Abstract

Random duty cycle (RDC) errors in quasi-phase-matching (QPM) gratings lead to a pedestal in the spatial-frequency spectrum that increases the conversion efficiency for nominally phase-mismatched processes. Here, we determine the statistical properties of the Fourier spectrum of the QPM grating in the presence of RDC errors. We illustrate these properties with examples corresponding to periodic gratings with parameters typical for continuous-wave interactions, and chirped gratings with parameters typical for devices involving broad optical bandwidths. We show how several applications are sensitive to RDC errors by calculating the conversion efficiency of relevant nonlinear-optical processes. Last, we propose a method to efficiently incorporate RDC errors into coupled-wave models of nonlinear-optical interactions while still retaining only a small number of QPM grating orders.

© 2013 Optical Society of America

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2012

2011

2010

2009

2008

M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phase-matching gratings I: practical design formulas,” J. Opt. Soc. Am. B 25, 463–480 (2008).
[CrossRef]

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[CrossRef]

2007

2006

2005

C. Langrock, E. Diamanti, R. V. Roussev, Y. Yamamoto, M. M. Fejer, and H. Takesue, “Highly efficient single-photon detection at communication wavelengths by use of upconversion in reverse-proton-exchanged periodically poled LiNbO3 waveguides,” Opt. Lett. 30, 1725–1727 (2005).
[CrossRef]

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quantum Electron. 41, 1540–1547 (2005).
[CrossRef]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef]

T. Fuji, J. Rauschenberger, A. Apolonski, V. S. Yakovlev, G. Tempea, T. Udem, C. Gohle, T. W. Haensch, W. Lehnert, M. Scherer, and F. Krausz, “Monolithic carrier-envelope phase-stabilization scheme,” Opt. Lett. 30, 332–334 (2005).
[CrossRef]

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[CrossRef]

2004

M. A. Albota and F. C. Wong, “Efficient single-photon counting at 1.55 μm by means of frequency upconversion,” Opt. Lett. 29, 1449–1451 (2004).
[CrossRef]

M. Baudrier-Raybaut, R. Haidar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432, 374–376 (2004).
[CrossRef]

2001

K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett. 88, 023903 (2001).
[CrossRef]

G. Imeshev, M. M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B 18, 534–539 (2001).
[CrossRef]

1999

K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. 35, 1649–1656 (1999).
[CrossRef]

1998

G. Rosenman, K. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, “Domain broadening in quasi-phase-matched nonlinear optical devices,” Appl. Phys. Lett. 73, 865–867 (1998).
[CrossRef]

G. Imeshev, A. Galvanauskas, D. Harter, M. A. Arbore, M. Proctor, and M. M. Fejer, “Engineerable femtosecond pulse shaping by second-harmonic generation with Fourier synthetic quasi-phase-matching gratings,” Opt. Lett. 23, 864–866 (1998).
[CrossRef]

1997

L. E. Myers and W. R. Bosenberg, “Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators,” IEEE J. Quantum Electron. 33, 1663–1672 (1997).
[CrossRef]

I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B 14, 2268–2294 (1997).
[CrossRef]

S. Zhu, Y. Zhu, and N. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

1996

R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron. 32, 1324–1333 (1996).
[CrossRef]

1993

1992

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

1991

1990

1984

R. Eckardt and J. Reintjes, “Phase matching limitations of high efficiency second harmonic generation,” IEEE J. Quantum Electron. 20, 1178–1187 (1984).
[CrossRef]

1968

R. G. Smith, J. E. Geusic, H. J. Levinstein, J. J. Rubin, S. Singh, and L. G. Van Uitert, “Continuous optical parametric oscillation in Ba2NaNb5O15,” Appl. Phys. Lett. 12, 308–310 (1968).
[CrossRef]

R. L. Byer, M. K. Oshman, J. F. Young, and S. E. Harris, “Visible CW parametric oscillator,” Appl. Phys. Lett. 13, 109 (1968).
[CrossRef]

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Afeyan, B.

Albota, M. A.

