Abstract

By an extension of ordinary geometrical optics (or acoustics) the intensity of the reflected and transmitted fields due to a point source in the presence of an arbitrary interface between two media is found. Particular consequences of the solution are the general lens and mirror law and the equations for the caustic surfaces.

© 1950 Optical Society of America

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References

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  1. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), Chapter IX.
  2. Rayleigh, The Theory of Sound (MacMillan Company, Ltd., London, 1937) Vol. II, Chapters IV and V.
  3. H. Primakoff and J. B. Keller, J. Acous. Soc. Am. 19, 820 (1947).
    [Crossref]
  4. J. B. Keller, Reflection and Transmission of Electromagnetic Waves by Thin Curved Shells, New York University, Washington Square College Mathematics Research Group, .
  5. R. K. Luneberg, “Mathematical theory of optics,” Brown University Graduate School (Providence, 1944), notes.
  6. J. B. Keller (Lecture notes, New York University).
  7. C. B. Barker and H. Riblet, Reflections from Curved SurfacesM. I. T. Radiation Laboratory, .
  8. J. B. Keller, Reflection of Electromagnetic Waves, New York University, Washington Square College Mathematics Research Group, .
  9. L. P. Eisenhart, Differential Geometry (Princeton University Press, Princeton, 1940), Chapter IV.
  10. van der Pol and Bremmer, Phil. Mag. [7] 24, 825 (1937), Eqs. 91–93.

1947 (1)

H. Primakoff and J. B. Keller, J. Acous. Soc. Am. 19, 820 (1947).
[Crossref]

1937 (1)

van der Pol and Bremmer, Phil. Mag. [7] 24, 825 (1937), Eqs. 91–93.

Barker, C. B.

C. B. Barker and H. Riblet, Reflections from Curved SurfacesM. I. T. Radiation Laboratory, .

Bremmer,

van der Pol and Bremmer, Phil. Mag. [7] 24, 825 (1937), Eqs. 91–93.

Eisenhart, L. P.

L. P. Eisenhart, Differential Geometry (Princeton University Press, Princeton, 1940), Chapter IV.

Keller, J. B.

H. Primakoff and J. B. Keller, J. Acous. Soc. Am. 19, 820 (1947).
[Crossref]

J. B. Keller, Reflection and Transmission of Electromagnetic Waves by Thin Curved Shells, New York University, Washington Square College Mathematics Research Group, .

J. B. Keller (Lecture notes, New York University).

J. B. Keller, Reflection of Electromagnetic Waves, New York University, Washington Square College Mathematics Research Group, .

Luneberg, R. K.

R. K. Luneberg, “Mathematical theory of optics,” Brown University Graduate School (Providence, 1944), notes.

Primakoff, H.

H. Primakoff and J. B. Keller, J. Acous. Soc. Am. 19, 820 (1947).
[Crossref]

Rayleigh,

Rayleigh, The Theory of Sound (MacMillan Company, Ltd., London, 1937) Vol. II, Chapters IV and V.

Riblet, H.

C. B. Barker and H. Riblet, Reflections from Curved SurfacesM. I. T. Radiation Laboratory, .

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), Chapter IX.

van der Pol,

van der Pol and Bremmer, Phil. Mag. [7] 24, 825 (1937), Eqs. 91–93.

J. Acous. Soc. Am. (1)

H. Primakoff and J. B. Keller, J. Acous. Soc. Am. 19, 820 (1947).
[Crossref]

Phil. Mag. [7] (1)

van der Pol and Bremmer, Phil. Mag. [7] 24, 825 (1937), Eqs. 91–93.

Other (8)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), Chapter IX.

Rayleigh, The Theory of Sound (MacMillan Company, Ltd., London, 1937) Vol. II, Chapters IV and V.

J. B. Keller, Reflection and Transmission of Electromagnetic Waves by Thin Curved Shells, New York University, Washington Square College Mathematics Research Group, .

R. K. Luneberg, “Mathematical theory of optics,” Brown University Graduate School (Providence, 1944), notes.

J. B. Keller (Lecture notes, New York University).

C. B. Barker and H. Riblet, Reflections from Curved SurfacesM. I. T. Radiation Laboratory, .

J. B. Keller, Reflection of Electromagnetic Waves, New York University, Washington Square College Mathematics Research Group, .

