Abstract

A description is given of the manner in which two and three internal reflection roof prisms bring about a deterioration in the diffraction pattern form. Assuming a rectangular limiting aperture, the deterioration consists of a doubling of the diffraction pattern in a direction perpendicular to the roof edge. Parallel to the roof edge no deterioration of the diffraction pattern takes place so that in this direction it is always of the Fraunhofer form. If plane polarized light is allowed to enter these prisms, the form of the doubling in a direction perpendicular to the roof edge can be changed by rotating the incident azimuth angle of the entering plane polarized light. Usually such a doubling of the diffraction pattern is associated with an error in the roof angle of the prism. The theory presented both in this paper and the previous paper predicts a doubling of the diffraction pattern in a direction perpendicular to the roof edge even for a prism with a perfect 90° roof angle. As compared with the earlier paper on this same subject, the theory presented in this paper is considerably simplified so that for prisms with uncoated reflecting surfaces the diffraction pattern form can now be calculated for a two or three internal reflection roof prism from a knowledge only of the angular deviation at the roof edge, the prism refractive index, and the entering azimuth angle of the plane polarized light if the prism is illuminated with plane polarized light. Experimentally it was found that silvering the roof surfaces for two different forms of prisms decreased the doubling of the diffraction pattern in a direction perpendicular to the roof edge. The theory presented in this paper makes it possible to calculate also the diffraction pattern forms for roof prisms with coated reflecting surfaces providing the refractive index and absorption index of the coating material are known. In this paper the theory will be carefully tested on the 90° deviation Amici roof prism by comparing the microphotometer tracings of the photographed diffraction patterns with those to be expected from the theory. Several conclusions about the forms of the diffraction patterns for roof prisms having a wide range in geometry will be presented both when these prisms have coated and uncoated reflecting roof surfaces.

© 1950 Optical Society of America

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References

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  1. A. I. Mahan, J. Opt. Soc. Am. 35, 623 (1945).
    [CrossRef]
  2. G. Joos, Zeiss Nachrichten 4, 9 (1943).
  3. The procedure for combining these waves two at a time is shown theoretically and experimentally in A. I. MahanJ. Opt. Soc. Am. 37, 852 (1947).
    [CrossRef] [PubMed]
  4. M. Born, Optik (Verlagsbuchhandlung, Julius Springer, Berlin, 1933), p. 43.
  5. W. König, Handbuch der Physik, Vol. 20 (Julius SpringerBerlin; 1928), pp. 202–207; 240–243.
  6. J. T. Tate, Phys. Rev. 34, 327 (1912).

1947 (1)

1945 (1)

1943 (1)

G. Joos, Zeiss Nachrichten 4, 9 (1943).

1912 (1)

J. T. Tate, Phys. Rev. 34, 327 (1912).

Born, M.

M. Born, Optik (Verlagsbuchhandlung, Julius Springer, Berlin, 1933), p. 43.

Joos, G.

G. Joos, Zeiss Nachrichten 4, 9 (1943).

König, W.

W. König, Handbuch der Physik, Vol. 20 (Julius SpringerBerlin; 1928), pp. 202–207; 240–243.

Mahan, A. I.

Tate, J. T.

J. T. Tate, Phys. Rev. 34, 327 (1912).

J. Opt. Soc. Am. (2)

Phys. Rev. (1)

J. T. Tate, Phys. Rev. 34, 327 (1912).

Zeiss Nachrichten (1)

G. Joos, Zeiss Nachrichten 4, 9 (1943).

Other (2)

M. Born, Optik (Verlagsbuchhandlung, Julius Springer, Berlin, 1933), p. 43.

W. König, Handbuch der Physik, Vol. 20 (Julius SpringerBerlin; 1928), pp. 202–207; 240–243.

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Figures (22)

Fig. 1
Fig. 1

Three-dimensional ray diagram for a 90° deviation Amici roof prism.

Fig. 2
Fig. 2

Ray diagram for 90° deviation Amici roof prism.

Fig. 3
Fig. 3

Directions of vibration of the perpendicular and parallel components of the electric vector at the two second reflecting roof surfaces for a 90° deviation Amici roof prism and the final directions about which they were resolved in the emerging aperture of the prism.

Fig. 4
Fig. 4

Ray diagram for a generalized roof prism.