Alibart, O.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[CrossRef]

Apolonski, A.

Arbore, M. A.

Arie, A.

G. Porat, Y. Silberberg, A. Arie, and H. Suchowski, “Two photon frequency conversion,” Opt. Express 20, 3613–3619 (2012).
[CrossRef]

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[CrossRef]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef]

K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman, “Multiple nonlinear optical interactions with arbitrary wave vector differences,” Phys. Rev. Lett. 88, 023903 (2001).
[CrossRef]

K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. 35, 1649–1656 (1999).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Arvidsson, G.

Asobe, M.

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quantum Electron. 41, 1540–1547 (2005).
[CrossRef]

Bahabad, A.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef]

Baldi, P.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[CrossRef]

Baronio, F.

Baudrier-Raybaut, M.

M. Baudrier-Raybaut, R. Haidar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432, 374–376 (2004).
[CrossRef]

Bhatia, A. B.

M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Born, M.

M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Bosenberg, W. R.

L. E. Myers and W. R. Bosenberg, “Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators,” IEEE J. Quantum Electron. 33, 1663–1672 (1997).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed (Academic, 2008).

Breunig, I.

Buse, K.

Byer, R. L.

S. T. Yang, R. C. Eckardt, and R. L. Byer, “Power and spectral characteristics of continuous-wave parametric oscillators: the doubly to singly resonant transition,” J. Opt. Soc. Am. B 10, 1684–1695 (1993).
[CrossRef]

S. T. Yang, R. C. Eckardt, and R. L. Byer, “Continuous-wave singly resonant optical parametric oscillator pumped by a single-frequency resonantly doubled Nd:YAG laser,” Opt. Lett. 18, 971–973 (1993).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

R. L. Byer, M. K. Oshman, J. F. Young, and S. E. Harris, “Visible CW parametric oscillator,” Appl. Phys. Lett. 13, 109 (1968).
[CrossRef]

Cerullo, G.

Chang, D.

Charbonneau-Lefort, M.

Chemla, D. S.

D. S. Chemla and J. Zyss, Nonlinear Optical Properties of Organic Molecules and Crystals (Academic, 1987).

Conforti, M.

De Angelis, C.

DeSalvo, R.

R. DeSalvo, A. Said, D. Hagan, E. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron. 32, 1324–1333 (1996).
[CrossRef]

Diamanti, E.

Dierolf, V.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Ebrahimzadeh, M.

Eckardt, R.

R. Eckardt and J. Reintjes, “Phase matching limitations of high efficiency second harmonic generation,” IEEE J. Quantum Electron. 20, 1178–1187 (1984).
[CrossRef]

Eckardt, R. C.

Eger, D.

G. Rosenman, K. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, “Domain broadening in quasi-phase-matched nonlinear optical devices,” Appl. Phys. Lett. 73, 865–867 (1998).
[CrossRef]

Fejer, M. M.

C. Heese, C. R. Phillips, B. W. Mayer, L. Gallmann, M. M. Fejer, and U. Keller, “75 MW few-cycle mid-infrared pulses from a collinear apodized APPLN-based OPCPA,” Opt. Express 20, 26888–26894 (2012).
[CrossRef]

C. R. Phillips and M. M. Fejer, “Adiabatic optical parametric oscillators: steady-state and dynamical behavior,” Opt. Express 20, 2466–2482 (2012).
[CrossRef]

C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Role of apodization in optical parametric amplifiers based on aperiodic quasi-phasematching gratings,” Opt. Express 20, 18066–18071 (2012).
[CrossRef]

J. S. Pelc, Q. Zhang, C. R. Phillips, L. Yu, Y. Yamamoto, and M. M. Fejer, “Cascaded frequency upconversion for high-speed single-photon detection at 1550 nm,” Opt. Lett. 37, 476–478 (2012).
[CrossRef]

J. S. Pelc, L. Ma, C. R. Phillips, Q. Zhang, C. Langrock, O. Slattery, X. Tang, and M. M. Fejer, “Long-wavelength-pumped upconversion single-photon detector at 1550 nm: performance and noise analysis,” Opt. Express 19, 21445–21456 (2011).
[CrossRef]

