Abstract

Orthogonally polarized four-wave mixing is studied theoretically in a configuration involving two coupling coefficients with different magnitudes, one of which is effectively zero. The configuration is studied to reduce the complexity of the feedback and to achieve a unidirectional power flow from one of the pumps to the diffracted wave. The analytical solution of the coupled-wave equations is derived for both a transmission-type grating and a reflection-type grating. The solution clearly explains the relative phases between the two components of a grating and its effect on the performance of the system. Phase matching is also studied. For characterization of its operation, diffraction efficiency and diffracted-beam reflectivity are calculated, plotted, and analyzed as functions of different quantities, such as coupling strength, probe ratio, pump ratio, and angular difference of the waves. A signal distributor as a possible application is also discussed.

[Optical Society of America ]

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References

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  1. S. I. Stepanov and M. P. Petrov , Opt. Commun. OPCOB8 53 , 64 ( 1985
    [CrossRef]
  2. K. D. Shaw and M. Cronin-Golomb , Opt. Commun. OPCOB8 65 , 301 ( 1988
    [CrossRef]
  3. T. K. Das and G. C. Bhar , Opt. Quantum Electron. OQELDI 26 , 1019 ( 1994
    [CrossRef]
  4. M. Cronin-Golomb , B. Fisher , J. O. White , and A. Yariv , IEEE J. Quantum Electron. IEJQA7 QE-20 , 12 ( 1984
    [CrossRef]
  5. M. Belic and M. Petrovic , J. Opt. Soc. Am. B JOBPDE 11 , 481 ( 1994
    [CrossRef]
  6. M. R. Belic and M. Lax , Opt. Commun. OPCOB8 56 , 197 ( 1985
    [CrossRef]
  7. P. E. Andersen , P. Buchhave , P. M. Petersen , and M. V. Vasnetsov , J. Opt. Soc. Am. B JOBPDE 12 , 1422 ( 1995
    [CrossRef]
  8. A. Roy and K. Singh , J. Mod. Opt. JMOPEW 41 , 987 ( 1994
    [CrossRef]
  9. Hongzhi Kong , C. Lin , A. M. Biernacki , and M. Cronin-Golomb , Opt. Lett. OPLEDP 13 , 324 ( 1988
    [CrossRef] [PubMed]
  10. N. C. Kothari , J. Opt. Soc. Am. B JOBPDE 13 , 2320 ( 1996
    [CrossRef]
  11. N. C. Kothari and B. E. A. Saleh , Phys. Rev. A PLRAAN 46 , 2896 ( 1992
    [CrossRef] [PubMed]
  12. M. Zgonik , K. Nakagawa , and P. Gunter , J. Opt. Soc. Am. B JOBPDE 12 , 1416 ( 1995
    [CrossRef]

Other (12)

S. I. Stepanov and M. P. Petrov , Opt. Commun. OPCOB8 53 , 64 ( 1985
[CrossRef]

K. D. Shaw and M. Cronin-Golomb , Opt. Commun. OPCOB8 65 , 301 ( 1988
[CrossRef]

T. K. Das and G. C. Bhar , Opt. Quantum Electron. OQELDI 26 , 1019 ( 1994
[CrossRef]

M. Cronin-Golomb , B. Fisher , J. O. White , and A. Yariv , IEEE J. Quantum Electron. IEJQA7 QE-20 , 12 ( 1984
[CrossRef]

M. Belic and M. Petrovic , J. Opt. Soc. Am. B JOBPDE 11 , 481 ( 1994
[CrossRef]

M. R. Belic and M. Lax , Opt. Commun. OPCOB8 56 , 197 ( 1985
[CrossRef]

P. E. Andersen , P. Buchhave , P. M. Petersen , and M. V. Vasnetsov , J. Opt. Soc. Am. B JOBPDE 12 , 1422 ( 1995
[CrossRef]

A. Roy and K. Singh , J. Mod. Opt. JMOPEW 41 , 987 ( 1994
[CrossRef]

Hongzhi Kong , C. Lin , A. M. Biernacki , and M. Cronin-Golomb , Opt. Lett. OPLEDP 13 , 324 ( 1988
[CrossRef] [PubMed]

N. C. Kothari , J. Opt. Soc. Am. B JOBPDE 13 , 2320 ( 1996
[CrossRef]

N. C. Kothari and B. E. A. Saleh , Phys. Rev. A PLRAAN 46 , 2896 ( 1992
[CrossRef] [PubMed]

M. Zgonik , K. Nakagawa , and P. Gunter , J. Opt. Soc. Am. B JOBPDE 12 , 1416 ( 1995
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Beam geometry of orthogonally polarized FWM in a PR crystal for a transmission grating.

