Abstract

A three-lens achromatic Fourier transform system is analyzed in the context of paraxial Fresnel diffraction theory. From the analysis a general solution for the required wavelength dependence of the various lenses is found. A particular arrangement of the general system is then considered. Using first-order lens design principles, it is shown that each dispersive lens can be fabricated using a holographic zone lens and glass element cascade. The paraxial chromatic aberrations of the resulting system are calculated. It is found that this system design yields an achromatic transformation that is well corrected (paraxially) over the entire visible spectrum.

© 1981 Optical Society of America

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References

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  1. E. N. Leith, J. A. Roth, Appl. Opt. 18, 2803 (1979).
    [CrossRef] [PubMed]
  2. P. Chavel, J. Opt. Soc. Am. 70, 935 (1980).
    [CrossRef]
  3. L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 94.
  4. J. M. Richardson, “Device for producing identifiable sine and cosine (Fourier) transforms of input signals by means of noncoherent optics,” U.S. Patent3,669,528 (13June1972).
  5. G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977), Chap. 5.
  6. G. L. Rogers, Appl. Opt. 18, 3152 (1979).
    [CrossRef] [PubMed]
  7. R. H. Katyl, Appl. Opt. 11, 1255 (1972).
    [CrossRef] [PubMed]
  8. C. G. Wynne, Opt. Commun. 28, 21 (1979).
    [CrossRef]
  9. G. M. Morris, N. George, Opt. Lett. 5, 446 (1980).
    [CrossRef] [PubMed]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.
  11. R. Kingslake, Lens Design Fundamentals (Academic, New York; 1978), p. 201.
  12. Schott Optical Glass, Inc., 400 York Ave., Duryea, Penn. 18642.
  13. This transform design is based on Eq. (6) and Eqs. (10)–(15) of Ref. 7. The following parameters were used to attain the lens powers in Eq. (34): N = −1; M0 = −1, f0 = a = S = 0.5 m, and z = −a.

1980 (2)

1979 (3)

1972 (1)

Chavel, P.

George, N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.

Katyl, R. H.

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, New York; 1978), p. 201.

Leith, E. N.

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 94.

Morris, G. M.

Richardson, J. M.

J. M. Richardson, “Device for producing identifiable sine and cosine (Fourier) transforms of input signals by means of noncoherent optics,” U.S. Patent3,669,528 (13June1972).

Rogers, G. L.

G. L. Rogers, Appl. Opt. 18, 3152 (1979).
[CrossRef] [PubMed]

G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977), Chap. 5.

Roth, J. A.

Wynne, C. G.

C. G. Wynne, Opt. Commun. 28, 21 (1979).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

C. G. Wynne, Opt. Commun. 28, 21 (1979).
[CrossRef]

Opt. Lett. (1)

Other (7)

L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 94.

J. M. Richardson, “Device for producing identifiable sine and cosine (Fourier) transforms of input signals by means of noncoherent optics,” U.S. Patent3,669,528 (13June1972).

G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977), Chap. 5.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.

R. Kingslake, Lens Design Fundamentals (Academic, New York; 1978), p. 201.

Schott Optical Glass, Inc., 400 York Ave., Duryea, Penn. 18642.

This transform design is based on Eq. (6) and Eqs. (10)–(15) of Ref. 7. The following parameters were used to attain the lens powers in Eq. (34): N = −1; M0 = −1, f0 = a = S = 0.5 m, and z = −a.

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Figures (7)

Fig. 1
Fig. 1

General three-lens optical system. Fi(λ) denotes the focal length of the ith lens (i = 1,2,3). Input is located in plane I. Output plane II is located directly behind lens F3(λ).

Fig. 2
Fig. 2

Theoretical curves for the lens power vs wavelength for (a) the first and third lenses [Eqs. (18) and (20)] and (b) the second lens [Eq. (19)]. With reference to Fig. 1, the distance between elements is A = 0, B = C = 0.5 m.

