Abstract

In the polarizing microscope set for extinction, only that image whose polarization has been altered is available to form an image. The lenses themselves introduce such an alternation by rotation of the plane of polarization of rays having oblique incidence. This paper shows that the diffraction image of a pinhole has the form sin2θ·J3(r)/r, where θ and r are polar coordinates in the image plane. The image has four bright zones separated by a dark cross and the central pattern becomes a four-leaf clover form. The diffraction image of a point source through a plate of uniaxial crystal cut perpendicular to its optic axis (z-cut) is also a four-leaf clover when the polarizers are crossed. When the polarizers are parallel, the diffraction image becomes similar to that of an astigmatic system.

© 1959 Optical Society of America

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References

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  1. B. R. A. Nijboer, Thesis, Groningen (1942).
  2. A. I. Mahan, J. Opt. Soc. Am. 40, 664 (1950), etc.; F. Zernike, J. Opt. Soc. Am. 47, 466 (1957).Dr. R. E. Hopkins kindly called our attention to this literature.
    [Crossref]
  3. H. Gamoh, Ôyö Butsuri 26, 102 (1957) (in Japanese); H. Slevogt, Optik 14, 383 (1957).
  4. F. Zernike, Physica 1, 689 (1934).
    [Crossref]
  5. C. J. Koester (private communication).
  6. E.g., M. Françon, Handbuch der Physik (Berlin, 1956), Vol. 24, p. 280.
  7. E.g., reference 6, p. 443.
  8. E.g., reference 1, p. 44.
  9. F. E. Wright, Am. J. Sci. 31, 157 (1911); J. Opt. Am. 7, 779 (1923); F. Rinne, Zentr. Mineral. Geol. 1, 88 (1900); S. Inoué, Exptl. Cell Research 3, 311 (1952); N. Rosenbusch and E. A. Würfing, Mikroskopische Physiographie (Stuttgart, 1921/4), Vol. 1, p. 640.
    [Crossref]
  10. S. Inoué and W. L. Hyde, J. Biophys. Biochem. Cytol. 3, 391 (1957).
    [Crossref]
  11. H. H. Hopkins, Proc. Phys. Soc. (London) 55, 116 (1943); B. Richards and E. Wolf, Proc. Phys. Soc. B114, 854(L) (1956); R. Burtin, Optica Acta 3, 104 (1956).
    [Crossref]
  12. H. Wolter, Handbuch der Physik (Berlin, 1956), Vol. 24, p. 582.

1957 (2)

H. Gamoh, Ôyö Butsuri 26, 102 (1957) (in Japanese); H. Slevogt, Optik 14, 383 (1957).

S. Inoué and W. L. Hyde, J. Biophys. Biochem. Cytol. 3, 391 (1957).
[Crossref]

1950 (1)

1943 (1)

H. H. Hopkins, Proc. Phys. Soc. (London) 55, 116 (1943); B. Richards and E. Wolf, Proc. Phys. Soc. B114, 854(L) (1956); R. Burtin, Optica Acta 3, 104 (1956).
[Crossref]

1934 (1)

F. Zernike, Physica 1, 689 (1934).
[Crossref]

1911 (1)

F. E. Wright, Am. J. Sci. 31, 157 (1911); J. Opt. Am. 7, 779 (1923); F. Rinne, Zentr. Mineral. Geol. 1, 88 (1900); S. Inoué, Exptl. Cell Research 3, 311 (1952); N. Rosenbusch and E. A. Würfing, Mikroskopische Physiographie (Stuttgart, 1921/4), Vol. 1, p. 640.
[Crossref]

Françon, M.

E.g., M. Françon, Handbuch der Physik (Berlin, 1956), Vol. 24, p. 280.

Gamoh, H.

H. Gamoh, Ôyö Butsuri 26, 102 (1957) (in Japanese); H. Slevogt, Optik 14, 383 (1957).

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc. (London) 55, 116 (1943); B. Richards and E. Wolf, Proc. Phys. Soc. B114, 854(L) (1956); R. Burtin, Optica Acta 3, 104 (1956).
[Crossref]

Hyde, W. L.

S. Inoué and W. L. Hyde, J. Biophys. Biochem. Cytol. 3, 391 (1957).
[Crossref]

Inoué, S.

S. Inoué and W. L. Hyde, J. Biophys. Biochem. Cytol. 3, 391 (1957).
[Crossref]

Koester, C. J.

C. J. Koester (private communication).

Mahan, A. I.

Nijboer, B. R. A.

B. R. A. Nijboer, Thesis, Groningen (1942).

Wolter, H.

H. Wolter, Handbuch der Physik (Berlin, 1956), Vol. 24, p. 582.

Wright, F. E.

