935 3566088849 86063477	0.9264086485 4849783331 65202495 E-02
0.7044835673 8293455484 58207044	0.8255827510 8878508270 69568103 E-02
0.7277694566 4029349689 04441159	0.7306466256 2173184923 01686737 E-02
0.7504488161 4060444129 26572297	0.6417971934 6374123518 26380248 E-02
0.7724610153 0435804028 93767166	0.5591745006 7047158592 38924556 E-02
0.7937472011 1529142145 35345932	0.4828628988 4274780594 39472622 E-02
0.8142504563 0906633852 22587269	0.4128924063 9033490732 27806408 E-02
0.8339159523 5142074989 45672602	0.3492402505 7411225839 44405848 E-02
0.8526910967 9590805452 69922043	0.2918325890 6919565532 41258132 E-02
0.8705256746 2611647352 99979058	0.2405464083 5959599528 52300011 E-02
0.8873719832 0303087598 55773764	0.1952115949 7066545695 73897805 E-02
0.9031849604 5482295781 03094573	0.1556131743 5442202694 35263614 E-02
0.9179223059 6359360185 33336050	0.1214937111 5354054258 44980658 E-02
0.9315445946 2097135151 50164879	0.9255586357 3497100062 22011536 E-03
0.9440153825 4095212996 78639651	0.6846508368 2151985039 66829933 E-03
0.9553013049 3140747807 62997290	0.4885245462 6620370345 95480646 E-03
0.9653721656 2844728181 58690176	0.3331765500 4587598419 89636978 E-03
0.9742010179 6975726056 16350959	0.2143203994 8545842187 36594045 E-03
0.9817642365 5412274904 05066658	0.1274182787 6086249031 91376689 E-03
0.9880415789 2436265047 76861980	0.6771381333 3216217294 81754795 E-04
0.9930162339 2887477895 28321127	0.3026569953 0513963209 42851280 E-04
0.9966748400 1858749516 02439285	0.9982035828 0485052441 39485304 E-05
0.9990073230 8937402711 32293595	0.1655280633 5491778653 48936290 E-05

$f (x)$	$\frac{1}{π} P \int_{- 1}^{1} \frac{f (s) d s}{x - s}$	Numerical Quadrature	From the Exact
	(Exact Result)	Result	Result
x	$\frac{x}{π} \log ∣ \frac{1 + x}{1 - x} ∣ - \frac{2}{π}$	$- 0.4617701961$	$- 0.4617701961$
$e^{x}$	$\frac{e^{x}}{π} {E_{1} (1 - x) - E_{1} (1 + x) - 2 Shi (1 - x)}$	$- 0.2908672551$	$- 0.2908672551$
$\sin x$	$\frac{\sin x}{π} {Ci (1 + x) - Ci (1 - x)} - \frac{\cos x}{π} {Si (1 + x) + Si (1 - x)}$	$- 0.4088775094$	$- 0.4088775094$

