Abstract

Fresnel diffraction is described by replacing the Fresnel-Kirchhoff integral, the lens transfer factor, and other operations by operators. The resulting operator algebra leads to the description of Fourier optics in a simple and compact way, bypassing the cumbersome integral calculus. Aberration effects and Gaussian beam illumination are also treated as a simple extension of the present theory.

© 1980 Optical Society of America

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References

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  1. A. Vander Lugt, “Operational notation for the analysis and synthesis of optical data processing systems,” Proc. IEEE 54, 1055–1063 (1966).
    [Crossref]
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  3. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  4. M. V. Klein, Optics (Wiley, New York, 1970).
  5. J. Knopp and M. F. Becker, “Virtual Fourier transform as an analytical tool in Fourier optics,” Appl. Opt. 17, 1669–1670 (1978).
    [Crossref] [PubMed]
  6. J. Shamir and G. Krieger, “High resolution detection of defects by one-dimensional spatial filtering,” Appl. Phys. 18, 363–373 (1979).
    [Crossref]
  7. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [Crossref] [PubMed]
  8. W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

1979 (1)

J. Shamir and G. Krieger, “High resolution detection of defects by one-dimensional spatial filtering,” Appl. Phys. 18, 363–373 (1979).
[Crossref]

1978 (1)

1966 (2)

A. Vander Lugt, “Operational notation for the analysis and synthesis of optical data processing systems,” Proc. IEEE 54, 1055–1063 (1966).
[Crossref]

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
[Crossref] [PubMed]

Becker, M. F.

Cathey, W. T.

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

Goodman, J. W.

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

Klein, M. V.

M. V. Klein, Optics (Wiley, New York, 1970).

Knopp, J.

Kogelnik, H.

Krieger, G.

J. Shamir and G. Krieger, “High resolution detection of defects by one-dimensional spatial filtering,” Appl. Phys. 18, 363–373 (1979).
[Crossref]

Li, T.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Shamir, J.

J. Shamir and G. Krieger, “High resolution detection of defects by one-dimensional spatial filtering,” Appl. Phys. 18, 363–373 (1979).
[Crossref]

Vander Lugt, A.

A. Vander Lugt, “Operational notation for the analysis and synthesis of optical data processing systems,” Proc. IEEE 54, 1055–1063 (1966).
[Crossref]

Appl. Opt. (2)

Appl. Phys. (1)

J. Shamir and G. Krieger, “High resolution detection of defects by one-dimensional spatial filtering,” Appl. Phys. 18, 363–373 (1979).
[Crossref]

Proc. IEEE (1)

A. Vander Lugt, “Operational notation for the analysis and synthesis of optical data processing systems,” Proc. IEEE 54, 1055–1063 (1966).
[Crossref]

Other (4)

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

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

M. V. Klein, Optics (Wiley, New York, 1970).

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

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Figures (7)

FIG. 1
FIG. 1

Definition of coordinates. The vector ρ defines transversal position.

FIG. 2
FIG. 2

Definition of distances involved in a transformation by a lens of focal length f.

FIG. 3
FIG. 3

Lensless FT of a transfer function t(ρ). (a) virtual FT, (b) real FT.

FIG. 4
FIG. 4

Systems diagram to illustrate that the multiplicative phase distortions introduced by the multipliers are the same for real and ideal systems.

FIG. 5
FIG. 5

Lens waveguide representation of a Fabry-Perot resonator.

FIG. 6
FIG. 6

“Lensless” Gaussian beam FT.

FIG. 7
FIG. 7

Lens transformations with Gaussian beam.

Equations (121)

