Abstract

The diffraction of Gaussian beams on intracavity Bragg gratings is analyzed theoretically. For reasonable waists the associated beam divergence does not significantly influence the diffraction efficiency of such devices. Nevertheless, the tilt angle of the incident beam, imposed by the Bragg resonance condition, strongly reduces the diffraction efficiency at short grating periods. However, the angular selectivity can be maintained if the Fabry–Perot cavity is tuned to the incident beam direction, which allows the use of small-volume holograms together with a dense angular multiplex. This theoretical analysis can be applied to the optimization of the diffraction properties of Gaussian beams on any intracavity Bragg grating, which could then be used for free-space parallel signal processing.

© 2005 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. L. Menez, I. Zaquine, A. Maruani, and R. Frey, "Intracavity Bragg gratings," J. Opt. Soc. Am. B 16, 1849-1855 (1999).
    [CrossRef]
  4. L. Menez, I. Zaquine, A. Maruani, and R. Frey, "Bragg thickness criterion for intracavity diffraction gratings," J. Opt. Soc. Am. B 19, 965-972 (2002).
    [CrossRef]
  5. L. Menez, I. Zaquine, A. Maruani, and R. Frey, "Intracavity refractive index Bragg gratings in absorbing and amplifying media," Opt. Commun. 204, 267-275 (2002).
    [CrossRef]
  6. L. Menez, I. Zaquine, A. Maruani, and R. Frey, "Experimental investigation of intracavity Bragg gratings," Opt. Lett. 27, 479-481 (2002).
    [CrossRef]
  7. K. Sundar, N. Mukunda, and R. Simon, "Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams," J. Opt. Soc. Am. A 12, 560-569 (1995).
    [CrossRef]
  8. X. Xue, H. Wei, and A. G. Kirk, "Intensity-based modal decomposition of optical beams in terms of Hermite-Gaussian functions," J. Opt. Soc. Am. A 17, 1086-1091 (2000).
    [CrossRef]
  9. J.-S. Lee and C.-S. Shim, "Characteristics of a spectrum-slicing filter composed of an angle-tuned Fabry-Perot etalon and a Gaussian input beam," IEEE Photonics Technol. Lett. 7, 905-907 (1995).
    [CrossRef]
  10. P. L. Penna, A. D. Virgilio, M. Fiorentino, A. Porzio, and S. Solimeno, "Transmittivity profile of high finesse plane parallel Fabry-Perot cavities illuminated by Gaussian beams," Opt. Commun. 162, 267-279 (1999).
    [CrossRef]
  11. O. Mata-Mendez and F. Chavez-Rivas, "Diffraction of Gaussian and Hermite-Gaussian beams by finite gratings," J. Opt. Soc. Am. A 18, 537-545 (2001).
    [CrossRef]
  12. S.-D. Wu and E. N. Glytsis, "Finite-number-of-periods holographic gratings with finite-width incident beams: analysis using the finite-difference frequency-domain method," J. Opt. Soc. Am. A 19, 2018-2029 (2002).
    [CrossRef]
  13. T. K. Gaylord and M. G. Moharam, "Planar dielectric grating diffraction theories," Appl. Phys. B 28, 1-14 (1982).
    [CrossRef]
  14. R. E. Collin, Field Theory of Guided Waves (Mc Graw-Hill, New York, 1960), p. 368.
  15. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif. 2003), chap. 2, p. 73.

2002 (4)

2001 (1)

2000 (1)

1999 (2)

L. Menez, I. Zaquine, A. Maruani, and R. Frey, "Intracavity Bragg gratings," J. Opt. Soc. Am. B 16, 1849-1855 (1999).
[CrossRef]

P. L. Penna, A. D. Virgilio, M. Fiorentino, A. Porzio, and S. Solimeno, "Transmittivity profile of high finesse plane parallel Fabry-Perot cavities illuminated by Gaussian beams," Opt. Commun. 162, 267-279 (1999).
[CrossRef]

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)

K. Sundar, N. Mukunda, and R. Simon, "Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams," J. Opt. Soc. Am. A 12, 560-569 (1995).
[CrossRef]

J.-S. Lee and C.-S. Shim, "Characteristics of a spectrum-slicing filter composed of an angle-tuned Fabry-Perot etalon and a Gaussian input beam," IEEE Photonics Technol. Lett. 7, 905-907 (1995).
[CrossRef]

1982 (1)

T. K. Gaylord and M. G. Moharam, "Planar dielectric grating diffraction theories," Appl. Phys. B 28, 1-14 (1982).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif. 2003), chap. 2, p. 73.

Burckhardt, C. B.

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

Chavez-Rivas, F.

Collier, R. J.

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

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (Mc Graw-Hill, New York, 1960), p. 368.

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]

Fiorentino, M.

