Second-harmonic generation in fibers

R. W. Terhune; D. A. Weinberger

doi:10.1364/JOSAB.4.000661

Journal of the Optical Society of America B
Vol. 4,
Issue 5,
pp. 661-674
(1987)
•https://doi.org/10.1364/JOSAB.4.000661

Second-harmonic generation in fibers

R. W. Terhune and D. A. Weinberger

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R. W. Terhune and D. A. Weinberger, "Second-harmonic generation in fibers," J. Opt. Soc. Am. B 4, 661-674 (1987)

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Abstract

In this analytical paper, a formalismfor second-harmonic and sum-frequency generation in glass fibers is developed. It considers only the lowest-order nonlinearities in regular, step-index fibers. Plots showing the dependence of index-matching frequencies on the core radius and index difference across the core-cladding interface are given. Estimates are given for the conversion efficiencies due to the nonlinear polarization at the core-cladding interface and the bulk nonlinear polarization proportional to E ∇E, which includes quadrupolar terms. It is concluded that the interface effects would be dominant. Calculations for the longitudinal-electric-field-induced nonlinear polarization proportional to E² that involve the same mode coupling are also included. The calculated conversion efficiencies are small and cannot explain the recently observed high conversion efficiencies that were obtained under nonindex-matched conditions. A maximum conversion efficiency of around 10⁻⁵ is predicted under index-matching conditions.

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Modes	Frequency (cm⁻¹)	ρ (10⁻⁶ m)	Mole Fraction GeO₂	∂Δβ/∂ In ρ (10⁴ m⁻¹)	2π∂Δβ/∂ω (m⁻¹/cm⁻¹)	Bulk (1/ρ²) O_δ₋_β₋₂₂_γ	Overlap Integrals

							Interface (1/ρ²)			Field Induced (1/iβ₁ρ²)

							O_A	O_B	O_C	O₁₁₂₂	O₁₂₂₁
LP₀₁ + LP₀₁ → LP₃₁
HE₁₁ + HE₁₁ → EH₂₁							0.062	−0.064	−0.003	0.002	0.000
LP₀₁ + LP₀₁ → LP₁₂
HE₁₁ + HE₁₁ → HE₂₂							0.053	0.054	−0.003	0.002	0.036
→TM₀₂	11108	1.5	0.164	21.8	49	−0.002	0.109	0.001	0.216	0.064	0.035
→TM₀₂	5602	3.5758	0.12	6.6	0^a	−0.002	0.101	0.000	0.201	0.056	0.030
LP₀₁ + LP₀₁ → LP₃₂
HE₁₁ + HE₁₁ → EH₂₂	14403	1.5	0.212	74.3	119	0.002	0.037	−0.040	−0.003	0.002	0.000
LP₀₁ + LP₀₁ → LP₁₃
HE₁₁ + HE₁₁ → TM₀₃	14721	1.5	0.212	70.1	117	0.002	0.057	0.002	0.113	−0.058	−0.031
→ TM₀₃	10593	3.0	0.14	23.8	56	0.001	0.043	0.001	0.086	−0.047	−0.025
→TM₀₃	6464	6.0	0.116	13.8	18	0.001	0.035	0.001	0.070	−0.039	−0.020
LP₀₁ + LP₁₁ → LP₂₂
HE₁₁ + TM₀₁ → EH₁₂	11897	1.5	0.24	42.8	77	−0.001	0.257	−0.143	0.251	−0.086	−0.061
+ TM₀₁ → EH₁₂	8071	3.0	0.20	19.6	34	−0.003	0.186	−0.103	0.182	−0.068	−0.040
+ HE₂₁ → EH₁₂	8074	3.0	0.20	19.5	34	0.022	0.090	0.002	0.178	−0.071	−0.034
+ TE₀₁ → EH₁₂^b	8089	3.0	0.20	19.6	34	0.008	0.000	0.000	−0.191	0.100	−0.042
LP₀₁ + LP₁₁ → LP₀₃
HE₁₁ + TM₀₁ → HE₁₃	8371	3.0	0.20	20.5	37	−0.008	0.192	0.097	0.186	−0.090	−0.153
+ HE₂₁ → HE₁₃	8374	3.0	0.20	20.6	37	0.013	0.092	0.002	0.182	−0.094	−0.045
+ TE₀₁ → HE₁₃^b	8388	3.0	0.20	20.8	37	0.008	0.000	0.000	−0.191	0.100	−0.042
LP₀₁ +LP₁₁ → LP₄₁
HE₁₁ + HE₂₁ → EH₃₁	10786	1.5	0.24	40.0	69	0.000	0.143	−0.153	−0.010	0.010	0.001
LP₁₁ + LP₁₁ → LP₁₁
TE₀₁ + TE₀₁ → TM₀₂	5013	1.65	0.80	14.9	19	0.042	0.000	0.000	0.473	0.465	0.000
TE₀₁ + TE₀₁ → TM₀₃	11444	1.65	0.80	75.6	144	0.026	0.000	0.000	0.141	0.115	0.000