Fig. 5
Fig. 5

Forms of diffraction patterns in unpolarized light in a direction perpendicular to the roof edge for roof prisms with uncoated roof surfaces having different angular deviations D at the roof edge.

Fig. 6
Fig. 6

Difference in phase between the perpendicular and parallel components of the electric vector for different angles of incidence on a glass-silver reflecting surface.

Fig. 7
Fig. 7

Fractions of the incident intensity reflected at a glass-silver reflecting surface for different angles of incidence for directions of polarization of the electric vector parallel and perpendicular to the plane of incidence.

Fig. 8
Fig. 8

Form of diffraction patterns in unpolarized light in a direction perpendicular to the roof edge for roof prisms with silver-coated roof surfaces having different angular deviations D at the roof edge.

Fig. 9
Fig. 9

Form of diffraction pattern in a direction parallel to the roof edge.

Fig. 10
Fig. 10

Forms of diffraction patterns in a direction perpendicular to the roof edge for a 90° deviation Amici roof prism with uncoated roof surfaces illuminated with plane polarized light of different entering azimuth angles.

Fig. 11
Fig. 11

Form of diffraction pattern in a direction perpendicular to the roof edge for a 90° deviation Amici roof prism with uncoated roof surfaces illuminated with unpolarized light.

Fig. 12
Fig. 12

Form of diffraction pattern in a direction perpendicular to the roof edge for a 90° deviation Amici roof prism having silver-coated roof surfaces and illuminated by unpolarized light.

Fig. 13
Fig. 13

Schematic diagram of optical system.

Fig. 14
Fig. 14

Amici roof prism surface contour photographs.

Fig. 15
Fig. 15

Fresnel diffraction pattern of Amici roof prism in vicinity of restricting aperture.

Fig. 16
Fig. 16

A typical microphotometer trace of a diffraction pattern and density calibration.

Fig. 17
Fig. 17

Method of converting microphotometered diffraction pattern to log-E pattern.

Fig. 18
Fig. 18

Experimental and theoretical forms of Fraunhofer pattern plotted on log basis.

Fig. 19
Fig. 19

Experimental and theoretical forms of diffraction patterns in a direction perpendicular to the roof edge plotted on log basis for 90° deviation Amici roof prism with uncoated roof surfaces when illuminated with plane polarized light of entering azimuth angles α.

Fig. 20
Fig. 20

Experimental and theoretical forms of diffraction patterns in a direction perpendicular to the roof edge plotted on log basis for 90° deviation Amici roof prism with uncoated roof surfaces when illuminated with unpolarized light.

Fig. 21
Fig. 21

Experimental and theoretical forms of the diffraction patterns in a direction perpendicular to the roof edge plotted on a log basis for the 90° deviation Amici roof prism with silver-coated roof surfaces when illuminated with unpolarized light.

Fig. 22
Fig. 22

Photographs of the diffraction patterns.

Tables (2)

Tables Icon

Table I Direction cosines of incident rays, reflected rays, “s” and “p” vibrations and equations of planes of incidence for rays entering upper half of prism aperture and emerging from lower.

Tables Icon

Table II Direction cosines of incident rays, reflected rays, “s” and “p” vibrations and equations of planes of incidence for rays entering lower half of prism aperture and emerging from upper.

Equations (46)

