Abstract

Coherent optical feedback in the form of a simple confocal resonator has been used to solve partial differential and integral equations. However, due to aberrations, the simple feedback system has a limited space–bandwidth product. An attempt is made here to increase the space–bandwidth product and the accuracy of the optical solution by introducing optical elements with spherical surfaces and negative lens effects into the simple confocal system.

© 1983 Optical Society of America

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References

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  1. J. Cederquist, S. H. Lee, J. Opt. Soc. Am. 70, 944 (1980).
    [CrossRef]
  2. J. Cederquist, S. H. Lee, J. Opt. Soc. Am. 71, 643 (1981).
    [CrossRef]
  3. J. Cederquist, J. Opt. Soc. Am. 71, 651 (1981).
    [CrossRef]
  4. W. J. Smith, Modern Optical Engineering; The Design of Optical Systems (McGraw-Hill, New York, 1966), pp. 254–257.
  5. Ref. 4, pp. 29 and 46.

1981 (2)

1980 (1)

J. Opt. Soc. Am. (3)

Other (2)

W. J. Smith, Modern Optical Engineering; The Design of Optical Systems (McGraw-Hill, New York, 1966), pp. 254–257.

Ref. 4, pp. 29 and 46.

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Figures (11)

Fig. 1
Fig. 1

Simple confocal feedback system.

Fig. 2
Fig. 2

Mangin mirror system.

Fig. 3
Fig. 3

Midplane negative lens system.

Fig. 4
Fig. 4

Feedback phase variations of the simple confocal system (R = 1.06 m, θ = 6.3 mrad) vs yB with as a parameter.

Fig. 5
Fig. 5

Feedback phase variations of the simple confocal system (R = 1.06 m, = −140 μm) vs yB with θ as a parameter.

Fig. 6
Fig. 6

Paraxial quantities of Eqs. (10) and (11). All quantities shown are positive, and primes indicate quantities after reflection or refraction.

Fig. 7
Fig. 7

Feedback phase variations of the mangin mirror system for R = 1.06 m, 1/r = 1.4110 m−1, = 0.0 μm, and different values of yB and θ.

Fig. 8
Fig. 8

Feedback phase variations of the midplane negative lens system for R = 1.06 m, 1/r = 1.1730 m−1, /2 = −33.0 μm, and different values of yB and θ.

Fig. 9
Fig. 9

Three-dimensional feedback phase error lot vs xB and yB for the simple confocal system (R = 1.06 m, = −140 μm, θ = 6.3 mrad).

Fig. 10
Fig. 10

Three-dimensional feedback phase error plot vs xB and yB for the mangin mirror system (R = 1.06 m, 1/r = 1.4110m−1, = 0.0 μm, θ = 6.3 mrad).

Fig. 11
Fig. 11

Three-dimensional feedback phase error plot vs xB and yB for the midplane negative lens system (R = 1.06 m, 1/r = 1.1730 m−1, /2 = −33.0 μm, θ = 6.3 mrad).

Tables (3)

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Table I Simple Confocal System with R = 1.06 m, = −140 μm, and θ = 6.3 mrad a

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Table II Mangin Mirror System with R = 1.06 m, 1/r = 1.4110 m−1, = 0.0 μm, and θ = 6.3 mrada

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Table III Midplane Negative Lens System with R = 1.06 m, 1/r = 1.1750m−1, and = −66.0 μma

Equations (47)