F. E. Wright, Am. J. Sci. 31, 157 (1911); J. Opt. Am. 7, 779 (1923); F. Rinne, Zentr. Mineral. Geol. 1, 88 (1900); S. Inoué, Exptl. Cell Research 3, 311 (1952); N. Rosenbusch and E. A. Würfing, Mikroskopische Physiographie (Stuttgart, 1921/4), Vol. 1, p. 640.
[Crossref]

Zernike, F.

F. Zernike, Physica 1, 689 (1934).
[Crossref]

Am. J. Sci. (1)

F. E. Wright, Am. J. Sci. 31, 157 (1911); J. Opt. Am. 7, 779 (1923); F. Rinne, Zentr. Mineral. Geol. 1, 88 (1900); S. Inoué, Exptl. Cell Research 3, 311 (1952); N. Rosenbusch and E. A. Würfing, Mikroskopische Physiographie (Stuttgart, 1921/4), Vol. 1, p. 640.
[Crossref]

J. Biophys. Biochem. Cytol. (1)

S. Inoué and W. L. Hyde, J. Biophys. Biochem. Cytol. 3, 391 (1957).
[Crossref]

J. Opt. Soc. Am. (1)

Ôyö Butsuri (1)

H. Gamoh, Ôyö Butsuri 26, 102 (1957) (in Japanese); H. Slevogt, Optik 14, 383 (1957).

Physica (1)

F. Zernike, Physica 1, 689 (1934).
[Crossref]

Proc. Phys. Soc. (London) (1)

H. H. Hopkins, Proc. Phys. Soc. (London) 55, 116 (1943); B. Richards and E. Wolf, Proc. Phys. Soc. B114, 854(L) (1956); R. Burtin, Optica Acta 3, 104 (1956).
[Crossref]

Other (6)

H. Wolter, Handbuch der Physik (Berlin, 1956), Vol. 24, p. 582.

B. R. A. Nijboer, Thesis, Groningen (1942).

C. J. Koester (private communication).

E.g., M. Françon, Handbuch der Physik (Berlin, 1956), Vol. 24, p. 280.

E.g., reference 6, p. 443.

E.g., reference 1, p. 44.

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Figures (13)

Fig. 1
Fig. 1

Optical system.

Fig. 2
Fig. 2

Diffraction image (crossed polarizers).

Fig. 3
Fig. 3

Examples of the value of (kpkn).

Fig. 4
Fig. 4

Contours of equal intensity of the diffraction image.

Fig. 5
Fig. 5

Intensity distribution along θ=π/4.

Figs. 6, 7, and 8
Figs. 6, 7, and 8

Diffraction images (a); contours of equal intensity (b); and intensity distribution along θπ/4 (c) (g=α1/2γ, γ is the off-cross angle). Fig. 6. g=5. Fig. 7. g=2.5. Fig. 8. g=1.75.

Fig. 9
Fig. 9

Maxima and zeros of the diffraction images.

Fig. 10
Fig. 10

Intensity distribution of the diffraction image along θ=π/4 (annular aperture, s=a0/a).

Fig. 11
Fig. 11

Diffraction image of a pinhole through a crystal plate.

Fig. 12
Fig. 12

Observed values of the maxima and zeros (z-cut uniaxial crystal) in arbitrary scale.

Fig. 13
Fig. 13

z-cut uniaxial crystal (crossed polarizers), conoscopic image.

Figs. 14 and 15
Figs. 14 and 15

z-cut uniaxial crystal (parallel polarizers). (a) Diffraction image. (b) Contours of equal intensity.

Fig. 16
Fig. 16

Intensity along θ=0.

Tables (2)

Tables Icon

Table I Maxima of {J3(az)/az}2.

Tables Icon

Table II Values of az.

Equations (37)

