Abstract

We develop an analytic description of continuous-wave four-wave mixing in silicon-on-insulator (SOI) waveguides including linear loss, two-photon absorption, and free-carrier absorption. Under the undepleted pump approximation, the pump equation decouples from the signal and idler equations and becomes a nonlinear differential equation that we solve analytically without further approximations. The signal and idler equations have no known solutions for arbitrary pump power evolution, but we calculate approximate field expressions based on a Magnus expansion, which has been used to study time-ordering effects in quantum optics. Lastly, we show that the phase-matching condition changes through the waveguide and that this explains the shape of the wavelength-conversion-efficiency spectrum in SOI waveguides and why it differs from that of highly nonlinear silica fibers.

© 2018 Optical Society of America

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2017 (4)

2016 (1)

2015 (1)

2014 (4)

2013 (1)

2012 (1)

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron. 18, 1016–1032 (2012).
[Crossref]

2011 (2)

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
[Crossref]

B. Kuyken, S. Clemmen, S. K. Selvaraja, W. Bogaerts, D. Van Thourhout, P. Emplit, S. Massar, G. Roelkens, and R. Baets, “On-chip parametric amplification with 26.5  dB gain at telecommunication wavelengths using CMOS-compatible hydrogenated amorphous silicon waveguides,” Opt. Lett. 36, 552–554 (2011).
[Crossref]

2010 (9)

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Analytical study of optical bistability in silicon ring resonators,” Opt. Lett. 35, 55–57 (2010).
[Crossref]

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18, 2884–2893 (2010).
[Crossref]

A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express 18, 3582–3591 (2010).
[Crossref]

J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Detailed characterization of a fiber-optic parametric amplifier in phase-sensitive and phase-insensitive operation,” Opt. Express 18, 4130–4137 (2010).
[Crossref]

I. D. Rukhlenko and M. Premaratne, “Spectral compression and group delay of optical pulses in silicon Raman amplifiers,” Opt. Lett. 35, 3138–3140 (2010).
[Crossref]

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[Crossref]

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4, 535–544 (2010).
[Crossref]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref]

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Nonlinear silicon photonics: analytical tools,” IEEE J. Sel. Top. Quantum Electron. 16, 200–215 (2010).
[Crossref]

2009 (2)

2008 (1)

2007 (4)

2006 (3)

2005 (3)

H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J.-I. Takahashi, and S.-I. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express 13, 4629–4637 (2005).
[Crossref]

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[Crossref]

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[Crossref]

2004 (1)

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[Crossref]

2002 (2)

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5  μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

2001 (1)

1957 (1)

1954 (1)

W. Magnus, “On the exponential solution of differential equations for a linear operator,” Commun. Pure Appl. Math. 7, 649–673 (1954).
[Crossref]

1949 (1)

F. J. Dyson, “The radiation theories of Tomonaga, Schwinger, and Feynman,” Phys. Rev. 75, 486–502 (1949).
[Crossref]

Afshar, S.

Agrawal, G. P.

Andrekson, P. A.

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron. 18, 1016–1032 (2012).
[Crossref]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
[Crossref]

J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Detailed characterization of a fiber-optic parametric amplifier in phase-sensitive and phase-insensitive operation,” Opt. Express 18, 4130–4137 (2010).
[Crossref]

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[Crossref]

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18, 2884–2893 (2010).
[Crossref]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[Crossref]

Asghari, M.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5  μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]

Bacco, D.

K. Guo, E. N. Christensen, J. B. Christensen, J. G. Koefoed, D. Bacco, Y. Ding, H. Ou, and K. Rottwitt, “High coincidence-to-accidental ratio continuous-wave photon-pair generation in a grating-coupled silicon strip waveguide,” Appl. Phys. Express 10, 062801 (2017).
[Crossref]

Baets, R.

Bell, B. A.

Blessing, D. J.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430–436 (2011).
[Crossref]

Bogaerts, W.

Bogris, A.

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron. 18, 1016–1032 (2012).
[Crossref]

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[Crossref]

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18, 2884–2893 (2010).
[Crossref]

Bristow, A. D.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200  nm,” Appl. Phys. Lett. 90, 191104 (2007).
[Crossref]

Chen, X.

Chou, C.-Y.

Christensen, E. N.

K. Guo, S. M. M. Friis, J. B. Christensen, E. N. Christensen, X. Shi, Y. Ding, H. Ou, and K. Rottwitt, “Full-vectorial propagation model and modified effective mode area of four-wave mixing in straight waveguides,” Opt. Lett. 42, 3670–3673 (2017).
[Crossref]

K. Guo, E. N. Christensen, J. B. Christensen, J. G. Koefoed, D. Bacco, Y. Ding, H. Ou, and K. Rottwitt, “High coincidence-to-accidental ratio continuous-wave photon-pair generation in a grating-coupled silicon strip waveguide,” Appl. Phys. Express 10, 062801 (2017).
[Crossref]

Christensen, J. B.