P. L. Penna, A. D. Virgilio, M. Fiorentino, A. Porzio, and S. Solimeno, "Transmittivity profile of high finesse plane parallel Fabry-Perot cavities illuminated by Gaussian beams," Opt. Commun. 162, 267-279 (1999).
[CrossRef]

Frey, R.

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, "Planar dielectric grating diffraction theories," Appl. Phys. B 28, 1-14 (1982).
[CrossRef]

Glytsis, E. N.

Kirk, A. G.

Lee, J.-S.

J.-S. Lee and C.-S. Shim, "Characteristics of a spectrum-slicing filter composed of an angle-tuned Fabry-Perot etalon and a Gaussian input beam," IEEE Photonics Technol. Lett. 7, 905-907 (1995).
[CrossRef]

Lin, L. H.

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

Maruani, A.

Mata-Mendez, O.

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]

Menez, L.

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, "Planar dielectric grating diffraction theories," Appl. Phys. B 28, 1-14 (1982).
[CrossRef]

Mukunda, N.

Nolte, D. D.

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]

Penna, P. L.

P. L. Penna, A. D. Virgilio, M. Fiorentino, A. Porzio, and S. Solimeno, "Transmittivity profile of high finesse plane parallel Fabry-Perot cavities illuminated by Gaussian beams," Opt. Commun. 162, 267-279 (1999).
[CrossRef]

Porzio, A.

P. L. Penna, A. D. Virgilio, M. Fiorentino, A. Porzio, and S. Solimeno, "Transmittivity profile of high finesse plane parallel Fabry-Perot cavities illuminated by Gaussian beams," Opt. Commun. 162, 267-279 (1999).
[CrossRef]

Shim, C.-S.

J.-S. Lee and C.-S. Shim, "Characteristics of a spectrum-slicing filter composed of an angle-tuned Fabry-Perot etalon and a Gaussian input beam," IEEE Photonics Technol. Lett. 7, 905-907 (1995).
[CrossRef]

Simon, R.

Solimeno, S.

P. L. Penna, A. D. Virgilio, M. Fiorentino, A. Porzio, and S. Solimeno, "Transmittivity profile of high finesse plane parallel Fabry-Perot cavities illuminated by Gaussian beams," Opt. Commun. 162, 267-279 (1999).
[CrossRef]

Sundar, K.

Virgilio, A. D.

P. L. Penna, A. D. Virgilio, M. Fiorentino, A. Porzio, and S. Solimeno, "Transmittivity profile of high finesse plane parallel Fabry-Perot cavities illuminated by Gaussian beams," Opt. Commun. 162, 267-279 (1999).
[CrossRef]

Wei, H.

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]

Wu, S.-D.

Xue, X.

Zaquine, I.

Appl. Phys. B (1)

T. K. Gaylord and M. G. Moharam, "Planar dielectric grating diffraction theories," Appl. Phys. B 28, 1-14 (1982).
[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]

IEEE Photonics Technol. Lett. (1)

J.-S. Lee and C.-S. Shim, "Characteristics of a spectrum-slicing filter composed of an angle-tuned Fabry-Perot etalon and a Gaussian input beam," IEEE Photonics Technol. Lett. 7, 905-907 (1995).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (2)

Opt. Commun. (2)

P. L. Penna, A. D. Virgilio, M. Fiorentino, A. Porzio, and S. Solimeno, "Transmittivity profile of high finesse plane parallel Fabry-Perot cavities illuminated by Gaussian beams," Opt. Commun. 162, 267-279 (1999).
[CrossRef]

L. Menez, I. Zaquine, A. Maruani, and R. Frey, "Intracavity refractive index Bragg gratings in absorbing and amplifying media," Opt. Commun. 204, 267-275 (2002).
[CrossRef]

Opt. Lett. (1)

Other (3)

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

R. E. Collin, Field Theory of Guided Waves (Mc Graw-Hill, New York, 1960), p. 368.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif. 2003), chap. 2, p. 73.

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Figures (9)

Fig. 1
Fig. 1

Setup of the asymmetric intracavity Bragg grating of thickness l and grating period Λ = 2 π K . The read wave of amplitude R I ( r ) and mean wave vector k I is incident at the Bragg angle, and the diffracted wave vector k SF ( 0 ) is therefore symmetric with respect to the incident intracavity wave vector k RF ( 0 ) about the z axis.

Fig. 2
Fig. 2

Grating wave vector conservation and phase mismatch Δ k for the forward and backward read and diffracted waves when θ θ B and therefore θ θ .

Fig. 3
Fig. 3

Normalized (a) y and (b) x profiles of the diffracted beam compared with those of the incident beam at various tilt angles. The finesse of the Fabry–Perot cavity is 28, the sample thickness is 1 mm , and the incident beam waist is 300 μ m .

