Abstract

Quadrature-phase squeezing by optical parametric amplification in an arbitrary photonic crystal was studied by use of a perturbation theory based on a Green function formalism. It is shown that the perturbative calculation can be performed to an infinite order. The result shows that quadrature-phase squeezing is enhanced by small group velocities of electromagnetic eigenmodes that are easily achieved in two- and three-dimensional photonic crystals.

© 2002 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J. 1995).
  2. C. M. Soukoulis, ed., Photonic Band Gaps and Localization (Plenum, New York, 1993).
  3. C. M. Soukoulis, ed., Photonic Band Gap Materials (Kluwer, Dordrecht, The Netherlands, 1996).
  4. K. Sakoda, “Photonic crystals,” in Optical Properties of Low-Dimensional Materials, T. Ogawa and Y. Kanemitsu, eds. (World Scientific, Singapore, 1998), Vol. 2, pp. 402–450.
  5. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, 2001).
  6. K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices. Solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732–5741 (1996).
    [CrossRef]
  7. K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742–5749 (1996).
    [CrossRef]
  8. K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express 4, 167–176 (1999), http://www.opticsexpress.org.
    [CrossRef] [PubMed]
  9. K. Sakoda, K. Ohtaka, and T. Ueta, “Low-threshold laser oscillation due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express 4, 481–489 (1999), http://www.opticsexpress.org.
    [CrossRef] [PubMed]
  10. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser—a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
    [CrossRef]
  11. See, for example, M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, Cambridge, 1997), Chap. 2.
  12. R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
    [CrossRef] [PubMed]

1999 (2)

1996 (2)

K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices. Solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732–5741 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742–5749 (1996).
[CrossRef]

1994 (1)

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser—a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

1991 (1)

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[CrossRef] [PubMed]

Bloemer, M. J.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser—a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Bowden, C. M.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser—a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Dowling, J. P.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser—a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Glauber, R. J.

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[CrossRef] [PubMed]

Lewenstein, M.

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[CrossRef] [PubMed]

Ohtaka, K.

K. Sakoda, K. Ohtaka, and T. Ueta, “Low-threshold laser oscillation due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express 4, 481–489 (1999), http://www.opticsexpress.org.
[CrossRef] [PubMed]

K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices. Solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732–5741 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742–5749 (1996).
[CrossRef]

Sakoda, K.

K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express 4, 167–176 (1999), http://www.opticsexpress.org.
[CrossRef] [PubMed]

K. Sakoda, K. Ohtaka, and T. Ueta, “Low-threshold laser oscillation due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals,” Opt. Express 4, 481–489 (1999), http://www.opticsexpress.org.
[CrossRef] [PubMed]

K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742–5749 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices. Solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732–5741 (1996).
[CrossRef]

Scalora, M.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser—a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Ueta, T.

J. Appl. Phys. (1)

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band-edge laser—a new approach to gain enhancement,” J. Appl. Phys. 75, 1896–1899 (1994).
[CrossRef]

Opt. Express (2)

Phys. Rev. A (1)

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[CrossRef] [PubMed]

Phys. Rev. B (2)

K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices. Solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B 54, 5732–5741 (1996).
[CrossRef]

K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B 54, 5742–5749 (1996).
[CrossRef]

Other (6)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J. 1995).

C. M. Soukoulis, ed., Photonic Band Gaps and Localization (Plenum, New York, 1993).

C. M. Soukoulis, ed., Photonic Band Gap Materials (Kluwer, Dordrecht, The Netherlands, 1996).

K. Sakoda, “Photonic crystals,” in Optical Properties of Low-Dimensional Materials, T. Ogawa and Y. Kanemitsu, eds. (World Scientific, Singapore, 1998), Vol. 2, pp. 402–450.

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, 2001).

See, for example, M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, Cambridge, 1997), Chap. 2.

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Figures (2)

Fig. 1
Fig. 1

Degenerate optical parametric amplification: schematic illustration of parametric amplification in a photonic crystal with second-order nonlinear susceptibility.

Fig. 2
Fig. 2

Contour map of the integration in Eq. (B8) for z>l.

Equations (60)

