Abstract

Amplitude squeezing of the second-harmonic generation in periodically poled quasi-phase-matched devices is analyzed with consideration for errors of the domain length. We show that the amount of squeezing is a complex function of the phase mismatch and of the input power and that it is practically impossible to maintain the perfect quasi-phase matching for an arbitrary input power. For evaluation of the availability of squeezing, we propose a contour map of squeezing that can visualize the tolerance of squeezing for the phase mismatch. It is shown that the effect of domain length error depends on the type of the error; the random duty-cycle error, where the mean domain period is precisely fixed, does not alter the squeezing performance, whereas the random period error, which fluctuates during the domain period, severely alters tuning characteristics. The available amount of squeezing is predicted to be determined by the tuning stability of the device.

© 2000 Optical Society of America

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References

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  1. D. K. Serkland, M. M. Fejer, R. L. Byer, and Y. Yamamoto, “Squeezing in a quasi-phase-matched LiNbO3 waveguide,” Opt. Lett. 20, 1649–1651 (1995).
    [CrossRef] [PubMed]
  2. D. K. Serkland, P. Kumar, M. A. Arbore, and M. M. Fejer, “Amplitude squeezing by means of quasi-phase-matched second-harmonic generation in a lithium niobate waveguide,” Opt. Lett. 22, 1497–1499 (1997).
    [CrossRef]
  3. Z. Y. Ou, “Propagation of quantum fluctuations in single-pass second-harmonic generation for arbitrary interaction length,” Phys. Rev. A 49, 2106–2116 (1994).
    [CrossRef] [PubMed]
  4. R.-D. Li and P. Kumar, “Quantum-noise reduction in traveling-wave second-harmonic generation,” Phys. Rev. A 49, 2157–2166 (1994).
    [CrossRef] [PubMed]
  5. T. Suhara, M. Fujimura, K. Kintaka, H. Nishihara, P. Kürz, and T. Mukai, “Theoretical analysis of squeezed-light generation by second-harmonic generation,” IEEE J. Quantum Electron. 32, 690–700 (1996).
    [CrossRef]
  6. L. Noirie, P. Bidaković, and J. A. Levenson, “Squeezing due to cascaded second-order nonlinearities in quasi-phase-matched media,” J. Opt. Soc. Am. B 14, 1–10 (1997).
    [CrossRef]
  7. S. Reynaud, C. Fabre, E. Giacobino, and A. Heidmann, “Photon noise reduction by passive optical bistable systems,” Phys. Rev. A 40, 1440–1446 (1989).
    [CrossRef] [PubMed]
  8. S. Helmfrid and G. Arvidsson, “Influence of randomly varying domain lengths and nonuniform effective index on second-harmonic generation in quasi-phase-matching waveguides,” J. Opt. Soc. Am. B 8, 797–804 (1991).
    [CrossRef]
  9. 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]
  10. J. Maeda and K. Kikuchi, “Squeezing characteristics analysis of a fundamental-confined second-harmonic generation system by means of a self-consistent method,” J. Opt. Soc. Am. B 14, 481–493 (1997).
    [CrossRef]

1997 (3)

1996 (1)

T. Suhara, M. Fujimura, K. Kintaka, H. Nishihara, P. Kürz, and T. Mukai, “Theoretical analysis of squeezed-light generation by second-harmonic generation,” IEEE J. Quantum Electron. 32, 690–700 (1996).
[CrossRef]

1995 (1)

1994 (2)

Z. Y. Ou, “Propagation of quantum fluctuations in single-pass second-harmonic generation for arbitrary interaction length,” Phys. Rev. A 49, 2106–2116 (1994).
[CrossRef] [PubMed]

R.-D. Li and P. Kumar, “Quantum-noise reduction in traveling-wave second-harmonic generation,” Phys. Rev. A 49, 2157–2166 (1994).
[CrossRef] [PubMed]

1992 (1)

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 (1)

1989 (1)

S. Reynaud, C. Fabre, E. Giacobino, and A. Heidmann, “Photon noise reduction by passive optical bistable systems,” Phys. Rev. A 40, 1440–1446 (1989).
[CrossRef] [PubMed]

Arbore, M. A.