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u 2 ( ρ 2 ) = exp j k z 21 j λ z 21 exp ( j k ( ρ 2 ρ 1 ) 2 2 z 21 ) u 1 ( ρ 1 ) d ρ 1 ,
ρ = x x ˆ + y y ˆ
z 21 = z 2 z 1
u 2 ( ρ 2 ) = exp ( j k z 21 ) j λ z 21 exp ( j k ρ 2 2 2 z 21 ) u 1 ( ρ 1 ) exp ( j k ρ 1 2 2 z 21 ) × exp ( j k ρ 2 z 21 · ρ 1 ) d ρ 1 = exp ( j k z 21 ) j λ z 21 exp ( j k ρ 2 2 2 z 21 ) F { u 1 ( ρ 1 ) exp ( j k ρ 1 2 2 z 21 ) } | ν = ρ 2 / λ z 21 ,
F [ f ( ρ ) ] f ( ρ ) exp ( j 2 π ν · ρ ) d ρ
V [ s ] f ( ρ ) = f ( s ρ )
F f ( ρ ) = F [ f ( ρ ) ]
Q [ a ] f ( ρ ) = exp [ j ( k / 2 ) a ρ 2 ] f ( ρ ) .
R [ z 21 ] = exp ( j k z 21 ) j λ z 21 Q [ 1 z 21 ] V [ 1 λ z 21 ] F Q [ 1 z 21 ] .
u 2 = R [ z 21 ] u 1 .
u 2 ( ρ ) = exp ( j k z 21 ) j λ z 21 exp ( j k ρ 2 2 z 21 ) u 1 ( ρ ) ,
F [ u 2 ( ρ ) ] = exp ( j k z 21 ) j λ z 21 F ( exp j k ρ 2 2 z 21 ) F [ u 1 ( ρ ) ] ,
F [ u 2 ( ρ ) ] = exp ( j k z 21 ) exp ( k 2 λ 2 z 21 ν 2 ) F [ u 1 ( ρ ) ]
F u 2 ( ρ ) = exp ( j k z 21 ) Q [ λ 2 z 21 ] F u 1 ( ρ ) .
u 2 ( ρ ) = F 1 F u 2 ( ρ ) = exp ( j k z 21 ) F 1 Q [ λ 2 z 21 ] F u 1 ( ρ ) .
R [ z ] = exp ( j k z ) F 1 Q [ λ 2 z ] F .
u 2 = L [ f ] u 1 ,
L [ f ] = Q [ 1 / f ] = Q [ 1 / | f | ] ,
Q [ a 1 ] Q [ a 2 ] = Q [ a 1 + a 2 ]
Q 1 [ a ] = Q [ a ]
Q [ 0 ] = 1
V [ s ] f = ( V [ s ] f ) V [ s ]
V [ s 1 ] V [ s 2 ] = V [ s 1 s 2 ] = V [ s 2 ] V [ s 1 ] ,
V 1 [ s ] = V [ 1 / s ] ,
V [ s ] Q [ a ] = Q [ s 2 a ] V [ s ] ,
F V [ s ] = ( 1 / s 2 ) V [ 1 / s ] F
F 1 = V [ 1 ] F = F V [ 1 ] , F = V [ 1 ] F 1 = F 1 V [ 1 ] ,
F F = V [ 1 ] ,
( F f g ) = ( F f ) F g
( F Q [ a ] f ) = { ( F Q [ a ] ) ( F f ) } = { j λ a Q [ λ 2 a ] ( F f ) } ,
( F Q [ a ] A ) = ( j λ a Q [ λ 2 a ] A ) .
R [ z ] = exp ( j k z ) F 1 Q [ λ 2 z ] F .
R [ z ] R [ z ] = exp ( j k z ) F 1 Q [ λ 2 z ] F × exp ( j k z ) F 1 Q [ λ 2 z ] F .
R [ z ] R ( z ) = F 1 Q [ λ 2 z ] Q [ λ 2 z ] F ,
R [ z ] R [ z ] = F 1 F = 1.
R 1 [ z ] = R [ z ] .
R 1 [ z 21 ] = R [ z 21 ] = R [ z 12 ]
R [ z j k ] = exp ( j k z j k ) F 1 Q [ λ 2 z j k ] F ,
z j k = z j i + z i k
Q [ λ 2 z j k ] = Q [ λ 2 z j i ] Q [ λ 2 z i k ] .
