Abstract

The degradation of second-harmonic generation in quasi-phase-matching waveguides, when random errors occur in the domain boundary position and when the effective index varies along the waveguide, is studied theoretically. Two models for random errors are used, one assuming independent shifts of the domain boundaries and one assuming independent domain lengths. Only the influence of random errors following the statistics in the second model might be of any significance in practical implementations. It is shown that in this case the normalized output power is a decreasing function of the product of the number of domains and a relative variance of the stochastic disturbances. If the difference in effective indices between the interacting modes varies along the waveguide, the same kind of ripple that has been observed in birefringence phase matching occurs.

© 1991 Optical Society of America

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References

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  1. E. J. Lim, M. M. Fejer, R. L. Byer, “Second-harmonic generation of blue and green light in periodically-poled planar lithium niobate waveguide,” in Digest of Topical Meeting on Nonlinear Guided-Wave Phenomena: Physics and Applications (Optical Society of America, Washington, D.C., 1989), postdeadline paper PD3.
  2. J. Webjörn, F. Laurell, G. Arvidsson, “Blue light generated by frequency doubling of laser diode light in a lithium niobate channel waveguide,” IEEE Photon. Technol. Lett. 1, 316–319 (1989).
    [CrossRef]
  3. G. Arvidsson, B. Jaskorzynska, “Periodically domain-inverted waveguides in lithium niobate for second harmonic generation: influence of the shape of the domain boundary on the conversion efficiency,” AIP Conf. Ser. 103, 47–52 (1989).
  4. M. M. Fejer, “Single crystal fibers: growth dynamics and nonlinear optical applications,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1986).
  5. F. Laurell, G. Arvidsson, “Frequency doubling in Ti:MgO:LiNbO3channel waveguides,” J. Opt. Soc. Am. B 5, 292–299 (1988).
    [CrossRef]
  6. W. Sohler, B. Hampel, R. Regener, R. Ricken, H. Suche, R. Volk, “Integrated optical parametric devices,” IEEE J. Lightwave Technol. LT-4, 772–777 (1986).
    [CrossRef]
  7. M. M. Fejer, G. A. Magel, E. J. Lim, “Quasi-phase-matched interactions in lithium niobate,” Proc. Soc. Photo-Opt. Instrum. Eng. 1148, 213–224 (1989).
  8. R. Kashyap, “Phase-matched periodic electric-field-induced second-harmonic generation in optical fibers,” J. Opt. Soc. Am. B 6, 313–328 (1989).
    [CrossRef]
  9. A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).
  10. J. Webjörn, F. Laurell, G. Arvidsson, “Fabrication of periodically domain-inverted channel waveguides in lithium niobate for second harmonic generation,” IEEE J. Lightwave Technol. LT-7, 1597–1600 (1989).
    [CrossRef]
  11. T. Suhara, H. Nishihara, “Theoretical analysis of waveguide second harmonic generation phase-matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
    [CrossRef]

1990

T. Suhara, H. Nishihara, “Theoretical analysis of waveguide second harmonic generation phase-matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

1989

R. Kashyap, “Phase-matched periodic electric-field-induced second-harmonic generation in optical fibers,” J. Opt. Soc. Am. B 6, 313–328 (1989).
[CrossRef]

J. Webjörn, F. Laurell, G. Arvidsson, “Blue light generated by frequency doubling of laser diode light in a lithium niobate channel waveguide,” IEEE Photon. Technol. Lett. 1, 316–319 (1989).
[CrossRef]

G. Arvidsson, B. Jaskorzynska, “Periodically domain-inverted waveguides in lithium niobate for second harmonic generation: influence of the shape of the domain boundary on the conversion efficiency,” AIP Conf. Ser. 103, 47–52 (1989).

M. M. Fejer, G. A. Magel, E. J. Lim, “Quasi-phase-matched interactions in lithium niobate,” Proc. Soc. Photo-Opt. Instrum. Eng. 1148, 213–224 (1989).

J. Webjörn, F. Laurell, G. Arvidsson, “Fabrication of periodically domain-inverted channel waveguides in lithium niobate for second harmonic generation,” IEEE J. Lightwave Technol. LT-7, 1597–1600 (1989).
[CrossRef]

1988

1986

W. Sohler, B. Hampel, R. Regener, R. Ricken, H. Suche, R. Volk, “Integrated optical parametric devices,” IEEE J. Lightwave Technol. LT-4, 772–777 (1986).
[CrossRef]

Arvidsson, G.

