Abstract

The dependence of the spectrometer scanning function on the spatial coherence of the illumination at the entrance slit implies that the spatial coherence of laser radiation must be degraded before its narrow spectral bandwith can be used to advantage in scanning-function calibration. A conventional incoherent source can be simulated by using a source having a relatively large correlation interval, provided that its correlation function has a favorable form. Partial-coherence theory is used to obtain conditions on the correlation function resulting from the properties of Globar radiation. The correlation function at a spectrometer entrance slit is experimentally determined using a method described by Beran and Parrent. The results for an incoherent Hg source are in agreement with Hopkins’s concept of an effective source. Results using laser light reflected from an integrating sphere indicate that such a source may be used interchangeably with an ordinary incoherent source for scanning-function calibration.

© 1967 Optical Society of America

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References

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  1. Hajime Sakai, Dissertation, The Johns Hopkins University, (1963).
  2. E. Marchand and R. Phillips, Appl. Opt. 2, 359 (1963).
    [Crossref]
  3. A. M. Goodbody, Proc. Phys. Soc. (London) 72, 411 (1958).
    [Crossref]
  4. R. Barakat, J. Opt. Soc. Am. 54, 38 (1964).
    [Crossref]
  5. G. H. Godfrey, Australian J. Sci. Res. 1, 1 (1948).
  6. P. H. van Cittert, Z. Physik 65, 547 (1930).
    [Crossref]
  7. Hajime Sakai, Japan. J. Appl. Phys. 4, Suppl. I, 308 (1965).
    [Crossref]
  8. W. J. Herget, W. E. Deeds, N. M. Gailar, R. J. Lovell, and A. H. Nielsen, J. Opt. Soc. Am. 52, 1113 (1962).
    [Crossref]
  9. Hajime Sakai and G. A. Vanasse, J. Opt. Soc. Am. 56, 357 (1966).
    [Crossref]
  10. L. Delbouille and G. Roland, J. Opt. Soc. Am. 56, 547A (1966).
  11. M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).
  12. W. H. Steel, Proc. Roy. Soc. (London) A249, 574 (1959).
    [Crossref]
  13. G. L. Rogers, Proc. Phys. Soc. (London) 81, Pt. 2. 323 (1963).
    [Crossref]
  14. H. H. Hopkins, Proc. Phys. Soc. (London) B69, Pt. 5, 562 (1956).
    [Crossref]
  15. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2nd ed., p. 510.
  16. M. V. R. K. Murty, J. Opt. Soc. Am. 54, 1187 (1964).
    [Crossref]
  17. W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964),
    [Crossref]
  18. P. S. Considine, J. Opt. Soc. Am. 56, 1001 (1966).
    [Crossref]
  19. H. H. Hopkins, Proc. Roy. Soc. (London) A208, 263 (1951).
    [Crossref]
  20. M. Beran and G. B. Parrent, Nuovo Cimento 27, 1049 (1963a).
    [Crossref]
  21. H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).
    [Crossref]
  22. besinc(x) = J1(x)/x.
  23. J. A. Ratcliffe, Rept. Prog. Phys. 19, 188 (1956).
    [Crossref]

1966 (3)

1965 (1)

Hajime Sakai, Japan. J. Appl. Phys. 4, Suppl. I, 308 (1965).
[Crossref]

1964 (3)

1963 (2)

G. L. Rogers, Proc. Phys. Soc. (London) 81, Pt. 2. 323 (1963).
[Crossref]

E. Marchand and R. Phillips, Appl. Opt. 2, 359 (1963).
[Crossref]

1962 (1)

1959 (1)

W. H. Steel, Proc. Roy. Soc. (London) A249, 574 (1959).
[Crossref]

1958 (1)

A. M. Goodbody, Proc. Phys. Soc. (London) 72, 411 (1958).
[Crossref]

1956 (2)

H. H. Hopkins, Proc. Phys. Soc. (London) B69, Pt. 5, 562 (1956).
[Crossref]

J. A. Ratcliffe, Rept. Prog. Phys. 19, 188 (1956).
[Crossref]

1953 (1)

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).
[Crossref]

1951 (1)

H. H. Hopkins, Proc. Roy. Soc. (London) A208, 263 (1951).
[Crossref]

1948 (1)

G. H. Godfrey, Australian J. Sci. Res. 1, 1 (1948).

1930 (1)

P. H. van Cittert, Z. Physik 65, 547 (1930).
[Crossref]

Barakat, R.

Beran, M.

M. Beran and G. B. Parrent, Nuovo Cimento 27, 1049 (1963a).
[Crossref]

Beran, M. J.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2nd ed., p. 510.

Considine, P. S.

Deeds, W. E.

Delbouille, L.