R [ z j k ] = exp [ j k ( z j i + z i k ) ] F 1 Q [ λ 2 z j i ] Q [ λ 2 z i k ] F .
R [ z j k ] = exp [ j k ( z j i + z i k ) ] F 1 Q [ λ 2 z j i ] F F 1 Q [ λ 2 z i k ] F
R [ z j k ] = R [ z j i ] R [ z i k ] .
u o = T u i .
T = R [ d 2 ] L [ f ] R [ d 1 ]
T = exp j k d 2 j λ d 2 Q [ 1 d 2 ] V [ 1 λ d 2 ] F Q [ 1 d 2 ] Q [ 1 f ] × exp j k d 1 j λ d 1 Q [ 1 d 1 ] V [ 1 λ d 1 ] F Q [ 1 d 1 ] = exp j k ( d 1 + d 2 ) λ 2 d 1 d 2 Q [ 1 d 2 ] V [ 1 λ d 2 ] × F Q [ 1 d 1 + 1 d 2 1 f ] V [ 1 λ d 1 ] F Q [ 1 d 1 ]
1 / d 1 + 1 / d 2 1 / f = 0 ,
T = exp j k ( d 1 + d 2 ) λ 2 d 1 d 2 Q [ 1 d 2 ] V [ 1 λ d 2 ] F λ 2 d 1 2 F V [ λ d 1 ] Q [ 1 d 1 ] .
T = exp j k ( d 1 + d 2 ) ( d 1 d 2 ) Q [ 1 d 2 ] V [ 1 λ d 2 ] × V [ 1 ] V [ λ d 1 ] Q [ 1 d 1 ] = exp j k ( d 1 + d 2 ) ( d 1 d 2 ) Q [ 1 d 2 ] V [ d 1 d 2 ] Q [ 1 d 1 ] ,
T = exp j k ( d 1 + d 2 ) ( d 1 d 2 ) Q [ 1 d 2 + ( d 1 d 2 ) 2 1 d 1 ] V [ d 1 d 2 ]
T = exp j k ( d 1 + d 2 ) ( d 1 d 2 ) Q [ d 2 + d 1 d 2 2 ] V [ d 1 d 2 ] .
M = d 2 / d 1 .
d 1 / d 2 = 1 / | M | .
T = exp j k ( d 1 + d 2 ) j λ d 2 Q [ 1 d 2 ] V [ 1 λ d 2 ] F Q [ 1 d 2 ] Q [ 1 f ] × F 1 Q [ λ 2 d 1 ] F .
T = exp j k ( d 1 + f ) j λ f Q [ 1 f ] V [ 1 λ f ] Q [ λ 2 d 1 ] F .
T = exp j k ( d 1 + f ) j λ f Q [ 1 f + ( 1 λ f ) 2 ( λ 2 d 1 ) ] V [ 1 λ f ] F = exp j k ( d 1 + f ) j λ f Q [ 1 f ( 1 d 1 f ) ] V [ 1 λ f ] F .
T = R [ f ] L [ f ] R [ d 1 ] = exp j k ( d 1 f ) j λ f Q [ 1 f ] V [ 1 λ f ] F Q [ 1 f ] × Q [ 1 f ] F 1 Q [ λ 2 d 1 ] F .
T = exp j k ( d 1 f ) / j λ f Q [ ( 1 / f ) ( 1 + d 1 / f ) ] V [ 1 / λ f ] F .
u ( ρ ) = R [ d ] Q [ 1 / d ] t ( ρ ) ,
u ( ρ ) = exp ( j k d ) j λ d Q [ 1 d ] V [ 1 λ d ] F Q [ 1 d ] Q [ 1 d ] t ( ρ ) = exp ( j k d ) j λ d Q [ 1 d ] V [ 1 λ d ] F t ( ρ ) .
u ( ρ ) = R [ d ] Q [ 1 / d ] t ( ρ ) = exp j k d j λ d Q [ 1 d ] V [ 1 λ d ] F Q [ 1 d ] Q [ 1 d ] t ( ρ ) = exp j k d / j λ d Q [ 1 / d ] V [ 1 / λ d ] F t ( ρ ) .
L = P ( ρ ) Q [ 1 / f ] .
T = exp j k ( d 1 + d 2 ) j λ d 2 Q [ 1 d 2 ] V [ 1 λ d 2 ] F Q [ 1 d 2 ] Q [ 1 f ] × P ( ρ ) F 1 Q [ λ 2 d 1 ] F .
T = exp j k ( d 1 + f ) j λ f Q [ 1 f ] V [ 1 λ f ] F P ( ρ ) F 1 Q [ λ 2 d 1 ] F .
