Abstract

A method is described for determining the distribution of the radiation reflected from a mirror due to point source. In particular, the distribution from an elliptical cylinder, which apparently has not previously analyzed, is investigated and examples are given of the irradiance on surfaces of various configurations. In addition, the distribution from mirrors of ellipsoidal, paraboloidal, and parabolic cylinder form is considered. The method used may be extended to other types of mirrors.

© 1958 Optical Society of America

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References

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  1. A. Blondel and C. Lavanchy, Ann. phys. 8, 51 (1917).
  2. M. Herzberger, J. Opt. Soc. Am. 37, 485 (1947).
    [Crossref] [PubMed]
  3. W. H. Steel, Australian J. Sci. Research A2, 335 (1949).
  4. A. Thomescheit, Optik 10, 221 (1953).
  5. F. Wachendorf, J. Opt. Soc. Am. 43, 1205 (1953).
    [Crossref]
  6. A. Blondel and C. Lavanchy, Ann. phys. 7, 249 (1917).
  7. F. Benford, Gen. Elec. Rev. 26 to 29 (1923 to 1926). A series of 22 articles.
  8. C. Févrot, Rev. opt. 23, 121 and 261 (1944).
  9. L. Dunoyer, Rev. opt. 27, 399 (1948).
  10. H. J. Hentschel, Optik 8, 517 (1951).
  11. F. Cabonnes and A. L. Vink, J. phys. radium 15, 817 (1954).
    [Crossref]
  12. J. Farber and B. I. Davis, J. Opt. Soc. Am. 47, 216 (1957).
    [Crossref]
  13. P. Kirkpatrick and A. V. Baez, J. Opt. Soc. Am. 38, 766 (1948).
    [Crossref] [PubMed]
  14. J. Shearer, Australian J. Sci. Research A3, 532 (1950).
  15. F. Benford, Gen. Elec. Rev. 26, 160–167 (1923).
  16. B. Goldberg and N. C. Benson, J. Opt. Soc. Am. 39, 497 (1949).
    [Crossref]

1957 (1)

1954 (1)

F. Cabonnes and A. L. Vink, J. phys. radium 15, 817 (1954).
[Crossref]

1953 (2)

A. Thomescheit, Optik 10, 221 (1953).

F. Wachendorf, J. Opt. Soc. Am. 43, 1205 (1953).
[Crossref]

1951 (1)

H. J. Hentschel, Optik 8, 517 (1951).

1950 (1)

J. Shearer, Australian J. Sci. Research A3, 532 (1950).

1949 (2)

B. Goldberg and N. C. Benson, J. Opt. Soc. Am. 39, 497 (1949).
[Crossref]

W. H. Steel, Australian J. Sci. Research A2, 335 (1949).

1948 (2)

1947 (1)

1944 (1)

C. Févrot, Rev. opt. 23, 121 and 261 (1944).

1923 (1)

F. Benford, Gen. Elec. Rev. 26, 160–167 (1923).

1917 (2)

A. Blondel and C. Lavanchy, Ann. phys. 8, 51 (1917).

A. Blondel and C. Lavanchy, Ann. phys. 7, 249 (1917).

Baez, A. V.

Benford, F.

F. Benford, Gen. Elec. Rev. 26 to 29 (1923 to 1926). A series of 22 articles.

F. Benford, Gen. Elec. Rev. 26, 160–167 (1923).

Benson, N. C.

Blondel, A.

A. Blondel and C. Lavanchy, Ann. phys. 8, 51 (1917).

A. Blondel and C. Lavanchy, Ann. phys. 7, 249 (1917).

Cabonnes, F.

F. Cabonnes and A. L. Vink, J. phys. radium 15, 817 (1954).
[Crossref]

Davis, B. I.

Dunoyer, L.

L. Dunoyer, Rev. opt. 27, 399 (1948).

Farber, J.

Févrot, C.

C. Févrot, Rev. opt. 23, 121 and 261 (1944).

Goldberg, B.

Hentschel, H. J.

H. J. Hentschel, Optik 8, 517 (1951).

Herzberger, M.

Kirkpatrick, P.

Lavanchy, C.

A. Blondel and C. Lavanchy, Ann. phys. 7, 249 (1917).

A. Blondel and C. Lavanchy, Ann. phys. 8, 51 (1917).

Shearer, J.

J. Shearer, Australian J. Sci. Research A3, 532 (1950).

Steel, W. H.

W. H. Steel, Australian J. Sci. Research A2, 335 (1949).

Thomescheit, A.

A. Thomescheit, Optik 10, 221 (1953).

Vink, A. L.

F. Cabonnes and A. L. Vink, J. phys. radium 15, 817 (1954).
[Crossref]

Wachendorf, F.

Ann. phys. (2)

A. Blondel and C. Lavanchy, Ann. phys. 8, 51 (1917).

A. Blondel and C. Lavanchy, Ann. phys. 7, 249 (1917).

Australian J. Sci. Research (2)

W. H. Steel, Australian J. Sci. Research A2, 335 (1949).

J. Shearer, Australian J. Sci. Research A3, 532 (1950).

Gen. Elec. Rev. (2)

F. Benford, Gen. Elec. Rev. 26, 160–167 (1923).

F. Benford, Gen. Elec. Rev. 26 to 29 (1923 to 1926). A series of 22 articles.

J. Opt. Soc. Am. (5)

J. phys. radium (1)

F. Cabonnes and A. L. Vink, J. phys. radium 15, 817 (1954).
[Crossref]

Optik (2)

H. J. Hentschel, Optik 8, 517 (1951).

A. Thomescheit, Optik 10, 221 (1953).

Rev. opt. (2)

C. Févrot, Rev. opt. 23, 121 and 261 (1944).

L. Dunoyer, Rev. opt. 27, 399 (1948).

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Figures (13)

Fig. 1
Fig. 1

Cross section of an elliptical mirror showing the path of a ray originating at F1.

