Abstract

The special affine Fourier transformation (SAFT) is a generalization of the fractional Fourier transformation (FRT) and represents the most general lossless inhomogeneous linear mapping, in phase space, as the integral transformation of a wave function. Here we first summarize the most well-known optical operations on light-wave functions (i.e., the FRT, lens transformation, free-space propagation, and magnification), in a unified way, from the viewpoint of the one-parameter Abelian subgroups of the SAFT. Then we present a new operation, which is the Lorentz-type hyperbolic transformation in phase space and exhibits squeezing. We also show that the SAFT including these five operations can be generated from any two independent operations.

© 1994 Optical Society of America

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References

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  2. A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
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  3. D. Mendlovic, H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
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  6. S. Abe, J. T. Sheridan, J. Phys. A 27, 4179 (1994).
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  7. S. Lang, SL2(R) (Addison-Wesley, Reading, Mass., 1975), Chap. V, p. 83.
  8. E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), Chap. 6, p. 172.
  9. A. R. Mickelson, Physical Optics (Van Nostrand Reinhold, New York, 1992), Chap. 5, p. 203.
  10. J. L. Synge, Relativity: The Special Theory (North-Holland, Amsterdam, 1955), Chap. IV, p. 74.
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1994 (2)

1993 (4)

1987 (1)

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

1980 (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

Abe, S.

S. Abe, J. T. Sheridan, J. Phys. A 27, 4179 (1994).
[CrossRef]

Barshan, B.

Hecht, E.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), Chap. 6, p. 172.

Kerr, F. H.

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

Lang, S.

S. Lang, SL2(R) (Addison-Wesley, Reading, Mass., 1975), Chap. V, p. 83.

Lohmann, A. W.

McBride, A. C.

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

Mendlovic, D.

Mickelson, A. R.

A. R. Mickelson, Physical Optics (Van Nostrand Reinhold, New York, 1992), Chap. 5, p. 203.

Namias, V.

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

Onural, L.

Ozaktas, H. M.

Sheridan, J. T.

S. Abe, J. T. Sheridan, J. Phys. A 27, 4179 (1994).
[CrossRef]

Synge, J. L.

J. L. Synge, Relativity: The Special Theory (North-Holland, Amsterdam, 1955), Chap. IV, p. 74.

Zajac, A.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), Chap. 6, p. 172.

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

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

J. Phys. A (1)

S. Abe, J. T. Sheridan, J. Phys. A 27, 4179 (1994).
[CrossRef]

Opt. Lett. (1)

Other (4)

S. Lang, SL2(R) (Addison-Wesley, Reading, Mass., 1975), Chap. V, p. 83.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), Chap. 6, p. 172.

A. R. Mickelson, Physical Optics (Van Nostrand Reinhold, New York, 1992), Chap. 5, p. 203.

J. L. Synge, Relativity: The Special Theory (North-Holland, Amsterdam, 1955), Chap. IV, p. 74.

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Equations (33)