C. R. Phillips, J. S. Pelc, and M. M. Fejer, “Continuous wave monolithic quasi-phase-matched optical parametric oscillator in periodically poled lithium niobate,” Opt. Lett. 36, 2973–2975 (2011).
[CrossRef]

J. S. Pelc, C. R. Phillips, D. Chang, C. Langrock, and M. M. Fejer, “Efficiency pedestal in quasi-phase-matching devices with random duty-cycle errors,” Opt. Lett. 36, 864–866 (2011).
[CrossRef]

C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, J. Jiang, M. E. Fermann, and I. Hartl, “Supercontinuum generation in quasi-phase-matched LiNbO3 waveguide pumped by a Tm-doped fiber laser system,” Opt. Lett. 36, 3912–3914 (2011).
[CrossRef]

C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, I. Hartl, and M. E. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express 19, 18754–18773 (2011).
[CrossRef]

C. Heese, C. R. Phillips, L. Gallmann, M. M. Fejer, and U. Keller, “Ultrabroadband, highly flexible amplifier for ultrashort midinfrared laser pulses based on aperiodically poled Mg:LiNbO3,” Opt. Lett. 35, 2340–2342 (2010).
[CrossRef]

J. S. Pelc, C. Langrock, Q. Zhang, and M. M. Fejer, “Influence of domain disorder on parametric noise in quasi-phase-matched quantum frequency converters,” Opt. Lett. 35, 2804–2806(2010).
[CrossRef]

C. R. Phillips and M. M. Fejer, “Stability of the singly resonant optical parametric oscillator,” J. Opt. Soc. Am. B 27, 2687–2699 (2010).
[CrossRef]

C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasi-phase-matched gratings,” Opt. Lett. 35, 3093–3095 (2010).
[CrossRef]

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C. Langrock, M. M. Fejer, I. Hartl, and M. E. Fermann, “Generation of octave-spanning spectra inside reverse-proton-exchanged periodically poled lithium niobate waveguides,” Opt. Lett. 32, 2478–2480 (2007).
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J. S. Pelc, C. R. Phillips, D. Chang, C. Langrock, and M. M. Fejer, “Efficiency pedestal in quasi-phase-matching devices with random duty-cycle errors,” Opt. Lett. 36, 864–866 (2011).
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J. S. Pelc, L. Ma, C. R. Phillips, Q. Zhang, C. Langrock, O. Slattery, X. Tang, and M. M. Fejer, “Long-wavelength-pumped upconversion single-photon detector at 1550 nm: performance and noise analysis,” Opt. Express 19, 21445–21456 (2011).
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Figures (5)

Fig. 1.
Fig. 1.

Schematic of a QPM grating with RDC errors. Vertical dashed lines indicate the ideal, equally spaced domain boundary positions. Elements of the domain boundary vectors z (random) and z 0 (ideal) are indicated.

Fig. 2.
Fig. 2.

Fourier spectra | g ˜ ( k ) | 2 with parameters σ z = 1 μm and a 5 cm crystal, and K g 0 = 2 × 10 5 m 1 ( 31.4 μm period). The inset shows | g ˜ ( k ) | 2 on a linear scale in the vicinity of first-order QPM; the effects of RDC errors cannot be seen on this linear scale.

Fig. 3.
Fig. 3.

Fourier spectra for a large, 3 μm, mean error in the domain boundaries. The grating is 1 cm long and the grating k -vector is K g = 2 × 10 5 m 1 (QPM period 31.4 μm ). For the red curve, the spectrum | g ˜ ( k ) | 2 was averaged over 50 gratings. The black curve corresponds to the analytical ensemble average from Eq. (9). The dashed blue curve shows a suitably normalized 1 / k 2 profile, which is the asymptotic functional form of the noise pedestal in Eq. (9) for large k .

Fig. 4.
Fig. 4.

