Abstract

A lens-evaluation method based on the dynamic properties of laser-produced speckles is investigated and developed into a real-time modulation-transfer-function measuring system controlled by a microprocessor. One of the most important advantages of this method is the redundancy that permits rough adjustment of the lateral position of the light detecting point in the focal plane. Experimental results applied to a grin-rod lens as well as to an ordinary camera lens are obtained. For detection of the best focal position, the method of counting the zero-cross distribution along the optical axis is found to be effective.

© 1983 Optical Society of America

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References

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  1. T. Sawatari, A. C. Elek, Appl. Opt. 12, 881 (1973).
    [CrossRef] [PubMed]
  2. L. H. Tanner, Appl. Opt. 13, 2026 (1974).
    [CrossRef] [PubMed]
  3. C. P. Grover, H. M. van Driel, Appl. Opt. 19, 900 (1980).
    [CrossRef] [PubMed]
  4. I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. Suppl. 1 14, 301 (1975).
    [CrossRef]
  5. S. Komatsu, T. Morioka, H. Ohzu, J. Opt. Soc. Am. 72, 1743 (1982).
  6. L. Mandel, Prog. Opt. 2, 181 (1963).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 57–100.
  8. K. A. O'Donnell, J. Opt. Soc. Am. 72, 191 (1982).
    [CrossRef]
  9. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1981), p. 485.
  10. A. Ghatak, K. Thyagarajan, Prog. Opt. 18, 12 (1980).

1982 (2)

S. Komatsu, T. Morioka, H. Ohzu, J. Opt. Soc. Am. 72, 1743 (1982).

K. A. O'Donnell, J. Opt. Soc. Am. 72, 191 (1982).
[CrossRef]

1980 (2)

A. Ghatak, K. Thyagarajan, Prog. Opt. 18, 12 (1980).

C. P. Grover, H. M. van Driel, Appl. Opt. 19, 900 (1980).
[CrossRef] [PubMed]

1975 (1)

I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. Suppl. 1 14, 301 (1975).
[CrossRef]

1974 (1)

1973 (1)

1963 (1)

L. Mandel, Prog. Opt. 2, 181 (1963).
[CrossRef]

Elek, A. C.

Ghatak, A.

A. Ghatak, K. Thyagarajan, Prog. Opt. 18, 12 (1980).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 57–100.

Grover, C. P.

Komatsu, S.

S. Komatsu, T. Morioka, H. Ohzu, J. Opt. Soc. Am. 72, 1743 (1982).

I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. Suppl. 1 14, 301 (1975).
[CrossRef]

Mandel, L.

L. Mandel, Prog. Opt. 2, 181 (1963).
[CrossRef]

Morioka, T.

S. Komatsu, T. Morioka, H. Ohzu, J. Opt. Soc. Am. 72, 1743 (1982).

O'Donnell, K. A.

Ohzu, H.

S. Komatsu, T. Morioka, H. Ohzu, J. Opt. Soc. Am. 72, 1743 (1982).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1981), p. 485.

Saito, H.

I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. Suppl. 1 14, 301 (1975).
[CrossRef]

Sawatari, T.

Tanner, L. H.

Thyagarajan, K.

A. Ghatak, K. Thyagarajan, Prog. Opt. 18, 12 (1980).

van Driel, H. M.

Yamaguchi, I.

I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. Suppl. 1 14, 301 (1975).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (2)

S. Komatsu, T. Morioka, H. Ohzu, J. Opt. Soc. Am. 72, 1743 (1982).

K. A. O'Donnell, J. Opt. Soc. Am. 72, 191 (1982).
[CrossRef]

Jpn. J. Appl. Phys. Suppl. 1 (1)

I. Yamaguchi, S. Komatsu, H. Saito, Jpn. J. Appl. Phys. Suppl. 1 14, 301 (1975).
[CrossRef]

Prog. Opt. (2)

L. Mandel, Prog. Opt. 2, 181 (1963).
[CrossRef]

A. Ghatak, K. Thyagarajan, Prog. Opt. 18, 12 (1980).

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 57–100.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1981), p. 485.

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Figures (11)

Fig. 1
Fig. 1

Speckle pattern appearing in the focal plane of a lens with pupil function P(ξ). The speckle pattern reveals dynamic behavior caused by in-plane motion of a diffuser inserted between a monochromatic light source and the lens.

Fig. 2
Fig. 2

Schematic diagram of the MTF measuring system based on the dynamic properties of laser-produced speckles.

Fig. 3
Fig. 3

Typical photocurrent due to dynamic speckles, and the autocorrelation function of the current which is calculated with 130 × 384 samples of data.

Fig. 4
Fig. 4

MTF curves measured with the system shown in Fig. 2. Dependence on the aperture size is shown for F-Nos. 2, 4, 5.6, and 8. The intensity fluctuations are detected on the best focal plane for F/5.6 of an ordinary camera lens (F:2/f = 55 mm). The vertical lines represent the standard deviations of the measured values, each of which is calculated from 100 × 256 sampled data.

Fig. 5
Fig. 5

Dependence of the average and the standard deviation of the measured MTF on the number N of averaging.

Fig. 6
Fig. 6

Measured MTFs at two points A and B in the focal plane.

Fig. 7
Fig. 7

Variation along the optical axis of a number of zero crossings of the photocurrent (− ○ −), that of the differentiated current (− △ −), and the correlation value (− □ −).

Fig. 8
Fig. 8

Experimental confirmation of the relation in Eq. (5).

Fig. 9
Fig. 9

Searching algorithm for the minimum point of the zero-crossing distribution.

Fig. 10
Fig. 10

Spot diagrams of a GRIN-rod lens at the paraxial focus (P = 0), the marginal focus (P = 1), and near the best focus (P = 0.75).

Fig. 11
Fig. 11

Measured MTF of a GRIN-rod lens and its geoemtrical MTFs calculated from the spot diagrams in Fig. 10.

Tables (1)

Tables Icon

Table I Standard Deviations and Processing Times of Three Methods for Finding the Best Focal Point; (the Far Right-Hand Side Column Shows the Expected Values of the Standard Deviation Which Would be Obtained for Measurements of 1-sec Duration)

Equations (10)

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C i ( τ ; r ) = | U ( t ; r ) U ( t + τ ; r ) | 2 ,
U ( t ; r ) = exp { jk / ( 2 b ) [ 1 ( a / b ) ( 1 + a / R ) ] | r | 2 } × d ( x + v t ) exp [ j 2 π / ( λ b ) ( 1 + a / R ) r x ] d x ,
C i ( τ ) = | P ( ξ ) P * [ ξ + ( 1 + a / R ) v τ ] d ξ | 2 .
ν = s ( 1 + a / R ) / ( λ b ) = α τ
cos [ π p ( τ ) ] = C ( τ ) ,
λ z = p ( τ ) / τ = 1 / π C ( 2 ) ( 0 ) / C ( 0 ) ( 0 ) = 1 / π ω 2 S ( ω ) d ω / S ( ω ) d ω ,
C i ( τ ) = | H ( α τ ) | 2 ;
S ( ω ) = F { | H ( α τ ) | 2 } = l s ( ω / α ) * l s ( ω / α ) ,
n ( r ) = n 0 ( 1 ½ A r 2 ) ,
σ T

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