Abstract

Accurate photometric measurements depend on the linearity of the detection system, i.e., whether the output is strictly proportional to the incident light flux. The usual method for checking linearity is to introduce filters of known absorption into the optical path. Unfortunately, the many possible errors inherent in this method make it difficult to determine linearity in this way to better than 1%. By using three polarizers in series, keeping the axes of the outer two parallel and rotating the middle polarizer, it is possible to eliminate most of these sources of error. If polarizers of the highest quality are used, photometric linearity may be determined to better than 0.1%. Accurate values for the transmission of standard filters can also be determined with this instrument. The technique is particularly useful for calibrating filters having large optical densities. An error analysis and some experimental results obtained using a three-polarizer system are given.

© 1966 Optical Society of America

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References

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  1. A. L. Olsen, K. B. LaBaw, L. W. Nichols, J. Opt. Soc. Am. 54, 813 (1964).
    [CrossRef]
  2. A. L. Olsen, W. R. McBride, J. Opt. Soc. Am. 53, 1003 (1963).
    [CrossRef]
  3. K. S. Gibson, in Precision Measurement and Calibration, Selected Papers on Optics, Metrology, and Radiation, S. F. Booth, Ed. (National Bureau of Standards Handbook 77—Vol. III, Washington, D. C., 1961), pp. 381–431.
  4. F. J. Howard, J. M. Hood, S. S. Ballard, J. Opt. Soc. Am. 45, 904 (1955); W. A. Shurcliff, S. S. Ballard, Polarized Light (D. Van Nostrand Company, Inc., New York, 1964), p. 107.
  5. R. Clark Jones, J. Opt. Soc. Am. 46, 528 (1956).
    [CrossRef]
  6. W. A. Shurcliff, Polarized Light (Harvard University Press, Cambridge, Mass., 1962), p. 53.
  7. H. E. Bennett, W. F. Koehler, J. Opt. Soc. Am. 50, 1 (1960).
    [CrossRef]
  8. R. W. Engstrom, J. Opt. Soc. Am. 37, 420 (1947).
    [CrossRef]
  9. W. Hermann, Z. Naturforsch. 12a, 1006 (1957).
  10. D. J. Baker, C. L. Wyatt, Appl. Opt. 3, 89 (1964).
    [CrossRef]
  11. R. L. Williams, J. Opt. Soc. Am. 52, 1237 (1962).
    [CrossRef]
  12. J. M. Vandenbelt, J. Opt. Soc. Am. 52, 284 (1962).
    [CrossRef]

1964 (2)

1963 (1)

1962 (2)

1960 (1)

1957 (1)

W. Hermann, Z. Naturforsch. 12a, 1006 (1957).

1956 (1)

1955 (1)

F. J. Howard, J. M. Hood, S. S. Ballard, J. Opt. Soc. Am. 45, 904 (1955); W. A. Shurcliff, S. S. Ballard, Polarized Light (D. Van Nostrand Company, Inc., New York, 1964), p. 107.

1947 (1)

Baker, D. J.

Ballard, S. S.

F. J. Howard, J. M. Hood, S. S. Ballard, J. Opt. Soc. Am. 45, 904 (1955); W. A. Shurcliff, S. S. Ballard, Polarized Light (D. Van Nostrand Company, Inc., New York, 1964), p. 107.

Bennett, H. E.

Clark Jones, R.

Engstrom, R. W.

Gibson, K. S.

K. S. Gibson, in Precision Measurement and Calibration, Selected Papers on Optics, Metrology, and Radiation, S. F. Booth, Ed. (National Bureau of Standards Handbook 77—Vol. III, Washington, D. C., 1961), pp. 381–431.

Hermann, W.

W. Hermann, Z. Naturforsch. 12a, 1006 (1957).

Hood, J. M.

F. J. Howard, J. M. Hood, S. S. Ballard, J. Opt. Soc. Am. 45, 904 (1955); W. A. Shurcliff, S. S. Ballard, Polarized Light (D. Van Nostrand Company, Inc., New York, 1964), p. 107.

Howard, F. J.

F. J. Howard, J. M. Hood, S. S. Ballard, J. Opt. Soc. Am. 45, 904 (1955); W. A. Shurcliff, S. S. Ballard, Polarized Light (D. Van Nostrand Company, Inc., New York, 1964), p. 107.

Koehler, W. F.

LaBaw, K. B.

McBride, W. R.

Nichols, L. W.

Olsen, A. L.

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard University Press, Cambridge, Mass., 1962), p. 53.

Vandenbelt, J. M.

Williams, R. L.

Wyatt, C. L.

Appl. Opt. (1)

J. Opt. Soc. Am. (8)

Z. Naturforsch. (1)

W. Hermann, Z. Naturforsch. 12a, 1006 (1957).

Other (2)

W. A. Shurcliff, Polarized Light (Harvard University Press, Cambridge, Mass., 1962), p. 53.

K. S. Gibson, in Precision Measurement and Calibration, Selected Papers on Optics, Metrology, and Radiation, S. F. Booth, Ed. (National Bureau of Standards Handbook 77—Vol. III, Washington, D. C., 1961), pp. 381–431.

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Figures (6)

Fig. 1
Fig. 1

Schematic view of the three-polarizer system. The transmission axes of the outer two polarizers, indicated by the parallel lines, are parallel. The transmission is decreased by rotating the middle polarizer through an angle θ.

Fig. 2
Fig. 2

Transmittance of the three-polarizer system as a function of middle polarizer angle (solid curve), and transmittance errors ΔI/I0 and ΔI/I (dashed and dashed-dotted curves) caused by a 1-min-of-arcsetting error for the middle polarizer.

Fig. 3
Fig. 3

Exploded view of three-polarizer system with fixed polarizer A, middle polarizer B, coarse angular adjustment knob C, fine adjustment knob D, and brake E. Dimensions of the unit are 15 cm × 12.5 cm × 6.5 cm excluding the knobs.

Fig. 4
Fig. 4

Nonlinearity of a 5-cm diam, end-on type, photomultiplier tube. Even if used at low light levels, a photomultiplier must be selected if linearity of better than 1% is desired.

Fig. 5
Fig. 5

Nonlinear response expected from a saturable detector such as the diffused junction silicon detector tested.

Fig. 6
Fig. 6

Observed nonlinearity of a 0.2-μ diffusion depth silicon detector. The shape of the nonlinearity curve is as predicted from Fig. 5.

Equations (6)

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I ( θ ) = 1 2 k 1 k 2 ( k 1 + k 2 ) + 1 2 ( k 1 + k 2 ) ( k 1 - k 2 ) 2 cos 4 θ + 1 4 ( k 1 k 2 ) ½ ( k 1 - k 2 ) 2 cos δ sin 2 2 θ ,
I ( θ ) 1 2 k 1 2 k 2 + 1 2 k 1 3 cos 4 θ + 1 4 k 1 / k 2 ½ cos δ sin 2 2 θ .
I I 0 = 1 2 k 1 3 [ cos 4 θ + 1 2 ( k 2 / k 1 ) ½ cos δ sin 2 2 θ + ( k 2 / k 1 ) ] 1 2 k 1 2 ( k 1 + k 2 ) ,
I / I 0 = cos 4 θ + 1 2 ( k 2 / k 1 ) ½ cos δ sin 2 2 θ + k 2 / k 1 .
I / I 0 = cos 2 θ cos 2 ( θ ± α ) ,
Δ I / I 0 = - 4 ( I / I 0 ) ( tan θ ) Δ θ .

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