K. Guo, E. N. Christensen, J. B. Christensen, J. G. Koefoed, D. Bacco, Y. Ding, H. Ou, and K. Rottwitt, “High coincidence-to-accidental ratio continuous-wave photon-pair generation in a grating-coupled silicon strip waveguide,” Appl. Phys. Express 10, 062801 (2017).
[Crossref]

K. Guo, S. M. M. Friis, J. B. Christensen, E. N. Christensen, X. Shi, Y. Ding, H. Ou, and K. Rottwitt, “Full-vectorial propagation model and modified effective mode area of four-wave mixing in straight waveguides,” Opt. Lett. 42, 3670–3673 (2017).
[Crossref]

Claps, R.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[Crossref]

Clemmen, S.

Cohen, O.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature 433, 725–728 (2005).
[Crossref]

Da Ros, F.

Dadap, J. I.

Dalgaard, K.

Dasgupta, S.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[Crossref]

Day, I. E.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5  μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]

Dimitropoulos, D.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. C. S. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[Crossref]

Ding, Y.

K. Guo, S. M. M. Friis, J. B. Christensen, E. N. Christensen, X. Shi, Y. Ding, H. Ou, and K. Rottwitt, “Full-vectorial propagation model and modified effective mode area of four-wave mixing in straight waveguides,” Opt. Lett. 42, 3670–3673 (2017).
[Crossref]

K. Guo, E. N. Christensen, J. B. Christensen, J. G. Koefoed, D. Bacco, Y. Ding, H. Ou, and K. Rottwitt, “High coincidence-to-accidental ratio continuous-wave photon-pair generation in a grating-coupled silicon strip waveguide,” Appl. Phys. Express 10, 062801 (2017).
[Crossref]

Dissanayake, C.

Drake, J.

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5  μm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]

Dyson, F. J.

F. J. Dyson, “The radiation theories of Tomonaga, Schwinger, and Feynman,” Phys. Rev. 75, 486–502 (1949).
[Crossref]

Ebendorff-Heidepriem, H.

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Figures (7)

Fig. 1.
Fig. 1. Frequency configuration of modulation instability. A frequency-degenerate pump (p) amplifies a signal (s) and an idler (i), together called sidebands; the idler may or may not be present at input.
Fig. 2.
Fig. 2. Pump power versus waveguide length for four different input pump powers. The numeric solutions to Eq. (9) are shown with P0=15  dBm (blue dots), P0=18  dBm (red triangles), P0=20  dBm (green squares), and P0=30  dBm (turquoise asterisks). The solid lines of each color are plots of the analytic expression Eq. (14) at the respective pump powers. The number N in each simulation is the number of numeric steps required to get a maximum error of 105 between the numeric and analytic results.
Fig. 3.
Fig. 3. Signal (blue) and idler (red) powers in FWM simulation in an example waveguide. Numeric curves (markers) are compared to analytic curves to second order (lines).
Fig. 4.
Fig. 4. Error of the analytic solutions of Eq. (42) (top plot) and Eq. (43) (bottom plot) to first (solid blue lines) and second (red dashed lines) order relative to a numeric solution in the same example waveguide as used in Fig. 3.
Fig. 5.
Fig. 5. Comparison of CE versus signal wavelength of a numeric solution to Eqs. (3) and (4) and the analytic solution of Eq. (44) with a pump wavelength of λp=1550  nm in the same example waveguide as used in Fig. 3 evaluated at length L=1.5  cm.
Fig. 6.
Fig. 6. PS signal gain versus input idler phase for two different input pump powers, P0=14  dBm and P0=20  dBm; results of both numeric simulations (blue dots) and the analytic result of Eq. (47) (red solid line) are shown.
Fig. 7.
Fig. 7. CE spectrum in PI operation at different positions in a waveguide. Numeric (blue dots) and full analytic (solid red line) solutions are compared to an unphysical analytic solution where the phase-matching condition (PMC) is forced constant at all positions through the waveguide (dashed red line). For the latter solution, the vertical dotted lines denote the constant wavelength of phase matching.