Fourier spectra for an example linearly chirped QPM grating with σ z = 1 μm , a chirp rate of d K g / d z = 2.5 × 10 6 m 2 , grating length 1 cm, and mean grating k -vector K g 0 = 2 × 10 5 m 1 .

Fig. 5.
Fig. 5.

Fourier spectrum arising from domain boundary variations defined by a grating phase perturbation δ ϕ ( z ) defined by applying a sinc filter to white noise.

Equations (45)

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d A i d z = i ω i d ( z ) n i c A s * A p e i Δ k z ,
A i = i 2 L π ω i d 0 n i c A s * A p a i ,
d a i d z = π 2 L d ( z ) d 0 e i Δ k z g ( z ) e i Δ k z ,
g ˜ z ( k ) = i π 2 k L n = 1 N ( 1 ) n ( e i k z [ n ] e i k z [ n 1 ] ) = i π k L [ e i k z [ 0 ] + ( 1 ) N e i k z [ N ] 2 + n = 1 N 1 ( 1 ) n e i k z [ n ] ] ,
g ˜ z ( k ) i π k L n = 1 N ( 1 ) n e i k z [ n ] .
g ˜ z ( k ) = e k 2 σ z 2 / 2 g ˜ z 0 ( k ) ,
e i k z = e k 2 σ z 2 / 2 e i k z
| g ˜ z ( k ) | 2 = ( π k L ) 2 m = 1 N n = 1 N ( 1 ) n m e i k ( z [ n ] z [ m ] ) = ( π k L ) 2 [ N ( 1 e k 2 σ z 2 ) + e k 2 σ z 2 n = 1 N m = 1 N ( 1 ) n m e i k ( z 0 [ n ] z 0 [ m ] ) ] ,
| g ˜ z ( k ) | 2 = e k 2 σ z 2 | g ˜ z 0 ( k ) | 2 + N ( π k L ) 2 ( 1 e k 2 σ z 2 ) ,
| g ˜ z 0 ( k ) | 2 = [ π 2 sinc ( k Λ D 2 ) ] 2 [ sin ( N Δ k 1 Λ D / 2 ) N sin ( Δ k 1 Λ D / 2 ) ] 2 .
d ( z ) d 0 sgn [ cos ( ϕ ( z ) ) cos ( π D ( z ) ) ] ( 2 D ( z ) 1 ) + m = m 0 2 sin ( π m D ( z ) ) π m exp ( i m ϕ ( z ) ) .
g ˜ z ( m ) ( k ) = 1 m L sin ( π m D ) 0 L exp [ i ( m ϕ ( z ) k z ) ] d z ,
g ˜ z ( d c ) ( k ) = π 2 ( 2 D 1 ) e i k L / 2 sinc ( k L 2 ) .
E ˜ ( z , ω ) u ( ω ) = 1 2 j A ˜ j ( z , ω ) exp ( i k ( ω ) z ) ,
d A ˜ i ( ω ) d z = i ω d ( z ) n i c 0 A ˜ p ( ω + ω ) A ˜ s * ( ω ) e i Δ k ( ω , ω ) z d ω 2 π ,
Δ k ( ω , ω ) = k ( ω + ω ) k ( ω ) k ( ω ) .
Δ k ( ω , ω ) k p k s + ω ω i v p k ( ω ) ,
A ˜ i ( ω ) = i 2 L π ω d 0 n ( ω ) c g ˜ z ( Δ k ( ω ) ) × F [ A p ( z = 0 , t ) A s ( z = 0 , t ) * ] ( ω ) .
W i 2 ϵ 0 ω i 2 d 0 2 L Λ D n i c 1 e Δ k ( ω i ) 2 σ z 2 Δ k ( ω i ) 2 Λ D 2 × | A p ( z = 0 , t ) A s ( z = 0 , t ) * | 2 d t .
