Abstract

The Fresnel–Kirchhoff free-space diffraction transform is generalized to describe paraxial-beam propagation in homogeneous birefringent media, extending Fourier optics to anisotropic homogeneous media. Canonical operator formalism is used for the derivation. The method is applied to designing an optical Fourier transformer in a biaxial crystal.

© 1986 Optical Society of America

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References

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  1. E. Ochoa, L. Hesselink, J. W. Goodman, “Real-time intensity inversion using two-wave and four-wave mixing in photorefractive Bi12GeO20,” Appl. Opt. 24, 1826–1832 (1985).
    [Crossref] [PubMed]
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  3. M. Lax, D. F. Nelson, “Imaging through a surface of an anisotropic medium with application to light scattering,”J. Opt. Soc. Am. 66, 694–704 (1976).
    [Crossref]
  4. J. J. Stamnes, G. C. Sherman, “Reflection and refraction of an arbitrary wave at a plane interface separating two uniaxial crystals,”J. Opt. Soc. Am. 67, 683–695 (1977).
    [Crossref]
  5. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  6. L. Thylen, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
    [Crossref]
  7. L. Thylen, D. Yevick, “Beam propagation method in anisotropic media,” Appl. Opt. 21, 2751–2754 (1982).
    [Crossref] [PubMed]
  8. J. A. Fleck, M. D. Feit, “Beam propagation in uniaxial anisotropic media,”J. Opt. Soc. Am. 73, 920–926 (1983).
    [Crossref]
  9. M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,”J. Opt. Soc. Am. 70, 150–158 (1980).
    [Crossref]
  10. M. Nazarathy, J. Shamir, “Holography described by operator algebra,”J. Opt. Soc. Am. 71, 529–541 (1981).
    [Crossref]
  11. M. Nazarathy, J. Shamir, “First order optics—a canonical operator representation: lossless systems,”J. Opt. Soc. Am. 72, 356–364 (1982).
    [Crossref]
  12. A. D. Boardman, Electromagnetic Surface Modes (Wiley, New York, 1982).
  13. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).
  14. A. Yariv, Optical Waves in Crystals (Wiley, New York, 1984).

1985 (1)

1983 (2)

L. Thylen, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
[Crossref]

J. A. Fleck, M. D. Feit, “Beam propagation in uniaxial anisotropic media,”J. Opt. Soc. Am. 73, 920–926 (1983).
[Crossref]

1982 (2)

1981 (1)

1980 (1)

1977 (1)

1976 (1)

Boardman, A. D.

A. D. Boardman, Electromagnetic Surface Modes (Wiley, New York, 1982).

Feit, M. D.

Fleck, J. A.

Goodman, J. W.

Hesselink, L.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).

Lax, M.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).

Nazarathy, M.

Nelson, D. F.

Ochoa, E.

Shamir, J.

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

Sherman, G. C.

Stamnes, J. J.

Thylen, L.

L. Thylen, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
[Crossref]

L. Thylen, D. Yevick, “Beam propagation method in anisotropic media,” Appl. Opt. 21, 2751–2754 (1982).
[Crossref] [PubMed]

Yariv, A.

A. Yariv, Optical Waves in Crystals (Wiley, New York, 1984).

Yevick, D.

Appl. Opt. (2)

J. Opt. Soc. Am. (6)

Opt. Quantum Electron. (1)

L. Thylen, “The beam propagation method: an analysis of its applicability,” Opt. Quantum Electron. 15, 433–439 (1983).
[Crossref]

Other (5)

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

A. D. Boardman, Electromagnetic Surface Modes (Wiley, New York, 1982).

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960).

A. Yariv, Optical Waves in Crystals (Wiley, New York, 1984).

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Figures (6)

Fig. 1
Fig. 1

Geometry of beam propagation in anisotropic homogeneous media. Notice that the beam wave fronts are approximately perpendicular to the ray direction rather than to the direction of beam propagation.

Fig. 2
Fig. 2

Block diagram for propagation in an anisotropic medium.

Fig. 3
Fig. 3

The wave-vector ellipse for TM polarization. Indicated are the beam and ray directions and three coordinate systems associated with the boundary, the ray and the ellipse, respectively. The incidence principal plane is the plane of the paper, and the x and x′ axes are normal to this plane.

Fig. 4
Fig. 4

Rotation to a ray-based coordinate system.

Fig. 5
Fig. 5

Geometry of isotropic propagation, alternatively considered on-axis along z ¯ and tilted in the xyz reference system.

Fig. 6
Fig. 6

Optical Fourier transformation in a biaxial crystal.

