Abstract

The theory of Bragg diffraction of finite beams by thick planar gratings is developed, using coupled-wave theory. Simple analytical expressions for the profiles of the transmitted and diffracted beams in the near field are obtained. Detailed diffraction characteristics for the important case of Gaussian-profile beams are presented. It is shown that the diffraction characteristics depend only on two normalized parameters, the grating strength, and a geometry parameter. The diffraction efficiency and the profiles of the transmitted and diffracted beams are calculated as functions of these two controlling parameters.

© 1980 Optical Society of America

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References

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  1. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [CrossRef]
  2. R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).
  3. H. Kogelnik, “Coupled-wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  4. F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectric constant,” J. Opt. Soc. Am. 63, 37–45 (1973).
    [CrossRef]
  5. L. E. Hargrove, “Effect of ultrasonic waves on Gaussian light beams with diameter comparable to ultrasonic wavelength,” J. Acoust. Soc. Am. 43, 847–851 (1968).
    [CrossRef]
  6. D. H. McMahon, “Relative efficiency of optical Bragg diffraction as a function of interaction geometry,” IEEE Trans. Sonic Ultrason. SU-16, 41–44 (1969).
  7. R. Guther and S. Kusch, “Kinematic theory of Gaussian beams in volume holography,” Sov. J. Quantum Electron. 6, 509–514 (1976).
    [CrossRef]
  8. R. S. Chu and T. Tamir, “Bragg diffraction of Gaussian beams by periodically modulated media,” J. Opt. Soc. Am. 66, 220–226 (1976).
    [CrossRef]
  9. R. S. Chu and T. Tamir, “Diffraction of Gaussian beams by periodically modulated media for incidence close to Bragg angle,” J. Opt. Soc. Am. 66, 1438–1440 (1976).
    [CrossRef]
  10. R. S. Chu, J. A. Kong, and T. Tamir, “Diffraction of Gaussian beams by a periodically modulated layer,” J. Opt. Soc. Am. 67, 1555–1561 (1977).
    [CrossRef]
  11. L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
    [CrossRef]
  12. W. E. Parry and L. Solymar, “A general solution for the two-dimensional volume holograms,” Opt. Quantum Electron. 9, 527–531 (1977).
    [CrossRef]
  13. R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley-Interscience, New York, 1962), Vol. II, pp. 450–459.
  14. A. E. Siegman, “Bragg diffraction of a Gaussian beam by a crossed-Gaussian volume grating,” J. Opt. Soc. Am. 67, 545–550 (1977).
    [CrossRef]
  15. B. W. Batterman and H. Cole, “Dynamic diffraction of x-rays by perfect crystals,” Rev. Mod. Phys. 36, 681–717 (1964).
    [CrossRef]
  16. P. St, J. Russell, L. Solymar, and M. P. Jordan, “Borrmann-like effects in volume holography,” Proc. ICO-11 Conf., 635–638 (1978).
  17. M. R. B. Forshaw, “Diffraction of a narrow laser beam by a thick hologram: experimental results,” Opt. Commun. 12, 279–281 (1974).
    [CrossRef]

1977 (4)

L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
[CrossRef]

W. E. Parry and L. Solymar, “A general solution for the two-dimensional volume holograms,” Opt. Quantum Electron. 9, 527–531 (1977).
[CrossRef]

A. E. Siegman, “Bragg diffraction of a Gaussian beam by a crossed-Gaussian volume grating,” J. Opt. Soc. Am. 67, 545–550 (1977).
[CrossRef]

R. S. Chu, J. A. Kong, and T. Tamir, “Diffraction of Gaussian beams by a periodically modulated layer,” J. Opt. Soc. Am. 67, 1555–1561 (1977).
[CrossRef]

1976 (3)

1974 (1)

M. R. B. Forshaw, “Diffraction of a narrow laser beam by a thick hologram: experimental results,” Opt. Commun. 12, 279–281 (1974).
[CrossRef]

1973 (1)

1970 (1)

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).