L. Delbouille and G. Roland, J. Opt. Soc. Am. 56, 547A (1966).

Gailar, N. M.

Godfrey, G. H.

G. H. Godfrey, Australian J. Sci. Res. 1, 1 (1948).

Goodbody, A. M.

A. M. Goodbody, Proc. Phys. Soc. (London) 72, 411 (1958).
[Crossref]

Herget, W. J.

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc. (London) B69, Pt. 5, 562 (1956).
[Crossref]

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).
[Crossref]

H. H. Hopkins, Proc. Roy. Soc. (London) A208, 263 (1951).
[Crossref]

Lovell, R. J.

Marchand, E.

Martienssen, W.

W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964),
[Crossref]

Murty, M. V. R. K.

Nielsen, A. H.

Parrent, G. B.

M. Beran and G. B. Parrent, Nuovo Cimento 27, 1049 (1963a).
[Crossref]

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).

Phillips, R.

Ratcliffe, J. A.

J. A. Ratcliffe, Rept. Prog. Phys. 19, 188 (1956).
[Crossref]

Rogers, G. L.

G. L. Rogers, Proc. Phys. Soc. (London) 81, Pt. 2. 323 (1963).
[Crossref]

Roland, G.

L. Delbouille and G. Roland, J. Opt. Soc. Am. 56, 547A (1966).

Sakai, Hajime

Hajime Sakai and G. A. Vanasse, J. Opt. Soc. Am. 56, 357 (1966).
[Crossref]

Hajime Sakai, Japan. J. Appl. Phys. 4, Suppl. I, 308 (1965).
[Crossref]

Hajime Sakai, Dissertation, The Johns Hopkins University, (1963).

Spiller, E.

W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964),
[Crossref]

Steel, W. H.

W. H. Steel, Proc. Roy. Soc. (London) A249, 574 (1959).
[Crossref]

van Cittert, P. H.

P. H. van Cittert, Z. Physik 65, 547 (1930).
[Crossref]

Vanasse, G. A.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2nd ed., p. 510.

Am. J. Phys. (1)

W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964),
[Crossref]

Appl. Opt. (1)

Australian J. Sci. Res. (1)

G. H. Godfrey, Australian J. Sci. Res. 1, 1 (1948).

J. Opt. Soc. Am. (6)

Japan. J. Appl. Phys. (1)

Hajime Sakai, Japan. J. Appl. Phys. 4, Suppl. I, 308 (1965).
[Crossref]

Nuovo Cimento (1)

M. Beran and G. B. Parrent, Nuovo Cimento 27, 1049 (1963a).
[Crossref]

Proc. Phys. Soc. (London) (3)

G. L. Rogers, Proc. Phys. Soc. (London) 81, Pt. 2. 323 (1963).
[Crossref]

H. H. Hopkins, Proc. Phys. Soc. (London) B69, Pt. 5, 562 (1956).
[Crossref]

A. M. Goodbody, Proc. Phys. Soc. (London) 72, 411 (1958).
[Crossref]

Proc. Roy. Soc. (London) (3)

H. H. Hopkins, Proc. Roy. Soc. (London) A208, 263 (1951).
[Crossref]

W. H. Steel, Proc. Roy. Soc. (London) A249, 574 (1959).
[Crossref]

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).
[Crossref]

Rept. Prog. Phys. (1)

J. A. Ratcliffe, Rept. Prog. Phys. 19, 188 (1956).
[Crossref]

Z. Physik (1)

P. H. van Cittert, Z. Physik 65, 547 (1930).
[Crossref]

Other (4)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., New York, 1964), 2nd ed., p. 510.

Hajime Sakai, Dissertation, The Johns Hopkins University, (1963).

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1964).

besinc(x) = J1(x)/x.

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Figures (9)

Fig. 1
Fig. 1

Coordinates for Eqs. (3) and (4). Z denotes the plane of the effective lens exit-pupil, (ξ1, ξ2) denotes the plane of the partially coherent source, (Z1,Z2) denotes the plane of the real lens exit pupil, and (X1,X2) denotes the final image plane.

Fig. 2
Fig. 2

Arrangement of apparatus for measurement of far-field pattern of slit. P. M. is a photomultiplier, S is a slit, M is a rectangular mask, O. G. is a piece of opal glass, and L focuses an image of the opal glass on the slit.

Fig. 3
Fig. 3

Far-field pattern of slit for x0 = 20 using Hg source. Aperture = f/8.97, slit width = 62.4 μ. Theoretical curve ——; experimental points ⊙.

Fig. 4
Fig. 4

Far-field pattern of slit for x0 = 6 using Hg source. Aperture = f/29.9, slit width = 62.4 μ. Theoretical curve ——; experimental points ⊙.

