Efficient numerical approach to the evaluation of Kramers–Kronig transforms

Frederick W. King

doi:10.1364/JOSAB.19.002427

Journal of the Optical Society of America B
Vol. 19,
Issue 10,
pp. 2427-2436
(2002)
•https://doi.org/10.1364/JOSAB.19.002427

Efficient numerical approach to the evaluation of Kramers–Kronig transforms

Frederick W. King

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Frederick W. King, "Efficient numerical approach to the evaluation of Kramers–Kronig transforms," J. Opt. Soc. Am. B 19, 2427-2436 (2002)

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Abstract

A method is presented to deal with the numerical evaluation of Kramers–Kronig transforms (the Hilbert transforms of even and odd functions on the positive real axis). The general Hilbert transform is also treated. The functions involved must be continuous on the integration interval with suitable asymptotic behavior for large values of the argument and must have an appropriate functional form in the vicinity of the singularity of the integrand of the transform. The approach is based on a specialized Gaussian quadrature technique that uses the weight function $log x^{- 1} .$ This choice allows the region in the vicinity of the singularity to be swept into the quadrature weights and abscissa values. Application to the Lorentzian and Gaussian line profiles is discussed.

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Integration Range	Weight Function $W (x)$	Name
[-1, 1]	1	Gaussian quadrature (Gauss–Legendre quadrature)
[-1, 1]	$(1 - x)^{α} (1 + x)^{β}$	Gauss–Jacobi quadrature
[-1, 1]	$\frac{1}{\sqrt{1 - x^{2}}}$	Gauss–Chebyshev quadrature
[0, ∞)	$exp (- x)$	Gauss–Laguerre quadrature
(-∞, ∞)	$exp (- x^{2})$	Gauss–Hermite quadrature

Abscissas $(x_{i})$ for $N = 20$	Weights $(w_{i})$
0.0025883279559219554283327359	0.0431427521332080785789708432
0.0152096623495602317206559633	0.0753837099085893595504548599
0.0385365503721653279598479658	0.0930532674516630513726903187
0.0721816138158739064349676493	0.1014567118498297544369183081
0.1154605264876331505588940442	0.1032017620560720690578209545
0.1674428562753296857182838946	0.1000225498052731665327959061
0.2269837872602025033612950778	0.0932597993002976780836606811
0.2927549609415458329919802565	0.0840289528719410564970846278
0.3632774298578589045379836519	0.0732855891300307409628311569
0.4369571400907683184866372473	0.0618503369137302899572280804
0.5121225946789673361956659079	0.0504166044383746776370507997
0.5870640449144099151324930349	0.0395513700052983853329369672
0.6600734133149094139120980731	0.0296940778958128448046071929
0.7294840839296874988710464156	0.0211563153554270976730226771
0.7937096719870858177436895133	0.0141237329389640204365980865
0.8512808927891257272221674697	0.0086609745043354986282325129
0.9008796808544175942234387574	0.0047199401462036049543670820
0.9413697491290916763026513697	0.0021513974039652061146779200
0.9718227410752631937384773794	0.0007197282146532026463582613
0.9915380814387119726524613572	0.0001204276763302167416927636

Abscissas $(x_{i})$ for $N = 30$	Weights $(w_{i})$
0.0012147717930219078619721006	0.0226427880155717972364752744
0.0070698848398636750737533940	0.0413409551797399820101542980
0.0179065334209946570966305841	0.0535552862121838184181036241
0.0336625807723838111165915285	0.0616404701977886751557191408
0.0542024908397753460463388698	0.0666716469645710224033528535
0.0793322130781204850643511402	0.0692992302274823246760522915
0.1088050595268703156314527676	0.0699838673153801675089576039
0.1423259410593709649922480915	0.0690857692423597837279540197
0.1795553150184500145917898783	0.0669051377878338536260187226
0.2201132403152004107157416530	0.0637023827725008780859398918
0.2635836721058316158484568971	0.0597087974372192066609431041
0.3095190319213846653744364405	0.0551323069967228090289563000
0.3574450468663050737184337872	0.0501605203287082618537455726
0.4068658301337656204348113582	0.0449622584725675350110335564
0.4572691630337391855682951966	0.0396882215927176291916741416
0.5081319313878266644607028144	0.0344711889762481862846590559
0.5589256644938356104249742630	0.0294259983406547029653365544
0.6091221219482568961309204283	0.0246494636759520711105755976
0.6581988719544272807805487755	0.0202203370228654626740508914
0.7056448040796570904501244317	0.0161993846047650720413967623
0.7509655196063949323128239950	0.0126296238529002917479516805
0.7936885435542358419120213755	0.0095367507774674102056836543
0.8333683040568877979879093287	0.0069297743569083090191822777
0.8695908269833030438714047134	0.0048018646776125538937283664
0.9019780963694003887372300190	0.0031314135724879546569499634
0.9301920340588491078199217901	0.0018832999405013986144201825
0.9539380538003178065748955753	0.0010103464495323391500696887
0.9729681404440586949363011266	0.0004549497337611650082475757
0.9870833569545012580278362274	0.0001508624177249058278381933
0.9961351660193061567458035849	0.0000251028572704322048291612

