Abstract

The theory of diffraction by gratings formed by the interference of two crossed three-dimensional “arbitrary-profile plane waves” in a photosensitive medium is developed. The detailed diffraction characteristics for the case of crossed-beam gratings formed by two three-dimensional “Gaussian plane waves” are presented. The diffraction efficiencies of these gratings and the profiles of the transmitted and diffracted beams are calculated as functions of the grating strength. The influence of the relative size (the Gaussian beam diameter) of the two writing beams on the diffraction efficiency is determined. Diffraction characteristics for readout with beams of profiles different from those used for writing are presented.

© 1980 Optical Society of America

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References

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  1. A. E. Siegman, “Bragg diffraction of a Gaussian beam by a crossed-Gaussian volume grating,” J. Opt. Soc. Am. 67, 545–550 (1977).
    [Crossref]
  2. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [Crossref]
  3. L. Solymar and M. P. Jordan, “Finite beams in large volume holograms,” Microwaves, Opt., Acoust. 1, 89–92 (1977).
    [Crossref]
  4. R. P. Kenan, “Theory of crossed-beam diffraction gratings,” IEEE J. Quantum Electron. QE-14, 924–930 (1978).
    [Crossref]
  5. L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
    [Crossref]
  6. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 10.
  7. W. E. Parry and L. Solymar, “A general solution for two-dimensional volume holograms,” Opt. Quantum Electron. 9, 527–531 (1977).
    [Crossref]
  8. R. Courant and D. Hubert, Methods of Mathematical Physics (Wiley Interscience, New York, 1962), Vol. II, pp. 450–459.

1978 (1)

R. P. Kenan, “Theory of crossed-beam diffraction gratings,” IEEE J. Quantum Electron. QE-14, 924–930 (1978).
[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 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]

L. Solymar and M. P. Jordan, “Finite beams in large volume holograms,” Microwaves, Opt., Acoust. 1, 89–92 (1977).
[Crossref]

1969 (1)

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

Courant, R.

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

Hubert, D.

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

Jordan, M. P.

L. Solymar and M. P. Jordan, “Finite beams in large volume holograms,” Microwaves, Opt., Acoust. 1, 89–92 (1977).
[Crossref]

Kenan, R. P.

R. P. Kenan, “Theory of crossed-beam diffraction gratings,” IEEE J. Quantum Electron. QE-14, 924–930 (1978).
[Crossref]

Kogelnik, H.

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

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 10.

Parry, W. E.

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

Siegman, A. E.

Solymar, L.

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

L. Solymar and M. P. Jordan, “Finite beams in large volume holograms,” Microwaves, Opt., Acoust. 1, 89–92 (1977).
[Crossref]

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

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 J. Quantum Electron. (1)

R. P. Kenan, “Theory of crossed-beam diffraction gratings,” IEEE J. Quantum Electron. QE-14, 924–930 (1978).
[Crossref]

J. Opt. Soc. Am. (1)

Microwaves, Opt., Acoust. (1)

L. Solymar and M. P. Jordan, “Finite beams in large volume holograms,” Microwaves, Opt., Acoust. 1, 89–92 (1977).
[Crossref]

Opt. Quantum Electron. (1)

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

Other (2)

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

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 10.

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Figures (4)

FIG. 1
FIG. 1

Configuration and coordinate system of a crossed-beam volume grating.

FIG. 2
FIG. 2

Spatial profiles of the transmitted R and the diffracted S beams for several values of the grating strength γ (and the corresponding diffraction efficiencies). The vertical scale is the same in all figures to allow direct comparison of results.

FIG. 3
FIG. 3

Diffraction efficiency as a function of the grating strength γ for several values of the ratio wry/wsy when reading with the reference R beam. The quantities wry and wsy are the Gaussian beam diameters of the R and the S beams, respectively.

FIG. 4
FIG. 4

Diffraction efficiency as a function of the grating γ strength for several values of the ratio wrr,r/wrr,w when reading with a beam of different Gaussian diameter than that of the writing beam. The quantities wrr,r and wrr,w are the Gaussian diameters in the plane of the grating of the reading and the writing beams, respectively. In Fig. 4(a) wry,r/wry,w = 1 and in Fig. 4(b) w r y , r / w r y , w = w r r , r / w r r , w where wry,r and wry,w are the reading and writing Gaussian beam diameters in the y direction.

Equations (32)

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E r = E r 0 A r ( x , y , z ) exp ( j k r · r ) ,
E s = E s 0 A s ( x , y , z ) exp ( j k s · r ) ,
n ( x , y , z ) = n 0 + n 1 A r ( x , y , z ) A s ( x , y , z ) cos ( 2 β x sin θ ) ,
2 E + k 2 E = 0 ,
E ( x , y , z ) = R ( x , y , z ) A r ( x , y , z ) exp ( j k r · r ) + S ( x , y , z ) A s ( x , y , z ) exp ( j k s · r ) .
cos θ R z + sin θ R x + j κ A s 2 S = 0 ,
cos θ S z sin θ S x + j κ A r 2 R = 0.
R ( s , r , y ) s + j κ A s 2 ( s , y ) S = 0 ,
S ( s , r , y ) r + j κ A r 2 ( r , y ) R = 0 ,
κ = κ / sin 2 θ .
S ( s , r , y ) = j R 0 ( u / υ ) 1 / 2 J 1 [ 2 κ ( u υ ) 1 / 2 ] ,
R ( s , r , y ) = R 0 J 0 [ 2 κ ( u υ ) 1 / 2 ] ,
u ( r , y ) = r A r 2 ( r , y ) d r
υ ( s , y ) = s A s 2 ( s , y ) d s .
D E 2 D = 1 J 0 2 [ 2 κ ( u υ ) 1 / 2 ] J 1 2 [ 2 κ ( u υ ) 1 / 2 ] ,
u = u ( )
υ = υ ( ) .
D E 3 D = { 1 J 0 2 [ f ( y ) ] J 1 2 [ f ( y ) ] } u ( , y ) d y / u ( , y ) d y ,
f ( y ) = 2 κ [ u ( , y ) υ ( , y ) ] 1 / 2 ,
u ( , y ) = A r 2 ( r , y ) d r ,
υ ( , y ) = A s 2 ( s , y ) d s .
S ( s , r , y ) = j κ r R 0 ( r , y ) A r ( r , y ) × J 0 ( 2 κ { υ [ u ( r , y ) u ( r , y ) ] } 1 / 2 ) d r
R ( s , r , y ) = R 0 ( , y ) J 0 [ 2 κ ( u υ ) 1 / 2 ] + r R 0 ( r , y ) r J 0 ( 2 κ { υ [ u ( r , y ) u ( r , y ) ] } 1 / 2 ) d r .
P d = η 1 A s 2 ( s , y ) | S ( s , , y ) | 2 d s d y ,
P t = η 1 A r 2 ( r , y ) | R ( , r , y ) | 2 d r d y ,
D E 3 D = P d / ( P d + P t ) .
A r ( r , y ) = exp ( r 2 / w r r 2 y 2 / w r y 2 ) ,
A s ( s , y ) = exp ( s 2 / w s s 2 y 2 / w s y 2 ) .
u ( r , y ) = ( π / 2 ) 1 / 2 w r r exp ( 2 y 2 / w r y 2 ) erf ( 2 r / w r r )
υ ( s , y ) = ( π / 2 ) 1 / 2 w s s exp ( 2 y 2 / w s y 2 ) erf ( 2 s / w s s ) ,
γ = ( 1 2 ) f ( 0 ) = π n 1 d / λ cos θ
d = ( π w r r w s s / 2 ) 1 / 2 2 sin θ .