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[ x k ] [ a b c d ] [ x k ] + [ m n ] ,
a d b c = 1 .
u SAFT ( x ) = 1 2 π | b | exp { i 2 b [ d x 2 + 2 ( b n d m ) x ] } × d x exp { i 2 b [ a x 2 2 ( x m ) x ] } u ( x ) .
g 1 ( θ ) = [ cos θ sin θ sin θ cos θ ] ( rotation ) ,
g 2 ( ξ ) = [ 1 0 ξ 1 ] ( lens transformation ) ,
g 3 ( η ) = [ 1 η 0 1 ] ( free-space propagation ) ,
g 4 ( α ) = [ e α 0 0 e α ] ( magnification ) .
g A ( t 1 ) g A ( t 2 ) = g A ( t 2 ) g A ( t 1 ) = g A ( t 1 + t 2 ) , g A ( t ) = g A 1 ( t ) , g A ( 0 ) = I ( A = 1 , 2 , 3 , 4 ) ,
u FRT ( x ) = 1 2 π | sin θ | exp [ ( i / 2 ) x 2 cot θ + i x m cos e c θ ) ] × d x exp [ ( i / 2 ) x 2 cot θ i ( x cos e c θ + m cot θ + n ) x ] u ( x ) ( the FRT , i . e . , rotation ) ,
u LENS ( x ) = exp [ 1 2 ξ x 2 i ( m ξ n ) x ] u ( x m ) ( lens transformation ) ,
u FSP ( x ) = 1 2 π | η | exp ( i n x ) d x × exp [ i 2 η ( x x + m ) 2 ] u ( x ) ( free - space propagation ) ,
u MGN ( x ) = exp ( α 2 + i n x ) u [ e α ( x m ) ] ( magnification ) ,
lim b 0 1 2 π i b exp [ i ( x 1 x 2 ) 2 2 b ] = δ ( x 1 x 2 ) ,
L 1 = [ 0 1 1 0 ] , L 2 = [ 0 1 1 0 ] , L 3 = [ 1 0 0 1 ] ,
[ L 1 , L 2 ] = 2 L 3 , [ L 2 , L 3 ] = 2 L 1 , [ L 3 , L 1 ] = 2 L 2
{ L 1 , L 1 } = { L 2 , L 2 } = { L 3 , L 3 } = 2 I , { L i , L j } = 0 ( i j ; i , j = 1 , 2 , 3 ) .
L ± = 1 2 ( L 1 L 2 ) ,
[ L + , L ] = L 3 , [ L 3 , L ± ] = ± 2 L ± ,
{ L ± , L } = I , { L 3 , L ± } = 0 .
g 1 ( θ ) = exp ( θ L 2 ) ,
g 2 ( ξ ) = I + ξ L = exp ( ξ L ) ,
g 3 = I + η L + = exp ( η L + ) ,
g 4 ( α ) = exp ( α L 3 ) .
g 5 ( ϕ ) = exp ( ϕ L 1 ) = [ cosh ϕ sinh ϕ sinh ϕ cosh ϕ ] .
u HYP ( x ) = 1 2 π | sinh ϕ | exp [ i 2 x 2 coth ϕ + i ( n m coth ϕ ) x ] d x exp [ i 2 x 2 coth ϕ i ( x m ) x cos e c h ϕ ] u ( x ) .
g = [ a b c d ] = [ s 0 0 s 1 ] [ 1 η 0 1 ] [ cos θ sin θ sin θ cos θ ] = sgn ( s ) g 4 ( ln | s | ) g 3 ( η ) g 1 ( θ ) ,
a = s ( cos θ η sin θ ) , b = s ( sin θ + η cos θ ) , c = sin θ / s , d = cos θ / s .
g 3 ( η ) g 2 ( ξ ) g 3 ( η ) = [ 1 + ξ η η ( 2 + ξ η ) ξ 1 + ξ η ] .
g 3 ( η ) g 2 ( ξ ) g 3 ( η ) = g 1 ( θ ) [ ξ = sin θ , η = sin θ / ( 1 + cos θ ) ] ,
g 3 ( η ) g 2 ( ξ ) g 3 ( η ) = g 5 ( ϕ ) [ ξ = sinh ϕ , η = sinh ϕ / ( 1 + cosh ϕ ) ] .
g 4 ( α ) = g 1 ( π / 4 ) g 5 ( ϕ ) g 1 1 ( π / 4 ) ( α = ϕ ) .
g 4 ( α / 2 ) g 1 ( θ ) g 4 1 ( α / 2 ) = [ cos θ e α sin θ e α sin θ cos θ ] ,
g 4 ( α / 2 ) g 5 ( ϕ ) g 4 1 ( α / 2 ) = [ cosh ϕ e α sinh ϕ e α sinh ϕ cosh ϕ ] ,

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