T = exp j k ( d 1 + f ) j λ f Q [ 1 f ] F { V [ λ f ] P ( ρ ) } × F 1 ( V [ 1 λ f ] Q [ λ 2 d 1 ] ) V [ 1 λ f ] F .
T = exp j k ( d 1 + f ) j λ f Q [ 1 f ] [ { F ( V [ λ f ] P ( ρ ) ) } F F 1 Q [ d 1 f 2 ] V [ 1 λ f ] F .
T = exp j k ( d 1 + f ) j λ f Q [ 1 f ] [ { F ( V [ λ f ] P ( ρ ) ) } Q [ d 1 f 2 ] V [ 1 λ f ] F .
T = R [ d 2 ] P ( ρ ) Q [ 1 / f ] R [ d 1 ] .
T = exp j k ( d 1 + d 2 ) λ 2 d 1 d 2 Q [ 1 d 2 ] V [ 1 λ d 2 ] F Q [ 1 d 1 + 1 d 2 1 f ] × P ( ρ ) V [ 1 λ d 1 ] F Q [ 1 d 1 ]
T = exp j k ( d 1 + d 2 ) λ 2 d 1 d 2 Q [ 1 d 2 ] V [ 1 λ d 2 ] F P ( ρ ) V [ 1 λ d 1 ] F Q [ 1 d 1 ] .
T = [ exp j k ( d 1 + d 2 ) ] ( d 2 d 1 ) Q [ 1 d 2 ] × F ( V [ λ d 2 ] P ) V [ d 2 d 1 ] F Q [ 1 d 1 ] = [ exp j k ( d 1 + d 2 ) ] ( d 1 d 2 ) Q [ 1 d 2 ] × F ( V [ λ d 2 ] P ) F V [ d 1 d 2 ] Q [ 1 d 1 ] .
T = [ exp j k ( d 1 + d 2 ) ] d 1 d 2 Q [ 1 d 2 ] [ { F ( V [ λ ( d 2 ) ] P } F F V [ d 1 d 2 ] Q [ 1 d 1 ]
T = [ exp j k ( d 1 + d 2 ) ] d 1 d 2 Q [ d 2 d 2 2 ] [ ( F V [ λ ( d 2 ) ] P ) Q [ d 1 d 2 2 ] V [ d 1 d 2 ] .
u = u 0 exp j [ p ( z ) + k ρ 2 2 q ( z ) ] ,
1 / q = 1 / R + j ( λ / π w 2 ) ,
u = A exp ( j k ρ 2 / z R ) exp ( ρ 2 / w 2 ) ,
u = Q [ 1 / R ] Q [ j ( λ / π w 2 ) ] A = Q [ 1 / q ] A .
u 1 = Q [ 1 / q 1 ] A ,
u 2 = R [ z 21 ] u 1 .
u 2 = exp j k z 21 j λ z 21 Q [ 1 z 21 ] V [ 1 λ z 21 ] F Q [ 1 z 21 ] Q [ 1 q 1 ] A
u 2 = exp j k z 21 j λ z 21 Q [ 1 z 21 ] V [ 1 λ z 21 ] F Q [ 1 z 21 + 1 q 1 ] A .
u 2 = exp j k z 21 ( 1 / z 21 + 1 / q 1 ) z 21 Q [ 1 z 21 ] V [ 1 λ z 21 ] × Q [ λ 2 / ( 1 z 21 + 1 q 1 ) ] A ,
u 2 = exp j k z 21 ( 1 / z 21 + 1 / q 1 ) z 21 Q [ 1 z 21 + ( 1 λ z 21 ) 2 λ 2 / ( 1 z 21 + 1 q 1 ) ] A
u 2 = exp j k z 21 1 + z 21 / q 1 Q [ 1 z 21 + q 1 ] A .
q 2 = q 1 + z 21 .
1 / q 1 = j ( λ / π w 1 2 ) .
1 q 1 + z 21 = 1 R 2 + j λ π w 1 2 .
w 2 = w 1 [ 1 + ( z 21 λ / π w 1 2 ) 2 ] 1 / 2 ,
R 2 = z 21 [ 1 + ( π w 1 2 / λ z 21 ) 2 ] ,
1 1 + z 21 / q 1 = 1 1 + j ( z 21 λ / π w 1 2 ) = 1 1 + ( z 21 λ , π w 1 2 ) 2 e j ϕ = w 1 w 2 e j ϕ ,
ϕ = t g 1 ( z 21 λ / π w 1 2 ) .
u 2 = w 1 w 2 exp j ( k z 21 ϕ ) exp ρ 2 ( j k 2 R 2 1 w 2 2 ) .