Arvidsson, G.

Bidakovic, P.

Byer, R. L.

D. K. Serkland, M. M. Fejer, R. L. Byer, and Y. Yamamoto, “Squeezing in a quasi-phase-matched LiNbO3 waveguide,” Opt. Lett. 20, 1649–1651 (1995).
[CrossRef] [PubMed]

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]

Fabre, C.

S. Reynaud, C. Fabre, E. Giacobino, and A. Heidmann, “Photon noise reduction by passive optical bistable systems,” Phys. Rev. A 40, 1440–1446 (1989).
[CrossRef] [PubMed]

Fejer, M. M.

Fujimura, M.

T. Suhara, M. Fujimura, K. Kintaka, H. Nishihara, P. Kürz, and T. Mukai, “Theoretical analysis of squeezed-light generation by second-harmonic generation,” IEEE J. Quantum Electron. 32, 690–700 (1996).
[CrossRef]

Giacobino, E.

S. Reynaud, C. Fabre, E. Giacobino, and A. Heidmann, “Photon noise reduction by passive optical bistable systems,” Phys. Rev. A 40, 1440–1446 (1989).
[CrossRef] [PubMed]

Heidmann, A.

S. Reynaud, C. Fabre, E. Giacobino, and A. Heidmann, “Photon noise reduction by passive optical bistable systems,” Phys. Rev. A 40, 1440–1446 (1989).
[CrossRef] [PubMed]

Helmfrid, S.

Jundt, D. H.

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]

Kikuchi, K.

Kintaka, K.

T. Suhara, M. Fujimura, K. Kintaka, H. Nishihara, P. Kürz, and T. Mukai, “Theoretical analysis of squeezed-light generation by second-harmonic generation,” IEEE J. Quantum Electron. 32, 690–700 (1996).
[CrossRef]

Kumar, P.

Kürz, P.

T. Suhara, M. Fujimura, K. Kintaka, H. Nishihara, P. Kürz, and T. Mukai, “Theoretical analysis of squeezed-light generation by second-harmonic generation,” IEEE J. Quantum Electron. 32, 690–700 (1996).
[CrossRef]

Levenson, J. A.

Li, R.-D.

R.-D. Li and P. Kumar, “Quantum-noise reduction in traveling-wave second-harmonic generation,” Phys. Rev. A 49, 2157–2166 (1994).
[CrossRef] [PubMed]

Maeda, J.

Magel, G. A.

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]

Mukai, T.

T. Suhara, M. Fujimura, K. Kintaka, H. Nishihara, P. Kürz, and T. Mukai, “Theoretical analysis of squeezed-light generation by second-harmonic generation,” IEEE J. Quantum Electron. 32, 690–700 (1996).
[CrossRef]

Nishihara, H.

T. Suhara, M. Fujimura, K. Kintaka, H. Nishihara, P. Kürz, and T. Mukai, “Theoretical analysis of squeezed-light generation by second-harmonic generation,” IEEE J. Quantum Electron. 32, 690–700 (1996).
[CrossRef]

Noirie, L.

Ou, Z. Y.

Z. Y. Ou, “Propagation of quantum fluctuations in single-pass second-harmonic generation for arbitrary interaction length,” Phys. Rev. A 49, 2106–2116 (1994).
[CrossRef] [PubMed]

Reynaud, S.

S. Reynaud, C. Fabre, E. Giacobino, and A. Heidmann, “Photon noise reduction by passive optical bistable systems,” Phys. Rev. A 40, 1440–1446 (1989).
[CrossRef] [PubMed]

Serkland, D. K.

Suhara, T.

T. Suhara, M. Fujimura, K. Kintaka, H. Nishihara, P. Kürz, and T. Mukai, “Theoretical analysis of squeezed-light generation by second-harmonic generation,” IEEE J. Quantum Electron. 32, 690–700 (1996).
[CrossRef]

Yamamoto, Y.