Abscissas $({\bar{x}}_{i})$	Weights $({\bar{w}}_{i})$
0.3242993314 4248107289 12086753 E-03	0.7155181881 1300426417 62569777 E-02
0.1860416905 5784504158 15950880 E-02	0.1373694600 1689768187 39119679 E-01
0.4696573639 0991652078 39421141 E-02	0.1869537888 1776323261 40629494 E-01
0.8833904836 9572925024 05254807 E-02	0.2263474845 1526893488 03176064 E-01
0.1426594295 0014852993 59075483 E-01	0.2582207148 4993127288 09788143 E-01
0.2098127619 9176948993 75918976 E-01	0.2840945573 6376243830 23740854 E-01
0.2896429015 1974968507 77796613 E-01	0.3049634644 8417452363 64635613 E-01
0.3819550862 7577167845 76524517 E-01	0.3215378277 2867450134 83128685 E-01
0.4865179933 4554167127 17892587 E-01	0.3343582004 8381905227 23927553 E-01
0.6030652447 3874537777 99581031 E-01	0.3438560534 1658051990 39257271 E-01
0.7312966757 2377715272 19560949 E-01	0.3503889570 1292293530 72188840 E-01
0.8708795071 6818465500 47868478 E-01	0.3542622661 8137415111 60834543 E-01
0.1021449492 7718703334 37534345	0.3557431317 9722807461 68939944 E-01
0.1182612079 0207493527 67365360	0.3550698901 9998310718 19109188 E-01
0.1353943598 7942803410 62150527	0.3524585360 1232078426 25832897 E-01
0.1534992510 2791649982 44103932	0.3481072831 2964683853 77116284 E-01
0.1725280687 4247676881 07516152	0.3421998324 7375104565 33520258 E-01
0.1924304764 8440543587 94060473	0.3349077416 9485313548 99206955 E-01
0.2131537537 9162642166 49791475	0.3263921574 4805421225 31608779 E-01
0.2346429417 4030179976 88820397	0.3168050869 5948533893 63340905 E-01
0.2568409936 8793538512 01640973	0.3062903316 6486108988 64536888 E-01
0.2796889310 5491968614 55351222	0.2949841701 0523366161 83963033 E-01
0.3031260038 4673172558 90597421	0.2830158532 0746897358 54189801 E-01
0.3270898555 7690101950 71618612	0.2705079584 7117274794 61080971 E-01
0.3515166922 1778264157 37389705	0.2575766378 9968035821 51906590 E-01
0.3763414547 7968952572 17397767	0.2443317861 4624767855 67389969 E-01
0.4014979950 9749646457 95562510	0.2308771492 5795409049 59389643 E-01
0.4269192543 8636326868 53936516	0.2173103899 0143052525 78216135 E-01
0.4525374441 1401036027 93196121	0.2037231215 8329769188 99028894 E-01
0.4782842287 2551210517 52776207	0.1902009218 1711193980 25609730 E-01
0.5040909097 4749546810 89911828	0.1768233322 1730712784 24231726 E-01
0.5298886107 9168916656 71621170	0.1636638519 6378294852 73754391 E-01
0.5556084629 7277580090 61005602	0.1507899298 6756158028 59216140 E-01
0.5811817902 5231879837 89849734	0.1382629592 9824698944 60722438 E-01
0.6065402942 1906302504 31149301	0.1261382794 4941953790 76135105 E-01
0.6316162378 1606124476 92522564	0.1144651857 7548128854 46957183 E-01
0.6563426275 2679276478 30505384	0.1032869519 0076314604 93804968 E-01
0.6806533935 3566088849 86063477	0.9264086485 4849783331 65202495 E-02
0.7044835673 8293455484 58207044	0.8255827510 8878508270 69568103 E-02
0.7277694566 4029349689 04441159	0.7306466256 2173184923 01686737 E-02
0.7504488161 4060444129 26572297	0.6417971934 6374123518 26380248 E-02
0.7724610153 0435804028 93767166	0.5591745006 7047158592 38924556 E-02
0.7937472011 1529142145 35345932	0.4828628988 4274780594 39472622 E-02
0.8142504563 0906633852 22587269	0.4128924063 9033490732 27806408 E-02
0.8339159523 5142074989 45672602	0.3492402505 7411225839 44405848 E-02
0.8526910967 9590805452 69922043	0.2918325890 6919565532 41258132 E-02
0.8705256746 2611647352 99979058	0.2405464083 5959599528 52300011 E-02
0.8873719832 0303087598 55773764	0.1952115949 7066545695 73897805 E-02
0.9031849604 5482295781 03094573	0.1556131743 5442202694 35263614 E-02
0.9179223059 6359360185 33336050	0.1214937111 5354054258 44980658 E-02
0.9315445946 2097135151 50164879	0.9255586357 3497100062 22011536 E-03
0.9440153825 4095212996 78639651	0.6846508368 2151985039 66829933 E-03
0.9553013049 3140747807 62997290	0.4885245462 6620370345 95480646 E-03
0.9653721656 2844728181 58690176	0.3331765500 4587598419 89636978 E-03
0.9742010179 6975726056 16350959	0.2143203994 8545842187 36594045 E-03
0.9817642365 5412274904 05066658	0.1274182787 6086249031 91376689 E-03
0.9880415789 2436265047 76861980	0.6771381333 3216217294 81754795 E-04
0.9930162339 2887477895 28321127	0.3026569953 0513963209 42851280 E-04
0.9966748400 1858749516 02439285	0.9982035828 0485052441 39485304 E-05
0.9990073230 8937402711 32293595	0.1655280633 5491778653 48936290 E-05

Abstract

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Figures (2)

Tables (2)

Equations (38)

Journal of the Optical Society of America B