Abstract

A diffraction-limited, electromagnetic theory of image formation is presented for a point-reference hologram whose recording arrangement consists of a surface of arbitrary shape, a point-reference source, and the object. The hologram is illuminated by a spherical electromagnetic wave during reconstruction. The electromagnetic hologram is assumed to have recorded two components of the field scattered from the object so that the vector field is completely reconstructed. The vector hologram is modeled by electric and magnetic surface currents determined from the irradiance of each of two orthogonal components of the object field on the film. The image field is described by a dyadic kernel, the system response to a point object, which is related to the scalar kernel by Π(r,r′) = DK(r,r′), where D is the dyadic operator D= (I+k−2∇∇). It is shown that the conjugate-image field produced by a point-reference electromagnetic hologram approximates the field produced by the ideal system, which forms the image of a point object by launching a spherically converging wave.

© 1971 Optical Society of America

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References

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  1. R. P. Porter, J. Opt. Soc. Am. 60, 1051 (1970).
    [Crossref]
  2. R. Barer, Nature 167, 642 (1951).
    [Crossref] [PubMed]
  3. G. L. Rogers, J. Opt. Soc. Am. 56, 831 (1966).
    [Crossref]
  4. M. De and L. Sévigny, J. Opt. Soc. Am. 57, 110 (1967).
    [Crossref]
  5. A. W. Lohmann, Appl. Opt. 4, 1667 (1965).
    [Crossref]
  6. O. Bryngdahl, J. Opt. Soc. Am. 57, 545 (1967).
    [Crossref] [PubMed]
  7. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 143.
  8. H. M. Smith, Principles of Holography (Wiley, New York, 1969), pp. 20–21.The conjugate-image field arises from the transmittance term proportional to the complex conjugate of the object field; the primary-image field arises from the term proportional to the object field. Note the inversion of the words primary and conjugate on p. 21 of Smith.
  9. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw–Hill, New York, 1953), pp. 54–92.
  10. H. Levine and J. Schwinger, Commun. Pure Appl. Math 3, 355 (1950).
    [Crossref]
  11. R. P. Porter, Phys. Letters 29A, 193 (1969).
  12. A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), p. 89.
  13. H. Hönl, A. W. Maue, and K. Westpfahl, in Handbuch Der Physik, Vol. XXV/1, edited by S. Flügge (Springer, Berlin, 1961), p. 241.
  14. C. Müller, Arch. Math 1, 296 (1948–1949).
    [Crossref]
  15. S. Silver, Microwave Antenna Theory and Design (McGraw–Hill, New York, 1949).
  16. C. H. Wilcox, Commun. Pure Appl. Math 9, 115 (1956).
    [Crossref]
  17. The result is valid for an open surface Sh if Je=Jm=0 everywhere except on Sh.

1970 (1)

1969 (1)

R. P. Porter, Phys. Letters 29A, 193 (1969).

1967 (2)

1966 (1)

1965 (1)

1956 (1)

C. H. Wilcox, Commun. Pure Appl. Math 9, 115 (1956).
[Crossref]

1951 (1)

R. Barer, Nature 167, 642 (1951).
[Crossref] [PubMed]

1950 (1)

H. Levine and J. Schwinger, Commun. Pure Appl. Math 3, 355 (1950).
[Crossref]

Barer, R.

R. Barer, Nature 167, 642 (1951).
[Crossref] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 143.

Bryngdahl, O.

De, M.

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw–Hill, New York, 1953), pp. 54–92.

Hönl, H.

H. Hönl, A. W. Maue, and K. Westpfahl, in Handbuch Der Physik, Vol. XXV/1, edited by S. Flügge (Springer, Berlin, 1961), p. 241.

Levine, H.

H. Levine and J. Schwinger, Commun. Pure Appl. Math 3, 355 (1950).
[Crossref]

Lohmann, A. W.

Maue, A. W.

H. Hönl, A. W. Maue, and K. Westpfahl, in Handbuch Der Physik, Vol. XXV/1, edited by S. Flügge (Springer, Berlin, 1961), p. 241.

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw–Hill, New York, 1953), pp. 54–92.

Müller, C.

C. Müller, Arch. Math 1, 296 (1948–1949).
[Crossref]

Porter, R. P.

R. P. Porter, J. Opt. Soc. Am. 60, 1051 (1970).
[Crossref]

R. P. Porter, Phys. Letters 29A, 193 (1969).

Rogers, G. L.

Schwinger, J.

H. Levine and J. Schwinger, Commun. Pure Appl. Math 3, 355 (1950).
[Crossref]

Sévigny, L.