1969 (2)

H. Kogelnik, “Coupled-wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

D. H. McMahon, “Relative efficiency of optical Bragg diffraction as a function of interaction geometry,” IEEE Trans. Sonic Ultrason. SU-16, 41–44 (1969).

1968 (1)

L. E. Hargrove, “Effect of ultrasonic waves on Gaussian light beams with diameter comparable to ultrasonic wavelength,” J. Acoust. Soc. Am. 43, 847–851 (1968).
[CrossRef]

1966 (1)

1964 (1)

B. W. Batterman and H. Cole, “Dynamic diffraction of x-rays by perfect crystals,” Rev. Mod. Phys. 36, 681–717 (1964).
[CrossRef]

Batterman, B. W.

B. W. Batterman and H. Cole, “Dynamic diffraction of x-rays by perfect crystals,” Rev. Mod. Phys. 36, 681–717 (1964).
[CrossRef]

Burckhardt, C. B.

Chu, R. S.

Cole, H.

B. W. Batterman and H. Cole, “Dynamic diffraction of x-rays by perfect crystals,” Rev. Mod. Phys. 36, 681–717 (1964).
[CrossRef]

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley-Interscience, New York, 1962), Vol. II, pp. 450–459.

Forshaw, M. R. B.

M. R. B. Forshaw, “Diffraction of a narrow laser beam by a thick hologram: experimental results,” Opt. Commun. 12, 279–281 (1974).
[CrossRef]

Guther, R.

R. Guther and S. Kusch, “Kinematic theory of Gaussian beams in volume holography,” Sov. J. Quantum Electron. 6, 509–514 (1976).
[CrossRef]

Hargrove, L. E.

L. E. Hargrove, “Effect of ultrasonic waves on Gaussian light beams with diameter comparable to ultrasonic wavelength,” J. Acoust. Soc. Am. 43, 847–851 (1968).
[CrossRef]

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley-Interscience, New York, 1962), Vol. II, pp. 450–459.

Jordan, M. P.

P. St, J. Russell, L. Solymar, and M. P. Jordan, “Borrmann-like effects in volume holography,” Proc. ICO-11 Conf., 635–638 (1978).

Kaspar, F. G.

Kogelnik, H.

H. Kogelnik, “Coupled-wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Kong, J. A.

Kusch, S.

R. Guther and S. Kusch, “Kinematic theory of Gaussian beams in volume holography,” Sov. J. Quantum Electron. 6, 509–514 (1976).
[CrossRef]

McMahon, D. H.

D. H. McMahon, “Relative efficiency of optical Bragg diffraction as a function of interaction geometry,” IEEE Trans. Sonic Ultrason. SU-16, 41–44 (1969).

Parry, W. E.

W. E. Parry and L. Solymar, “A general solution for the two-dimensional volume holograms,” Opt. Quantum Electron. 9, 527–531 (1977).
[CrossRef]

Russell, J.

P. St, J. Russell, L. Solymar, and M. P. Jordan, “Borrmann-like effects in volume holography,” Proc. ICO-11 Conf., 635–638 (1978).

Siegman, A. E.

Solymar, L.

W. E. Parry and L. Solymar, “A general solution for the two-dimensional volume holograms,” Opt. Quantum Electron. 9, 527–531 (1977).
[CrossRef]

L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
[CrossRef]

P. St, J. Russell, L. Solymar, and M. P. Jordan, “Borrmann-like effects in volume holography,” Proc. ICO-11 Conf., 635–638 (1978).

St, P.

P. St, J. Russell, L. Solymar, and M. P. Jordan, “Borrmann-like effects in volume holography,” Proc. ICO-11 Conf., 635–638 (1978).

Tamir, T.

Appl. Phys. Lett. (1)

L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled-wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).

IEEE Trans. Sonic Ultrason. (1)

D. H. McMahon, “Relative efficiency of optical Bragg diffraction as a function of interaction geometry,” IEEE Trans. Sonic Ultrason. SU-16, 41–44 (1969).