Fig. 5
Fig. 5

Far-field of slit for x0 = 3 using Hg source. Aperture = f/59.8, slit width = 62.4 μ. Theoretical curve ——; experi mental points ⊙.

Fig. 6
Fig. 6

Far-field pattern of slit for x0 = 16.7 using laser ground-glass source.

Fig. 7
Fig. 7

Far-field pattern of slit for x0 = 16 using laser integrating sphere source. Aperture = f/8.74, slit width = 56.4 μ. Theoretical curve ——; experimental points ⊙.

Fig. 8
Fig. 8

Far-field pattern of slit for x0 = 10 using laser integrating-sphere source. Aperture = f/14, slit width = 56.4 μ. Theo retical curve ——; experimental points ⊙.

Fig. 9
Fig. 9

Far-field pattern of slit for x0 = 6 using laser integrating sphere source. Aperture = f/23.2; slit width = 56.4 μ. Theoretical curve ——; experimental points ⊙,

Equations (11)

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I ˜ ( z x ) = Γ ˆ x ( z x + z x / 2 , z x z x / 2 , z y ) P ( z x + z x / 2 , z y ) × P * ( z x z x / 2 , z y ) d z y d z x .
Γ ˆ ( x 1 x 2 ) = P ( z ) P * ( z ) × exp [ ( 2 π i ν z / R ) · ( x 1 x 2 ) ] d z ,
Γ ˆ ( x 2 , x 1 , ν ) = K ( x 1 ξ 1 , ν ) K * ( x 2 ξ 2 , ν ) × Γ ˆ 0 ( ξ 1 , ξ 2 , ν ) d ξ 1 d ξ 2 ,
K ( x ξ ) = P 1 ( z 1 ) exp [ 2 π i ν z 1 R 2 · ( x ξ ) ] d z 1 ,
Γ ˆ ( x 2 , x 1 , ν ) = | P 0 ( z ) | 2 exp [ 2 π i ν z M R 0 · ( ξ 1 ξ 2 ) ] d z × P 1 * ( z 2 ) exp [ 2 π i ν z 2 R 2 · ( x 1 ξ 2 ) ] d z 2 × P 1 ( z 1 ) exp [ 2 π i ν z 1 R 2 · ( x 1 ξ 1 ) ] d z 1 d ξ 1 d ξ 2 ,
Γ ˆ ( x 2 , x 1 , ν ) = | P 0 ( z ) | 2 P 1 * ( R 2 z 2 ) P 1 ( R 2 z 1 ) × exp [ 2 π i ν ( z 1 · x 1 z 2 · x 2 ) ] × ( R 2 2 / ν 2 ) δ [ ( z / M R 0 ) z 1 ] δ [ z 2 ( z / M R 0 ) ] × d z 1 d z 2 d z .
Γ ˆ ( x 2 , x 1 , ν ) = R 0 R 2 2 R 1 ν 2 | P 0 ( z R 0 / R 1 ) | 2 | P 1 ( z ) | 2 × exp [ 2 π i ν z R 2 · ( x 1 x 2 ) ] d z ,
Γ ˆ ( l 1 , l 2 , ν ) = Γ ˆ 0 ( ξ 1 ξ 2 ) g ( ξ 2 ) × exp [ 2 π i ν ( l · ξ 1 l 2 · ξ 2 ) ] d ξ 1 d ξ 2 ,
Γ ˆ ( l 1 , l 2 ) = 2 w 2 w 2 s 2 s 2 ( s | ξ x | / 2 ) × sinc [ 2 π ν ( s | ξ x | / 2 ) ( l 1 x l 2 x ) ] × 2 ( w | ξ y | / 2 ) sinc [ 2 π ν ( w | ξ y | / 2 ) ( l 1 y l 2 y ) ] × Γ ˆ 0 ( ξ x , ξ y ) exp [ π i ν ξ x ( l 1 x + l 2 x ) ] × exp [ π i ν ξ y ( l 1 y + l 2 y ) ] d ξ x d ξ y .
Γ ˆ ( l 1 , l 2 ) 2 w sinc 2 π ν w ( l 1 y l 2 y ) 2 s sinc 2 π ν s ( l 1 x l 2 x ) × ( R 2 2 / ν 2 ) I e [ R 2 ( l 1 x + l 2 x ) / 2 , R 2 ( l 1 y + l 2 y ) / 2 ] ,
I ( l 1 x ) Si 4 π s ν ( a / R 2 + l 1 x ) Si 4 π s ν ( a / R 2 + l 1 x ) sin 2 2 π s ν ( a / R 2 + l 1 x ) 2 π s ν ( a / R 2 + l 1 x ) + sin 2 2 π s ν ( a / R 2 + l 1 x ) 2 π s ν ( a / R 2 + l 1 x ) .