Abstract

The noise-suppression characteristics of an achromatic optical processing system when operated with a broad-spectrum light source are analyzed. It is shown that such systems produce considerable noise improvement. Both signal-dependent and signal-independent noise are considered. In each case, we find that the achromatic coherent system behaves much like an incoherent imaging system.

© 1979 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. Leith, J. Roth, Appl. Opt. 16, 2565 (1977).
    [Crossref] [PubMed]
  2. For justification of Eqs. (3)–(5), see J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chap. 4.

1977 (1)

Goodman, J. W.

For justification of Eqs. (3)–(5), see J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chap. 4.

Leith, E.

Roth, J.

Appl. Opt. (1)

Other (1)

For justification of Eqs. (3)–(5), see J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), Chap. 4.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Holographic configuration.

Fig. 2
Fig. 2

Achromatic holographic configuration.

Fig. 3
Fig. 3

Alternate holographic configuration.

Fig. 4
Fig. 4

Configuration for noise analysis.

Fig. 5
Fig. 5

Achromatic coherent transfer function for signal-independent noise; zn in mm, fx in lines/mm, Δλ = 2 × 10−4 mm.

Fig. 6
Fig. 6

Achromatic coherent transfer function for signal-dependent noise; σn → 0. Parameters: zs = 250 mm, λ0 = 5 × 10−4 mm, Δλ = 2 × 10−4 mm, α = 250 lines/mm, zn in mm, fx in lines/mm.

Equations (49)

Equations on this page are rendered with MathJax. Learn more.