Fig. 4
Fig. 4

Diffraction efficiency of the Bragg device under Gaussian illumination (a) versus cavity thickness for various tilt angles (solid curve for θ B = 0.1 and dashed curve for θ B = 0.01 ), (b) versus cavity thickness for various beam waists (solid curve for w 0 = 300 μ m and dashed curve for w 0 = 30 μ m ), (c) versus beam waist for various tilt angles (solid curve for θ B = 0.1 and dashed curve for θ B = 0.01 ), (d) versus cavity thickness (dotted-dashed curve for a plane wave at θ B = 0.1 , solid curve for a beam waist of w 0 = 300 μ m at θ B = 0.01 and dotted curve for a beam waist of w 0 = 300 μ m at θ B = 0.1 . In (a)–(c) the refractive-index modulation is adjusted at every data point for maximum diffraction efficiency in the plane-wave approximation ( ρ = 1 ) , whereas in (d) the refractive-index modulation is fixed for a maximum value at l 0 = 300 μ m in the plane-wave case.

Fig. 5
Fig. 5

(a) Normalized transmission ρ T of the empty Fabry–Perot cavity and (b) diffraction efficiency ρ DR of the intracavity Bragg grating plotted as a function of the Bragg detuning for various beam waists (solid, dashed, dotted-dashed and dotted curves for plane waves, w 0 = 1000 , 300, and 100 μ m , respectively). The Fabry–Perot cavity is tuned to the Bragg angle of the grating θ B .

Fig. 6
Fig. 6

(a) Maximum transmission ρ T and angular width FWHM T of the empty Fabry–Perot cavity and (b) maximum diffraction efficiency ρ DR and angular width FWHM DR of the intracavity Bragg device plotted as a function of the read beam waist; the Fabry–Perot cavity is tuned to the Bragg angle of the grating θ B .

Fig. 7
Fig. 7

Normalized diffraction efficiency ρ DR of the intracavity Bragg grating plotted as a function of the Bragg detuning for various beam waists (solid, dashed, dotted-dashed and dotted curves for plane waves, w 0 = 1000 , 300, and 100 μ m , respectively); the Fabry–Perot cavity is tuned to the incident beam direction θ = θ B + Δ θ .

Fig. 8
Fig. 8

Maximum diffraction efficiency ρ DR and angular width FWHM DR for the intracavity Bragg device plotted as a function of the read beam waist; the Fabry–Perot cavity is tuned to the incident beam direction θ = θ B + Δ θ .

Fig. 9
Fig. 9

Normalized transmission ρ T of the empty Fabry–Perot cavity tuned (a) to the Bragg angle θ B and (b) to the read beam direction θ B + Δ θ . The coupling to the Fabry–Perot cavity of the diffracted beam at θ B Δ θ is much less in the latter case.

Equations (42)