IEEE J. Quantum Electron. (2)

T. Suhara, M. Fujimura, K. Kintaka, H. Nishihara, P. Kürz, and T. Mukai, “Theoretical analysis of squeezed-light generation by second-harmonic generation,” IEEE J. Quantum Electron. 32, 690–700 (1996).
[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]

J. Opt. Soc. Am. B (3)

Opt. Lett. (2)

Phys. Rev. A (3)

S. Reynaud, C. Fabre, E. Giacobino, and A. Heidmann, “Photon noise reduction by passive optical bistable systems,” Phys. Rev. A 40, 1440–1446 (1989).
[CrossRef] [PubMed]

Z. Y. Ou, “Propagation of quantum fluctuations in single-pass second-harmonic generation for arbitrary interaction length,” Phys. Rev. A 49, 2106–2116 (1994).
[CrossRef] [PubMed]

R.-D. Li and P. Kumar, “Quantum-noise reduction in traveling-wave second-harmonic generation,” Phys. Rev. A 49, 2157–2166 (1994).
[CrossRef] [PubMed]

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Figures (13)

Fig. 1
Fig. 1

Walk-off of squeezed axis from phase of mean amplitude.

Fig. 2
Fig. 2

Model of device and error. (a) Random duty-cycle error, (b) random period error. Up arrows and down arrows indicate noninverted and inverted polarization, respectively.

Fig. 3
Fig. 3

Spectral density of noise of outputs from a perfectly QPM device as a function of input power. (a) Fundamental output, (b) harmonic output. SNL, shot-noise level.

Fig. 4
Fig. 4

Contour map of squeezing in a perfectly QPM device. (a) Fundamental output, (b) harmonic output.

Fig. 5
Fig. 5

Precise plot of Fig. 4(a) in the vicinity of perfect quasi-phase matching.

Fig. 6
Fig. 6

Contour map of squeezing in a QPM device with random duty-cycle error of 10%. (a) Fundamental output, (b) harmonic output.

Fig. 7
Fig. 7

Precise plot of Fig. 6(a) in the vicinity of perfect quasi-phase matching.

Fig. 8
Fig. 8

Contour map of squeezing in a QPM device with random period error of 1%. (a) Fundamental output, (b) harmonic output.

Fig. 9
Fig. 9

Precise plot of Fig. 8(a) in the vicinity of perfect quasi-phase matching.

Fig. 10
Fig. 10

Margin of domain length variation for 10-dB squeezing as a function of domain period error.

Fig. 11
Fig. 11

The same as Fig. 8 but with another error pattern.

Fig. 12
Fig. 12

Accumulated phase mismatch of error pattern for Fig. 8. Deviation of domain length variation: 320 ppm.

Fig. 13
Fig. 13

Spatial evolution of phases and spectral density of noise of fundamental amplitude for error pattern of Fig. 8. Input fundamental power: 1.25 W.

Equations (92)

