Comparison of error properties of techniques used for measuring second-order nonlinear optical coefficients with least-squares fitting

Masashi Kiguchi

doi:10.1364/JOSAB.12.000871

Comparison of error properties of techniques used for measuring second-order nonlinear optical coefficients with least-squares fitting

Masashi Kiguchi

Journal of the Optical Society of America B
Vol. 12,
Issue 5,
pp. 871-875
(1995)
•https://doi.org/10.1364/JOSAB.12.000871

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Masashi Kiguchi, "Comparison of error properties of techniques used for measuring second-order nonlinear optical coefficients with least-squares fitting," J. Opt. Soc. Am. B 12, 871-875 (1995)

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History
- Original Manuscript: August 13, 1993
- Revised Manuscript: January 17, 1995
- Published: May 1, 1995

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Abstract

The technique of second-harmonic generation with an evanescent wave (SHEW) and the Maker fringe technique were examined by simulation of least-squares fitting. The calculations were made for correlation among parameters in a fit and error-propagation coefficients between parameters. Correlation among parameters for the SHEW technique is lower and the parameter value for the second-order nonlinear optical coefficient d is less sensitive to errors of refractive indices than for the Maker fringe technique. As a result, use of the SHEW technique can provide more-accurate values for the d coefficient than use of the Maker fringe technique.

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Coefficients	Maker Fringe		SHEW

	Case M1	Case M2	Case S1	Case S2	Case S3
w_i = 1/S_i
L (μm)	500	4	–	–	–
θ_in (deg)	−40–40	−40–40	5–85	38–85	50–80
r(n_s^ω − n_s²^ω)^b	−0.9999923	−0.9999917	0.029323	−0.022701	−0.999874
r(n_s^ω − d)	−0.9999973	−0.9999968	−0.384741	−0.872498	−0.999796
r(n_s²^ω − d)	0.9999976	0.9999988	0.146696	−0.140668	0.999352
r(d − L)	−0.9999994	−0.9999999	–	–	–
r(n_s^ω − L)	0.9999979	0.9999972	–	–	–
r(n_s²^ω − L)	−0.9999982	−0.9999985	–	–	–
Δd/Δn_s^ω^c	−20.42	−24.02	−2.445	−3.021	−0.4365
Δd/Δn_s²^ω	19.04	17.16	1.354	−3.087	0.5341
Δd/ΔL	−0.001970	−0.2509	–	–	–
w_i = 1/S_i²
L (μm)	500	4	–	–	–
θ_in(deg)	−40–40	−40–40	5–85	38–85	50–80
r(n_s^ω − n_s^2ω)^b	−0.9999998	−0.9999917	−0.253062	−0.384959	−0.999867
r(n_s^ω − d)	−0.9999802	−0.9999968	−0.024039	−0.84201	−0.999833
r(n_s²^ω − d)	0.9999804	0.9999988	0.010383	0.145052	0.999405
r(d − L)	−0.9999803	−0.9999999	–	–	–
r(n_s^ω − L)	0.9999999	0.9999972	–	–	–
r(n_s^2ω − L)	−0.9999999	−0.9999985	–	–	–
Δd/Δn_s^ω^c	−20.43	−24.03	−0.06035	−1.041	−0.4269
Δd/Δn_s²^ω	19.04	17.15	0.02484	0.6686	0.5132
Δd/ΔL	−0.001971	−0.2510	–	–	–

Coefficients	Maker Fringe		SHEW

	Case M3	Case M4	Case S4	Case S5	Case S6
w_i = 1/S_i
L (μm)	500	4	–	–	–
θ_in (deg)	−40–40	−40–40	5–85	50–80	60–80
r(n_s^ω − n_s^2ω)^b	−0.9999807	−0.7779881	0.032841	0.446852	−0.996954
r(n_s^ω − d)	−0.9998847	−0.8025341	−0.242716	−0.988666	−0.998944
r(n_s²^ω − d)	0.9998916	0.9991969	0.311140	−0.419253	0.992349
r(d − L)	−0.9998933	−0.9980364	–	–	–
r(n_s^ω − L)	0.9999941	0.8383258	–	–	–
r(n_s²^ω − L)	−0.9999961	−0.9947261	–	–	–
Δd/Δn_s^ω^c	−5.397	−15.03	−0.9837	−1.694	−0.7780
Δd/Δn_s²^ω	4.330	3.525	1.574	−17.74	2.178
Δd/ΔL	−0.001922	−0.3054	–	–	–
w_i = 1/S_i²
L (μm)	500	4	–	–	–
θ_in (deg)	−40–40	−40–40	5–85	50–80	60–80
r(n_s^ω − n_s^2ω)^b	−0.9999707	−0.9520684	−0.116194	0.385007	−0.996868
r(n_s^ω − d)	−0.9663214	−0.9602349	−0.003312	−0.981521	−0.998946
r(n_s²^ω − d)	0.9663332	0.9996102	0.089070	−0.415965	0.992228
r(d − L)	−0.9663349	−0.9992665	–	–	–
r(n_s^ω − L)	0.9999910	0.9702212	–	–	–
r(n_s²^ω − L)	−0.9999942	−0.9978086	–	–	–
Δd/Δn_s^ω^c	−5.575	−12.86	−0.008335	−1.358	−0.7432
Δd/Δn_s²^ω	4.481	3.661	0.2031	−6.921	1.930
Δd/ΔL	−0.001987	−0.2897	–	–	–