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R I ( r 0 ) = y ̂ R I ( r 0 ) exp ( i k I r 0 ) = y ̂ R I ( r 0 ) exp ( i k RF ( 0 ) r 0 ) ,
R I ( r 0 ) = 1 2 π + d δ k x cos θ d δ k y R ̃ I ( δ k x , δ k y , 0 ) × exp ( i δ k z sin θ x ) exp [ i ( δ k x x + δ k y y ) ] ,
R ̃ I ( δ k x , δ k y , 0 ) = ( 2 P I w 0 2 c n 0 ) 1 2 exp [ w 0 2 2 ( k x 2 cos 2 θ + δ k y 2 ) ]
R F , B ( r ) = y ̂ R F , B ( r ) exp ( i k RF , RB ( 0 ) r ) ,
S F , B ( r ) = y ̂ S F , B ( r ) exp ( i k SF , SB ( 0 ) r ) ,
k RF ( 0 ) = k ( sin θ x ̂ + cos θ z ̂ ) ,
k RB ( 0 ) = k ( sin θ x ̂ cos θ z ̂ ) ,
k SF ( 0 ) = k ( sin θ x ̂ + cos θ z ̂ ) ,
k SB ( 0 ) = k ( sin θ x ̂ cos θ z ̂ ) ,
δ k RF = ( δ k z R sin θ + δ k x ) x ̂ + δ k y y ̂ ( δ k z R cos θ + δ k x tan θ ) z ̂ ,
δ k RB = ( δ k z R sin θ + δ k x ) x ̂ + δ k y y ̂ + ( δ k z R cos θ + δ k x tan θ ) z ̂ ,
δ k SF = ( δ k z S sin θ + δ k x ) x ̂ + δ k y y ̂ + ( δ k z S cos θ + δ k x tan θ ) z ̂ ,
δ k SB = ( δ k z S sin θ + δ k x ) x ̂ + δ k y y ̂ + ( δ k z S cos θ + δ k x tan θ ) z ̂
R F ( r ) = 1 2 π + d δ k x cos θ d δ k y R ̃ F ( δ k x , δ k y , z ) × exp ( i δ k RF r ) ,
S F ( r ) = 1 2 π + d δ k x cos θ d δ k y S ̃ F ( δ k x , δ k y , z ) × exp ( i δ k SF r ) exp ( i Δ k z ) ,
R B ( r ) = 1 2 π + d δ k x cos θ d δ k y R ̃ B ( δ k x , δ k y , z ) × exp ( i δ k RB r ) ,
S B ( r ) = 1 2 π + d δ k x cos θ d δ k y S ̃ B ( δ k x , δ k y , z ) × exp ( i δ k SB r ) exp ( i Δ k z ) ,
Δ E ( r ) + ( k + i k ) E ( r ) = [ 4 π 2 λ 2 n 0 Δ n cos ( K x ) ] E ( r )
R ̃ F z + k cos θ R ̃ F = i π Δ n λ cos θ S ̃ F ,
S ̃ F z + ( k cos θ i Δ k ) S ̃ F = i π Δ n λ cos θ R ̃ F ,
R ̃ B z + k cos θ R ̃ B = i π Δ n λ cos θ S ̃ B ,
S ̃ B z + ( k cos θ i Δ k ) S ̃ B = i π Δ n λ cos θ R ̃ B .
R ̃ F ( δ k x , δ k y , z ) = A F + exp ( r F + z ) + A F exp ( r F z ) ,
R ̃ B ( δ k x , δ k y , z ) = A B + exp ( r B + z ) + A B exp ( r B z ) ,
S ̃ F ( δ k x , δ k y , z ) = i λ cos θ π Δ n [ ( r F + + k cos θ ) A F + exp ( r F + z ) + ( r F + k cos θ ) A F exp ( r F z ) ] ,
S ̃ B ( δ k x , δ k y , z ) = i λ cos θ π Δ n [ ( r B + k cos θ ) A B + exp ( r B + z ) + ( r B k cos θ ) A B exp ( r B z ) ] ,
r F ± = r B = k 2 ( 1 cos θ + 1 cos θ ) + i Δ k 2 ± i Δ ,
Δ = [ π Δ n λ ( cos θ cos θ ) 1 2 ] 2 + [ Δ k 2 + i k 2 ( 1 cos θ 1 cos θ ) ] 2 .
R F δ k x δ k y ( x , y , 0 ) = r 1 R B δ k x δ k y ( x , y , 0 ) + t 1 R I δ k x δ k y ( x , y , 0 ) ,
S F δ k x δ k y ( x , y , 0 ) = r 1 S B δ k x δ k y ( x , y , 0 ) ,
R B δ k x δ k y ( x , y , l ) = r 2 R F δ k x δ k y ( x , y , l ) ,
S B δ k x δ k y ( x , y , l ) = r 2 S F δ k x δ k y ( x , y , l ) ,
R F δ k x , δ k y ( r ) = y ̂ R ̃ F ( δ k x , δ k y , z ) exp [ i ( k RF ( 0 ) + δ k RF ) r ] ,
R B δ k x , δ k y ( r ) = y ̂ R ̃ B ( δ k x , δ k y , z ) exp [ i ( k RB ( 0 ) + δ k RB ) r ] ,
S F δ k x , δ k y ( r ) = y ̂ S ̃ F ( δ k x , δ k y , z ) exp ( i Δ k z ) × exp [ i ( k SF ( 0 ) + δ k SF ) r ] ,
S B δ k x , δ k y ( r ) = y ̂ S ̃ B ( δ k x , δ k y , z ) exp ( i Δ k z ) × exp [ i ( k SB ( 0 ) + δ k SB ) r ]
A F ± = ± i t 1 ( r + k cos θ ) 2 Δ [ 1 r 1 r 2 exp ( 2 r ± L ) ] e I ,
A B ± = i t 1 ( r ± + k cos θ ) r 2 exp ( 2 r l ) 2 Δ [ 1 r 1 r 2 exp ( 2 r l ) ] e I ,
ρ T = c n ( 1 R 2 ) 2 π P I + R ̃ F ( δ k x , δ k y , l ) 2 d δ k x cos θ d δ k y ,
ρ R = c n 2 π P I + R ̃ B ( δ k x , δ k y , 0 ) 1 R 1 R ̃ I ( δ k x , δ k y , 0 ) R 1 2 d δ k x cos θ d δ k y ,
ρ DT = c n ( 1 R 2 ) 2 π P I + S ̃ F ( δ k x , δ k y , l ) 2 d δ k x cos θ d δ k y ,
ρ DR = c n ( 1 R 1 ) 2 π P I + S ̃ B ( δ k x , δ k y , 0 ) 2 d δ k x cos θ d δ k y .

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