Abstract

The grating thickness limit lFP between the Raman–Nath and the Bragg diffraction regimes is calculated for an index grating placed in an asymmetric Fabry–Perot resonator with a totally reflecting back mirror and compared with that which was obtained for the same grating with no cavity lM. Owing to the increase of the effective interaction length inside the Fabry–Perot cavity, the stronger the front mirror reflectivity R1 of the cavity, the smaller the thickness above which the whole diffracted intensity can be concentrated into one unique diffracted beam: lFP=[(1-R1)/(1+R1)]lM/2.

© 2002 Optical Society of America

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References

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  1. R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, Orlando, Fla., 1971).
  2. Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
    [CrossRef]
  3. M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
    [CrossRef]
  4. M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman–Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19–23 (1980).
    [CrossRef]
  5. R. Magnusson and T. K. Gaylord, “Analysis of multiwave diffraction of thick gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
    [CrossRef]
  6. T. K. Gaylord and M. G. Moharam, “Thin and thick gratings: terminology clarification,” Appl. Opt. 20, 3271–3273 (1981).
    [CrossRef] [PubMed]
  7. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 9, pp. 354–358.
  8. S. Mallick, “Effets d’paisseur dans les reseaux,” presented at Ecole d’t d’Optolectronique Cargèse, France, summer 1990.
  9. L. Menez, I. Zaquine, A. Maruani, and R. Frey, “Intracavity Bragg gratings,” J. Opt. Soc. Am. B 16, 1849–1855 (1999).
    [CrossRef]
  10. J. H. Collet, R. Buhleier, and J. O. White, “Enhanced diffraction of light in GaAs microcavities,” J. Opt. Soc. Am. B 12, 2439–2444 (1995).
    [CrossRef]
  11. K. M. Kwolek, M. R. Melloch, D. D. Nolte, and G. A. Brost, “Photorefractive asymmetric Fabry–Perot quantum wells: transverse-field geometry,” Appl. Phys. Lett. 67, 736–738 (1995).
    [CrossRef]
  12. D. D. Nolte, K. M. Kwolek, C. Lenox, and B. Streetman, “Dynamic holography in a broad-area optically pumped vertical GaAs microcavity,” J. Opt. Soc. Am. B 18, 257–263 (2001).
    [CrossRef]
  13. L. A. Pipes, Applied Mathematics for Engineers and Physicists (McGraw-Hill, New York, 1958), Chap. 4, pp. 101–102.
  14. M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1970), Chap. 12, pp. 604–607.
  15. D. D. Nolte, D. H. Olson, G. E. Doran, W. H. Knox, and A. M. Glass, “Resonant photodiffractive effect in semi-insulating multiple quantum wells,” J. Opt. Soc. Am. B 7, 2217–2225 (1990).
    [CrossRef]
  16. Q. Wang, R. M. Brubaker, D. D. Nolte, and M. R. Melloch, “Photorefractive quantum wells: transverse Franz–Keldysh geometry,” J. Opt. Soc. Am. B 9, 1626–1641 (1992).
    [CrossRef]

2001 (1)

1999 (1)

1998 (1)

Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
[CrossRef]

1995 (2)

J. H. Collet, R. Buhleier, and J. O. White, “Enhanced diffraction of light in GaAs microcavities,” J. Opt. Soc. Am. B 12, 2439–2444 (1995).
[CrossRef]

K. M. Kwolek, M. R. Melloch, D. D. Nolte, and G. A. Brost, “Photorefractive asymmetric Fabry–Perot quantum wells: transverse-field geometry,” Appl. Phys. Lett. 67, 736–738 (1995).
[CrossRef]

1992 (1)

1990 (1)

1981 (1)

1980 (2)

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman–Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19–23 (1980).
[CrossRef]

1977 (1)

Brost, G. A.

K. M. Kwolek, M. R. Melloch, D. D. Nolte, and G. A. Brost, “Photorefractive asymmetric Fabry–Perot quantum wells: transverse-field geometry,” Appl. Phys. Lett. 67, 736–738 (1995).
[CrossRef]

Brubaker, R. M.

Buhleier, R.

Collet, J. H.

Ding, Y.

Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
[CrossRef]

Doran, G. E.

Frey, R.

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, “Thin and thick gratings: terminology clarification,” Appl. Opt. 20, 3271–3273 (1981).
[CrossRef] [PubMed]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman–Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19–23 (1980).
[CrossRef]

R. Magnusson and T. K. Gaylord, “Analysis of multiwave diffraction of thick gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
[CrossRef]

Glass, A. M.