J. Webjörn, F. Laurell, G. Arvidsson, “Blue light generated by frequency doubling of laser diode light in a lithium niobate channel waveguide,” IEEE Photon. Technol. Lett. 1, 316–319 (1989).
[CrossRef]

J. Webjörn, F. Laurell, G. Arvidsson, “Fabrication of periodically domain-inverted channel waveguides in lithium niobate for second harmonic generation,” IEEE J. Lightwave Technol. LT-7, 1597–1600 (1989).
[CrossRef]

G. Arvidsson, B. Jaskorzynska, “Periodically domain-inverted waveguides in lithium niobate for second harmonic generation: influence of the shape of the domain boundary on the conversion efficiency,” AIP Conf. Ser. 103, 47–52 (1989).

F. Laurell, G. Arvidsson, “Frequency doubling in Ti:MgO:LiNbO3channel waveguides,” J. Opt. Soc. Am. B 5, 292–299 (1988).
[CrossRef]

Byer, R. L.

E. J. Lim, M. M. Fejer, R. L. Byer, “Second-harmonic generation of blue and green light in periodically-poled planar lithium niobate waveguide,” in Digest of Topical Meeting on Nonlinear Guided-Wave Phenomena: Physics and Applications (Optical Society of America, Washington, D.C., 1989), postdeadline paper PD3.

Fejer, M. M.

M. M. Fejer, G. A. Magel, E. J. Lim, “Quasi-phase-matched interactions in lithium niobate,” Proc. Soc. Photo-Opt. Instrum. Eng. 1148, 213–224 (1989).

E. J. Lim, M. M. Fejer, R. L. Byer, “Second-harmonic generation of blue and green light in periodically-poled planar lithium niobate waveguide,” in Digest of Topical Meeting on Nonlinear Guided-Wave Phenomena: Physics and Applications (Optical Society of America, Washington, D.C., 1989), postdeadline paper PD3.

M. M. Fejer, “Single crystal fibers: growth dynamics and nonlinear optical applications,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1986).

Hampel, B.

W. Sohler, B. Hampel, R. Regener, R. Ricken, H. Suche, R. Volk, “Integrated optical parametric devices,” IEEE J. Lightwave Technol. LT-4, 772–777 (1986).
[CrossRef]

Jaskorzynska, B.

G. Arvidsson, B. Jaskorzynska, “Periodically domain-inverted waveguides in lithium niobate for second harmonic generation: influence of the shape of the domain boundary on the conversion efficiency,” AIP Conf. Ser. 103, 47–52 (1989).

Kashyap, R.

Laurell, F.

J. Webjörn, F. Laurell, G. Arvidsson, “Blue light generated by frequency doubling of laser diode light in a lithium niobate channel waveguide,” IEEE Photon. Technol. Lett. 1, 316–319 (1989).
[CrossRef]

J. Webjörn, F. Laurell, G. Arvidsson, “Fabrication of periodically domain-inverted channel waveguides in lithium niobate for second harmonic generation,” IEEE J. Lightwave Technol. LT-7, 1597–1600 (1989).
[CrossRef]

F. Laurell, G. Arvidsson, “Frequency doubling in Ti:MgO:LiNbO3channel waveguides,” J. Opt. Soc. Am. B 5, 292–299 (1988).
[CrossRef]

Lim, E. J.

M. M. Fejer, G. A. Magel, E. J. Lim, “Quasi-phase-matched interactions in lithium niobate,” Proc. Soc. Photo-Opt. Instrum. Eng. 1148, 213–224 (1989).

E. J. Lim, M. M. Fejer, R. L. Byer, “Second-harmonic generation of blue and green light in periodically-poled planar lithium niobate waveguide,” in Digest of Topical Meeting on Nonlinear Guided-Wave Phenomena: Physics and Applications (Optical Society of America, Washington, D.C., 1989), postdeadline paper PD3.

Magel, G. A.

M. M. Fejer, G. A. Magel, E. J. Lim, “Quasi-phase-matched interactions in lithium niobate,” Proc. Soc. Photo-Opt. Instrum. Eng. 1148, 213–224 (1989).

Nishihara, H.

T. Suhara, H. Nishihara, “Theoretical analysis of waveguide second harmonic generation phase-matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

Regener, R.