Modes	Frequency (cm⁻¹)	ρ (10⁻⁶ m)	Mole Fraction GeO₂	Interaction Lengths (10⁻⁴ m⁻¹ W^−1/2)		Width^aδω_match/2π (cm⁻¹)	Peak Conversion Efficiency (10⁶)^a

				Interface^b 1/l_interface	Field Induced^c 1/l_E		Interface	Field Induced
LP₀₁+LP₀₁ → LP₃₁
HE₁₁ + HE₁₁ → EH₂₁	10335	1.5	0.164	1.5	−0.2	1.6	0.2	–
LP₀₁ +LP₀₁ → LP₁₂
HE₁₁ + HE₁₁ → HE₂₂	11113	1.5	0.164	2.0	−4.3	1.3	0.9	3.9
→ TM₀₂	11108	1.5	0.164	3.4	−8.2	1.3	2,6	14.5
→ TM₀₂	5602	3.5758	0.12	0.5	−1.3	34^e	0.2	6.8
LP₀₁ + LP₀₁ → LP₃₂
HE₁₁+ HE₁₁ → EH₂₂	14403	1.5	0.212	1.5	0.16	1.9	0.2	0.00
LP₀₁ + LP₀₁ → LP₁₃
HE₁₁ + HE₁₁ → TM₀₃	14721	1.5	0.212	3.9	−5.0	1.8	1.0	1.7
→ TM₀₃	10593	3.0	0.14	0.4	−1.1	1.7	0.1	1.0
→ TM₀₃	6464	6.0	0.116	0.04	−0.3	2.3	0.00	0.4
LP₀₁ + LP₁₁ → LP₂₂
HF₁₁ + TM0₁ → EH₁₀	11897	1.5	0.24	11.0	11.6	1.7	13.3	14.9
+ TM₀₁ → EH₁₂	8071	3.0	0.20	1.1	−2.3	1.7	1.3	5.0
+ HE₂₁ → EH₁₂	8074	3.0	0.20	0.7	−2.1	1.7	0.4	4.4
+ TE₀₁ → EH₁₂^d	8089	3.0	0.20	0.2	2.7	L7	0.02	6.9
LP₀₁ + LP₁₁ → LP₀₃
HE₁₁ + TM₀₁ → HE₁₃	8371	3.0	0.20	1.5	−6.1	1.7	2.1	34.0
+ HE₂₁, → HE₁₃	8374	3.0	0.20	0.7	−2.8	1.7	0.4	7.2
+ TE₀₁ → HE₁₃^d	8388	3.0	0.20	0.1	−0.2	1.7	0.00	0.1
LP₀₁ + LP₁₁ → LP₄₁
HE₁₁ + HE₂₁ → EH₃₁	10786	1.5	0.24	5.1	0.7	1.7	3.0	0.06
LP₁₁ + LP₁₁ → LP_1n
TE₀₁ + TE₀₁ → TM₀₂	5013	1.65	0.80	0.3	−22.3	2.4	0.04	190.0
TE₀₁ + TE₀₁ → TM₀₃	11444	1.65	0.80	0.2	−5.1	1.6	0.00	2.0

Modes	Δβ (10⁵ m⁻¹)	Bulk (1/ρ²)		Interface (1/ρ²)