Silver, S.

S. Silver, Microwave Antenna Theory and Design (McGraw–Hill, New York, 1949).

Smith, H. M.

H. M. Smith, Principles of Holography (Wiley, New York, 1969), pp. 20–21.The conjugate-image field arises from the transmittance term proportional to the complex conjugate of the object field; the primary-image field arises from the term proportional to the object field. Note the inversion of the words primary and conjugate on p. 21 of Smith.

Sommerfeld, A.

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), p. 89.

Westpfahl, K.

H. Hönl, A. W. Maue, and K. Westpfahl, in Handbuch Der Physik, Vol. XXV/1, edited by S. Flügge (Springer, Berlin, 1961), p. 241.

Wilcox, C. H.

C. H. Wilcox, Commun. Pure Appl. Math 9, 115 (1956).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 143.

Appl. Opt. (1)

Arch. Math (1)

C. Müller, Arch. Math 1, 296 (1948–1949).
[Crossref]

Commun. Pure Appl. Math (2)

C. H. Wilcox, Commun. Pure Appl. Math 9, 115 (1956).
[Crossref]

H. Levine and J. Schwinger, Commun. Pure Appl. Math 3, 355 (1950).
[Crossref]

J. Opt. Soc. Am. (4)

Nature (1)

R. Barer, Nature 167, 642 (1951).
[Crossref] [PubMed]

Phys. Letters (1)

R. P. Porter, Phys. Letters 29A, 193 (1969).

Other (7)

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), p. 89.

H. Hönl, A. W. Maue, and K. Westpfahl, in Handbuch Der Physik, Vol. XXV/1, edited by S. Flügge (Springer, Berlin, 1961), p. 241.

The result is valid for an open surface Sh if Je=Jm=0 everywhere except on Sh.

S. Silver, Microwave Antenna Theory and Design (McGraw–Hill, New York, 1949).

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 143.

H. M. Smith, Principles of Holography (Wiley, New York, 1969), pp. 20–21.The conjugate-image field arises from the transmittance term proportional to the complex conjugate of the object field; the primary-image field arises from the term proportional to the object field. Note the inversion of the words primary and conjugate on p. 21 of Smith.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw–Hill, New York, 1953), pp. 54–92.

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Figures (4)

F. 1
F. 1

Geometry for a holographic surface Sh, of arbitrary shape. The r and υ spaces are, as indicated, inside and outside Sh. The unit normal n ̂ points into the υ space. The reference source is at R0; the object is at r′. During the reconstruction step, the inside sources are removed and reconstruction sources are placed either inside or outside.

F. 2
F. 2

Geometry for open hologram asymptotic to wedge of angle α.

F. 3
F. 3

Geometry of spherical coordinate system and coordinate system on Sh. The tangential unit vectors, û1 and û2, oriented such that u ̂ 1 × u ̂ 2 = r ̂ and ϕ ̂ · u ̂ 1 = θ ̂ · u ̂ 2 = 0.

F. 4
F. 4

Hologram illuminated by reconstruction source and by the field H′ diverging from the image at r′. The absorber for the field converging from the reconstruction source is shown at R0. In general, the converging wave illuminates the whole film.

Equations (82)