Fig. 3
Fig. 3

Lens power vs wavelength for the holographic zone lens 1/FH2(λ) [Eq. (27)] and the glass element 1/FL2(λ) [Eq. (28)] with the parameter β = 1.80. Note that 1/FH2(λ) + 1/FL2(λ) equals the required lens power 1/F2(λ) given in Eq. (19).

Fig. 4
Fig. 4

Lens power vs wavelength for the ideal lens in Eq. (19) (solid line) and for a thin-lens cascade consisting of a holographic zone lens [Eq. (27)] and a glass doublet [Eq. (33)] (dashed line). Various parameters are as follows: glass A = BK-7; glass B = SF-57; F = 0.5 m; λ0 = 514.5 nm; λ1 = 0.75λ0; λ2 = 1.25λ0; and β = 1.80.

Fig. 5
Fig. 5

Longitudinal chromatic aberration of (I) the present system design [Eqs. (35)(37)], and (II) a system based on Katyl's design requirements [Eq. (34)].

Fig. 6
Fig. 6

Chromatic variation of transform size obtained with (I) the present system design [Eqs. (35)(37)], (II) Katyl's system [Eq. (34)], and (III) a single achromatic lens. λ0 = 514.5 nm.

Fig. 7
Fig. 7

Phase curvature of the transform-plane field vs wavelength for the present system design [Eqs. (35)(37)].

Equations (64)

