Abstract

We investigate the properties of a common type of two-beam interferometer in which the recombined beams are focused onto a detector and in which the optical system acting on the separated beams is composed of plane mirrors. For an arbitrarily complicated system of plane mirrors the behavior may be analyzed in terms of the four parameters path difference, angle of tilt, lateral shear, and angle of rotary shear. Fourier spectrometers vary only path difference, while shearing interferometers vary only lateral shear. We show that from measurements of the detected interference as a function of both lateral shear and path difference one may compute the optical transfer function of the output optics as a function of both optical and spatial frequency.

© 1978 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, New York,
  2. W. H. Steel, Interferometry (Cambridge, London, 1967), p. 485.
  3. E. R. Peck, J. Opt. Soc. Am. 38, 66 (1948).
    [Crossref]
  4. E. R. Peck, J. Opt. Soc. Am. 47, 250 (1957).
    [Crossref]
  5. D. K. Lambert and P. L. Richards, Appl. Opt. (to be published).
  6. W. H. Steel, Opt. Acta 11, 9 (1964).
    [Crossref]
  7. H. H. Hopkins, Opt. Acta 2, 23 (1955).
    [Crossref]
  8. J. C. Wyant, Appl. Opt. 14, 1613 (1975).
    [Crossref] [PubMed]
  9. P. Dumontet, Opt. Acta 2, 53 (1955).
    [Crossref]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  11. Reference 1, Chap. 10.
  12. W. H. Carter, J. Opt. Soc. Am. 65, 1054 (1975).
    [Crossref]
  13. Reference 1, p. 544.
  14. H. H. Hopkins, J. Opt. Soc. Am. 47, 508 (1957).
    [Crossref]
  15. G. Schultz, Opt. Acta 11, 43 (1964).
    [Crossref]
  16. R. Landwehr, Opt. Acta 6, 52 (1959).
    [Crossref]
  17. W. H. Steel, J. Opt. Soc. Am. 54, 151 (1964).
    [Crossref]
  18. Reference 1, p. 482.
  19. W. H. Carter and E. Wolf, J. Opt. Soc. Am. 67, 785 (1977).
    [Crossref]
  20. A. Erdeyli and et al., Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2, pp. 335–336.

1977 (1)

1975 (2)

1964 (3)

W. H. Steel, Opt. Acta 11, 9 (1964).
[Crossref]

G. Schultz, Opt. Acta 11, 43 (1964).
[Crossref]

W. H. Steel, J. Opt. Soc. Am. 54, 151 (1964).
[Crossref]

1959 (1)

R. Landwehr, Opt. Acta 6, 52 (1959).
[Crossref]

1957 (2)

1955 (2)

H. H. Hopkins, Opt. Acta 2, 23 (1955).
[Crossref]

P. Dumontet, Opt. Acta 2, 53 (1955).
[Crossref]

1948 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York,

Carter, W. H.

Dumontet, P.

P. Dumontet, Opt. Acta 2, 53 (1955).
[Crossref]

Erdeyli, A.

A. Erdeyli and et al., Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2, pp. 335–336.

Goodman, J. W.

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

Hopkins, H. H.

Lambert, D. K.

D. K. Lambert and P. L. Richards, Appl. Opt. (to be published).

Landwehr, R.

R. Landwehr, Opt. Acta 6, 52 (1959).
[Crossref]

Peck, E. R.

Richards, P. L.

D. K. Lambert and P. L. Richards, Appl. Opt. (to be published).

Schultz, G.

G. Schultz, Opt. Acta 11, 43 (1964).
[Crossref]

Steel, W. H.

W. H. Steel, Opt. Acta 11, 9 (1964).
[Crossref]

W. H. Steel, J. Opt. Soc. Am. 54, 151 (1964).
[Crossref]

W. H. Steel, Interferometry (Cambridge, London, 1967), p. 485.

Wolf, E.

W. H. Carter and E. Wolf, J. Opt. Soc. Am. 67, 785 (1977).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Pergamon, New York,

Wyant, J. C.

Appl. Opt. (1)

J. Opt. Soc. Am. (6)

Opt. Acta (5)

G. Schultz, Opt. Acta 11, 43 (1964).
[Crossref]

R. Landwehr, Opt. Acta 6, 52 (1959).
[Crossref]

W. H. Steel, Opt. Acta 11, 9 (1964).
[Crossref]

H. H. Hopkins, Opt. Acta 2, 23 (1955).
[Crossref]

P. Dumontet, Opt. Acta 2, 53 (1955).
[Crossref]

Other (8)

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

Reference 1, Chap. 10.

D. K. Lambert and P. L. Richards, Appl. Opt. (to be published).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York,

W. H. Steel, Interferometry (Cambridge, London, 1967), p. 485.

Reference 1, p. 544.