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A s 3 = A 0 { cos 2 α ( cos 2 β cos 2 γ + sin 2 β sin 2 γ ) + sin 2 α ( sin 2 β cos 2 γ + cos 2 β sin 2 γ ) - 1 2 sin 2 α sin 2 β cos 2 γ cos ( d s 1 - d p 1 ) - sin 2 γ [ cos 2 α cos 2 β + sin 2 α sin 2 β - 1 2 sin 2 α sin 2 β cos ( d s 1 - d p 1 ) ] 1 2 · [ cos 2 α sin 2 β + sin 2 α cos 2 β + 1 2 sin 2 α sin 2 β cos ( d s 1 - d p 1 ) ] 1 2 · cos [ ( R s 2 - R p 2 ) + ( d s 2 - d p 2 ) ] } 1 2 · sin ( ω t + d i + d g + R s 3 ) ,
A p 3 = A 0 { cos 2 α ( cos 2 β sin 2 γ + sin 2 β cos 2 γ ) + sin 2 α ( sin 2 β sin 2 γ + cos 2 β cos 2 γ ) + 1 2 sin 2 α sin 2 β cos 2 γ cos ( d s 1 - d p 1 ) + sin 2 γ [ cos 2 α cos 2 β + sin 2 α sin 2 β - 1 2 sin 2 α sin 2 β cos ( d s 1 - d p 1 ) ] 1 2 · [ cos 2 α sin 2 β + sin 2 α cos 2 β + 1 2 sin 2 α sin 2 β cos ( d s 1 - d p 1 ) ] 1 2 · cos [ ( R s 2 - R p 2 ) + ( d s 2 - d p 2 ) ] } 1 2 · sin ( ω t + d i + d g + R p 3 ) .
tan R s 2 = cos α cos β sin d s 1 - sin α sin β sin d p 1 cos α cos β cos d s 1 - sin α sin β cos d p 1 ,
tan R p 2 = cos α sin β sin d s 1 + sin α cos β sin d p 1 cos α sin β cos d s 1 + sin α cos β cos d p 1 ,
tan R s 3 = C sin ( R s 2 + d s 2 ) - D sin ( R p 2 + d p 2 ) C cos ( R s 2 + d s 2 ) - D cos ( R p 2 + d p 2 ) ,
tan R p 3 = E sin ( R s 2 + d s 2 ) + F sin ( R p 2 + d p 2 ) E cos ( R s 2 + d s 2 ) + F cos ( R p 2 + d p 2 ) .
C = cos γ { cos 2 α cos 2 β + sin 2 α sin 2 β - 1 2 sin 2 α sin 2 β cos ( d s 1 - d p 1 ) } 1 2 ,
D = sin γ { cos 2 α sin 2 β + sin 2 α cos 2 β + 1 2 sin 2 α sin 2 β     cos ( d s 1 - d p 1 ) } 1 2 ,
E = sin γ { cos 2 α cos 2 β + sin 2 α sin 2 β - 1 2 sin 2 α sin 2 β cos ( d s 1 - d p 1 ) } 1 2 ,
F = cos γ { cos 2 α sin 2 β + sin 2 α cos 2 β + 1 2 sin 2 α sin 2 β cos ( d s 1 - d p 1 ) } 1 2 .
J = ( 4 a b ) 2 · sin 2 U U 2 · sin 2 V 2 2 ( V 2 ) 2 × { A 0 2 + Ā s 3 A ¯ s 3 cos [ V - ( R s 3 - s 3 ) ] + Ā p 3 A ¯ p 3 cos [ V - ( R p 3 - p 3 ) ] } ,
U = 2 π a ξ λ F ,             V = 2 π b μ λ F .
J = ( 4 a b ) 2 · sin 2 U U 2 · sin 2 V 2 2 ( V 2 ) 2 { A 0 2 + [ Ā s 3 A ¯ s 3 cos R s 3 cos s 3 + Ā s 3 A ¯ s 3 sin R s 3 sin s 3 + Ā p 3 A ¯ p 3 cos R p 3 cos p 3 + Ā p 3 A ¯ p 3 sin R p 3 sin p 3 ] cos V + [ Ā s 3 A ¯ s 3 sin R s 3 cos s 3 - Ā s 3 A ¯ s 3 cos R s 3 sin s 3 + Ā p 3 A ¯ p 3 sin R p 3 cos p 3 - Ā p 3 A ¯ p 3 cos R p 3 sin p 3 ] sin V } .