u 2 = L [ f ] Q [ 1 q 1 ] A = Q [ 1 f ] Q [ 1 q 1 ] A = Q [ 1 q 1 1 f ] A = Q [ 1 q 2 ] A ,
1 / q 2 = 1 / q 1 1 / f
T = L [ f 2 ] R [ d ] L [ f 1 ] R [ d ]
u 1 = Q [ 1 q 1 ] A t ( ρ ) = Q [ 1 R 1 ] Q [ j λ π w 1 2 ] A t ( ρ ) .
u 2 = T u 1 .
u 2 = R [ d ] u 1 ,
u 2 = exp j k d j λ d Q [ 1 d ] V [ 1 λ d ] F Q [ 1 d ] × Q [ 1 R 1 ] Q [ j λ π w 1 2 ] A t ( ρ ) .
u 2 = exp jkR 1 j λ R 12 Q [ 1 R 1 ] V [ 1 λ R 1 ] F Q [ j λ π w 1 2 ] A t ( ρ )
u 2 = A π w 1 2 j λ R 1 exp ( j k R 1 ) Q [ 1 R 1 ] V [ 1 λ R 1 ] × { Q [ j λ π w 1 2 ] F t ( ρ ) } .
u 2 = R [ d 2 ] L [ f ] R [ d 1 ] u 1 .
u 1 = Q [ 1 d 1 ] Q [ j λ π w 1 2 ] A t ( ρ ) .
R [ d 1 ] = R [ d 1 + d 1 ] R [ d 1 ] .
u 2 = R [ d 2 ] L [ f ] R [ d 1 + d 1 ] R [ d 1 ] Q [ 1 d 1 ] Q [ j λ π w 1 2 ] A t ( ρ ) ,
u 2 = T 2 T 1 Q [ j λ π w 1 2 ] A t ( ρ ) ,
T 1 = R [ d 1 ] Q [ 1 d 1 ] = exp ( j k d 1 ) j λ d 1 × Q [ 1 d 1 ] V [ 1 λ d 1 ] F ,
T 2 = R [ d 2 ] L [ f ] R [ d 1 + d 1 ]
T 2 = exp j k ( d 1 + d 1 + d 2 ) ( d 1 + d 1 d 2 ) × Q [ d 1 + d 1 + d 2 d 2 2 ] V [ d 1 + d 1 d 2 ] .
T 2 T 1 = exp j k ( d 1 + d 2 ) ( d 1 + d 1 j λ d 1 d 2 2 ) × Q [ d 1 + d 1 + d 2 d 2 2 + ( d 1 + d 2 d 2 ) 2 ( 1 d 1 ) ] V [ d 1 + d 1 λ d 1 d 2 ] F = [ exp j k ( d 1 + d 1 ) ] ( d 1 + d 1 j λ d 1 d 2 2 ) × Q [ 1 d 2 2 ( d 1 d 2 d 2 2 d 1 ) ] V ( d 1 + d 1 λ d 1 d 2 ) F .
u 2 = [ exp j k ( d 1 + d 2 ) ] ( ( d 1 + d 1 ) A π w 1 2 j λ d 1 d 2 ) × Q [ 1 d 2 2 ( d 1 d 2 d 2 2 d 1 ) ] V [ d 1 + d 1 λ d 1 d 2 ] . × { Q [ j λ π w 1 2 ] F t ( ρ ) } ,
u 2 = R [ d 2 ] L [ f ] R [ d 1 ] Q [ j λ π w 1 2 ] A t ( ρ ) .
u 2 = exp j k ( d 1 + d 2 ) ( d 1 d 2 ) Q ( d 2 + d 1 d 2 2 ) × V [ d 1 d 2 ] Q [ j λ π w 1 2 ] A t ( ρ ) .
Q [ 1 ; a ] f ( x , y ) = exp ( j k 2 a x 2 ) f ( x , y )
Q [ 2 ; a ] f ( x , y ) = exp ( j k 2 a y 2 ) f ( x , y ) .
Q [ a ] = Q [ 1 ; a ] Q [ 2 ; a ] .
L = Q [ i ; 1 / f ]
V [ s ] = V [ 1 ; s ] V [ 2 ; s ]
F = F [ 1 ] F [ 2 ] .
R [ z ] = F 1 Q [ λ 2 z ] F = F [ 1 ] 1 F [ 2 ] 1 Q [ 1 ; λ [ z ] × Q [ 2 ; λ [ z ] F [ 1 ] F [ 2 ] = F [ 1 ] 1 Q [ 1 ; λ [ z ] F [ 1 ] F [ 2 ] 1 × Q [ 2 ; λ [ z ] F [ 2 ] = R [ 1 ; z ] R [ 2 ; z ] .
u 0 = T u i = T [ 1 ] T [ 2 ] u i = T [ 2 ] T [ 1 ] u i .