Knox, W. H.

Kwolek, K. M.

D. D. Nolte, K. M. Kwolek, C. Lenox, and B. Streetman, “Dynamic holography in a broad-area optically pumped vertical GaAs microcavity,” J. Opt. Soc. Am. B 18, 257–263 (2001).
[CrossRef]

K. M. Kwolek, M. R. Melloch, D. D. Nolte, and G. A. Brost, “Photorefractive asymmetric Fabry–Perot quantum wells: transverse-field geometry,” Appl. Phys. Lett. 67, 736–738 (1995).
[CrossRef]

Lenox, C.

Magnusson, R.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman–Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19–23 (1980).
[CrossRef]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

R. Magnusson and T. K. Gaylord, “Analysis of multiwave diffraction of thick gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
[CrossRef]

Maruani, A.

Melloch, M. R.

Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
[CrossRef]

K. M. Kwolek, M. R. Melloch, D. D. Nolte, and G. A. Brost, “Photorefractive asymmetric Fabry–Perot quantum wells: transverse-field geometry,” Appl. Phys. Lett. 67, 736–738 (1995).
[CrossRef]

Q. Wang, R. M. Brubaker, D. D. Nolte, and M. R. Melloch, “Photorefractive quantum wells: transverse Franz–Keldysh geometry,” J. Opt. Soc. Am. B 9, 1626–1641 (1992).
[CrossRef]

Menez, L.

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, “Thin and thick gratings: terminology clarification,” Appl. Opt. 20, 3271–3273 (1981).
[CrossRef] [PubMed]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman–Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19–23 (1980).
[CrossRef]

Nolte, D. D.

Olson, D. H.

Streetman, B.

Wang, Q.

Weiner, A. M.

Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
[CrossRef]

White, J. O.

Zaquine, I.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

K. M. Kwolek, M. R. Melloch, D. D. Nolte, and G. A. Brost, “Photorefractive asymmetric Fabry–Perot quantum wells: transverse-field geometry,” Appl. Phys. Lett. 67, 736–738 (1995).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (5)

Opt. Commun. (2)

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Bragg regime diffraction by phase gratings,” Opt. Commun. 32, 14–18 (1980).
[CrossRef]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman–Nath regime diffraction by phase gratings,” Opt. Commun. 32, 19–23 (1980).
[CrossRef]

Other (5)

R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, Orlando, Fla., 1971).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 9, pp. 354–358.

S. Mallick, “Effets d’paisseur dans les reseaux,” presented at Ecole d’t d’Optolectronique Cargèse, France, summer 1990.

L. A. Pipes, Applied Mathematics for Engineers and Physicists (McGraw-Hill, New York, 1958), Chap. 4, pp. 101–102.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1970), Chap. 12, pp. 604–607.

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Figures (10)

Fig. 1
Fig. 1

Intracavity Bragg grating. The nonlinear medium where the grating is recorded is inserted into a Fabry–Perot resonator. An incident read wave EI gives rise to reflected ER, transmitted ET, and diffracted ERD and ETD waves, the numbers in whose subscripts indicate their diffraction order. The intracavity wave vectors of the read and diffracted waves are, respectively, k and k. They bear the subscripts F for forward propagating and B for backward propagating. For simplicity, refraction is not taken into account in this figure.

Fig. 2
Fig. 2

Phase matching. The forward and backward wave vectors of the read wave, kF and kB, and grating wave vector K are represented as well as all diffracted waves kF and kB (numbered by their diffraction orders), according to the Raman–Nath relation.

Fig. 3
Fig. 3

Bragg criterion in a bare grating. Relative diffracted intensity I1/I-1 of order 1 normalized to order -1 is plotted versus QM=πλl/Λ2. The reference level is plotted at 0.01 (dotted line).

Fig. 4
Fig. 4

Logarithmic plot of diffraction efficiency I-1/IR of order -1 versus Fabry–Perot interference order m for two reflection coefficients values, r1=0.6 and r1=0.9.

Fig. 5
Fig. 5

Logarithmic plot of relative intensity I-2/I-1 of order -2 normalized to order -1 versus interference order m for a bare grating (r1=0) and for an intracavity grating (r1=0.9).