Fig. 2
Fig. 2

Relationship between ϕ and α for different eccentricities of an ellipse.

Fig. 3
Fig. 3

Variation of / with the angle ϕ for various eccentricities.

Fig. 4
Fig. 4

Variation of / with the angle α for various eccentricities.

Fig. 5
Fig. 5

Relative irradiation, curves I, II, and III, of the surfaces shown at A, B, and C, respectively, due to a point source at the focus F1 of an ellipsoidal mirror of eccentricity 0.7. For curve III, R was taken equal to S.

Fig. 6
Fig. 6

Horizontal cross section of an elliptical cylinder showing the circle DD from which any ray originating at F1 and reflected from the cylinder appears to originate.

Fig. 7
Fig. 7

Vertical cross section of the elliptical cylinder including the path of a reflected ray.

Fig. 8
Fig. 8

Relative irradiation, curves I, II, and III, of the surfaces shown at A, B, and C, respectively, due to a point source on the line focus at F1 of an elliptical cylinder of eccentricity 0.7 for the case in which the inclination angle β is zero. The solid curves are for a distant mirror (2a/S=K=0) and the ordinate scale on the left applies; the dashed curves (including a dashed curve coincident with curve I) are for K=1, and the scale on the right applies.

Fig. 9
Fig. 9

Horizontal cross section showing a ray reflected from an elliptical cylinder and striking a symmetrically positioned circular cylinder.

Fig. 10
Fig. 10

Distribution of irradiance per unit of source intensity around the side of the circular cylinder in the plane shown in Fig. 9 for two eccentricities of the elliptical cylinder reflector.

Fig. 11
Fig. 11

Horizontal cross section of parabolic mirror.

Fig. 12
Fig. 12

Vertical cross section of parabolic mirror showing a ray inclined at angle β.

Fig. 13
Fig. 13

Relative irradiation of a surface normal to the x axis due to reflection from (I) a parabolic cylinder and (II) a paraboloid. The ordinate for curve I is fdϕ/dy and for curve II is f2(/dy)2.

Tables (1)

Tables Icon

Table I Showing the sequence of steps for calculating the irradiance around the side of a circular cylinder for the configuration shown in Fig. 9. The quantity E2/I1 is plotted against δ in Fig. 10.

Equations (26)

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r = l / ( 1 + e cos ϕ ) ,
sin α = [ ( 1 - e 2 ) sin ϕ ] / ( 1 + e 2 + 2 e cos ϕ ) ,
cos α = [ ( 1 + e 2 ) cos ϕ + 2 e ] / ( 1 + e 2 + 2 e cos ϕ ) .
sin ϕ = [ ( 1 - e 2 ) sin α ] / ( 1 + e 2 - 2 e cos α ) ,
cos ϕ = [ ( 1 + e 2 ) cos α - 2 e ] / ( 1 + e 2 - 2 e cos α ) .
d ϕ / d α = ( 1 + e 2 + 2 e cos ϕ ) / ( 1 - e 2 ) ,
d ϕ / d α = ( 1 - e 2 ) / ( 1 + e 2 - 2 e cos α ) .
e = sin ϕ - sin α sin ( ϕ + α ) = sin ( ϕ - α ) sin ϕ + sin α = 1 - cos ( ϕ - α ) cos α - cos ϕ .
I 2 = ( d Ω 1 / d Ω 2 ) I 1 = ( sin ϕ / sin α ) ( d ϕ / d α ) I 1 ,
I 2 = ( d ϕ / d α ) 2 I 1 .
E 2 = ( I 1 / s 2 ) ( d ϕ / d α ) 2 cos i ,
E 2 = ( I 1 / s h s ) ( d ϕ / d α ) cos i .
s h = s + 2 a / cos β ,
E 2 / I 1 = ( 1 / q h q ) ( d ϕ / d α ) cos 3 β cos i .
E 2 = ( I 1 / q h q ) ( d ϕ / d α ) cos 3 β cos ( α + δ ) ,
q = ( P 2 + R 2 - 2 P R cos δ ) 1 2 ,
q h = q + 2 a .
sin α = ( R sin δ ) / q ,
cos α = ( P - R cos δ ) / q ,
cos ( α + δ ) = ( P cos δ - R ) / q = ( R 2 - P 2 sin 2 α ) 1 2 / R .
sin α m = cos δ m = R / P .
E 2 / I 1 = [ cos ( α + δ ) / q h q ] ( d ϕ / d α ) .
r = 2 f / ( 1 + cos ϕ ) ,
E 2 = I ( d ϕ / d y ) 2 cos i .
E 2 = ( I / X D ) ( d ϕ / d y ) cos 2 β cos i ,
d ϕ / d y = 4 f / ( y 2 + 4 f 2 ) = ( 1 + cos ϕ ) / 2 f = 1 / r ,