W. Sohler, B. Hampel, R. Regener, R. Ricken, H. Suche, R. Volk, “Integrated optical parametric devices,” IEEE J. Lightwave Technol. LT-4, 772–777 (1986).
[CrossRef]

Ricken, R.

W. Sohler, B. Hampel, R. Regener, R. Ricken, H. Suche, R. Volk, “Integrated optical parametric devices,” IEEE J. Lightwave Technol. LT-4, 772–777 (1986).
[CrossRef]

Sohler, W.

W. Sohler, B. Hampel, R. Regener, R. Ricken, H. Suche, R. Volk, “Integrated optical parametric devices,” IEEE J. Lightwave Technol. LT-4, 772–777 (1986).
[CrossRef]

Suche, H.

W. Sohler, B. Hampel, R. Regener, R. Ricken, H. Suche, R. Volk, “Integrated optical parametric devices,” IEEE J. Lightwave Technol. LT-4, 772–777 (1986).
[CrossRef]

Suhara, T.

T. Suhara, H. Nishihara, “Theoretical analysis of waveguide second harmonic generation phase-matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

Volk, R.

W. Sohler, B. Hampel, R. Regener, R. Ricken, H. Suche, R. Volk, “Integrated optical parametric devices,” IEEE J. Lightwave Technol. LT-4, 772–777 (1986).
[CrossRef]

Webjörn, J.

J. Webjörn, F. Laurell, G. Arvidsson, “Fabrication of periodically domain-inverted channel waveguides in lithium niobate for second harmonic generation,” IEEE J. Lightwave Technol. LT-7, 1597–1600 (1989).
[CrossRef]

J. Webjörn, F. Laurell, G. Arvidsson, “Blue light generated by frequency doubling of laser diode light in a lithium niobate channel waveguide,” IEEE Photon. Technol. Lett. 1, 316–319 (1989).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).

AIP Conf. Ser.

G. Arvidsson, B. Jaskorzynska, “Periodically domain-inverted waveguides in lithium niobate for second harmonic generation: influence of the shape of the domain boundary on the conversion efficiency,” AIP Conf. Ser. 103, 47–52 (1989).

IEEE J. Lightwave Technol.

J. Webjörn, F. Laurell, G. Arvidsson, “Fabrication of periodically domain-inverted channel waveguides in lithium niobate for second harmonic generation,” IEEE J. Lightwave Technol. LT-7, 1597–1600 (1989).
[CrossRef]

W. Sohler, B. Hampel, R. Regener, R. Ricken, H. Suche, R. Volk, “Integrated optical parametric devices,” IEEE J. Lightwave Technol. LT-4, 772–777 (1986).
[CrossRef]

IEEE J. Quantum Electron.

T. Suhara, H. Nishihara, “Theoretical analysis of waveguide second harmonic generation phase-matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

IEEE Photon. Technol. Lett.

J. Webjörn, F. Laurell, G. Arvidsson, “Blue light generated by frequency doubling of laser diode light in a lithium niobate channel waveguide,” IEEE Photon. Technol. Lett. 1, 316–319 (1989).
[CrossRef]

J. Opt. Soc. Am. B

Proc. Soc. Photo-Opt. Instrum. Eng.

M. M. Fejer, G. A. Magel, E. J. Lim, “Quasi-phase-matched interactions in lithium niobate,” Proc. Soc. Photo-Opt. Instrum. Eng. 1148, 213–224 (1989).

Other

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).

E. J. Lim, M. M. Fejer, R. L. Byer, “Second-harmonic generation of blue and green light in periodically-poled planar lithium niobate waveguide,” in Digest of Topical Meeting on Nonlinear Guided-Wave Phenomena: Physics and Applications (Optical Society of America, Washington, D.C., 1989), postdeadline paper PD3.

M. M. Fejer, “Single crystal fibers: growth dynamics and nonlinear optical applications,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1986).

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Figures (8)

Fig. 1
Fig. 1

Orientation of the domains along the waveguide. The sign of the effective nonlinearity changes with the domain orientation. The average period Λ and the domain border coordinates zk are defined here. The propagation direction of the interacting modes is the z axis.

Fig. 2
Fig. 2

Normalized second-harmonic power as a function of the relative domain boundary rms deviation σΔ/(Λ/2).

