Abstract

A method is outlined by which the limiting resolving power of any prism may be readily determined. It has been necessary to redefine the limit of resolution as the minimum angular of the central maxima of the individual diffraction images at which there may exist a minimum of total intensity between the positions of the individual central maxima. Application to a 10-cm rocksalt prism is included.

© 1944 Optical Society of America

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References

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  1. The resolving power of spectrometers under conditions such that the limiting factor is the finite slit widths necessary to obtain the minimum energy required for measurement has been discussed by J. Strong, Phys. Rev. 37, 1661 (1931).
    [Crossref]
  2. Rayleigh’s statement in the Encyclopaedia Britannica, Vol.  XXIV (1888), is: “We conclude that a double line cannot be fairly resolved unless its components subtend an angle exceeding that subtended by the wave-length of light at a distance equal to the horizontal aperture.” Note that this statement contains no basis for distinguishing between the cases of the absorbing and the non-absorbing prisms.
  3. H. M. Reese, Astrophys. J. 13, 199 (1901).
    [Crossref]
  4. This is valid for all values of f since, as f→0, r→∞ in such a way that rf is finite.
  5. The limiting values given in (7) and (8) are obtained from the equations(6a)6+(r2f2w2-6) cos rfw-4rfw sin rfw=0and(6b)6r2-2=0to which (6) may be reduced in the limits fw→0 and fw→∞, respectively. Note that w is finite and ≫0 since W≫λ.
  6. It is assumed throughout that n, dn/dλ, and k are evaluated at the given wave-length λ.
  7. See, for example, R. W. Wood, Physical Optics (The Macmillan Company, New York, 1934), third edition, p. 251.
  8. H. M. Reese (reference 3) discusses in some detail the determination of minima and maxima in I(θ).
  9. In the calculation of λ/Δλ for rocksalt, n and dn/dλ were obtained from the data of F. Paschen [Ann. d. Physik 26, 120 (1908)] as corrected to 25° by P. C. Cross [Rev. Sci. Inst. 4, 197 (1933)]. The values of k were determined from the transmission data of H. Rubens and A. Trowbridge [Ann. d. Physik 60, 724 (1897)].
    [Crossref]

1931 (1)

The resolving power of spectrometers under conditions such that the limiting factor is the finite slit widths necessary to obtain the minimum energy required for measurement has been discussed by J. Strong, Phys. Rev. 37, 1661 (1931).
[Crossref]

1908 (1)

In the calculation of λ/Δλ for rocksalt, n and dn/dλ were obtained from the data of F. Paschen [Ann. d. Physik 26, 120 (1908)] as corrected to 25° by P. C. Cross [Rev. Sci. Inst. 4, 197 (1933)]. The values of k were determined from the transmission data of H. Rubens and A. Trowbridge [Ann. d. Physik 60, 724 (1897)].
[Crossref]

1901 (1)

H. M. Reese, Astrophys. J. 13, 199 (1901).
[Crossref]

1888 (1)

Rayleigh’s statement in the Encyclopaedia Britannica, Vol.  XXIV (1888), is: “We conclude that a double line cannot be fairly resolved unless its components subtend an angle exceeding that subtended by the wave-length of light at a distance equal to the horizontal aperture.” Note that this statement contains no basis for distinguishing between the cases of the absorbing and the non-absorbing prisms.

Paschen, F.

In the calculation of λ/Δλ for rocksalt, n and dn/dλ were obtained from the data of F. Paschen [Ann. d. Physik 26, 120 (1908)] as corrected to 25° by P. C. Cross [Rev. Sci. Inst. 4, 197 (1933)]. The values of k were determined from the transmission data of H. Rubens and A. Trowbridge [Ann. d. Physik 60, 724 (1897)].
[Crossref]

Rayleigh,

Rayleigh’s statement in the Encyclopaedia Britannica, Vol.  XXIV (1888), is: “We conclude that a double line cannot be fairly resolved unless its components subtend an angle exceeding that subtended by the wave-length of light at a distance equal to the horizontal aperture.” Note that this statement contains no basis for distinguishing between the cases of the absorbing and the non-absorbing prisms.

Reese, H. M.