Tables (1)

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Table 1 Operator Algebra

Equations (97)

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k 0 = ω / c ,
k = k 0 p = k p ^ = k 0 n ( p ^ ) p ^ ,
p ^ = s x x ^ + s y y ^ + s z z ^ .
p = n ( p ^ ) p ^ .
p x = n s x , p y = n s y , p z = n s z .
p z = p z ( p y ) .
U ( p y , z ) = F p y y u ( y , z ) = ( j λ 0 ) - 1 / 2 - d y × [ exp ( - j k 0 p y y ) ] u ( y , z ) .
u ( y , z ) = F y p y - 1 U ( p y , z ) = ( j λ 0 ) - 1 / 2 - d p y × [ exp ( j k 0 p y y ) ] U ( p y , z ) .
u ( y , z ) = ( j λ 0 ) - 1 / 2 - d p y U ( p y , 0 ) exp { j k 0 [ y p y + z p z ( p y ) ] } .
u ( y , z ) = F y p y - 1 exp [ j k 0 z p z ( p y ) ] U ( p y , 0 )
U ( p y , z ) = { exp [ j k 0 z p z ( p y ) ] } U ( p y , 0 ) .
u ( y , z ) = u ˜ ( y , z ) exp j k 0 ( p ¯ y y + p ¯ z z ) ,
p ¯ = p ¯ y y ^ + p ¯ z ( p y ) z ^
U ˜ ( p y , z ) = u ˜ p y F y ( y , z ) = exp [ - j k 0 p ¯ z z ] U ( p y + p ¯ y , z ) .
U ˜ ( p y , z ) = ( exp { j k 0 z [ p y + p ¯ y ) - p ¯ z ] } ) × U ˜ ( p y , 0 ) = H ˜ ( p y ) U ˜ ( p y , 0 ) ,
H ˜ ( p y ) = exp { j k 0 z [ p z ( p y + p ¯ y ) - p ¯ z ] }
u ˜ ( y , z ) = F y p y - 1 H ˜ ( p y ) F p y y u ˜ ( y , 0 )
u ˜ ( y , z ) = T ˜ u ˜ ( y , 0 ) ,
T ˜ = F y p y - 1 H ˜ ( p y ) F p y y
H ˜ ( p y ) = exp { j k 0 z [ - α ¯ p y - ½ β ¯ p y 2 + a ( p y ) ] } ,
α ¯ = - p z p y | p y = p ¯ y ,
β ¯ = - 2 p z p y 2 | p y = p ¯ y .
H ˜ ( p y ) = G p y [ - z α ¯ ] Q p y [ - z β ¯ ] exp [ j k 0 z a ( p y ) ] .
T ˜ = F - 1 G [ - z α ¯ ] Q [ - z β ¯ ] exp [ j k 0 z a ( p y ) ] F ,
T ˜ = S [ z α ¯ ] F - 1 Q [ - z β ¯ ] F F - 1 exp [ j k 0 z a ( p y ) ] F .
A = F - 1 exp [ j k 0 z a ( p y ) ] F
R [ z β ¯ ] = F - 1 Q [ - z β ¯ ] F
T ˜ = S [ - z α ¯ ] R [ - z β ¯ ] A ,
u ˜ a ( y ) = A u ˜ ( y , 0 ) = F - 1 exp [ j k 0 z a ( p y ) ] F u ˜ ( y , 0 ) = F - 1 exp [ j k 0 z a ( p y ) ] U ˜ ( p y , 0 ) .
u ˜ a ( y ) = F - 1 U ˜ ( p y , 0 ) = u ˜ ( y , 0 ) .
T ˜ = S [ z α ¯ ] R [ z β ¯ ] .
tan θ b = α ¯ .
u ˜ ( y , z ) = ( j λ 0 β ¯ z ) - 1 / 2 - d y × exp [ - j k 0 2 β ¯ z ( y - α ¯ z - y ) 2 u ˜ a ( y ) ] ,
u ˜ a ( y ) = u ˜ ( y , 0 ) ,
B = μ 0 H
D = 0 E .
n 2 ( x p x 2 + y p y 2 + z p z 2 ) - [ p x 2 x ( y + z ) + p y 2 y ( x + z ) + p z 2 z ( x + y ) ] + x y z = 0 ,
n 2 = p x 2 + p y 2 + p z 2 ,
p x = 0 ,
( n 2 - x ) ( y p y 2 + z p z 2 - y z ) = 0.
p y 2 + p z 2 = x
p y 2 z + p z 2 y = 1.
p z = y ( 1 - p y 2 z ) 1 / 2 .
p z = n ( θ ¯ ) cos θ ¯ , p y = n ( θ ¯ ) sin θ ¯ .
n ( θ ¯ ) = ( cos 2 θ ¯ y + sin 2 θ ¯ z ) - 1 / 2 .
u T M ( y , z ) = D y ( y , z ) .
α ¯ = y z p y p z