x	Quadrature Result for the Hilbert Transform	Percentage Error
0.1	$- 1.582 756 340 \times 10^{- 1}$	$2.1 \times 10^{- 8}$
0.2	$- 1.552 731 152 116 052 07 \times 10^{- 1}$	$- 8.0 \times 10^{- 16}$
0.5	$- 1.273 239 544 735 162 686 151 \dots \times 10^{- 1}$	$- 1.2 \times 10^{- 30}$
0.9	$- 3.151 583 031 522 679 916 215 \dots \times 10^{- 1}$	$- 3.1 \times 10^{- 30}$
1.0	$- 3 . \times 10^{- 30}$	—
2.0	$1.591 549 430 918 453 357 688 \dots \times 10^{- 1}$	0
5.0	$7.489 644 380 795 074 624 417 6 \times 10^{- 2}$	$- 6.2 \times 10^{- 21}$
10.0	$3.493 645 092 261 1 \times 10^{- 2}$	$- 4.1 \times 10^{- 13}$
20.0	$1.670 687 247 \times 10^{- 2}$	$- 4.0 \times 10^{- 8}$
30.0	$1.096 316 7 \times 10^{- 2}$	$3.7 \times 10^{- 6}$
40.0	$8.156 432 \times 10^{- 3}$	$3.0 \times 10^{- 5}$
50.0	$6.493 4 \times 10^{- 3}$	$4.6 \times 10^{- 4}$

N	Quadrature Result for the Hilbert Transform	Percentage Error
Test value: $x = 0.1,$ exact result, $- 1.582 756 340 140 395 604 331 441 \times 10^{- 1}$
10	$- 2 \times 10^{- 1}$	32
20	$- 1.575 \times 10^{- 1}$	$- 4.7 \times 10^{- 1}$
30	$- 1.582 79 \times 10^{- 1}$	$2.4 \times 10^{- 3}$
40	$- 1.582 757 0 \times 10^{- 1}$	$4.3 \times 10^{- 5}$
50	$- 1.582 756 32 \times 10^{- 1}$	$- 1.4 \times 10^{- 6}$
60	$- 1.582 756 340 5 \times 10^{- 1}$	$2.1 \times 10^{- 8}$
Test value: $x = 10,$ exact result, $3.493 645 092 261 117 126 634 033 830 128 \times 10^{- 2}$
10	$3.42 \times 10^{- 2}$	-2.2
20	$3.493 58 \times 10^{- 2}$	$- 1.8 \times 10^{- 3}$
30	$3.493 646 \times 10^{- 2}$	$3.0 \times 10^{- 5}$
40	$3.493 645 094 \times 10^{- 2}$	$3.8 \times 10^{- 8}$
50	$3.493 645 092 253 \times 10^{- 2}$	$2.2 \times 10^{- 10}$
60	$3.493 645 092 261 10 \times 10^{- 2}$	$- 4.1 \times 10^{- 13}$

x	Quadrature Result for the Hilbert Transform	Percentage Error
0.2	$6.3239258 \times 10^{- 2}$	$2.3 \times 10^{- 6}$
0.3	$1.2398235848 \times 10^{- 1}$	$- 4.4 \times 10^{- 8}$
0.4	$1.79928906255 \times 10^{- 1}$	$2.7 \times 10^{- 9}$
0.5	$2.291471375338 \times 10^{- 1}$	$- 3.6 \times 10^{- 10}$
0.6	$2.702045938494 \times 10^{- 1}$	$- 5.7 \times 10^{- 11}$
0.7	$3.0224364264440 \times 10^{- 1}$	$- 1.0 \times 10^{- 11}$
0.8	$3.24996976916080 \times 10^{- 1}$	$- 1.6 \times 10^{- 12}$
0.9	$3.387464676226255 \times 10^{- 1}$	$2.0 \times 10^{- 13}$
1.0	$3.442357927011524 \times 10^{- 1}$	$1.8 \times 10^{- 13}$
1.5	$2.90621533638526725 \times 10^{- 1}$	$2.0 \times 10^{- 15}$
2.0	$2.056118376478668203 \times 10^{- 1}$	$- 1.7 \times 10^{- 16}$
3.0	$1.18128131171603032992 \times 10^{- 1}$	$- 8.9 \times 10^{- 18}$
4.0	$8.462414178948227469695 \times 10^{- 2}$	$- 3.6 \times 10^{- 19}$
5.0	$6.640891109983551851412 \times 10^{- 2}$	$- 1.3 \times 10^{- 18}$

Abstract

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Tables (6)

Equations (64)

Journal of the Optical Society of America B