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G ( f x , f y ; λ ) = [ i / ( F λ 0 ) ] dxdyg ( x , y ) × exp [ i 2 π ( f x x + f y y ) ] ,
t i ( x , y ) = exp { i π ( x 2 + y 2 ) / [ λ F i ( λ ) ] } .
E I I ( u , υ ; λ ) = P ( λ ) exp { i π ( u 2 + υ 2 ) / [ λ R ( λ ) ] } dxdyg ( x , y ) × exp { i π ( x 2 + y 2 ) / [ λ S ( λ ) ] } × exp { i 2 π ( u x + υ y ) / [ λ T ( λ ) ] } ,
P ( λ ) = i λ A B C α 1 ( λ ) [ α 2 ( λ ) 1 B 2 α 1 ( λ ) ] ,
1 R ( λ ) = 1 C 1 F 3 ( λ ) 1 C 2 [ α 2 ( λ ) 1 B 2 α 1 ( λ ) ] ,
1 S ( λ ) = 1 A 1 A 2 α 1 ( λ ) 1 [ A B α 1 ( λ ) ] 2 [ α 2 ( λ ) 1 B 2 α 1 ( λ ) ] ,
1 T ( λ ) = 1 A B C α 1 ( λ ) [ α 2 ( λ ) 1 B 2 α 1 ( λ ) ] .
α 1 ( λ ) = 1 A + 1 B 1 F 1 ( λ ) ,
α 2 ( λ ) = 1 B + 1 C 1 F 2 ( λ ) .
1 R ( λ ) = 0 ,
1 S ( λ ) = 0 ,
1 T ( λ ) = λ F λ 0 ,
α 2 ( λ ) = ( F λ 0 A B C λ + 1 B 2 ) 1 α 1 ( λ ) .
α 1 ( λ ) = 1 A + C λ B F λ 0 .
α 2 ( λ ) = F λ 0 B C λ .
1 F 1 ( λ ) = 1 B C λ B F λ 0 .
1 F 2 ( λ ) = 1 B + 1 C F λ 0 B C λ .
1 F 3 ( λ ) = 1 C B λ C F λ 0 + A λ 2 ( F λ 0 ) 2 .
A = 0 ;
B = C = F = 0.5 m ;
λ 0 = 514.5 nm .
1 F 1 ( λ ) = 1 F λ F λ 0 ,
1 F 2 ( λ ) = 2 F λ 0 F λ ,
1 F 3 ( λ ) = 1 F 1 ( λ ) ,
E 1 ( x , y ; λ ) = exp ( i 2 π x sin θ 0 / λ ) .
E I I ( u , υ ; λ ) = i F λ 0 δ ( υ ) δ [ u F sin θ 0 ( λ 0 / λ ) ] .
lateral color = F λ 0 sin θ 0 ( 1 λ F 1 λ C ) .
g ( x , y ) = ½ [ 1 + cos ( 2 π x sin θ 0 / λ 0 ) ] .
E II ( u , υ ; λ ) = ( i F λ 0 / 2 ) δ ( υ ) [ δ ( u ) + ½ δ ( u F sin θ 0 ) + ½ δ ( u + sin θ 0 ) .
T H 1 ( x 1 , y 1 ) = exp [ + i π ( x 1 2 + y 1 2 ) / ( λ 0 F ) ] .
T H 2 ( x 2 , y 2 ) = exp [ i π ( x 2 2 + y 2 2 ) / ( λ 0 β F ) ] .
1 F H 2 ( λ ) = λ β F λ 0 .
1 F L 2 ( λ ) = 1 F ( 2 λ 0 λ λ β λ 0 ) .
1 / F D ( λ ) = [ n A ( λ ) 1 ] c A + [ n B ( λ ) 1 ] c B ,
1 / F L 2 ( λ 1 ) = [ n A ( λ 1 ) 1 ] c A + [ n B ( λ 1 ) 1 ] c B ,
1 / F L 2 ( λ 2 ) = [ n A ( λ 2 ) 1 ] c A + [ n B ( λ 2 ) 1 ] c B .
c A = [ n B ( λ 1 ) 1 ] / F L 2 ( λ 2 ) [ n B ( λ 2 ) 1 ] / F L 2 ( λ 1 ) [ n A ( λ 2 ) 1 ] [ n B ( λ 1 ) 1 ] [ n A ( λ 1 ) 1 ] [ n B ( λ 2 ) 1 ] ,
c B = 1 [ n B ( λ 1 ) 1 ] { 1 F L 2 ( λ 1 ) [ n A ( λ 1 ) 1 ] c A } .
1 F D ( λ ) { [ n A ( λ ) 1 ] [ n B ( λ 1 ) 1 ] [ n A ( λ 1 ) 1 ] [ n B ( λ 1 ) 1 ] } c A + [ n B ( λ 1 ) 1 ] [ n B ( λ 1 ) 1 ] F L 2 ( λ 1 ) ,
1 F 1 ( λ ) = 1 F + 1 F λ 0 ,
1 F 2 ( λ ) = [ n ( λ ) 1 ] c ,
1 F 1 ( λ ) = 1 F 1 F λ 0 ,
1 F 2 ( λ ) = 1 F H 2 ( λ ) + 1 F D ( λ ) .
1 F 3 ( λ ) = 1 F 1 ( λ ) ,