		O₂_γ^b	O_δ₋_β₋₂_γ	O_A	O_B	O_C
LP₀₁ + LP₁₁ → LP₀₁
HE₁₁ + TE₀₁ → HE₁₁^c	1.8	−0.003	−0.728	0.000	−0.062	0.124
+ TM₀₁ → HE₁₁	1.8	−0.003	−0.801	0.122	0.063	0.122
+ HE₂₁ → HE₁₁	1.8	−0.004	0.726	0.062	0.000	0.122
LP₀₁ +LP₁₁ → LP₂₁
HE₁₁ +TE₀₁ → EH₁₁^c	1.3	0.004	−0.014	0.000	0.130	0.254
+ TM₀₁ → EH₁₁	1.3	0.004	−0.016	0.254	−0.130 0.252
+ HE₂₁ → EH₁₁	1.3	0.004	0.012	0.126	0.001	0.250
+ HE₂₁ → HE₃₁	1.3	0.000	0.001	0.126	0.129	0.000
LP₀₁ + LP₁₁ → LP₀₂
HE₁₁ + TE₀₁ → HE₁₂^c	1.2	0.003 0.000	0.000	−0.133	0.270
+ TM₀₁ → HE₁₂	1.2	0.003	0.088	0.268	0.135	0.268
+ HE₂₁ → HE₁₂	1.2	0.008	−0.113	0.134	0.001	0.266
LP₀₁ +LP₀₁ → LP₁₁
HE₁₁ + HE₁₁ → TM₀₁	1.3	0.002	−0.006	0.152	0.002	0.304
LP₀₁ + LP₀₁ → LP₁₂
HE₁₁ + HE₁₁ → HE₂₂	0.6	0.000	−0.001	0.104	0.104	−0.002
LP₀₁ + LP₀₁ → LP₃₁
HE₁₁ + HE₁₁ → EH₂₁	0.8	0.000	−0.002	0.118	−0.118	−0.002
LP₁₁ + LP₁₁ → LP₁₁
HE₂₁ + TM₀₁ → HE₂₁	1.9	−0.006	−0.058	0.262	0.134	0.262
HE₂₁ + HE₂₁ → TM₀₁	1.9	0.000	1.053	0.262	0.002	0.522
TE₀₁ +TE₀₁ →TM₀₁	1.9	0.000	−1.062	0.000	0.000	0.526
TM₀₁ + TM₀₁ → TM₀₁	1.9	0.000	−1.063	0.536	0.002	0.536
TM₀₁ + TE₀₁ → TE₀₁	1.9	−0.006	0.061	0.000	0.266	0.000
HE₂₁ +HE₂₁^c → TE₀₁	1.9	0.000	−0.008	0.000	0.000	0.000
HE₂₁ + TE₀₁ → HE₂₁^c	1.9	0.004	−0.003	0.000	−0.131	0.262

Modes	Frequency (cm⁻¹)	ρ (10⁻⁶ m)	Mole Fraction GeO₂	∂Δβ/∂ In ρ (10⁴ m⁻¹)	2π∂Δβ/∂ω (m⁻¹/cm⁻¹)	Bulk (1/ρ²) O_δ₋_β₋₂₂_γ	Overlap Integrals

							Interface (1/ρ²)			Field Induced (1/iβ₁ρ²)

							O_A	O_B	O_C	O₁₁₂₂	O₁₂₂₁
LP₀₁ + LP₀₁ → LP₃₁
HE₁₁ + HE₁₁ → EH₂₁							0.062	−0.064	−0.003	0.002	0.000
LP₀₁ + LP₀₁ → LP₁₂
HE₁₁ + HE₁₁ → HE₂₂							0.053	0.054	−0.003	0.002	0.036
→TM₀₂	11108	1.5	0.164	21.8	49	−0.002	0.109	0.001	0.216	0.064	0.035
→TM₀₂	5602	3.5758	0.12	6.6	0^a	−0.002	0.101	0.000	0.201	0.056	0.030
LP₀₁ + LP₀₁ → LP₃₂
HE₁₁ + HE₁₁ → EH₂₂	14403	1.5	0.212	74.3	119	0.002	0.037	−0.040	−0.003	0.002	0.000
LP₀₁ + LP₀₁ → LP₁₃
HE₁₁ + HE₁₁ → TM₀₃	14721	1.5	0.212	70.1	117	0.002	0.057	0.002	0.113	−0.058	−0.031
→ TM₀₃	10593	3.0	0.14	23.8	56	0.001	0.043	0.001	0.086	−0.047	−0.025
→TM₀₃	6464	6.0	0.116	13.8	18	0.001	0.035	0.001	0.070	−0.039	−0.020
LP₀₁ + LP₁₁ → LP₂₂
HE₁₁ + TM₀₁ → EH₁₂	11897	1.5	0.24	42.8	77	−0.001	0.257	−0.143	0.251	−0.086	−0.061
+ TM₀₁ → EH₁₂	8071	3.0	0.20	19.6	34	−0.003	0.186	−0.103	0.182	−0.068	−0.040
+ HE₂₁ → EH₁₂	8074	3.0	0.20	19.5	34	0.022	0.090	0.002	0.178	−0.071	−0.034
+ TE₀₁ → EH₁₂^b	8089	3.0	0.20	19.6	34	0.008	0.000	0.000	−0.191	0.100	−0.042
LP₀₁ + LP₁₁ → LP₀₃
HE₁₁ + TM₀₁ → HE₁₃	8371	3.0	0.20	20.5	37	−0.008	0.192	0.097	0.186	−0.090	−0.153
+ HE₂₁ → HE₁₃	8374	3.0	0.20	20.6	37	0.013	0.092	0.002	0.182	−0.094	−0.045
+ TE₀₁ → HE₁₃^b	8388	3.0	0.20	20.8	37	0.008	0.000	0.000	−0.191	0.100	−0.042
LP₀₁ +LP₁₁ → LP₄₁
HE₁₁ + HE₂₁ → EH₃₁	10786	1.5	0.24	40.0	69	0.000	0.143	−0.153	−0.010	0.010	0.001
LP₁₁ + LP₁₁ → LP₁₁
TE₀₁ + TE₀₁ → TM₀₂	5013	1.65	0.80	14.9	19	0.042	0.000	0.000	0.473	0.465	0.000
TE₀₁ + TE₀₁ → TM₀₃	11444	1.65	0.80	75.6	144	0.026	0.000	0.000	0.141	0.115	0.000