Coefficients	Maker Fringe		SHEW

	Case M1	Case M2	Case S1	Case S2	Case S3
w_i = 1/S_i
L (μm)	500	4	–	–	–
θ_in (deg)	−40–40	−40–40	5–85	38–85	50–80
r(n_s^ω − n_s²^ω)^b	−0.9999923	−0.9999917	0.029323	−0.022701	−0.999874
r(n_s^ω − d)	−0.9999973	−0.9999968	−0.384741	−0.872498	−0.999796
r(n_s²^ω − d)	0.9999976	0.9999988	0.146696	−0.140668	0.999352
r(d − L)	−0.9999994	−0.9999999	–	–	–
r(n_s^ω − L)	0.9999979	0.9999972	–	–	–
r(n_s²^ω − L)	−0.9999982	−0.9999985	–	–	–
Δd/Δn_s^ω^c	−20.42	−24.02	−2.445	−3.021	−0.4365
Δd/Δn_s²^ω	19.04	17.16	1.354	−3.087	0.5341
Δd/ΔL	−0.001970	−0.2509	–	–	–
w_i = 1/S_i²
L (μm)	500	4	–	–	–
θ_in(deg)	−40–40	−40–40	5–85	38–85	50–80
r(n_s^ω − n_s^2ω)^b	−0.9999998	−0.9999917	−0.253062	−0.384959	−0.999867
r(n_s^ω − d)	−0.9999802	−0.9999968	−0.024039	−0.84201	−0.999833
r(n_s²^ω − d)	0.9999804	0.9999988	0.010383	0.145052	0.999405
r(d − L)	−0.9999803	−0.9999999	–	–	–
r(n_s^ω − L)	0.9999999	0.9999972	–	–	–
r(n_s^2ω − L)	−0.9999999	−0.9999985	–	–	–
Δd/Δn_s^ω^c	−20.43	−24.03	−0.06035	−1.041	−0.4269
Δd/Δn_s²^ω	19.04	17.15	0.02484	0.6686	0.5132
Δd/ΔL	−0.001971	−0.2510	–	–	–

Coefficients	Maker Fringe		SHEW

	Case M3	Case M4	Case S4	Case S5	Case S6
w_i = 1/S_i
L (μm)	500	4	–	–	–
θ_in (deg)	−40–40	−40–40	5–85	50–80	60–80
r(n_s^ω − n_s^2ω)^b	−0.9999807	−0.7779881	0.032841	0.446852	−0.996954
r(n_s^ω − d)	−0.9998847	−0.8025341	−0.242716	−0.988666	−0.998944
r(n_s²^ω − d)	0.9998916	0.9991969	0.311140	−0.419253	0.992349
r(d − L)	−0.9998933	−0.9980364	–	–	–
r(n_s^ω − L)	0.9999941	0.8383258	–	–	–
r(n_s²^ω − L)	−0.9999961	−0.9947261	–	–	–
Δd/Δn_s^ω^c	−5.397	−15.03	−0.9837	−1.694	−0.7780
Δd/Δn_s²^ω	4.330	3.525	1.574	−17.74	2.178
Δd/ΔL	−0.001922	−0.3054	–	–	–
w_i = 1/S_i²
L (μm)	500	4	–	–	–
θ_in (deg)	−40–40	−40–40	5–85	50–80	60–80
r(n_s^ω − n_s^2ω)^b	−0.9999707	−0.9520684	−0.116194	0.385007	−0.996868
r(n_s^ω − d)	−0.9663214	−0.9602349	−0.003312	−0.981521	−0.998946
r(n_s²^ω − d)	0.9663332	0.9996102	0.089070	−0.415965	0.992228
r(d − L)	−0.9663349	−0.9992665	–	–	–
r(n_s^ω − L)	0.9999910	0.9702212	–	–	–
r(n_s²^ω − L)	−0.9999942	−0.9978086	–	–	–
Δd/Δn_s^ω^c	−5.575	−12.86	−0.008335	−1.358	−0.7432
Δd/Δn_s²^ω	4.481	3.661	0.2031	−6.921	1.930
Δd/ΔL	−0.001987	−0.2897	–	–	–

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