J = ( 4 a b ) 2 · sin 2 U U 2 · sin 2 V 2 2 ( V 2 ) 2 { A 0 2 + G cos V + K sin V } ,
G = A 0 2 { [ C cos ( R s 2 + d s 2 ) - D cos ( R p 2 + d p 2 ) ] [ C cos ( s 2 + d s 2 ) - D cos ( p 2 + d p 2 ) ] + [ C sin ( R s 2 + d s 2 ) - D sin ( R p 2 + d p 2 ) ] [ C sin ( s 2 + d s 2 ) - D sin ( p 2 + d p 2 ) ] + [ E cos ( R s 2 + d s 2 ) + F cos ( R p 2 + d p 2 ) ] [ E cos ( s 2 + d s 2 ) + F cos ( p 2 + d p 2 ) ] + [ E sin ( R s 2 + d s 2 ) + F sin ( R p 2 + d p 2 ) ] [ E sin ( s 2 + d s 2 ) + F sin ( p 2 + d p 2 ) ] } ,
K = A 0 2 { [ C sin ( R s 2 + d s 2 ) - D sin ( R p 2 + d p 2 ) ] [ C cos ( s 2 + d s 2 ) - D cos ( p 2 + d p 2 ) ] - [ C cos ( R s 2 + d s 2 ) - D cos ( R p 2 + d p 2 ) ] [ C sin ( s 2 + d s 2 ) - D sin ( p 2 + d p 2 ) ] + [ E sin ( R s 2 + d s 2 ) + F sin ( R p 2 + d p 2 ) ] [ E cos ( s 2 + d s 2 ) + F cos ( p 2 + d p 2 ) ] - [ E cos ( R s 2 + d s 2 ) + F cos ( R p 2 + d p 2 ) ] [ E sin ( s 2 + d s 2 ) + F sin ( p 2 + d p 2 ) ] } .
G = A 0 2 { cos ( α - δ ) cos ( β - ) cos ( γ - η ) - sin ( α - δ ) sin ( β - ) cos ( γ - η ) cos ( d s 1 - d p 1 ) - cos ( α - δ ) sin ( β - ) sin ( γ - η ) cos ( d s 2 - d p 2 ) - sin ( α - δ ) cos ( β - ) sin ( γ - η ) · cos ( d s 1 - d p 1 ) cos ( d s 2 - d p 2 ) + sin ( α - δ ) cos ( β + ) sin ( γ - η ) sin ( d s 1 - d p 1 ) sin ( d s 2 - d p 2 ) } ,
K = A 0 2 { sin ( α + δ ) sin ( β - ) cos ( γ - η ) sin ( d s 1 - d p 1 ) + sin ( α + δ ) cos ( β - ) sin ( γ - η ) · sin ( d s 1 - d p 1 ) cos ( d s 2 - d p 2 ) + cos ( α + δ ) sin ( β + ) sin ( γ - η ) sin ( d s 2 - d p 2 ) + sin ( α + δ ) cos ( β + ) sin ( γ - η ) cos ( d s 1 - d p 1 ) sin ( d s 2 - d p 2 ) } .
= - β ,             η = - γ .
δ - α = H .
d s 1 - d p 1 = d s 2 - d p 2 = Φ .
G = A 0 2 { sin H [ sin 2 β cos 2 γ cos Φ + cos 2 β sin 2 γ cos 2 Φ - sin 2 γ sin 2 Φ + cos H [ cos 2 β cos 2 γ - sin 2 β sin 2 γ cos Φ ] } ,
K = A 0 2 { sin ( 2 α + H ) sin Φ [ sin 2 β cos 2 γ + cos 2 β sin 2 γ cos Φ + sin 2 γ cos Φ ] } .
sin β = 2 cos 1 2 D cos 2 1 2 D + 1 , cos β = sin 2 1 2 D cos 2 1 2 D + 1 , γ = 90 ° + β / 2 , H = 180 ° - β .
J = ( 4 a b ) 2 · sin 2 U U 2 · sin 2 V 2 2 ( V 2 ) 2 A 0 2 × { 1 + [ 1 - 32 cos 2 1 2 D sin 4 1 2 D ( cos 2 1 2 D + 1 ) 4 cos 4 Φ 2 ] cos V + 8 sin 4 1 2 D cos 1 2 D sin Φ cos 2 Φ 2 ( cos 2 1 2 D + 1 ) 4 × [ sin 2 α sin 2 1 2 D - 2 cos 2 α cos 1 2 D ] sin V } .