Fig. 2
Fig. 2

Beam geometry of orthogonally polarized FWM in a PR crystal for a reflection grating.  

Fig. 3
Fig. 3

rOrd.eff (dotted) and rExt. ord.eff (solid), as functions of grating-vector angle in BaTiO3 crystal. The thick solid curve is for the reflection grating (ϕ1-ϕp=-165°), and the thin solid curve is for the transmission grating (ϕ1-ϕp=-15°).

Fig. 4
Fig. 4

Numerical solution for ϕ1 and ϕp of Eq. (2) for BaTiO3 and KNbO3 crystals. The arrow roughly indicates the regions for transmission and reflection gratings.

Fig. 5
Fig. 5

Phase matching in a PR crystal in the case of orthogonal polarization for a transmission grating.

Fig. 6
Fig. 6

Phase matching in a PR crystal in the case of orthogonal polarization for a reflection grating.

Fig. 7
Fig. 7

DE plotted as a function of coupling strength (γr) in the case of a transmission grating for different values of pump ratio ψ, where probe ratio σ=1 and |ϕ1-ϕp|=15°.

Fig. 8
Fig. 8

DE plotted as a function of coupling strength (-γr) in the case of a reflection grating, for different values of pump ratio ψ, where probe ratio σ=1 and |ϕ1-ϕp|=165°.

Fig. 9
Fig. 9

DE as a function of angular difference, for different values of ψ, including transmission and reflection gratings (dotted curve for the OPC and solid curve for the IPC). σ=1; absolute coupling strength is 0.4.

Fig. 10
Fig. 10

Contour plots of DE as a function of log ψ and log σ: (a) transmission grating, γr=0.4, and |ϕ1-ϕp|=15°; (b) reflection grating, γr=0.4, and |ϕ1-ϕp|=165°.

Fig. 11
Fig. 11

Signal distributor: The angles are shown according to the earlier calculations.  

Fig. 12
Fig. 12

Contour plots of log of DBR for a transmission grating as a function of log ψ and log σ: (a) IPC, γr=3, and |ϕ1-ϕp|=15°; (b) OPC, γr=-3 and |ϕ1-ϕp|=15°.

Equations (101)