E A ( x 1 , y 1 ; λ ) = ( i λ A ) exp { i π λ [ 1 A 1 F 1 ( λ ) ] ( x 1 2 + y 1 2 ) } × dxdyg ( x , y ) × exp [ i π ( x 2 + y 2 ) / ( λ A ) ] × exp [ i 2 π ( x 1 x + y 1 y ) / ( λ A ) ] ,
E B ( x 2 , y 2 ; λ ) = ( i λ B ) exp { i π λ [ 1 B 1 F 2 ( λ ) ] ( x 2 2 + y 2 2 ) } × d x 1 d y 1 E A ( x 1 , y 1 ; λ ) exp [ i π ( x 1 2 + y 1 2 ) / ( λ B ) ] × exp [ i 2 π ( x 2 x 1 + y 2 y 1 ) / ( λ B ) ] .
E B ( x 2 , y 2 ; λ ) = ( 1 λ 2 A B ) exp { i π λ [ 1 B 1 F 2 ( λ ) ] ( x 2 2 + y 2 2 ) } dxdyg ( x , y ) × exp [ i π ( x 2 + y 2 ) / ( λ A ) ] d x 1 d y 1 exp { i π λ [ 1 A + 1 B 1 F 1 ( λ ) ] ( x 1 2 + y 1 2 ) } × exp ( i 2 π λ [ x 1 ( x A + x 2 B ) + y 1 ( y A + y 2 B ) ] ) .
E B ( x 2 , y 2 ; λ ) = [ i λ A B α 1 ( λ ) ] exp { i π λ [ 1 B 1 F 2 ( λ ) 1 B 2 α 1 ( λ ) ] ( x 2 2 + y 2 2 ) } × dxdyg ( x , y ) exp { i π λ [ 1 A 1 A 2 α 1 ( λ ) ] ( x 2 + y 2 ) } exp { i 2 π ( x 2 x + y 2 y ) / [ λ A B α 1 ( λ ) ] } ,
α 1 ( λ ) = 1 A + 1 B 1 F 1 ( λ ) .
E II ( u , υ , λ ) = ( i λ C ) exp { i π λ [ 1 C 1 F 3 ( λ ) ] ( u 2 + υ 2 ) } d x 2 d y 2 E B ( x 2 , y 2 ; λ ) × exp [ i π ( x 2 2 + y 2 2 ) / ( λ C ) ] exp [ i 2 π ( u x 2 + υ y 2 ) / ( λ C ) ] .
E II ( u , υ ; λ ) = [ 1 λ 2 A B C α 1 ( λ ) ] exp { i π λ [ 1 C 1 F 3 ( λ ) ] ( u 2 + υ 2 ) } dxdyg ( x , y ) × exp { i π λ [ 1 A 1 A 2 α 1 ( λ ) ] ( x 2 + y 2 ) } d x 2 d y 2 × exp { i π λ [ α 2 ( λ ) 1 B 2 α 1 ( λ ) ] ( x 2 2 + y 2 2 ) } × exp ( i 2 π λ { x 2 ( u C + x A B α 1 ( λ ) ) + y 2 [ υ C + y A B α 1 ( λ ) ] } ) ,
α 2 ( λ ) = 1 B + 1 C 1 F 2 ( λ ) ,
E II ( u , υ ; λ ) = { i λ A B C α 1 ( λ ) [ α 2 ( λ ) 1 B 2 α 1 ( λ ) ] } exp ( i π λ { 1 C 1 F 3 ( λ ) 1 C 2 [ α 2 ( λ ) 1 B 2 α 1 ( λ ) ] } ( u 2 + υ 2 ) ) × dxdyg ( x , y ) exp ( i π λ { 1 A 1 A 2 α 1 ( λ ) 1 [ A B α 1 ( λ ) ] 2 [ α 2 ( λ ) 1 B 2 α 1 ( λ ) ] } ( x 2 + y 2 ) ) × exp ( i 2 π λ { 1 [ A B C α 1 ( λ ) ] [ α 2 ( λ ) 1 B 2 α 1 ( λ ) ] } ( u x + υ y ) ) .
α 2 ( λ ) 1 B 2 α 1 ( λ ) 0
L ch = C L ( λ ) ,
α 2 ( λ ) = A B 2 [ 1 A α 1 ( λ ) ] .
α 2 = F 1 ( λ ) B [ F 1 ( λ ) B ] .
1 B + 1 L ( λ ) 1 F 2 ( λ ) = F 1 ( λ ) B [ F 1 ( λ ) B ] .
L ( λ ) = 1 B F 1 ( λ ) 1 F 2 ( λ ) + 1 F 1 ( λ ) [ 1 B F 2 ( λ ) ] .
u ( λ ) = f x λ T ( λ ) ,
u ( λ ) u ( λ 0 ) = λ T ( λ ) λ 0 T ( λ 0 ) ,
T ( λ ) = A B C α 1 ( λ ) [ α 2 ( λ ) 1 B 2 α 1 ( λ ) ]
T ( λ ) = [ ( A + B ) F 1 ( λ ) A B ] [ ( B + C ) F 2 ( λ ) B C ] B F 1 ( λ ) F 2 ( λ ) A C B .
T ( λ ) = A [ 1 B F 1 ( λ ) C F 1 ( λ ) C F 2 ( λ ) + B C F 1 ( λ ) F 2 ( λ ) ] + B C [ 1 B + 1 C 1 F 2 ( λ ) ] .

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