Reference 1, p. 482.

A. Erdeyli and et al., Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2, pp. 335–336.

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Figures (3)

FIG. 1
FIG. 1

A schematic diagram of the system in which two beam interference takes place. Throughout this manuscript the basis vectors and origin 0 of the three-dimensional vector field x will remain the same. Note that the coordinate systems on P and P′ are defined so that the line along the normal vector to the plane through the origin in the plane also passes through the origin of x.

FIG. 2
FIG. 2

The planes used in our analysis of interference in a plane mirror two-beam interferometer.

FIG. 3
FIG. 3

Diagram illustrating the parameters used to discuss the geometric relationship between Ps and Pt.

Equations (47)

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i ( r , ν ) = j = 1 2 K i j ( r , r , ν ) j ( r , ν ) d 2 r .
E ( x , t ) 2 = E s ( x , t ) 2 + E t ( x , t ) 2 + 2 E s ( x , t ) · E t ( x , t ) .
g ( ν ) = - e - 2 π i ν t G ( t ) dt ,
G ( t ) · H ( t ) = - g ( ν ) h * ( ν ) d ν = 2 Re 0 g ( ν ) h * ( ν ) d ν .
F s t = Re ν = 0 i = 1 2 j = 1 2 A i j ( r s , r t , ν ) × i s ( r s , ν ) j t * ( r t , ν ) d 2 r s d 2 r t d ν .
F s t = Re ν = 0 k = 1 2 l = 1 2 A k l ( r s , r t , ν ) × J k l s t ( r s , r t , ν ) d 2 r s d 2 r t d ν , J k l s t ( r s , r t , ν ) = i = 1 2 j = 1 2 K k i s ( r s , r s , ν ) × K i j t * ( r t , r t , ν ) J i j s t ( r s , r t , ν ) d 2 r s d 2 r t d ν .
F s t = Re ν = 0 A ( r s , r t , ν ) J s t ( r s , r t , ν ) d 2 r s d 2 r t d ν , J s t ( r s , r t , ν ) = K s ( r s , r s , ν ) K t * ( r t , r t , ν ) × J s t ( r s , r t , ν ) d 2 r s d 2 r t .
F s t = Re ν = 0 A ( r s , ν ) J s t ( r s , r s , ν ) d 2 r s d ν .
W s t ( r s , r t , ν ) = J s t ( r s , r t , ν ) / [ s ( r s , ν ) · t ( r t , ν ) ] ,
k ˆ o = k ˆ i - 2 ( k i · n ˆ m ) n ˆ m , p ˆ o = - p ˆ i + 2 ( p ˆ i · n ˆ m ) n ˆ m .
J s t ( r s , r t , ν ) = J u ( r s , r t , ν ) × exp [ - 2 π i σ Δ x ( 1 + r t - r a 2 / f 2 ) 1 / 2 ] .
F s t = Re ν = 0 A ( r s , ν ) J u ( r s , r s , ν ) × exp [ - 2 π i σ Δ x ( 1 + r s - r 2 / f 2 ) 1 / 2 ] d 2 r s d ν .
J s t ( r s , r t , ν ) = J u ( r s , r t - Δ r , ν ) .
F s t = Re ν = 0 A ( r s , ν ) J u ( r s , r s - Δ r , ν ) d 2 r s d ν .
J s t ( r s , r t , ν ) = C J v ( r s , r t + l , ν ) × G ( r s , r s , ν ) G * ( r t , r t , ν ) d 2 r s d 2 r t .
J s t ( r s , r t , ν ) = C J ( r s - r t - l , ν ) × G ( r s , r s , ν ) G * ( r t , r t , ν ) d 2 r s d 2 r t .
L ( s , ν , r s , r t ) = G ( r s , r s , ν ) G * ( r s - s , r t , ν ) d 2 r s .
J s t ( r s , r t , ν ) = C J ( s - l , ν ) L ( s , ν , r s , r t ) d 2 s .
L ( s , ν , r , r ) = L ( 0 , ν , r , r ) ( σ s / f , ν ) .
J s t ( r , r , ν ) = C 1 ( ν ) J ( s - l , ν ) ( σ s / f , ν ) d 2 s .