Fig. 6
Fig. 6

Logarithmic plot of order 1 diffraction efficiency I1/IR versus interference order m for a bare grating (r1=0) and for an intracavity grating (r1=0.9).

Fig. 7
Fig. 7

Logarithmic plot of relative intensity I1/I-1 versus interference order m for a bare grating (r1=0) and for an intracavity grating (r1=0.9).

Fig. 8
Fig. 8

Logarithmic plot of relative intensity I1/I-1 versus modified Q parameter QFP=[2(1+r1)/(1-r1)]QM for three reflection coefficient values (r1=0.9, r1=0.5 and r1=0, the bare grating).

Fig. 9
Fig. 9

Double-resonance intracavity refraction angle θDR versus relative difference Δm/m between the Fabry–Perot interference orders of diffraction orders 1 and -1.

Fig. 10
Fig. 10

Linear plots of (a) order 1 diffraction efficiency I1/IR versus the interference order m and (b) order -1 diffraction efficiency I-1/IR versus interference order m.

Equations (41)

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n=n0+Δn sin(K·r)=n0+Δn2i[exp(iK·r)-exp(-iK·r)].
k(sin θp-sin θ)=pK,
sin θp=(2p+1)sin θ.
|(2pmax+1)sin θ|1.
RF·kˆF exp(ikF·r)+RB·kˆB exp(ikB·r)
+p0p=-pmaxpmax[SFp·kˆFp
×exp(ikFp·r)SBp·kˆBp exp(ikBp·r)]
=2iπλ0 n0+Δn2i[exp(iK·r)-exp(-iK·r)]×RF·kˆF×exp(ikF·r)+RB·kˆB×exp(ikB·r)+p0p=-pmaxpmax[SFp·kˆFp×exp(ikFp·r)+SBp·kˆBp exp(ikBp·r)],
cos θdRFdz=πΔnλ0[SF-1-SF1 exp(-iΔkz)],
cos θ dSF-1dz=πΔnλ0[-RF+SF-2 exp(-iΔkz)],
cos θ1 dSF1dz=πΔnλ0[RF exp(iΔkz)],
cos θ1 dSF-2dz=πΔnλ0[-SF-1 exp(iΔkz)];
-cos θ dRBdz=πΔnλ0[SB-1-SB1 exp(iΔkz)],
-cos θ dSB-1dz=πΔnλ0[-RB+SB-2 exp(iΔkz)],
-cos θ1 dSB1dz=πΔnλ0[RB exp(-iΔkz)],
-cos θ1 dSB-2dz=πΔnλ0[-SB-1 exp(-iΔkz)].
SF1(0)=r1SB1(0),
SF-2(0)=r1SB-2(0),
r2SF1(l)=SB1(l)exp(-2ikl cos θ1),
r2SF-2(l)=SB-2(l)exp(-2ikl cos θ1).
ddz SF1RFSF-1SF-2=[u]SF1RFSF-1SF-2,
I1/I-1=sinc2(Δkl)sinc2(QM).
2lm cos θ=mλ.
lopt=λ0 cos θπΔn arccos2r11+r12
I-1FP=(1-r12)(1+r12-2r1 cos 2β)2I-1bare.
I-1FP=(1+r1)(1-r1)2I-1bare.
ifl>lM2,then I1bareI-1bare<0.01.
ifl>lM2,then I1FPI-1FP<0.01(1-r1)(1+r1)2.
ifl>lM2 (1-r1)(1+r1)=lFP,thenI1FP/I-1FP<0.01.
I-1nomirβ2=πΔnlmλ cos θ2,
I-1FP(1+r1)(1-r1)2(2β)2=(1+r1)(1-r1)22 πΔnlmλ cos θ2=πΔnleffλ cos θ2,
leff=2(1+r1)(1-r1)lm.
sin θ1=3 sin θ
2lDR cos θ=mλ(man integer)
2lDR cos θ1=(m-Δm)λ
(Δman integerandΔm<m)
cos θDR=1+Δm4m-Δm28m2-1/2.
SB1(0)-rF+ sin(2β)+Δk[cos(2β)-r1](1+r12-2r1 cos 2β)-Δk exp(2ikl cos θ1)1-r1 exp(2ikl cos θ1),
SB-1(0)sin(2β)(1+r12-2r1 cos 2β),
dSB-1(0)dβcos(2β)(1+r12)/2-r1(1+r12-2r1 cos 2β)2.
Δmπ(1-r1)lDR.

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