Abstract

The behavior of orthotomic systems of rays, rays associated with a system of wavefronts, is analyzed from the point of view of classical geometrical optics. The rays themselves are described in terms of the ray equation derived from Fermat’s principle. A condition for an aggregate of rays to comprise an orthotomic system is found. Some consequences of this condition on the geometric properties of wavefronts are found. The resemblance of some of these to the Maxwell equations is noted.

© 1971 Optical Society of America

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References

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  1. D. J. Struik, Lectureson Classical Differential Geometry (Addison-Wesley, Reading, Mass., 1961), Chap. 1.
  2. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 159.
  3. M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), p. 421.
  4. G. A. Bliss, Lectures on the Calculus of Variations (University of Chicago Press, Chicago, 1946), Chap. 1.
  5. A. R. Forsyth, A Treatise on Differential Equations (Macmillan, London, 1929), pp. 309–338.
  6. Ref. 1, Chap. 3.
  7. M. Klein, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965), pp. 185–186.

Bliss, G. A.

G. A. Bliss, Lectures on the Calculus of Variations (University of Chicago Press, Chicago, 1946), Chap. 1.

Forsyth, A. R.

A. R. Forsyth, A Treatise on Differential Equations (Macmillan, London, 1929), pp. 309–338.

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), p. 421.

Kay, I. W.

M. Klein, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965), pp. 185–186.

Klein, M.

M. Klein, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965), pp. 185–186.

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 159.

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 159.

Struik, D. J.

D. J. Struik, Lectureson Classical Differential Geometry (Addison-Wesley, Reading, Mass., 1961), Chap. 1.

Other (7)

D. J. Struik, Lectureson Classical Differential Geometry (Addison-Wesley, Reading, Mass., 1961), Chap. 1.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), p. 159.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), p. 421.

G. A. Bliss, Lectures on the Calculus of Variations (University of Chicago Press, Chicago, 1946), Chap. 1.

A. R. Forsyth, A Treatise on Differential Equations (Macmillan, London, 1929), pp. 309–338.

Ref. 1, Chap. 3.

M. Klein, I. W. Kay, Electromagnetic Theory and Geometrical Optics (Interscience, New York, 1965), pp. 185–186.

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Equations (41)

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n = n ( P ) .
P ( s ) .
P ( s ) = t .
n = ρ P , 1 / ρ 2 = P 2 ,
b = t × n .
1 / τ = ρ 2 ( P × P ) · P .
t = n × b ,             n = b × t ,             b = t × n , t · n = t · b = n · b = 0.
t = ( 1 / ρ ) n , n = - ( 1 / ρ ) t + ( 1 / τ ) b , b = - ( 1 / τ ) n .
( V · ) f ,
= ( x , y , z ) .
V · f ,
( t · ) t = ( 1 / ρ ) n , ( t · ) n = - ( 1 / ρ ) t + ( 1 / τ ) b , ( t · ) b = - ( 1 / τ ) n .
( d / d s ) ( n P ) = n ,
n P + ( n · P ) P = n ;
( n / ρ ) n + ( n · t ) t = n .
n = ( n · t ) = ( t · ) n .
( b · ) n = b · n = 0 , ( n · ) n = n · n = n ρ .
n P · d W ,
n P · d W = 0 ,
× ( n P ) = 0.
n × P + n × P = 0 ,
× P = P × P .
· ( P × P ) = 0.
× t = ( 1 / ρ ) b , · [ ( 1 / ρ ) b ] = 0.
t n = n .
( t n · ) t n = ( 1 / ρ n ) n n , ( t n · ) n n = - ( 1 / ρ n ) t n + ( 1 / τ n ) b n , ( t n · ) b n = - ( 1 / τ n ) n n ,
n n = t .
( n · ) n = ( 1 / ρ n ) t , ( n · ) t = - ( 1 / ρ n ) n + ( 1 / τ n ) ( - b ) , ( n · ) ( - b ) = - ( 1 / τ n ) t .
( t b · ) t b = ( 1 / ρ b ) n b , ( t b · ) n b = - ( 1 / ρ b ) t b + ( 1 / τ b ) b b , ( t b · ) b b = - ( 1 / τ b ) n b .
t b = b ,             n b = t , b b = t b × n b = b × t = n .
( b · ) b = ( 1 / ρ b ) t , ( b · ) t = - ( 1 / ρ b ) b + ( 1 / τ b ) n , ( b · ) n = - ( 1 / τ b ) t .
1 / τ n = - ( 1 / τ b ) = σ .
( t · ) t = ( 1 / ρ ) n ,             ( n · ) n = ( 1 / ρ n ) t ,             ( b · ) t = ( 1 / ρ b ) t , ( t · ) n = - ( 1 / ρ ) t + ( 1 / τ ) b ,             ( n · ) t = - ( 1 / ρ n ) n - σ b , ( b · ) t = - ( 1 / ρ b ) b - σ n , ( t · ) b = - ( 1 / τ ) n ,             ( n · ) b = σ t ,             ( b · ) n = σ t .
( V 2 ) = 2 ( V · ) V + 2 V × ( × V ) = 0 ,
V × ( × V ) = - ( V · ) V .
· t = · ( n × b ) = b · ( × n ) - n · ( × b ) = ( t × n ) · ( × n ) - ( b × t ) · ( × b ) = t · [ n × ( × n ) + b × ( × b ) ] = - t · [ ( n · ) n + ( b · ) b ] = - t · [ ( 1 / ρ n ) t + ( 1 / ρ b ) t ] = - [ ( 1 / ρ n ) + ( 1 / ρ b ) ] .
· n = 1 / ρ , · b = 0.
× t = × ( n × b ) = ( b · ) n - ( n · ) b + n ( · b ) - b ( · n ) = ( 1 / ρ ) b ,
× n = × ( b × t ) = ( t · ) b - ( b · ) t + b ( · t ) - t ( · b ) = [ - ( 1 / τ ) + σ ] n - ( 1 / ρ n ) b , × b = × ( t × n ) = ( n · ) t - ( t · ) n + t ( · n ) - n ( · t ) = [ - ( 1 / τ ) - σ ] b + ( 1 / ρ b ) n .
ρ · b = ( b · ) ρ = 0.
· ( n n ) = n · n + n · n = 0 , · ( n b ) = n · b = 0 , × ( n n ) = n × n + n × n = ( n · t ) b + n [ ( - 1 τ + σ ) n - 1 ρ n b ] = n b - n τ n + n [ σ n - 1 ρ n b ] = ( n b ) + n [ σ n - 1 ρ n b ] , × ( n b ) = n × b + n × b = n ρ t - ( n · t ) n + n [ ( - 1 τ - σ ) b + 1 ρ b n ] = - n n - n [ - 1 ρ t + 1 τ b ] + n [ 1 ρ b n - σ b ] = - ( n n ) + n [ 1 ρ b n - σ b ] .

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