Abstract

Measurement of optical transfer functions making use of a wavefront-shearing interferometer usually requires the path difference between the two arms of the interferometer to be varied.

The same effect can be achieved by a polarizing technique and, depending on the polarizing arrangement, either the modulus, or the real and imaginary parts of the optical transfer function can be recorded.

The theory of these two methods is given and the experimental procedure is described. Transfer functions for a defocused optical system are measured and are in good agreement with calculated results. Some of the problems involved in the measurement of the argument of the transfer function with an interferometer are also briefly discussed.

© 1966 Optical Society of America

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References

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  1. H. H. Hopkins, Opt. Acta 2, 23 (1955).
    [CrossRef]
  2. L. R. Baker, Proc. Phys. Soc. (London) 68B, 871 (1955).
  3. H. H. Hopkins, Proc. Roy. Soc. (London) 231A, 91 (1955).
  4. D. Kelsall, Proc. Phys. Soc. (London) 73, 465 (1959).
    [CrossRef]
  5. P. Hariharen and D. Sen, Proc. Phys. Soc. (London) 76, 434 (1960).
    [CrossRef]
  6. A. J. Montgomery, J. Opt. Soc. Am. 54, 191 (1964).
    [CrossRef]
  7. A. M. Goodbody, Proc. Phys. Soc. (London) 75, 677 (1960).
    [CrossRef]
  8. W. H. Steel, Opt. Acta 11, 9 (1964).
    [CrossRef]

1964 (2)

1960 (2)

P. Hariharen and D. Sen, Proc. Phys. Soc. (London) 76, 434 (1960).
[CrossRef]

A. M. Goodbody, Proc. Phys. Soc. (London) 75, 677 (1960).
[CrossRef]

1959 (1)

D. Kelsall, Proc. Phys. Soc. (London) 73, 465 (1959).
[CrossRef]

1955 (3)

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

L. R. Baker, Proc. Phys. Soc. (London) 68B, 871 (1955).

H. H. Hopkins, Proc. Roy. Soc. (London) 231A, 91 (1955).

Baker, L. R.

L. R. Baker, Proc. Phys. Soc. (London) 68B, 871 (1955).

Goodbody, A. M.

A. M. Goodbody, Proc. Phys. Soc. (London) 75, 677 (1960).
[CrossRef]

Hariharen, P.

P. Hariharen and D. Sen, Proc. Phys. Soc. (London) 76, 434 (1960).
[CrossRef]

Hopkins, H. H.

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

H. H. Hopkins, Proc. Roy. Soc. (London) 231A, 91 (1955).

Kelsall, D.

D. Kelsall, Proc. Phys. Soc. (London) 73, 465 (1959).
[CrossRef]

Montgomery, A. J.

Sen, D.

P. Hariharen and D. Sen, Proc. Phys. Soc. (London) 76, 434 (1960).
[CrossRef]

Steel, W. H.

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

J. Opt. Soc. Am. (1)

Opt. Acta (2)

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

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

Proc. Phys. Soc. (London) (4)

L. R. Baker, Proc. Phys. Soc. (London) 68B, 871 (1955).

D. Kelsall, Proc. Phys. Soc. (London) 73, 465 (1959).
[CrossRef]

P. Hariharen and D. Sen, Proc. Phys. Soc. (London) 76, 434 (1960).
[CrossRef]

A. M. Goodbody, Proc. Phys. Soc. (London) 75, 677 (1960).
[CrossRef]

Proc. Roy. Soc. (London) (1)

H. H. Hopkins, Proc. Roy. Soc. (London) 231A, 91 (1955).

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Figures (4)

Fig. 1
Fig. 1

Sheared pupils. Arrows show directions of polarization. Analyzer direction is at angle θ to y axis.

Fig. 2
Fig. 2

Diagram showing state of polarization produced by a path difference between the two wavefronts. Path differences due to shearing are shown at top: (a) resulting polarizations without λ/4 plate; (b) right-hand circularly polarized component, when λ/4 plate is used; (c) left-hand circularly polarized component, when λ/4 plate is used; (d) resultant polarizations with λ/4 plate.

Fig. 3
Fig. 3

Schematic diagram of interferometer. M1, M2, mirrors; BS, beam splitter; S1, S2, shear plates; P1, P2, polarizers. Collimated beam from test lens enters from bottom left.

Fig. 4
Fig. 4

Transfer function of a defocused system. ×=Computed points, ——=measured curves (reproducibility 2%).

Equations (18)

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D ( s , ψ ) = T ( s , ψ ) e i θ ( s , ψ ) ,
D ( s , ψ ) = 1 A - + f ( x + s / 2 , y ) f * ( x - s / 2 , y ) d x d y .
A = - + f ( x , y ) 2 d x d y .
f ( x , y ) = τ ( x , y ) e i k W ( x , y ) ,
F ( θ ) = - + f ( x + s / 2 , y ) sin θ + e i k δ f ( x - s / 2 , y ) cos θ 2 d x d y ,
F ( θ ) = sin 2 θ - + f ( x + s / 2 , y ) 2 d x d y + cos 2 θ - + f ( x - s / 2 , y ) 2 d x d y + 2 sin θ cos θ { e i k δ - + f ( x + s / 2 , y ) × f * ( x - s / 2 , y ) d x d y } ,
- + f ( x + s / 2 , y ) 2 d x d y = - + f ( x , y ) 2 d x d y = A ,
F ( θ ) = A { 1 + sin 2 θ [ e i k δ D ( s ) ] } ,
e i k δ = 1
F ( θ ) = A { 1 + sin 2 θ T ( s ) cos θ ( s ) } .
F ( θ ) = A { 1 + sin 2 θ T ( s ) sin θ ( s ) } .
{ ( 1 / 2 ) f ( x + s / 2 , y ) + ( 1 / 2 ) f ( x + s / 2 , y ) e i k δ } .
{ ( 1 / 2 ) f ( x + s / 2 , y ) + ( i / 2 ) f ( x + s / 2 , y ) } .
F ( ϕ ) = - + | 1 2 f ( x - s / 2 , y ) e - i ϕ + 1 2 f ( x + s / 2 , y ) e i ϕ e i k δ | 2 d x d y ,
F ( ϕ ) = A + { - + f ( x + s / 2 , y ) × f * ( x - s / 2 , y ) e i k δ e i 2 ϕ d x d y } , = A { 1 + [ T ( s ) e i θ ( s ) e i k δ e i 2 ϕ ] } , = A { 1 + T ( s ) cos [ 2 ϕ + k δ + θ ( s ) ] } .
W 20 = n λ / π = 1 2 r 2 x / f ( f + x ) ,
θ ( s ) = a s + b s 3 + etc .
θ ( s ) measured = ( a + a ) s + b s 3 + ,