J. Acoust. Soc. Am. (1)

L. E. Hargrove, “Effect of ultrasonic waves on Gaussian light beams with diameter comparable to ultrasonic wavelength,” J. Acoust. Soc. Am. 43, 847–851 (1968).
[CrossRef]

J. Opt. Soc. Am. (6)

Opt. Commun. (1)

M. R. B. Forshaw, “Diffraction of a narrow laser beam by a thick hologram: experimental results,” Opt. Commun. 12, 279–281 (1974).
[CrossRef]

Opt. Quantum Electron. (1)

W. E. Parry and L. Solymar, “A general solution for the two-dimensional volume holograms,” Opt. Quantum Electron. 9, 527–531 (1977).
[CrossRef]

Rev. Mod. Phys. (1)

B. W. Batterman and H. Cole, “Dynamic diffraction of x-rays by perfect crystals,” Rev. Mod. Phys. 36, 681–717 (1964).
[CrossRef]

Sov. J. Quantum Electron. (1)

R. Guther and S. Kusch, “Kinematic theory of Gaussian beams in volume holography,” Sov. J. Quantum Electron. 6, 509–514 (1976).
[CrossRef]

Other (2)

P. St, J. Russell, L. Solymar, and M. P. Jordan, “Borrmann-like effects in volume holography,” Proc. ICO-11 Conf., 635–638 (1978).

R. Courant and D. Hilbert, Methods of Mathematical Physics (Wiley-Interscience, New York, 1962), Vol. II, pp. 450–459.

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Figures (6)

FIG. 1
FIG. 1

Diffraction geometry and coordinate systems.

FIG. 2
FIG. 2

Intensity profiles of the R beam as a function of the grating strength γ for several values of the geometry parameter g.

FIG. 3
FIG. 3

Intensity profiles of the S beam as a function of the grating strength γ for several values of the geometry parameter g. The vertical scale is 1.6 times the vertical scale in Fig. 2.

FIG. 4
FIG. 4

Intensity profiles of the R beam for several values of the grating strength γ for g = 3.0.

FIG. 5
FIG. 5

Intensity profiles of the S beam for several values of the grating strength γ for g = 3.0. The vertical scale is the same as in Fig. 4.

FIG. 6
FIG. 6

Diffraction efficiency as a function of the grating strength γ for several values of the geometry parameter g.

Equations (17)

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n ( x ) = n 0 + n 1 cos ( 2 π x / Λ ) ,
E ( x , z ) = R ( x , z ) exp ( j k r · r ) + S ( x , z ) exp ( j k s · r ) ,
2 E + k 2 E = 0
cos θ R ( x , z ) z + sin θ R ( x , z ) x + j κ S ( x , z ) = 0 ,
cos θ S ( x , z ) z sin θ S ( x , z ) x + j κ R ( x , z ) = 0 ,
κ = π n 1 / λ .
R ( r , s ) s + j κ S ( r , s ) = 0 ,
S ( r , s ) r + j κ R ( r , s ) = 0 ,
κ = κ / sin 2 θ .
R ( r ) = R o ( r ) 1 2 γ 1 + 1 R o [ r d ( 1 u ) sin θ ] × ( 1 + u 1 u ) 1 / 2 J 1 [ γ ( 1 u 2 ) 1 / 2 ] d u ,
S ( s ) = j 1 2 γ 1 + 1 R o [ d ( 1 u ) sin θ s ] × J 0 [ γ ( 1 u 2 ) 1 / 2 ] d u ,
P d = η 1 S ( s ) S * ( s ) d s
P t = η 1 R ( r ) R * ( r ) d r ,
D E = P d / ( P d + P t ) .
R o ( x , z ) = E o exp [ ( z sin θ x cos θ ) 2 / w 2 ] = R o ( r ) = E o exp ( r 2 / w 2 ) ,
R ( r ) = R o ( r ) 1 2 γ E o 1 + 1 exp { [ g ( 1 u ) r ] 2 } × ( 1 + u 1 u ) 1 / 2 J 1 [ γ ( 1 u 2 ) 1 / 2 ] d u ,
S ( s ) = j 1 2 γ E o 1 + 1 exp { [ g ( 1 u ) s ] 2 } × J o [ γ ( 1 u 2 ) 1 / 2 ] d u ,