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rExt. ord.eff=r11eff(n(O))4 cos ϕ1 cos ϕp+r33eff(n(E))4 sin ϕ1 sin ϕp-r13eff(n(O))2(n(E))2 sin(ϕ1+ϕp),
rOrd.eff=r22eff(n(O))4.
cos(ϕ1-ϕp)=[r11eff(n(O))4-r33eff(n(E))4]cos(ϕ1+ϕp)-2r13eff(n(O))2(n(E))2 sin(ϕ1+ϕp)r11eff(n(O))4+r33eff(n(E))4.
n(O)=2.489,n(E)=2.424,
r11eff=40 cos ϕg-18 cos 5ϕg-2 cos 7ϕg+1.5 cos 9ϕg,
r22eff=36.5 cos ϕg-21.5 cos 3ϕg+4.5 cos 5ϕg+2.5 cos 7ϕg,
r33eff=141 cos ϕg-102 cos 3ϕg+37 cos 5ϕg+14 cos 7ϕg,
r13eff=1160 sin ϕg-115 sin 3ϕg+70 sin 5ϕg+19 sin 7ϕg,
ϕg=ϕ1+ϕp2-π2 signsinϕ1-ϕp2.
signsinϕ1-ϕp2,
n(O)=2.334,n(E)=2.213,
r11eff=42 cos ϕg-12 cos 3ϕg+10 cos 5ϕg-2 cos 7ϕg+2 cos 9ϕg,
r22eff=11.5 cos ϕg-3.5 cos 3ϕg+3.5 cos 5ϕg,
r33eff=74 cos ϕg-17 cos 3ϕg+5 cos 5ϕg-3 cos 7ϕg,
r13eff=91 sin ϕg-18 sin 3ϕg+20 sin 5ϕg-3 sin 7ϕg+4 sin 9ϕg.
|K1-Kp|<|K1|+|Kp|<2 2πn(O)λ=|K2|+|Kd|.
dA1dr=dAp*dr=0,
dA2*dr=-γI0 [A1Ap* cos(ϕ1-ϕp)+AdA2*]Ad*,
dAddr=γI0 [A1Ap* cos(ϕ1-ϕp)+AdA2*]A2,
I0A1A1*+ApAp*+A2A2*+AdAd*,
γπλ rOrd.eff 1n(O) cosϕp-ϕ12 ED1+(ED/EN),
A1a1 exp(jθ1),
Apap exp(jθp),
A2a2(r)exp[jθ2(r)],
Adad(r)exp[jθd(r)],
α(γ/I0)a1ap cos(ϕ1-ϕp),
βγ/I0.
da2dr=-(α cos θ0+βada2)ad,
a2 dθ2dr=α sin θ0ad,
daddr=(α cos θ0+βada2)a2,
ad dθddr=α sin θ0a2,
θ0(θ1-θp)-(θd-θ2)ρI-ρII.
ddr (a22+ad2)=0.
a2I2+d cos t,
adI2+d sin t,
I2+dI2+Id=a22+ad2=const.
DEId(R=R)I2(R=0)=1[1-X2 cot(1-X2R)-X]2+1,
Rαr=γr a1ap cos(ϕ1-ϕp)I0,
XβI2+d2α=I2+d2I1Ip cos(ϕ1-ϕp).
DBRId(r=r)Ip=ad2(r=r)ap2,
ψI2(r=0)I1=a22(r=0)a12,
σIpI1=ap2a12.
dA1dr=dAp*dr=0,
dA2*dr=γI0 [A1Ap* cos(ϕ1-ϕp)+AdA2*]Ad*,
dAddr=γI0 [A1Ap* cos(ϕ1-ϕp)+AdA2*]A2,
γπλ rOrd.eff 1n(O) sinϕp-ϕ12 E01+(ED/EN);
da2dr=γI0 [a1ap cos(ϕ1-ϕp)cos θ0+ada2]ad,
a2 dθ2dr=-γI0 a1ap cos(ϕ1-ϕp)sin θ0ad,
daddr=γI0 [a1ap cos(ϕ1-ϕp)cos θ0+ada2]a2,
ad dθddr=γI0 a1ap cos(ϕ1-ϕp)sin θ0a2.
ddr (a22-ad2)=0.
a2I2-d cosh t,
adI2-d sinh t,
I2-dI2-Id=a22-ad2=const.
I0=a12+ap2+I2-d cosh 2t.
R=ln1+sinh 2tQ+P1+Q2 tanh-1 1+Q2Q coth t+1,
P2(a12+ap2)I2-d,
Q2a1ap cos(ϕ1-ϕp)cos θ0I2-d,
Rγr.
DEId(r=r)I2(r=r)=ad2(r=r)a22(r=r)=tanh2 t.
ψI2(r=r)I1=a22(r=r)a12.
θ0=(θ1-θp)-(θd-θ2)=ρI-ρII=0or±π(0tπ/2).
0tπ/2.
ad=0,a2=I2+d,
θ0=0,or±π.
ad=I2+d,a2=0,
θ0=0,or±π.
d(θ2-θd)dr=α sin θ0(cot t-tan t),
dtdr=-α cos θ0-βI2+d sin(2t)2.
tan θ02=expC1+2α+0r cot(2t)dr,
θ0=(θ1-θp)-(θd-θ2)=ρI-ρII=0or±π
(0tπ/2).
dtdR=cos θ0+X sin 2t,
Rαr=γr a1ap cos(ϕ1-ϕp)I0,
XβI2+d2α=I2+d2I1Ip cos(ϕ1-ϕp).
tan t=1cos θ0 {-X+1-X2 tan[1-X2(R+C2)]},
tan(1-X2C2)=X1-X2.
t(R=0)=0,
ad=0atr=0.
DEId(R=R)I2(R=0)=sin2 t=1[1-X2 cot(1-X2R)-X]2+1,
R=αr.
tant=RR-1,
DEId(R=R)I2(R=0)=R22R2-2R+1.
tan t=1cos θ0 {-X+X2-1 ×tanh[X2-1(C3-R)]},
tanh(X2-1C3)=XX2-1.
DEId(R=R)I2(R=0)=1[X2-1 coth(X2-1R)-X]2+1.
ad=0,a2=I2-d,
θ0=0or±π.
d(θ2-θd)dr=-2 γI0 a1ap cos(ϕ1-ϕp)sin θ0 coth 2t.
logtan θ02=-2γa1ap cos(ϕ1-ϕp)×+0r coth 2ta12+ap2+I2-d cosh 2t dr.
θ0=(θ1-θp)-(θd-θ2)=ρI-ρII=0or±π(0tπ/2).
P+2 cosh 2tQ+sinh 2t dt=dR,
P(2a12+ap2)I2-d,
Q2a1ap cos(ϕ1-ϕp)cos θ0I2-d,
Rγr.
P1+Q2 tanh-1-1+Q tanh t1+Q2+ln(Q+sinh 2t)
=R+C5,
t=0atr=0(R=0),
C5=P1+Q2 tanh-1-11+Q2+ln Q.
R=ln1+sinh 2tQ+P1+Q2 tanh-1 1+Q2Q coth t+1.
DEId(r=r)I2(r=r)=ad2(r=r)a22(r=r)=tanh2 t.

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