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l 2 = A C ¯ + C E ¯ = { [ ( R + ) - ( z 1 + z 2 ) ] 2 + ( ρ 2 - ρ 1 ) 2 } 1 / 2 + { [ ( R + ) - ( z 1 + z 2 ) ] 2 + ( ρ 1 + ρ 2 ) 2 } 1 / 2 .
l 2 d = [ 1 + ( ρ 2 - ρ 1 ) 2 d 2 ] 1 / 2 + [ 1 + ( ρ 1 + ρ 2 ) 2 d 2 ] 1 / 2 .
l 4 d + 2 ( ρ 1 2 + ρ 2 2 ) d - 1 2 d 3 ( ρ 1 4 + 6 ρ 1 2 ρ 2 2 + ρ 2 4 ) = 4 ( R + ) - 4 ( z 1 + z 2 ) + 2 ( ρ 1 2 + ρ 2 2 ) ( R + ) [ 1 - ( z 1 + z 2 ) ( R + ) ] - 1 - ( ρ 1 4 + 6 ρ 1 2 ρ 2 2 + ρ 2 4 ) 2 ( R + ) 3 [ 1 - ( z 1 + z 2 ) ( R + ) ] - 3 .
l 4 ( R + ) - 4 ( z 1 + z 2 ) + 2 ( ρ 1 2 + ρ 2 2 ) ( R + ) [ 1 + ( z 1 + z 2 ) ( R + ) + ] - ( ρ 1 4 + 6 ρ 1 2 ρ 2 2 + ρ 2 4 ) 2 ( R + ) 3 [ 1 + ] .
z = ρ 2 2 R + ρ 4 8 R 3 + .
l = 4 ( R + ) - 2 ( ρ 1 2 + ρ 2 2 ) R 2 ( 1 + / R ) - ρ 1 2 ρ 2 2 R 3 × [ 3 ( 1 + / R ) 3 - 2 ( 1 + / R ) 2 ] + ( ρ 1 4 + ρ 2 4 ) R 3 [ 1 ( 1 + / R ) 2 - 1 2 ( 1 + / R ) 3 - 1 2 ] .
l 4 ( R + ) - 2 ( ρ 1 2 + ρ 2 2 ) R 2 - ρ 1 2 ρ 2 2 R 3 ,
Δ β = 2 π ( Δ l ) λ
ρ 1 = y B - R θ 2 ,             ρ 2 = y B + R θ 2 .
Δ β - 2 π ( R λ ) ( y B R ) 2 [ ( y B R ) 2 + 4 ( R ) - θ 2 2 ] .
BW y = 2 y D λ ( EFL ) .
SBP y = 2 y D λ ( EFL ) ( y B max - 2 y D ) .
n 1 u 1 = ( n 1 - n 1 ) r 1 y 1 + n 1 u 1 ,
y 2 = y 1 - ( t 1 n 1 ) ( n 1 u 1 ) , n 2 = n 1 , u 2 = u 1 .
l i = y i / u i
EFL = y 1 / u k
r 1 = - r t 1 = t n 1 = 1 r 2 = - R t 2 = - t n 2 = N r 3 = - r n 3 = - N n 4 = - 1 ,
y 1 = y 1             u 1 = 0 ,
l 3 = - d             y 4 = 0.
N u 1 = ( N - 1 ) - r y 1 = - η y 1 ,
y 2 = y 1 - ( t N ) ( - η y 1 ) = y 1 ( 1 + t η N ) ,
- N u 2 = ( - N - N - R ) ( y 1 ) ( 1 + t η N ) - η y 1 , = ( 2 N R ) ( 1 + t η N ) y 1 - η y 1 ,
y 3 = y 2 - ( - t - N ) [ ( 2 N R ) ( 1 + t η N ) y 1 - η y 1 ] = [ ( 1 + t η N ) ( 1 - 2 t R ) + ( t η N ) ] y 1 ,
- u 3 = ( - 1 + N - r ) [ ( 1 + t η N ) ( 1 - 2 t R ) y 1 + ( t η N ) y 1 ] + ( 2 N R ) ( 1 + t η N ) y 1 - η y 1 , u 3 = [ η ( 1 + t η N ) ( 1 - 2 t R ) + ( t η 2 N ) - ( 2 N R ) ( 1 + t η N ) + η ] y 1 .
d = - l 3 = y 3 u 3 = ( 1 + t η N ) ( 1 - 2 t R ) + ( t η N ) ( 1 + t η N ) [ ( 2 N R ) + ( 2 t η R ) - 2 η ] ,
EFL = y 1 u 4 = { ( 1 + t η N ) [ ( 2 N R ) + ( 2 t η R ) - 2 η ] } - 1 .
1 / r = 1.4110 m - 1 , R = 1.06 m , d = 0.68875 m , EFL = 0.69855 m , = 0.0 μ m .
r 1 = r t 1 = ( d - t ) n 1 = N r 2 = - R t 2 = - ( d - t ) n 2 = 1 r 3 = r n 3 = - 1 n 4 = - N ,
y 1 = y 1             u 1 = - y 1 / t ,
u 3 = 0             y 3 = ?
u 1 = ( 1 - N ) r y 1 + N ( y 1 - t ) = - y 1 ( η + N t ) ,
y 2 = y 1 - ( d - t 1 ) ( - y 1 ) ( η + N t ) = y 1 [ 1 + ( d - t ) ( η + N t ) ] ,
- u 2 = ( - 1 - 1 ) - R ( y 1 ) [ 1 + ( d - t 1 ) ( η + N t ) ] - y 1 ( η + N t ) = 2 y 1 R [ 1 + ( d - t ) ( η + N t ) ] - y 1 ( η + N t ) ,
- N u 3 = 0 = - y 1 η [ 1 + ( d - t ) ( η + N t ) ] + 2 y 1 η ( d - t ) R [ 1 + ( d - t ) ( η + N t ) ] - y 1 η ( d - t ) ( η + N t ) + 2 y 1 R [ 1 + ( d - t ) ( η + N t ) ] - y 1 ( η + N d ) , u 3 = 0 = [ η t + N ] ( d - t ) 2 + [ 2 t - η t R - N R + N η ] ( d - t ) + [ t η - R t - R N 2 η ] ,
y 3 = y 2 - [ - ( d - t ) - 1 ] { 2 y 1 R [ 1 + ( d - t ) ( η + N t ) ] - y 1 ( η + N t ) } = { [ 1 + ( η + N / t ) ( d - t ) ] - 2 ( d - t ) R [ 1 + ( η + N t ) ( d - t ) ] + ( η + N t ) ( d - t ) } y 1 ,
a = η t + N ,
b = a ( 1 / η - R ) + t = 2 t - η t R - N R + N η ,
c = t / η - R t - R N / 2 η .
d - t = ( b 2 - 4 a c ) 1 / 2 - b 2 a ,
d - t = R 2 - t N ,
d = R / 2 ,
EFL = y 3 - u 1 = [ - 2 ( η t + N ) R ] ( d - t ) 2 + [ 2 ( η t + N ) - 2 t R ] ( d - t ) + t .
EFL = - 2 N R d 2 + 2 N d .
d F [ 1 + ½ ( F / f ) ] .
EFL N F [ 1 - ¼ ( F f ) 2 ] .
1 / r = 1.1750 m - 1 , R = 1.06 m , d = 0.68859 m , EFL = 0.73798 m , = - 66.0 μ m .
( 1.22 λ ) ( 2 EFL ) ( y B max - y B min ) .

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