β ¯ = y z p z [ 1 + ( p y p z ) 2 y z ] .
tan θ b = α ¯ = y z tan θ ¯
β ¯ = y / z n ( θ ¯ ) cos θ ¯ ( 1 + y ζ tan 2 θ ¯ ) ,
u ( y ) = cos θ ¯ exp ( j k 0 n y sin θ ¯ ) u ¯ ( y cos θ ¯ ) .
u ˜ ( y ) = s ¯ - 1 / 2 V [ s ¯ ] u ¯ ( y ) ,
s ¯ = cos θ ¯ .
u ¯ ( y ) = s ¯ 1 / 2 V [ 1 / s ¯ ] u ˜ ( y ) .
u ˜ 2 ( y 2 ) = ρ 1 / 2 V [ ρ ] u ˜ 1 ( y 1 ) ] = ρ u ˜ 1 ( ρ y 1 ) ,
ρ = cos θ 2 cos θ 1 .
x = y = n 2
tan θ b = α ¯ = tan θ ¯ ,
β ¯ = 1 n cos 3 θ ¯ = 1 n s ¯ - 3 .
u ˜ ( y , z ) = S [ z tan θ ¯ ] R [ z n s ¯ - 3 ] u ˜ ( y , 0 ) .
u ˜ ( y 3 , 0 ) = R [ z n s ¯ - 3 ] u ˜ ( y , 0 ) = R [ L n s ¯ - 2 ] u ˜ ( y , 0 ) ,
L = z / s ¯
u ¯ ( y ¯ , L ) = R [ L n ] u ¯ ( y ¯ , 0 ) = ( j λ 0 L / n ) - 1 / 2 - d y × exp [ j k 0 n 2 L ( y ¯ - y ) 2 ] u ¯ ( y , 0 ) .
R [ d ] V [ m ] = V [ m ] R [ m 2 d ] .
u ˜ ( y 3 , L ) = s - 1 / 2 V [ s ¯ ] u ( y ¯ , L ) = s - 1 / 2 V [ s ¯ ] R [ L n ] u ( y ¯ , 0 ) = s - 1 / 2 V [ s ¯ ] R [ L n ] s ¯ - 1 / 2 V [ 1 s ¯ ] u ˜ ( y , 0 ) = V [ s ¯ ] V [ 1 s ¯ ] R [ L n s - 2 ] u ˜ ( y , 0 ) = R [ L n s - 2 ] u ˜ ( y , 0 ) .
θ b = γ .
θ ¯ = arctan ( z y tan γ ) .
s i = sin θ i ,
s ¯ = sin ( γ - θ ¯ ) .
G [ s i ] Q [ - 1 / f ] t ( y ) = G [ n ( θ ¯ ) s ] Q [ - n ( θ ¯ ) / r ] t ( y ) ,
s i = n ( θ ¯ ) s ¯ ,
r = f / n ( θ ¯ ) .
u ˜ t = Q [ - 1 / f ] t ( y ) .
ρ = cos θ ¯ cos ( γ - θ ¯ ) .
u ˜ ( y , 0 ) = ρ 1 / 2 V [ ρ ] Q [ - 1 / f ] t ( y ) = ρ 1 / 2 Q [ - ρ 2 / f ] V [ ρ ] t ( y ) .
z = d cos γ
α ¯ z = z tan γ = d sin γ
u ˜ ( y , d cos γ ) = S [ d sin γ ] R [ β ¯ d cos γ ] u ˜ ( y , 0 ) .
u ˜ 3 ( y 3 , 0 ) = R [ β ¯ d cos γ ] u ˜ ( y , 0 ) = ρ 1 / 2 R [ β ¯ d cos γ ] × Q [ - ρ 2 / f ] V [ ρ ] t ( y ) ,
R [ l ] Q [ - 1 / l ] = Q [ 1 / l ] V [ 1 / l ] F ,
f / ρ 2 = β ¯ d cos γ .
f = ρ 2 β ¯ d cos γ ,
u ˜ 3 ( y 3 , 0 ) = ρ 1 / 2 Q [ 1 / l ] V [ 1 / l ] FV [ ρ ] t ( y ) .
u ˜ ( y , d ) = ρ - 1 / 2 V [ 1 / ρ ] u ˜ 3 ( y 3 , 0 ) = Q [ 1 / l ρ 2 ] V [ 1 / l ρ 2 ] F t ( y ) .
f = d / n .
Q [ c ] u ( x ) = exp [ j k 0 2 c x 2 ] u ( x ) ,
V [ a ] u ( x ) = a 1 / 2 u ( a x ) ,
F u ( x ) = ( j λ 0 ) - 1 / 2 - d x exp [ - j k 0 p x ] u ( x ) ,
S [ m ] u ( x ) = u ( x - m ) ,
G [ p ] u ( x ) = exp [ j k 0 p x ] u ( x ) ,
R [ d ] u ( x 1 ) = ( j λ 0 d ) - 1 / 2 - d x 1 exp [ j k 0 2 d ( x 2 - x 1 ) 2 ] u ( x 1 ) .
R [ d ] = Q [ 1 / d ] V [ 1 / d ] FQ [ 1 / d ]
R [ d ] = F - 1 Q [ - d ] F .
R [ d ] Q [ - 1 / d ] = Q [ 1 / d ] V [ 1 / d ] FQ [ 1 / d ] × Q [ - 1 / d ] = Q [ 1 / d ] V [ 1 / d ] F .
u 2 ( r ) = T r q u 1 ( q ) .
Q y [ c ] G x [ p ] = exp [ j k 0 ( p x + ½ c y 2 ) ] .
M u ( x ) = M ( x ) u ( x ) ,

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