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E1=iω020Sn1(ω0)c1/2dωa(z, ω)×exp{iω[n1(ω)z/c-t]}+h.c.,
E2=i 2ω020Sn2(2ω0)c1/2dωb(z, ω)×exp{iω[n2(ω)z/c-t]}+h.c.,
[x(z, ω), x(z, ω)]=[x(z, ω), x(z, ω)]=0,
[x(z, ω), x(z, ω)]=δ(ω-ω)(x=a, b).
da(z, ω)dz=pκdωa(z, ω)b(z, ω+ω)×exp[-iΔk1(ω, ω)z],
db(z, ω)dz=-p κ2dωa(z, ω)a(z, ω-ω)×exp[iΔk2(ω, ω)z],
Δk1(ω, ω)
=n2(ω+ω)(ω+ω)-n1(ω)ω-n1(ω)ωc,
Δk2(ω, ω)
=n2(ω)ω-n1(ω)ω-n1(ω-ω)(ω-ω)c,
κ=χ(2)ω0n1(ω0)cω02π0Sn2(2ω0)c1/2,
a(z, ω0+δω)=u(z)δ(δω)+Δa˜(z, δω),
b(z, 2ω0+δω)=v(z)δ(δω)+Δb˜(z, δω),
limT0 Δx˜(0, δω)Δx˜(0,-δω)T0T0=1,
limT0 Δx˜(0,-δω)Δx˜(0, δω)T0T0
=limT0 Δx˜(0, δω)Δx˜(0, δω)T0T0=limT0 Δx˜(0,-δω)Δx˜(0,-δω)T0T0
=0,
du(z)dz=pκu*(z)v(z)exp(-iΔkz),
dv(z)dz=-pκ2u2(z)exp(iΔkz),
dΔa˜(z, ω)dz=-pκ[Δa˜(z,-δω)v(z)-u*(z)Δb˜(z, δω)×exp(-iΔkz)]exp(-iΔkz),
dΔb˜(z, δω)dz=pκu(z)Δa˜(z, δω)×exp(iΔkz)×exp(iΔkz),
Δk=2ω0c[n2(2ω0)-n1(ω0)]
Δk=ω0c2 dn2(2ω0)dω-dn1(ω0)dωω0+n2(2ω0)-n1(ω0)δω
zj=πcjΔk+Δzj,
dj=zj-zj-1,
dj=πcΔk+Δdj,
zj=πcjΔk+m=0jdm(j=1, 2, ).
un-1=un-2-κun-2*vn-2f(n-1),
vn-1=vn-2+κ2un-22f*(n-1),
f(n)=exp(iΔkzn)-exp(iΔkzn-1)iΔk.
un=un-1+κun-1*vn-1f(n),
vn=vn-1-κ2un-12f*(n).
un=un-2-κun-2*vn-2[f(n-1)-f(n)],
vn=vn-2+κ2un-22[f*(n-1)-f*(n)].
Δa˜(zn, 0)=Δa˜(zn-2, 0)-κ[Δa˜(zn-2, 0)vn-2-un-2*Δb˜(zn-2, 0)][f(n-1)-f(n)],
Δb˜(zn, 0)=Δb˜(zn-2, 0)+κun-2Δa˜(zn-2, 0)×[f*(n-1)-f*(n)].
f(n-1)-f(n)=4iΔk,
un-un-2zn-zn-2=i 2κπun-2*vn-2,
vn-vn-2zn-zn-2=i κπun-22,
Δa˜(zn, 0)-Δa˜(zn-2, 0)zn-zn-2
=i 2κπ[Δa˜(zn-2, 0)vn-2-un-2*Δb˜(zn-2, 0)],
Δb˜(zn, 0)-Δb˜(zn-2, 0)zn-zn-2
=2κπun-2Δa˜(zn-2, 0).
f(n-1)-f(n)
=1iΔk[-exp(iΔkΔzn)-2 exp(iΔkΔzn-1)
-exp(iΔkΔzn-2)].
f(n-1)-f(n)i 4Δk-(2Δzn-1+Δzn+Δzn-1).
un-un-2zn-zn-2=-i 2κeffπun-2*vn-2 exp(iΔθn),
vn-vn-2zn-zn-2=-i κeffπun-22 exp(-iΔθn),
Δa˜(zn, 0)-Δa˜(zn-2, 0)zn-zn-2
=-i 2κeffπ[Δa˜(zn-2, 0)vn-2-un-2*Δb˜(zn-2, 0)]×exp(iΔθn),