Fig. 3
Fig. 3

Normalized output power for independent domain lengths as a function of a parameter α, which is defined in Eqs. (17) as the number of periods times the relative variance of the domain lengths. The solid curve shows the expectation for a fixed point with phase mismatch given by 2π over the average period. The squares represent simulations of this expectation for a grating with 1000 periods, which we obtained by taking the average of 500 realizations. The asterisks denote peak powers in the process simulation. To yield better agreement between the curve and the simulated points, one can replace α by α/2.5, which leads to the results shown by the dashed curve.

Fig. 4
Fig. 4

The y axis corresponds to N2 times the normalized output power and the x axis corresponds to N, where N is the number of domains. A constant relative random error variance of 0.001 is assumed. The slope in this log–log diagram equals 2 for small errors and 1 for large errors, which indicates a linear growth.

Fig. 5
Fig. 5

In (a) the bandwidth of the process is defined as the full width at half-maximum in μ of the expectation integral in Eq. (18). As we noted in the text, this bandwidth corresponds not to a smooth peak in a given realization but to a region with several individual peaks. On the x axis is plotted α. The solid curve in (b) shows the normalized output power bandwidth product and the dashed curve shows the output power of the process.

Fig. 6
Fig. 6

The solid curves in (a) and (b) show a realization of the process for α = 0.64 (a) and α = 4 (b). The normalized output power is on the y axis and the phase mismatch μ is on the x axis. We have assumed 1000 domain-inverted layers and a relative variance of 0.025 and 0.063 in (a) and (b), respectively. The dashed curves represent the conversion efficiency integral, also as a function of μ. We see that the highest peak is not always located at zero, which explains the difference between expected peak power and expected power at zero.

Fig. 7
Fig. 7

Output power as a function of phase mismatch δ0 when the effective index difference varies parabolically along the waveguide. We assume that δ1 = 0 and ζ2 = 0.5 in (a) and (b) and variations ζ2 = 40 (a) and δ2 = 160 (b). This results in a ripple to the left of the main peak. In (c) we retain δ2 = 40 but assume that ζ2 = 0.7, which decreases the modulation of the ripple but leaves the output power almost unchanged.

Fig. 8
Fig. 8

Two examples of output power as a function of phase mismatch δ0 for a linear variation of the effective index difference. In (a) δ1 = 10 and in (b) δ1 = 20. In both cases ζ1 = 0.5.

Tables (1)

Tables Icon

Table 1 Normalized Output Power for Some Sets of Domain Lengths

Equations (28)