I = u 0 + f ( x ) 2 = u 0 2 + f ( x ) 2 + u 0 * f ( x ) + u 0 f * ( x ) .
u 0 2 f ( x ) = f ( x ) .
f ( x ) = f ( x ) * h ( z s ) ,
h ( z s ) = 1 ( i λ z s ) 1 / 2 exp ( i π x 2 λ z s )
H ( f x ) = exp ( - i π λ z s f x 2 ) ,
n ( x ) = n ( x ) * h ( z n ) .
I = u 0 + f ( x ) 2 ,
λ 0 - Δ λ 2 λ λ 0 + Δ λ 2 .
I = u 0 + f ( x ) 2 d λ ,
u r ( x ) = u 0 2 f ( x ) d λ = f ( x ) d λ
u r ( x ) = λ 0 - Δ λ / 2 λ 0 + Δ λ / 2 d λ - d x ^ 1 ( i λ z s ) 1 / 2 f ( x ^ ) exp [ i π ( x ^ - x ) 2 λ z s ] ,
f * h = F { F H } ,
u r ( x ) = λ 0 - Δ λ / 2 λ 0 + Δ λ / 2 d λ - d f x F ( f x ) × exp ( - i π λ z s f x 2 ) exp ( - i 2 π f x x ) ,
u r = Δ λ - F ( f x ) H 1 ( f x ) exp ( - i 2 π f x x ) d f x ,
H 1 ( f x ) = exp ( - i π λ 0 z s f x 2 ) sinc ( z s f x 2 Δ λ ) .
S = δ ( z - z s λ 0 / λ ) ,
f ( x ) = F [ f ( x - x ) exp ( i π x 2 / λ 0 z s ) ] ,
f ( x ) = f ( x - x ) exp ( i π x 2 / λ 0 z s ) d x = f * h .
u n - ( x ) = exp [ i π x 2 / λ ( z + z n ) ] ,
u n - ( x ) = exp [ i π x 2 / ( λ 0 z s + λ z n ) ] .
u n + ( x ) = n ( x - x ) exp [ i π x 2 / ( λ 0 z s + λ z n ) ] ,
u out ( x , λ ) = n ( x - x ) exp [ i π x 2 / ( λ 0 z s + λ z n ) ] d x ,
u out ( x , λ ) = N ( f x ) × exp [ - i π ( λ 0 z s + λ z n ) f x 2 ] exp ( i 2 π f x x ) d f x .
u ¯ out ( x , Δ λ ) = λ 0 - Δ λ / λ 2 λ 0 + Δ λ / 2 d λ - d f x N ( f x ) × exp [ - i π ( λ 0 z s + λ z n ) f x 2 ] exp ( i 2 π f x x ) ,
u ¯ out ( x , Δ λ ) = - N ( f x ) H 2 ( f x ) exp ( i 2 π f x x ) d f x = F - 1 [ N ( f x ) H 2 ( f x ) ] ,
H 2 ( f x ) = Δ λ exp [ - i π ( λ 0 z s ) f x 2 ] sinc [ z n f x 2 Δ λ / 2 ] .
u 1 ( x 1 ) = exp ( i π x 1 2 λ 0 z s - λ z n ) ,
t 1 = 1 - n ( x 1 ) .
u 2 = [ 1 - n ( x 1 ) ] exp ( i π x 1 2 λ 0 z s - λ z n ) × 1 ( i λ z n ) 1 / 2 exp [ i π ( x 1 - x 2 ) 2 λ z n ] d x 1 .
f ( x 3 ) = f ( x 2 - x 3 ) u 2 ( x 2 ) d x 2 .
F = F U 2 .
U 2 = H = exp ( - i π λ 0 z s f x 2 ) ,
U 2 = R ( 1 - N ) ,
H achromatic = R ( 1 - N ϕ d λ ) ,
r ( x 2 , y 2 ) = exp ( i 2 π α x 2 ) exp [ i π ( x 2 2 + y 2 2 ) λ 0 z s ] ,
n ( x 1 , y 1 ) = exp [ - ( x 1 - x 0 ) 2 + ( y 1 - y 0 ) 2 σ n 2 ] ,
u 2 ( x 2 , y 2 ) = exp [ i π ( x 2 2 + y 2 2 ) λ 0 z s + i 2 π α x 2 ] - exp ( i 2 π α x 2 ) 1 - 1 i π σ n 2 ( λ z n λ 0 z s ) ( λ 0 z s - λ z n ) × exp i π ( - i π σ n 2 ( x 2 2 + y 2 2 ) + λ z n [ ( a λ z n + x 0 ) 2 + y 0 2 ] + ( λ 0 z s - λ z n ) { [ x 2 - ( α λ z n + x 0 ) ] 2 + ( y 2 - y 0 ) 2 } λ z n ( λ 0 z s - λ z n ) - i π σ n 2 λ 0 z s ) .
U 2 ( f x , f y ) = exp [ - i π λ 0 z s ( f x 2 + f y 2 ) ] - π σ n 2 λ 0 z s ( λ 0 z s - λ z n ) - i π σ n 2 exp [ - i π λ 0 z s ( f x 2 + f y 2 ) ] × exp i π { [ ( x 0 + α λ z n ) - f x ( λ 0 z s - λ z n ) ] 2 + [ y 0 - f y ( λ 0 z s - λ z n ) ] 2 ( λ 0 z s - λ z n ) - i π σ n 2 } .
λ 0 - Δ λ 2 λ λ 0 + Δ λ 2 .
H noise ( f x , f y ) = | 1 Δ λ λ 0 - Δ λ / 2 λ 0 + Δ λ / 2 π σ n 2 λ 0 z s ( λ 0 z s - λ z n ) - i π σ n 2 exp [ - i π λ 0 z s ( f x 2 + f y 2 ) ] × exp i π { [ ( x 0 + α λ z n ) - f x ( λ 0 z s - λ z n ) ] 2 + [ y 0 - f y ( λ 0 z s - ( z n ) ] 2 ( λ 0 z s - λ z n ) - i π σ n 2 } d λ | .
lim σ n 0 n ( x , y ) π σ n 2 = δ ( x 1 - x 0 , y 1 - y 0 ) .
H noise σ n 0 ( f x , f y ) = | 1 Δ λ λ 0 - Δ λ / 2 λ 0 + Δ λ / 2 λ 0 z s ( λ 0 z s - λ z n ) exp [ - i π λ 0 z s ( f x 2 + f y 2 ) ] × exp i π { [ ( x 0 + α λ z n ) - f x ( λ 0 z s - λ z n ) ] 2 , + [ y 0 - f y ( λ 0 z s - λ z n ) ] 2 ( λ 0 z s - λ z n ) } d λ | .
O ( x , y ) = f ( x + x , y + y ) r ^ ( x , y ) d x d y ,
n ( x 1 , y 1 ) = exp [ - ( x 1 - x 0 ) 2 + ( y 1 - y 0 ) 2 σ n 2 ] ,
r ( x , y ) = g 1 ( x , y ; x ^ , y ^ ) r ( x ^ , y ^ ) d x ^ d y ^ ,
g 1 ( x , y ; x ^ , y ^ ) = δ ( x + x ^ , y + y ^ ) - π σ n 2 ( λ F ) 2 exp ( - π 2 σ n 2 ( λ F ) 2 [ ( x + x ^ ) 2 + ( y + y ^ ) 2 ] + i π λ F { [ x ^ - ( x 0 - α λ F ) ] 2 - [ x + ( x 0 - α λ F ) ] 2 + ( y ^ - y 0 ) 2 - ( y + y 0 ) 2 } ) ,
λ 0 - Δ λ 2 λ λ 0 + δ λ 2 ,
g ¯ 1 = | 1 Δ λ λ 0 - Δ 1 λ / 2 λ 0 + δ λ / 2 π σ n 2 ( λ F ) 2 exp ( - π 2 σ n 2 ( λ F ) 2 [ ( x ^ + x ) 2 + ( y ^ + y ) 2 ] + i π λ F { [ x ^ - ( x 0 - α λ F ) ] 2 - [ x + ( x 0 - α λ F ) ] 2 + ( y ^ - y 0 ) 2 - ( y + y 0 ) 2 } ) d λ | ,
g 2 = δ ( x + x ^ , y + y ^ ) - π σ n 2 ( λ F ) 2 × exp ( - π 2 σ n 2 ( λ F ) 2 [ ( x + x ^ ) 2 + ( y + y ^ ) 2 ] + i π λ F { [ x - ( x 0 - α λ F ) ] 2 - [ x ^ 0 + ( x 0 - α λ F ) ] 2 + ( y - y 0 ) 2 - ( y ^ + y 0 ) 2 } ) .

Metrics