Δb˜(zn, 0)-Δb˜(zn-2, 0)zn-zn-2
=-i 2κeffπun-2Δa˜(zn-2, 0)exp(-iΔθn),
n=πΛ0(Δzn+2Δzn-1+Δzn-2),
Λ0=2πΔkzn-zn-2,
κeff=κ1+n2,
Δθn=tan-1 nn.
Δθn2π62Δzn2Λ0/2,
f(n-1)-f(n)=1iΔn{-exp[iΔk(dn+dn-1+dn-2)]-2 exp[iΔk(dn-1+dn-2)]-exp(iΔkdn-2)}×expiΔkm=1n-3Δdm.
un-un-2zn-zn-2=-i 2κeffπun-2*vn-2 exp(iΔθn+iΘn),
vn-vn-2zn-zn-2=-i κeffπun-22 exp(-iΔθn-iΘn),
Δa˜(zn, 0)-Δa˜(zn-2, 0)zn-zn-2
=-i 2κeffπ[Δa˜(zn-2, 0)vn-2-un-2*Δb˜(zn-2, 0)]
×exp(iΔθn+iΘn),
Δb˜(zn, 0)-Δb˜(zn-2, 0)zn-zn-2
=-i 2κeffπun-2Δa˜(zn-2, 0)×exp(-iΔθn-iΘn),
n=πΛ0(Δdn+3Δdn-1+4Δdn-2),
κeff=κ1+n2,
Δθn=tan-1 nn,
Θn=Δkm=1n-3Δdm.
Δθn2=π262Δdm2Λ0/2,
Θn2=πΛ0/2m=1n-3Δdm2=πn-3 Δdm2Λ0/2,
dΔa˜dz(z, 0)=-i 2κπ[Δa˜(z, 0)v(z)+Δb˜(z, 0)u(z)]exp[iΔθ(z)],
dΔa˜dz(z, 0)=2κπΔa˜(z, 0)v(z)×expiΔθ(z)+ϕ2(z)-π2,
dΔa˜dz(z, 0)=2κπΔa˜(z, 0)v(z)×exp-iΔθ(z)+ϕ2(z)-π2,
u(z)=u(z)exp[iϕ1(z)],
v(z)=v(z)exp[iϕ2(z)].
Δa˜1(z, 0)=12Δa˜(z, 0)exp i2Δθ(z)+ϕ2(z)-π2+Δa˜(z, 0)exp -i2Δθ(z)+ϕ2(z)-π2,
Δa˜2(z, 0)=12iΔa˜(z, 0)exp i2Δθ(z)+ϕ2(z)-π2-Δa˜(z, 0)exp -i2Δθ(z)+ϕ2(z)-π2,
dΔa˜1dz(z, 0)=2κπv(z)Δa˜1(z, 0)-12d[ϕ2(z)+Δθ(z)]dzΔa˜2(z, 0),
dΔa˜2dz(z, 0)=-2κπv(z)Δa˜2(z, 0)+12d[ϕ2(z)+Δθ(z)]dzΔa˜1(z, 0),
ϕSQ(z)=12ϕ2(z)+Δθ(z)-π2.
Δx(z, 0)=τ[Δa˜(z, 0), Δa˜(z, 0), Δb˜(z, 0), Δb˜(z, 0)]
Δx(z, 0)=T(z)Δx(0, 0).
Δu˜(z, 0)=12×[Δa˜(z, 0)exp(-iϕa)+Δa˜(z, 0)exp(iϕa)],
Δv˜(z, 0)=12×[Δb˜(z, 0)exp(-iϕb)+Δb˜(z, 0)exp(iϕb)],
Sf=limT0 Δu˜(l, 0)Δu˜(l, 0)T0T0=14×(T11T22+T21T12+T13T24+T23T14)+14×(T11T12+T13T14)exp(-2iϕa)+14×(T21T22+T23T24)exp(2iϕa),
Ss=limT0 Δv˜(l, 0)Δv˜(l, 0)T0T0=14×(T31T42+T41T32+T33T44+T43T34)+14×(T31T32+T33T34)exp(-2iϕb)+14×(T41T42+T43T44)exp(2iϕb),
Λ=2d=c2ω0[n1(ω0)-n2(2ω0)].
ΔΛ=-c2ω0[n1(ω0)-n2(2ω0)]2Δn1ΔT-Δn2ΔTΔT,
Δρ=ΔΛΛ0+αΔT=1(n0-n1)Δn0ΔT-Δn1ΔT+αΔT,
ΔρΔf=1(n0-n1)Δn0Δf-Δn1Δf,

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