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A ( r , θ ) = 0 a 0 2 π P ( ρ , φ ) exp { i z ρ cos ( θ - φ ) } ρ d ρ d φ ,
z = 2 π r / λ l .
A ( r , θ ) = 2 π 0 a J 0 ( z ρ ) ρ d ρ = 2 π a 2 J 1 ( a z ) / a z .
P ( ρ , φ ) = E ( ρ , φ ) exp { i V ( ρ , φ ) } .
E p = cos φ ,             E n = sin φ .
E A = K p cos φ · sin ( φ - γ ) - K n sin φ · cos ( φ - γ ) = 1 2 [ sin ( 2 φ - γ ) ( K p - K n ) - sin γ · ( K p + K n ) .
A ( r , θ ) = π sin ( γ - 2 θ ) 0 a ( K p - K n ) J 2 ( z ρ ) ρ d ρ - π sin γ 0 a ( K p + K n ) J 0 ( z ρ ) ρ d ρ .
( K p - K n ) = n = 1 α n R 2 n 2 ( ρ / a ) ,
( K p + K n ) = n = 0 β n R 2 n 0 ( ρ / a ) .
0 a R 2 n m ( ρ / a ) J m ( z ρ ) ρ d ρ = ( - 1 ) n - m / 2 a 2 J 2 n + 1 ( a z ) / a z ,
A ( r , θ ) = π a 2 { sin ( γ - 2 θ ) F 1 ( z ) + sin γ F 2 ( z ) } ,
F 1 ( z ) = { α 1 J 3 ( a z ) - α 2 J 5 ( a z ) + α 3 J 7 ( a z ) - } / a z , F 2 ( z ) = - { β 0 J 1 ( a z ) - β 2 J 3 ( a z ) + β 3 J 5 ( a z ) - } / a z .
J 2 n + 1 ( a z ) ~ ( - 1 ) n J 1 ( a z ) ;
A ( r , θ ) = π a 2 { α sin ( γ - 2 θ ) + β sin γ } J 1 ( a z ) / a z ,
α = - n = 1 α n ,             β = - n = 0 β n .
A ( r , θ ) = - π sin 2 θ 0 a ( k p - k n ) J 2 ( z ρ ) ρ d ρ = - π a 2 sin 2 θ · F 1 ( z ) .
( k p - k n ) = α ρ 2 ,
I = const × { sin 2 θ · J 3 ( a z ) / a z } 2 .
z { J 3 ( a z ) / a z } = 0 ,             e . g . ,             J 2 ( a z ) = 2 J 4 ( a z ) .
A ( r , θ ) = 2 π 0 a J 0 ( z ρ ) ρ d ρ .
A ( r , θ ) = - 2 π a 2 γ { J 1 ( a z ) + g sin 2 θ · J 3 ( a z ) } / a z ,
J 1 ( a z ) = g J 3 ( a z )             and             ( g + 3 ) J 2 ( a z ) = 2 g J 4 ( a z ) .
K p = k p exp ( i δ p )             and             K n = k n exp ( i δ n ) ,
A ( r , θ ) = - π sin 2 θ a 0 a { exp ( i δ p ) - exp ( i δ n ) } J 2 ( z ρ ) ρ d ρ ,
A ( r , θ ) = - i π a 2 α 1 sin 2 θ { J 3 ( a z ) a z - s 4 J 3 ( s a z ) s a z } ,
K p = exp ( i Δ 0 )             and             K n = exp ( i Δ e ) ,
Δ 0 = ( 2 π d / λ ) ( n 0 cos r 0 - n cos i ) , Δ e = ( 2 π d / λ ) ( n e cos r e - n cos i ) ,
n 0 cos r 0 = n 1 { 1 - ( n sin i / n 1 ) 2 } 1 2 , n e cos r e = n 1 { 1 - ( n sin i / n 2 ) 2 } 1 2 .
Δ 0 = Δ + B 0 ρ 2 + ,             Δ e = Δ + B e ρ 2 + ,
Δ = ( 2 π d / λ ) ( n 1 - n ) ,             B 0 = ( π d / λ ) ( 1 n 1 - 1 n ) ( n f ) 2 ,             B e = ( π d / λ ) { 1 n 1 ( n 1 n 2 ) 2 - 1 n } ( n f ) 2 ,
α 1 = i ( B 0 - B e ) ,             β 0 = 2 ,             β 1 = i ( B 0 + B e ) .
I = const × { sin 2 θ · J 3 ( a z ) / a z } 2 ,
const = π 2 a 4 ( B 0 - B e ) 2 .
A ( r , θ ) = π cos 2 θ 0 a exp ( i c ρ 2 ) ( K p - K n ) J 2 ( z ρ ) ρ d ρ - π 0 a exp ( i c ρ 2 ) ( K p + K n ) J 0 ( z ρ ) ρ d ρ .
A ( r , θ ) = π a 2 exp ( i Δ ) { J 1 ( a z ) + i [ ( B 0 - B e ) cos 2 θ + ( B 0 + B e + 2 c ) ] J 3 ( a z ) } / a z .
I ~ [ J 1 ( a z ) / a z ] 2 + G 2 [ J 3 ( a z ) / a z ] 2
G = ( B 0 - B e ) sin 2 θ ,             when             c = - B 0 , ( B 0 - B e ) cos 2 θ ,             when             c = - 1 2 ( B 0 + B e ) , ( B 0 - B e ) cos 2 θ , when             c = - B e .