L. P. Eisenhart, Differential Geometry (Princeton University Press, Princeton, 1940), Chapter IV.

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Figures (2)

Fig. 2
Fig. 2

Transformations for transmission.

Equations (44)

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E ( x , y , z ) = E i ( x , y , z ) ( d σ / d σ ) 1 2 ,
E ( x , y , z ) = J 1 2 ( p / p ) E i ( x , y , z ) .
T i ( x , y , z ) = - n i - 1 I + [ n i - 1 ( I · N ) - ( - 1 ) i { 1 - n 2 i - 1 [ 1 - ( I · N ) 2 ] } 1 2 ] N .             ( i = 1 , 2 )
I ( x , y , z ) = ( x 1 - x D 1 ( x , y , z ) , y 1 - y D 1 ( x , y , z ) , z 1 - z D 1 ( x , y , z ) ) N ( x , y , z ) = ( - z / x N ( x , y , z ) , - z / y N ( x , y , z ) , 1 N ( x , y , z ) ) ,
D 1 2 ( x , y , z ) = ( x 1 - x ) 2 + ( y 1 - y ) 2 + ( z 1 - z ) 2
N 2 ( x , y , z ) = 1 + ( z / x ) 2 + ( z / y ) 2 .
x - x T i x = y - y T i y = z - z T i z
x = x + ( z - z ) T i x ( x , y , z ) T i z ( x , y , z ) y = y + ( z - z ) T i y ( x , y , z ) T i z ( x , y , z ) ,
J ( s π ) = J - 1 ( π s ) = { x x y y - x y y x } - 1
J ( p / p ) = J ( p / π ) J ( π / s ) J ( s / π ) J ( π / p ) ,
J ( p / p ) = J ( π / s ) J ( s / π ) .
z = a ( x ) 2 + b ( y ) 2 + ,
x = x - z tan θ ( x , y ) y = y - z tan ϕ ( x , y ) ,
J ( π s ) = x x y y - y y y x .
J ( p / p ) = J ( s / π ) .
J ( p p ) = { 1 + z cos α i [ n i - 1 D 1 ( 1 + z 1 2 D 1 2 + n 2 ( i - 1 ) z 1 2 ( x 1 2 + y 1 2 ) D 1 4 cos 2 α i ) - 2 ( a + b ) A - 2 ( a x 1 2 + b y 1 2 D 1 2 ) n 2 ( i - 1 ) cos 2 α i A ] + z 2 cos 2 α i [ n 2 ( i - 1 ) D 1 2 ( z 1 2 D 1 2 + n 2 ( i - 1 ) z 1 2 ( x 1 2 + y 1 2 ) D 1 4 cos 2 α i ) - 2 ( a + b ) ( n i - 1 D 1 A ) - 2 ( a x 1 2 + b y 1 2 D 1 2 ) n 2 ( i - 1 ) cos 2 α i ( n i - 1 D 1 A ) + 4 a b ( 1 + n 2 ( i - 1 ) ( x 1 2 + y 1 2 ) D 1 2 cos 2 α i ) A 2 + 2 ( b x 1 2 + a y 1 2 D 1 2 ) × ( 1 - n 2 ( i - 1 ) z 1 2 D 1 2 cos 2 α i ) ( n i - 1 D 1 A ) ] } - 1 .
G m = ( a + b ) ;             G g = 4 a b
G 11 = 2 ( a x 1 2 + b y 1 2 D 1 2 ) 1 sin 2 γ 2 G m - G 11 = 2 ( b x 1 2 + a y 1 2 D 1 2 ) 1 sin 2 γ
D 2 = x 2 + y 2 + z 2 = z 2 / ( cos 2 α i ) .
J ( p p ) = { 1 + D [ n i - 1 D 1 ( 1 + cos 2 γ cos 2 α i ) - ( 2 G m + G 11 tan 2 α i ) ( n i - 1 cos γ + cos α i ) ] + D 2 [ n 2 ( i - 1 ) D 1 2 ( cos 2 γ cos 2 α i ) - ( 2 G m + G 11 tan 2 γ ) × ( cos 2 γ cos 2 α i ) ( n i - 1 cos γ + cos α i ) n i - 1 D 1 + G g sec 2 α i ( n i - 1 cos γ + cos α i ) 2 ] } - 1 .