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·(A)=0,
E(r, t)=-tA(r, t).
1(r)×[×Ekn(r)]=ωkn2c2Ekn(r),
Ekn(r+a)=exp(ik·a)Ekn(r),
V dr(r)Ekn*(r)·Ekn(r)=Vδkkδnn,
A(r, t)=1Vkn[qkn(t)Ekn(r)+qkn*(t)Ekn*(r)],
E(r, t)=iVkn ωkn[qkn(t)Ekn(r)-qkn*(t)Ekn*(r)].
qkn(t)20ωkn1/2aˆkn(t),
qkn*(t)20ωkn1/2aˆkn+(t),
Hˆ=kn ωknaˆkn+(t)aˆkn(t)+12.
[aˆkn(t), aˆkn+(t)]=1.
Eˆ(r, t)=kniωkn20V1/2[aˆkn(t)Ekn(r)-aˆkn+(t)Eˆkn*(r)].
kp=(0, 0, kp),ks=(0, 0, ks).
Eˆs(r, t)=iωs20V1/2[aˆs(0)Es(r)exp(-iωst)-aˆs+(0)Es*(r)exp(iωst)],
Pˆ(0)(r, t)=χ(2)(r):[Ep(r, t)+Eˆs(r, t)]22χ(2)(r):Ep(r, t)Eˆs(r, t)2A ωs20V1/2χ(2)(r):[Ep(r)Es*(r)×aˆs+(0)exp(-iωst+iθ)+H.c.],
Eˆind(1)(r, t)=-Pˆ(0)(r, t)0(r)+10Vkn ωknEkn(r)×V dr -t dtEkn*(r)·Pˆ(0)(r, t)sin ωkn(t-t).
Eˆind(1)(r, t)ilωs20V1/2[βaˆs+(0)Es(r)exp(-iωst+iθ)-β*aˆs(0)Es*(r)exp(iωst-iθ)],
β=βηξ,β=ωsAF(s, p, s)0vg,
η=sin(lΔkz/2)nz sin(aΔkz/2),Δkz=kp-2ks,
ξ=exp[ia(nz-1)Δkz/2],
F(s, p, s)=1V0V0drEs*(r)·χ(2)(r):Ep(r)Es*(r),
Δkz=2πma,
kp=2ks+G,
kp=2ks.
Pˆ(1)(r, t)2Alωs20V1/2χ(2)(r):[β*Ep(r)Es*(r)aˆs(0)×exp(-iωst)+H.c.],
Eˆind(2)(r, t)i|β|2l22!ωs20V1/2[aˆs(0)Es(r)exp(-iωst)-aˆs+(0)Es*(r)exp(iωst)].
Eˆ(r, t)=Eˆs(r, t)+Eˆind(1)(r, t)+Eˆind(2)(r, t)+iωs20V1/2[bˆEs(r)exp(-iωst)-bˆ+Es*(r)exp(iωst)],
bˆ=1+|β|2l22!+|β|4l44!+aˆs(0)+exp[i(θ+ϕ)]|β|l+|β|3l33!+aˆs+(0)=aˆs(0)cosh|β|l+exp[i(θ+ϕ)]×aˆs+(0)sinh|β|l,
bˆ+=aˆs+(0)cosh|β|l+exp[-i(θ+ϕ)]×aˆs(0)sinh|β|l,
β=|β|exp(iϕ).
E1(r, t)=-12i[Es(r)exp(-iωst)-Es*(r)exp(iωst)],
E2(r, t)=12[Es(r)exp(-iωst)+Es*(r)exp(iωst)].
Eˆ(r, t)=2ωs0V1/2[qˆE1(r, t)+pˆE2(r, t)],
qˆ=12(bˆ+bˆ+),pˆ=i2(bˆ-bˆ+).
[qˆ, pˆ]=-i2,ΔqΔp14.
aˆs|α=α|α.
(Δq)2=α|qˆ2|α-α|qˆ|α2=14cosh2|β|l+sinh2|β|l+2 cosθ+ϕ-π2cosh|β|l sinh|β|l,
(Δp)2=α|pˆ2|α-α|pˆ|α2=14cosh2|β|l+sinh2|β|l-2 cosθ+ϕ-π2cosh|β|l sinh|β|l,
Δq=½ exp(|β|l),Δp=½ exp(-|β|l),
Δq=½ exp(-|β|l),Δp=½ exp(|β|l).
Pˆ(+)(r, t)=Bˆχ(2)(r) : Ep(r)Es*(r)exp[(-iωs+δ)t],
Bˆ=2Aωs20V1/2 exp(iθ)aˆs+(0)
V drEkn*(r)·Pˆ(+)(r, t)
=Bˆ exp(-iωst)×jx=0Nx-1jy=0Ny-1jz=0nz-1 exp[ia(kp-ks-k)·j]×V0 drEkn*(r)·χ(2)(r):Ep(rˆ)Es*(r)
=NxNynzξkηkV0δkx0δky0Bˆ exp(-iωst)F(kn, p, s),
ξk=exp[ia(nz-1)(kp-ks-kz)/2],
ηk=sin[l(kp-ks-kz)/2]nz sin[a(kp-ks-kz)/2].
1VV dr -tdtEkn*(r)·Pˆ(+)(r, t)sin ωkn(t-t)=nzξkηkδkx0δky0BˆF(kn, p, s)exp(-iωst)2Nz×1ωs+ωkn+iδ-1ωs-ωkn+iδ.
1ωs+ωk+iδ-1ωs-ωk+iδ2πiδ(ωs-ωk),
vg=ωkkzωk=ωs,
Eˆind(1)(r, t)ianzξηωsBˆF(s, p, s)20vgEs(r)exp(-iωst)+H.c.,
Es(r)=essexp(iksz),
Ep(r)=eppexp(ikpz),
vg=cs,
F(s, p, s)=es·χ(2):epesV0spV0dr exp[i(kp-2ks)z]=es·χ(2):epesspexp(iaΔkz)-1iaΔkz.
Eˆind(1)(r, t)iAles·χ(2):epescsωs3203pV1/2×sin(lΔkz/2)lΔkz/2{aˆs+(0)es exp(ilΔkz/2)×exp[i(ksz-ωst+θ)]-H.c.}.
Eˆind(1)(r, t)=A2πiωs203psV1/2 - dkzωkek·χ(2):epeskek exp(ikzz)×aˆs+(0)exp(-iωst+iθ)exp[i(kp-ks-kz)l]-1kp-ks-kz1ωs+ωk+iδ-1ωs-ωk+iδ-aˆs(0)exp(iωst-iθ)exp[i(-kp+ks-kz)l]-1kp-ks+kz1ωs+ωk-iδ-1ωs-ωk-iδ.
ωk=ckz/k(kz>0)-ckz/k(kz<0),
Eˆind(1)(r, t)A2πciωs203psV1/2-dωkωkek·χ(2):epeskek expikcz×aˆs+(0)exp(-iωst+iθ)exp[i(kp-ks-kωk/c)l]-1kp-ks-kωk/c1ωk+ωs+iδ+1ωk-ωs-iδ-aˆs(0)exp(iωst-iθ)exp[-i(kp-ks+kωk/c)l]-1kp-ks+kωk/c1ωk+ωs-iδ+1ωk-ωs+iδ.
ω=±ck(kp-ks)

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