F s t = Re ν = 0 A ( r s , ν ) × J ( s - l , ν ) ( σ s / f , ν ) d 2 s d 2 r s d ν .
J s t ( r s , r t , ν ) = J u ( r s , w , ν ) .
F s t = Re ν = 0 A ( r s , ν ) J u ( r s , w , ν ) d 2 r s d ν .
J s t ( l , Δ x , r s 1 , r s 2 , ν ) = J v ( r s 1 - l , r t , ν ) K * ( r t , r s 2 , Δ x , ν ) d 2 r t
K * ( r t , r s 2 , Δ x , ν ) = [ U 1 ( u , v ) - i U 2 ( u , v ) ] 4 π Δ x r t - r s 2 ) , u = 8 π 2 σ 2 Δ x r t - r s 2 ,             v = 2 π σ r t - r s 2 .
J s t ( l , Δ x , r s , r t , ν ) = J s t ( l , Δ x , r s 1 , r s 2 , ν ) × G ( r s 1 , r s , ν ) G * ( r s 2 , r t , ν ) d 2 r s 1 d 2 r s 2 = J v ( r s 1 - l , r t , ν ) K * ( r t , r s 2 , Δ x , ν ) × G ( r s 1 , r s , ν ) G * ( r s 2 , r t , ν ) d 2 r t d 2 r s 1 d 2 r s 2 .
J s t ( l , Δ x , r s , r t , ν ) = J ( s - l - p , ν ) × K ( p , Δ x , ν ) L ( s , ν , r s , r t ) d 2 s d 2 p .
j s t ( K , Δ x , r s , r t , ν ) = J s t ( l , Δ x , r s , r t , ν ) e 2 π i K · l d 2 l ,
j s t ( K , Δ x , r s , r t , ν ) = j ( K , ν ) k ( K , Δ x , ν ) l ( K , ν , r s , r t ) .
F s t ( l , Δ x ) = ν = 0 C ( ν ) j ( K , ν ) l ( K , ν ) × exp { 2 π i [ Δ x ( σ 2 - K 2 ) 1 / 2 - K - l ] } d 2 k d ν .
f s t ( K , σ ) = F s t ( l , Δ x ) exp { 2 π i [ K - l - σ Δ x ] } d 2 l d x
f s t ( K , σ ) = C ( ν ) j ( K , ν ) l ( K , ν ) , where ν = c ( σ 2 + K 2 ) 1 / 2 .
l ( K , ν ) = f s t ( K , σ ) / f s t ( K , σ ) , where σ = [ ( ν / c ) 2 - K 2 ] .
( σ s / f , ν ) = ( 1 / C ) l ( K , ν ) e - 2 π i K · s d 2 K , where C = l ( K , ν ) d 2 K and σ = ν / c .
( K , ν ) = ( 1 / C ) F s t ( Δ x , c f K / ν ) exp { - 2 π i Δ x [ ( ν / c ) 2 - K 2 ] 1 / 2 } d ( Δ x ) , where C = F s t ( Δ x , 0 ) e - 2 π i ( ν / c ) Δ x d ( Δ x ) .
( K , ν ) = ( r , ν ) e 2 π i K · r d 2 r , ( r , ν ) = ( K , ν ) e - 2 π i K · r d 2 K .
k = 2 π [ - i = 1 2 K i u ˆ i + [ ( ν / c ) - K ] n ˆ ]
E = i = 1 2 i o u ˆ i .
E = [ ( k ˆ · n ˆ ) 2 - 1 ] ( k ˆ · n ˆ ) E n ˆ .
i ( r , ν ) = j = 1 2 K i j ( K , r , ν ) j ( K , ν ) d 2 K .
K i j ( K , K , ν ) = K i j ( K , r , ν ) e - 2 π i K · r d 2 r , K i j ( K , r , ν ) = K i j ( K , K , ν ) e - 2 π i K · r d 2 K .
K i j ( r , r , ν ) = K K i j ( K , r , ν ) e 2 π i K · r d 2 K .
K i j ( r , r , ν ) = K i j ( K , K , ν ) e 2 π i ( K · r - K · r ) d 2 K d 2 K , K i j ( K , K , ν ) = K i j ( r , r , ν ) × e - 2 π i ( K · r - K · r ) d 2 r d 2 r .
i ( r , ν ) = j = 1 2 K i j ( r , r , ν ) j ( r , ν ) d 2 r , i ( K , ν ) = j = 1 2 K I J ( K , K , ν ) j ( K , ν ) d 2 K .
K i j ( K , K , ν ) = e 2 π i Δ x ( σ 2 - K 2 ) 1 / 2 δ i j δ ( K - K ) .
K i j ( r , r , ν ) = δ i j K = 0 σ 2 π J o ( 2 π K r - r ) × e 2 π i Δ x ( σ 2 - K 2 ) K d K = δ i j [ U 1 ( u , v ) + i U 2 ( u , v ) / 4 π Δ x r - r , where u = 8 π 2 σ 2 Δ x r - r ,             v = 2 π σ r - r .
U i ( u , v ) = m = 0 ( - 1 ) m ( u / v ) i + 2 m J i + 2 m ( v ) .