tan Φ 2 = - cos i ( n g 2 sin 2 i - 1 ) 1 2 n g sin 2 i .
sin i = ( 3 + cos D ) 1 2 2 .
J = ( 4 a b ) 2 A 0 2 · sin 2 U U 2 · sin 2 V 2 2 ( V 2 ) 2 × { 1 + [ 1 - 8 n g 4 sin 4 1 2 D cos 2 1 2 D [ n g 2 ( 1 + cos 2 1 2 D ) - sin 2 1 2 D ] 2 ] cos V - 4 n g 3 sin 5 1 2 D cos 1 2 D [ n g 2 ( 1 + cos 2 1 2 D ) - 2 ] 1 2 [ n g 2 ( 1 + cos 2 1 2 D ) - sin 2 1 2 D ] 2 ( 1 + cos 2 1 2 D ) [ sin 2 α sin 2 1 2 D - 2 cos 2 α cos 1 2 D ] sin V } .
cos θ = 1 ( cos 2 1 2 D + 1 ) 1 2 .
J = ( 4 a b ) 2 A 0 2 · sin 2 U U 2 · sin 2 V 2 2 ( V 2 ) 2 × { 1 + [ 1 - 8 n g 4 sin 4 1 2 D cos 2 1 2 D [ n g 2 ( 1 + cos 2 1 2 D ) - sin 2 1 2 D ] 2 ] cos V } .
sin 1 2 D = { 3 n g 2 - n g ( 5 n g 2 - 4 ) 1 2 n g 2 + 1 } 1 2 .
tan α = - sin 2 1 2 D 2 cos 1 2 D ± ( sin 4 1 2 D 4 cos 2 1 2 D + 1 ) 1 2 .
J = ( 4 a b ) 2 A 0 2 · sin 2 U U 2 · sin 2 V 2 2 ( V 2 ) 2 × { 1 + 4 sin 4 1 2 D cos 2 1 2 D ( cos 2 1 2 D + 1 ) 4 [ sin 4 1 2 D 8 cos 2 1 2 D ( ρ s 4 + ρ p 4 ) - 2 cos Φ ( ρ s 3 ρ p + ρ s ρ p 3 ) - ρ s 2 ρ p 2 × ( 2 cos 2 Φ - 4 cos 2 1 2 D sin 4 1 2 D ) ] cos V } .
tan Φ = - 2 n g B sin i tan i n g 2 sin 2 i tan 2 i - ( A 2 + B 2 ) ,
ρ s 2 = A 2 + B 2 - 2 n g A cos i + n g 2 cos 2 i A 2 + B 2 + 2 n g A cos i + n g 2 cos 2 i ,
ρ p 2 = ρ s 2 A 2 + B 2 - 2 n g A sin i tan i + n g 2 sin 2 i tan 2 i A 2 + B 2 + 2 n g A sin i tan i + n g 2 sin 2 i tan 2 i .
A 2 = 1 2 { ( [ n m 2 ( 1 - χ m 2 ) - n g 2 sin 2 i ] 2 + 4 n m 4 χ m 2 ) 1 2 + n m 2 ( 1 - χ m 2 ) - n g 2 sin 2 i } ,
B 2 = 1 2 { ( [ n m 2 ( 1 - χ m 2 ) - n g 2 sin 2 i ] 2 + 4 n m 4 χ m 2 ) 1 2 - n m 2 ( 1 - χ m 2 ) + n g 2 sin 2 i } .
sin 2 V 2 2 ( V 2 ) 2 [ 1 + ρ s 2 ρ p 2 cos V ] , ( D = 0 ° ) sin 2 V 2 2 ( V 2 ) 2 [ 1 + ρ s 4 + ρ p 4 2 cos V ] . ( D = 180 ° )
sin 2 V V 2 , ( D = 0 ° , i = 90 ° ) sin 2 V 2 2 ( V 2 ) 2 [ 1 + ρ s 4 + ρ p 4 2 cos V ] . ( D = 180 ° , i = 45 ° )
J = ( 4 a b ) 2 A 0 2 · sin 2 U U 2 · sin 2 V 2 2 ( V 2 ) 2 × { 1 + 0.392759 cos V - ( 0.227767 cos 2 α - 0.080528 sin 2 α ) sin V } .
ξ = λ F 2 a · m .             m = 1 , 2 , 3 , etc .
μ = λ F b · n ,             n = 1 , 2 , 3 , etc .
n g = 1.5184 ,             n m = 0.530 ,             χ m = 8.62.
ρ s = 0.96749 ,             ρ p = 0.89042 ,             Φ = 226 ° 3 0 3 4 .
J = ( 4 a b ) 2 A 0 2 · sin 2 U U 2 · sin 2 V 2 2 ( V 2 ) 2 × { 1 + 0.728702 cos V } .