D 1 2 = ( x 1 - x ) 2 + ( y 1 - y ) 2 + ( z 1 - z ) 2 D 2 = ( x - x ) 2 + ( y - y ) 2 + ( z - z ) 2 = ( z - z ) 2 cos 2 α i .
E ( x , y , z ) = E i ( x , y , z ) × J 1 2 ( G m ( x , y ) , G g ( x , y ) , G 11 ( x , y ) , α i , γ , D ) ,
E ( x , y , z ) = E * = const .
D ( x , y , z ) = { 1 - ( E i 2 ( x , y , z ) E * 2 ) } { - [ H 2 ] ± [ ( H 2 4 - K ) + E i 2 ( x , y , z ) E * 2 K ] 1 2 } - 1 ,
D = { - H / 2 ± ( ( H / 2 ) 2 - K ) 1 2 } - 1 ;
[ H 2 - 4 K ] n 2 ( i - 1 ) D 1 2 [ cos 2 γ cos 2 α i - 1 ] 2 + 2 n i - 1 D 1 ( 2 G m - G 11 tan 2 α i ) [ cos 2 γ cos 2 α i - 1 ] ( n i - 1 cos γ + cos α i ) + [ ( 2 G m + G 11 tan 2 α i ) 2 - 4 G g sec 2 α i ] × ( n i - 1 cos γ + cos α i ) 2 = 0 ,
1 / D = - n / D 1 + 2 a ( n - 1 ) .
1 / D = - n / D 1 ± [ 1 - ( - 1 ) i n ] G g 1 2 ,
1 / f = ± [ 1 - ( - 1 ) i n ] G g 1 2 .
1 / D + n / D 1 = 1 / f .
I 2 = N 2 = T i 2 = 1 I · N = cos γ T i · N = cos α i .
T 1 · N = I · N ;
T 2 · N = - { 1 - n 2 [ 1 - ( I · N ) 2 ] } 1 2 ,
T i = A i I + B i N .
A 1 = - 1 B 1 = 2 ( I · N ) }             and             A 1 = + 1 B 1 = 0 } .
T 1 = - I + 2 ( I · N ) N .
A 2 = - n B 2 = n ( I · N ) - { 1 - n 2 [ 1 - ( I · N ) 2 ] } 1 2
A 2 = n B 2 = - n ( I · N ) - { 1 - n 2 [ 1 - ( I · N ) ] } 1 2 .
T 2 = - n I + ( n ( I · N ) - { 1 - n 2 [ 1 - ( I · N ) 2 ] } 1 2 ) N .
T i = - ( n i - 1 ) I + ( n i - 1 ( I · N ) - ( - 1 ) i { 1 - n 2 ( i - 1 ) [ 1 - ( I · N ) 2 ] } 1 2 ) N
T i = ( - n i - 1 ) I + ( n i - 1 cos γ + cos α i ) N .
J ( p p ) = J ( s π ) = { 1 + z [ x ( T i x T i z ) + y ( T i y T i z ) ] + z 2 [ x ( T i x T i z ) y ( T i y T i z ) - y ( T i x T i z ) x ( T i y T i z ) ] } - 1 ,
T i z = - ( - 1 ) i { 1 - n 2 ( i - 1 ) [ 1 - z 1 2 D 1 2 ] } 1 2 = cos α i T i x = - n i - 1 x 1 / D 1 T i y = - n i - 1 y 1 / D 1 T i x x = n i - 1 D 1 ( 1 - x 1 2 D 1 2 ) - 2 a A ,             T i x y = - n i - 1 x 1 y 1 D 1 3 T i y y = n i - 1 D 1 ( 1 - y 1 2 D 1 2 ) - 2 b A ,             T i y x = n i - 1 x 1 y 1 D 1 3 T i z x = [ - 2 a A n i - 1 x 1 D 1 + n 2 ( i - 1 ) x 1 z 1 2 D 1 4 ] 1 cos α i i T z y = [ - 2 b A n i - 1 y 1 D 1 + n 2 ( i - 1 ) y 1 z 1 2 D 1 4 ] 1 cos α i
A = n i - 1 z 1 D 1 - ( - 1 ) i { 1 - n 2 ( i - 1 ) [ 1 - z 1 2 D 1 2 ] } 1 2 = n i - 1 cos γ + cos α i ; D 1 2 = x 1 2 + y 1 2 + z 1 2 = z 1 2 / ( cos 2 γ ) ,