| g ˜ z ( k ) | 4 2 N 2 ( π k L ) 4 ( 1 e ( k σ z ) 2 ) 2 + 4 N ( π k L ) 2 ( 1 e ( k σ z ) 2 ) e ( k σ z ) 2 | g ˜ z 0 ( k ) | 2 + e 2 ( k σ z ) 2 | g ˜ z 0 ( k ) | 4 .
σ η N ( π k L ) 2 ( 1 e k 2 σ z 2 ) .
d A i ( ± ) d z = i ω i d ( z ) n i c A s * A p e i Δ k ± z ,
[ a i ( ) ( z = 0 ) a i ( + ) ( z = L ) ] = [ g ˜ z ( Δ k ) g ˜ z ( Δ k + ) , ] ,
| g ˜ ( k ) | 2 1 N ( 1 e k 2 σ z 2 ) + e k 2 σ z 2 sinc ( ( k K g ) L / 2 ) 2 .
I abs ( z ) = 0 z β TPA I SH ( z ) 2 d z ,
I abs q = 1 N Λ D , q β TPA I SH ( z [ q ] ) 2 β TPA I max 2 L N q = 1 N | h ˜ z ( q ) ( k ) | 4 ,
h ˜ z ( q ) ( k ) = π k L n = 1 q ( 1 ) n e i k z [ n ] ,
I max = n SH ϵ 0 c 2 | 2 π ω 1 d eff n SH c A FH 2 L | 2 .
| h ˜ z ( m ) ( k ) | 4 2 m 2 ( π k L ) 4 ( 1 e ( k σ z ) 2 ) 2 + 4 m ( π k L ) 2 ( 1 e ( k σ z ) 2 ) e ( k σ z ) 2 | h ˜ z 0 ( m ) ( k ) | 2 + e 2 ( k σ z ) 2 | h ˜ z 0 ( m ) ( k ) | 4 .
I abs 2 3 β TPA I Λ 2 Λ D ( L Λ D ) 3 ( π σ z Λ D ) 4 ( 1 e k 2 σ z 2 k 2 σ z 2 ) 2 ,
U abs = I abs ( pk ) I ¯ FH 4 ( x , y , t ) d x d y d t ,
I ¯ 1 4 ( x , y , t ) d x d y d t = ( π 8 ) 3 / 2 w 2 τ 2 ln ( 2 ) .
sinc 2 ( Γ s ) = N p 1 ,
Γ s 2 = ω i ω p ( 2 d 0 / π ) 2 | A s | 2 L 2 n i n p c 2 .
a NL = Γ s 2 N ( π σ z Λ D ) 2 1 e Δ k s 2 σ z 2 Δ k s 2 σ z 2 n i n p n s n SH ω s 2 ω i ω p ( d SHG d OPO ) 2 Γ s 2 a 0 ,
W SH = 2 ϵ 0 ω s 2 d 0 2 L Λ D n SH c 1 e Δ k s 2 σ z 2 Δ k s 2 Λ D 2 | A s ( z = 0 , t ) 2 | 2 d t .
a NL = 2 1 / 2 Γ s , pk 2 a 0 ,
d A SH d z = i ω s d ( z ) n SH c A s 2 e i Δ k s z ,
d A SF d z = i ω SF d ( z ) n SF c A p A SH e i Δ k SFG z ,
| a SF | 2 1 N 2 ( 2 π 2 Δ k SFG Δ k s Λ D 2 ) 2 × [ 1 + e ( Δ k SFG 2 + Δ k s 2 ) σ z 2 e Δ k s 2 σ z 2 e Δ k SFG 2 σ z 2 ] ,
A SH = 2 ω SF ω s d 0 2 L 2 π 2 n SH n SF c 2 A p ( 0 ) A s ( 0 ) 2 a SF .
I SF I p = n i 2 n p n SH n SF ( ω s ω SF ω i ω p ) 2 Γ s 4 I ¯ s 2 | a SF | 2 ,
δ ϕ ( z ) ϕ ( z ) ϕ 0 ( z ) .
δ ϕ ( z n 0 + δ z n ) = ( ϕ 0 ( z n 0 + δ z n ) ϕ 0 ( z n 0 ) ) K g ( z ) δ z n ,
σ δ ϕ K g w π ,

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