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d E ( 2 ω ) ( z ) d z = - i ω ɛ 0 d ( z ) n ( 2 ω ) ( μ ɛ 0 ) 1 / 2 ( E ( ω ) ) 2 exp { i [ β ( 2 ω ) - 2 β ( ω ) ] z } ,
P A OVL = 1 2 n ( μ ɛ 0 ) 1 / 2 E 2 ,
E ( 2 ω ) ( z k ) = E ( 2 ω ) ( z k - 1 ) - i ω ɛ 0 d ( z I k ) n ( 2 ω ) ( μ ɛ 0 ) 1 / 2 ( E ( ω ) ) 2 × exp ( i Δ β z k ) - exp ( i Δ β z k - 1 ) i Δ β ,
E ( 2 ω ) ( l ) = - i k = 0 2 N ( - 1 ) k a k ω ɛ 0 d n ( 2 ω ) ( μ ɛ 0 ) 1 / 2 ( E ( ω ) ) 2 exp ( i Δ β z k ) i Δ β , a k = { 1 for k = 0             and             k = 2 N 2 otherwise .
P ( 2 ω ) = P max ( 2 ω ) 16 N 2 [ k = 0 2 N a k ( - 1 ) k exp ( i Δ β z k ) ] × [ m = 0 2 N a m ( - 1 ) m exp ( i Δ β z m ) ] * ,
P max ( 2 ω ) = 2 ( μ ɛ 0 ) 3 / 2 2 2 ɛ 0 2 d 2 ω 2 l 2 π 2 q 2 ( n ( ω ) ) 2 n ( 2 ω ) ( P ( ω ) ) 2 A OVL .
E { exp ( i X ) } = exp ( i E { X } - V { X } / 2 ) ,             X N ,
P N = E { P ( 2 ω ) P max ( 2 ω ) } = 1 16 N 2 k = 0 2 N m = 0 2 N a k a m ( - 1 ) m + k × exp [ i π q ( k - m ) - σ Δ 2 ( Δ β ) 2 ( 1 - δ m k ) ] = 8 N - 2 16 N 2 + 16 N 2 - 8 N + 2 16 N 2 exp [ - σ Δ 2 ( Δ β ) 2 ] ,
P N = 1 2 N + ( 1 - 1 2 N ) exp [ - ( 2 π q σ Δ Λ ) 2 ] .
σ Δ Λ / 2 q = 1 π ln 2 .
Δ β = 2 π q T k N = 2 π q N l + T k - l 2 π q Λ [ 1 - 1 l ( T k - l ) ] .
Δ β = ( 1 + μ q N ) 2 π q T k N ( 1 + μ q N ) 2 π q Λ [ 1 - 1 l ( T k - l ) ] .
Δ β z k = Δ β [ k Λ 2 + ( n = 1 k T n - k Λ 2 ) ] ( 1 + μ q N ) ( 2 π q Λ n = 1 k T n - k 2 N 2 π q Λ n = 1 2 N T n + k q π ) .
E { Δ β z k - Δ β z m } = ( 1 + μ q N ) ( k - m ) π q
V { Δ β z k - Δ β z m } = ( 1 + μ q N ) 2 ( 2 π q Λ ) 2 σ T 2 × [ k - m ( 1 - k - m 2 N ) 2 + ( 2 N - k - m ) k - m 2 ( 2 N ) 2 ] ( 2 π q Λ ) 2 σ T 2 ( k - m - k - m 2 2 N ) ,
P N = E { 2 2 16 N 2 [ k = 1 2 N ( - 1 ) k exp ( i Δ β z k ) ] [ m = 1 2 N ( - 1 ) m exp ( i Δ β z m ) ] * } = 1 4 N 2 k , m exp [ i μ N ( k - m ) π ] exp [ - σ T 2 2 ( 2 π q Λ ) 2 ( k - m - k - m 2 2 N ) ] = 1 2 N + 1 2 N 2 r = 1 2 N ( 2 N - r ) cos ( μ r N π ) exp [ - σ T 2 2 ( 2 π q Λ ) 2 ( r - r 2 2 N ) ] 2 0 2 N ( 1 - r 2 N ) cos ( 2 π μ r 2 N ) exp { - σ T 2 2 ( 2 π q Λ ) 2 [ r 2 N - r 2 ( 2 N ) 2 ] 2 N } d r 2 N .
α = N ( σ T Λ / 2 q ) 2 ,             ρ = r 2 N ,
P N = 2 0 1 ( 1 - ρ ) cos ( 2 π μ ρ ) exp [ - π 2 α ( ρ - ρ 2 ) ] d ρ .
P N = 2 0 1 ( 1 - ρ ) exp [ - π 2 α ( ρ - ρ 2 ) ] d ρ .
N ( σ T Λ / 2 q ) 2 = 4.5 1 π 2 .
z m + 2 M = z m + M Λ .
k = 0 2 N a k ( - 1 ) k exp ( i Δ β z k ) = [ m = 0 2 M a m ( - 1 ) m exp ( i Δ β z m ) ] × [ k = 0 K - 1 exp ( i Δ β k M Λ ) ] .
P N = P ( 2 ω ) P max ( 2 ω ) = 1 4 M 2 | m = 1 2 M ( - 1 ) m exp ( i 2 π q Λ z m ) | 2 sin 2 ( δ β l / 2 ) ( δ β l / 2 ) 2 ,
Δ β = 2 π q Λ + δ β .
Δ β ( z ) = 2 π q / Λ + δ β ( z ) .
P N = P ( 2 ω ) P max ( 2 ω ) = 1 4 N 2 | k = 0 2 N exp [ i n = 0 k δ β ( z n ) ( z n - z n - 1 ) ] | 2 1 l 2 | 0 l exp [ i 0 z δ β ( ζ ) d ζ ] d z | 2 ,
δ β ( z ) = δ 0 2 π l + δ 1 2 π l 2 ( z - ζ 1 l ) + δ 2 2 π l 3 ( z - ζ 2 l ) 2 .
P N = | 0 1 exp { i ( 2 π ) × 0 z [ δ 0 + δ 1 ( ζ - ζ 1 ) + δ 2 ( ζ - ζ 2 ) 2 ] d ζ } d z | 2 .

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