Modes	Frequency (cm⁻¹)	ρ (10⁻⁶ m)	Mole Fraction GeO₂	Interaction Lengths (10⁻⁴ m⁻¹ W^−1/2)		Width^aδω_match/2π (cm⁻¹)	Peak Conversion Efficiency (10⁶)^a

				Interface^b 1/l_interface	Field Induced^c 1/l_E		Interface	Field Induced
LP₀₁+LP₀₁ → LP₃₁
HE₁₁ + HE₁₁ → EH₂₁	10335	1.5	0.164	1.5	−0.2	1.6	0.2	–
LP₀₁ +LP₀₁ → LP₁₂
HE₁₁ + HE₁₁ → HE₂₂	11113	1.5	0.164	2.0	−4.3	1.3	0.9	3.9
→ TM₀₂	11108	1.5	0.164	3.4	−8.2	1.3	2,6	14.5
→ TM₀₂	5602	3.5758	0.12	0.5	−1.3	34^e	0.2	6.8
LP₀₁ + LP₀₁ → LP₃₂
HE₁₁+ HE₁₁ → EH₂₂	14403	1.5	0.212	1.5	0.16	1.9	0.2	0.00
LP₀₁ + LP₀₁ → LP₁₃
HE₁₁ + HE₁₁ → TM₀₃	14721	1.5	0.212	3.9	−5.0	1.8	1.0	1.7
→ TM₀₃	10593	3.0	0.14	0.4	−1.1	1.7	0.1	1.0
→ TM₀₃	6464	6.0	0.116	0.04	−0.3	2.3	0.00	0.4
LP₀₁ + LP₁₁ → LP₂₂
HF₁₁ + TM0₁ → EH₁₀	11897	1.5	0.24	11.0	11.6	1.7	13.3	14.9
+ TM₀₁ → EH₁₂	8071	3.0	0.20	1.1	−2.3	1.7	1.3	5.0
+ HE₂₁ → EH₁₂	8074	3.0	0.20	0.7	−2.1	1.7	0.4	4.4
+ TE₀₁ → EH₁₂^d	8089	3.0	0.20	0.2	2.7	L7	0.02	6.9
LP₀₁ + LP₁₁ → LP₀₃
HE₁₁ + TM₀₁ → HE₁₃	8371	3.0	0.20	1.5	−6.1	1.7	2.1	34.0
+ HE₂₁, → HE₁₃	8374	3.0	0.20	0.7	−2.8	1.7	0.4	7.2
+ TE₀₁ → HE₁₃^d	8388	3.0	0.20	0.1	−0.2	1.7	0.00	0.1
LP₀₁ + LP₁₁ → LP₄₁
HE₁₁ + HE₂₁ → EH₃₁	10786	1.5	0.24	5.1	0.7	1.7	3.0	0.06
LP₁₁ + LP₁₁ → LP_1n
TE₀₁ + TE₀₁ → TM₀₂	5013	1.65	0.80	0.3	−22.3	2.4	0.04	190.0
TE₀₁ + TE₀₁ → TM₀₃	11444	1.65	0.80	0.2	−5.1	1.6	0.00	2.0

Abstract

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Figures (4)

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Equations (49)

Journal of the Optical Society of America B