Equations (50)

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{E(r,t)H(r,t)}=12nAn(z)Nn{F(x,y)G(x,y)}exp(iωnt+iβnz)+c.c.,
Nn2=14S[Fn×Gn*+Fn*×Gn]·zds
zAp=iγ[(|Ap|2+2|As|2+2|Ai|2)Ap+2AsAiAp*exp(iΔβz)],
zAj=iγ[(|Aj|2+2|Ap|2+2|Ak|2)Aj+Ap2Ak*exp(iΔβz)],
γ=ωn2cAeff+iβT2Aeff,
n2=3Re(χc(3))4nc2ε0c,βT=3ωIm(χc(3))2nc2ε0c2,
Aeff=3ng2nc2[Sdsnw(x,y)2|F|2]2cdsF*·[2|F|2F+(F·F)F*],
αj=αl,j+αf,jPp2(z)
zPp=αl,pPp2Im(γ)Pp2αf,pPp3,
zϕp=Re(γ)Pp,
w(ξ)ξw(ξ)=w(ξ)κξ,
ξ(τ)=C0τ2τ+κexp[tanh1(12τ14κ)14κ],
w(τ)=τξ(τ),
αl,pz+C1=12ln(ν2Pp2+νPp1+κ)+tanh1(1+2νPp114κ)14κ,
zAs=2iγPpAs+iγAp2Ai*exp(iΔβz)αs/2As,
zAi=2iγPpAi+iγAp2As*exp(iΔβz)αi/2Ai,
zBs=ts(z)Bi*,
zBi=ti(z)Bs*,
ts(z)=iγPpexp(Fi*Fs)exp(2iϕpiΔβz),
ti(z)=iγPpexp(Fs*Fi)exp(2iϕpiΔβz),
z[BsBi*]=[0t(z)t*(z)0][BsBi*],
t(z)=iγPpexp[4iRe(γ)0zdzPp(z)]×exp[2iϕpiΔβz],
A(z)=[0t(z)t*(z)0].
[BsBi*]=Sexp(0zdzA(z))[Bs(0)Bi*(0)],
exp(0zdzA(z))=n(0zdzA(z))nn!
=I+0zdz1A(z1)+120zdz10zdz2A(z1)A(z2)+,
Sexp(0zdzA(z))=I+0zdz1A(z1)+120zdz10z1dz2A(z1)A(z2)+120zdz1z1zdz2A(z2)A(z1)+,
120zdz10z1dz2A(z1)A(z2)+120zdz1z1zdz2A(z2)A(z1)=0zdz10z1dz2A(z1)A(z2),
Sexp(0zdzA(z))=exp(Ω1(z)+Ω2(z)+),
Ω1=0zdz1A(z1),
Ω2=120zdz10z1dz2[A(z1),A(z2)],
[A(zi),A(zj)]=iaij[1001],
Ω1=[0α1α1*0],
Ω2=[iα200iα2],
T(z)=exp(Ω1+Ω2)=Λ[exp(λ)00exp(λ)]Λ1
Ω1+Ω2=[iα2α1α1*iα2]
λ=|α1|2α22,
Λ=[α1α1λiα2λiα2].
T(z)=[coshλ+iα2λsinhλα1λsinhλα1*λsinhλcoshλiα2λsinhλ],
[Bs(z)Bi*(z)]=T(z)[Bs(0)Bi*(0)].
aij=2|γ|2Pp(zi)Pp(zj)×sin[Δβ(zizj)+2Re(γ)zjzidzPp(z)],
As(z)=As(0)[cosh(λ)+iα2λsinh(λ)]exp[2iγ0zdzPp(z)]×exp(120zdzαl,s+αf,sPp2(z)),
Ai(z)=As*(0)α1λsinh(λ)exp[2iγ0zdzPp(z)]×exp(120zdzαl,i+αf,iPp2(z)),
ηPI=|α1|2λsinh2(λ)exp[4Im(γ)0LdzPp(z)]×exp[0Ldzαl,i+αf,iPp2(z)].
Aj(z)=(Aj(0)eiϕj(0)cosh(|α1|)+Ak*(0)eiϕk(0)α1|α1|sinh(|α1|))×exp(120zdz4iγPp(z)αl,jαl,jPp2(z))
GPS=[cosh2(|α1|)+Pi(0)Ps(0)sinh2(|α1|)+2Pi(0)Ps(0)cosh(|α1|)sinh(|α1|)cos(ϕPS)]×exp(0Ldz4Im(γ)Pp(z)+αl,s+αf,sPp2(z)),
GPS=[cosh(2|α1|)+sinh(2|α1|)cos(ϕPS)]×exp(0Ldz4Im(γ)Pp(z)+αl,s+αf,sPp2(z)).
|α1|=|γ0zdz1Pp(z1)×exp[iΔβz2iRe(γ)0z1dzPp(z)]|,
Δβz=2Re(γ)0zdzPp(z).
α1(v)=γ0zdz1Pp(z1)exp[iΔβz12iRe(γ)P0z1],

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