H. M. Reese, Astrophys. J. 13, 199 (1901).
[Crossref]

H. M. Reese (reference 3) discusses in some detail the determination of minima and maxima in I(θ).

Strong, J.

The resolving power of spectrometers under conditions such that the limiting factor is the finite slit widths necessary to obtain the minimum energy required for measurement has been discussed by J. Strong, Phys. Rev. 37, 1661 (1931).
[Crossref]

Wood, R. W.

See, for example, R. W. Wood, Physical Optics (The Macmillan Company, New York, 1934), third edition, p. 251.

Ann. d. Physik (1)

In the calculation of λ/Δλ for rocksalt, n and dn/dλ were obtained from the data of F. Paschen [Ann. d. Physik 26, 120 (1908)] as corrected to 25° by P. C. Cross [Rev. Sci. Inst. 4, 197 (1933)]. The values of k were determined from the transmission data of H. Rubens and A. Trowbridge [Ann. d. Physik 60, 724 (1897)].
[Crossref]

Astrophys. J. (1)

H. M. Reese, Astrophys. J. 13, 199 (1901).
[Crossref]

Encyclopaedia Britannica (1)

Rayleigh’s statement in the Encyclopaedia Britannica, Vol.  XXIV (1888), is: “We conclude that a double line cannot be fairly resolved unless its components subtend an angle exceeding that subtended by the wave-length of light at a distance equal to the horizontal aperture.” Note that this statement contains no basis for distinguishing between the cases of the absorbing and the non-absorbing prisms.

Phys. Rev. (1)

The resolving power of spectrometers under conditions such that the limiting factor is the finite slit widths necessary to obtain the minimum energy required for measurement has been discussed by J. Strong, Phys. Rev. 37, 1661 (1931).
[Crossref]

Other (5)

This is valid for all values of f since, as f→0, r→∞ in such a way that rf is finite.

The limiting values given in (7) and (8) are obtained from the equations(6a)6+(r2f2w2-6) cos rfw-4rfw sin rfw=0and(6b)6r2-2=0to which (6) may be reduced in the limits fw→0 and fw→∞, respectively. Note that w is finite and ≫0 since W≫λ.

It is assumed throughout that n, dn/dλ, and k are evaluated at the given wave-length λ.

See, for example, R. W. Wood, Physical Optics (The Macmillan Company, New York, 1934), third edition, p. 251.

H. M. Reese (reference 3) discusses in some detail the determination of minima and maxima in I(θ).

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Figures (4)

Fig. 2
Fig. 2

Geometrical relations for various optical paths through the prism.

Equations (17)

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I ( θ 0 + d θ ) + I ( θ 0 - d θ ) 2 I ( θ 0 ) .
( d 2 I / d θ 2 ) θ 0 0.
I = K E / Z ,
d I / d θ = K ( 2 w e - f w Z sin w θ - 2 θ E ) / Z 2 ,
d 2 I / d θ 2 = K [ 2 w 2 e - f w Z 2 cos w θ + 8 θ 2 E - 2 Z ( 4 w θ e - f w sin w θ + E ) ] / Z 3 .
{ ( 2 + f 2 w 2 ) + r 2 [ r 2 f 2 w 2 + ( 2 f 2 w 2 - 6 ) ] } cos r f w - 4 r f w ( 1 + r 2 ) sin r f w + ( 6 r 2 - 2 ) cosh f w = 0.
Lim f w 0 r f w = 2.606 ,
Lim f w r f w = 3 - 1 2 f w .
Δ D = λ k M / 2 · 3 1 2 π W = λ k M / 10.883 W .
0 < k M < 1 :             L.R.P. = 1.2 W ( d D / d n ) ( d n / d λ ) = 1.2 R.R.P. ( 0 ) ,
k M > 16 :             L.R.P. = 10.88 W ( d D / d n ) ( d n / d λ ) / k M ,
0 < k L < 1 :             L.R.P. = 1.2 L ( d n / d λ )
k L > 16 :             L.R.P. = 10.88 ( d n / d λ ) / k .
2 cos s f w + ( s + 1 s ) f w sin s f w - 2 cosh f w = 0.
Lim f w 0 s f w = 2 π .
6+(r2f2w2-6)cosrfw-4rfwsinrfw=0
6r2-2=0