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× E = j ω μ H J m
× H = j ω E + J c
J m = j ω ρ m
J e = j ω ρ e .
× × H = k 2 H j ω J m + × J e ,
× × G x = k 2 G x + x ̂ u 0 ( r r ) ,
× × Γ ( r , r ) = k 2 Γ ( r , r ) + I u 0 ( r r ) ,
Γ f ( r , r ) = ( I + k 2 ) G f ( r , r ) ,
( 2 + k 2 ) G f ( r , r ) = u 0 ( r r )
G f ( r , r ) = exp ( j k | r r | ) / 4 π | r r | .
j ω J m ( r ) + × J e ( r ) = V I u 0 ( r r ) [ j ω J m ( r ) + × J e ( r ) ] d V ,
H ( r ) = V Γ f ( r , r ) [ j ω J m ( r ) + × J e ( r ) ] d V ,
n ̂ × ( H r H υ ) = J e
n ̂ × ( E r E υ ) = J m ,
H ( r ) = { H r ( r ) , r space H υ ( r ) , υ space } = S h { [ × Γ f ( r , r ) ] J e j ω Γ f ( r , r ) J m } d S ,
× J e j ω J m = c u 0 ( r r ) ,
H s = Γ f ( r , r ) c
H b r ( r ) = Γ f * ( r , r ) c * + H ,
× × H = k 2 H c * u 0 ( r r ) .
Π b ( r , r ) = { Π b r ( r , r ) Π b υ ( r , r ) } = { Γ f * ( r , r ) Γ f ( r , r ) , r space Γ f ( r , r ) , υ space .
Π b ( r , r ) = ( I + 1 k 2 ) × { G f * ( r , r ) G f ( r , r ) , r space G f ( r , r ) , υ space ,
Π b ( r , r ) = D K b ( r , r ) ,
K b r ( r , r ) = ( j / 2 π ) ( sin k | r r | ) / | r r | ,
J e = n ̂ × [ Γ f * ( r , r ) c * ]
J m = ( j ω ) 1 n ̂ × [ × Γ f * ( r , r ) c * ] ,
J e = n ̂ × H s * ( r )
J m = n ̂ × E s * ( r ) .
H b ( r ) = V Π b ( r , r ) { [ × J e * ( r ) ] j ω J m * ( r ) } d V .
Π b ° ( r , r ) = S h { [ × Γ f ( r , r ) ] [ n ̂ × Γ f * ( r , r ) ] + Γ f ( r , r ) [ n ̂ × ( × Γ f * ( r , r ) ) ] } d S .
S h { } d S = S { } d S
G f ( r , r ) = 1 4 j H 0 ( 2 ) ( k | r r | ) ,
G f ( r , r ) = 1 4 j ( 2 / π k | r r | ) 1 2 × exp [ j ( k | r r | π / 4 ) ] .
n ̂ × Γ f * ( r , r ) = n ̂ × I G f * ( r , r ) ,
× Γ f ( r , r ) = j k ( n ̂ × I ) G f ( r , r ) ,
n ̂ × [ × Γ f * ( r , r ) ] = j k ( n ̂ n ̂ I ) G f * ( r , r ) ,
Π b r ° ( r , r ) = S G f ( r , r ) G f * ( r , r ) [ j k ( n ̂ × I ) ( n ̂ × I ) + j k ( I n ̂ n ̂ ) ( I n ̂ n ̂ ) ] d S .
Π b r ° ( r , r ) = 2 j k r α 2 π G f ( r , r ) × G f * ( r , r ) ( I n ̂ n ̂ ) d θ ,
Π b r ° ( r , r ) = D α 2 π 2 j k r G f ( r , r ) G f * ( r , r ) d θ .
Π b r ° ( r , r ) = D α 2 π j 4 π exp [ j k d cos ( θ θ ) ] d θ .
H T = T Φ ,
T ll = A l | H 0 l | 2 γ ,
J e = j k 1 a 1 x ̂ 3 u 0 ( r )
J m = j ( ω ) 1 a 2 x ̂ 3 u 0 ( r ) ,
H i ( r ) = ( a 1 ϕ ̂ + a 2 θ ̂ ) G f ( r , 0 ) sin θ ,
H 0 l = [ a l sin θ exp ( j k r ) / 4 π r ] + H s l ,
T l j = { 1 + 4 π r | a l | 2 sin θ [ a l * H s l exp ( j k r ) + a l H s l * exp ( j k r ) ] } δ l j ,
δ l j = 1 , l = j , = 0 , l j .
H T = T H i * + H , for r , = H i * + H , for r + .
T H i * = H i * + Δ exp ( j k r ) ,
Δ = [ H s r ̂ ( r ̂ H s ) ] exp ( j k r ) + [ H s * r ̂ ( r ̂ H s * ) ] exp ( j k r )
H T r H T υ = Δ exp ( j k r ) .
[ Δ exp ( j k r ) ] j k r ̂ Δ = 0 ,
E T = ( μ ) 1 2 { r ̂ × Δ exp ( j k r ) + E + E i * E i * + E
E T r E T υ = ( μ / ) 1 2 r ̂ × Δ exp ( j k r ) .
H T ( r ) = H i * + H t ( r ) + H t ( r ) ,
H t r H t υ = H s * ( r ) = r ̂ [ r ̂ H s * ( r ) ] ,
E t r E t υ = ( μ / ) 1 2 r ̂ × H s * ( r ) ,
H t r H t υ = { H s ( r ) r ̂ [ r ̂ H s ( r ) ] } exp ( 2 j k r ) ,
E t r E t υ = ( μ / ) 1 2 r ̂ × H s ( r ) exp ( 2 j k r ) .
Γ f ( r , r ) = ( I r ̂ r ̂ ) G f ( r , r )
r ̂ Γ f ( r , r ) = 0 .
H t ( r ) = S h ( [ × Γ f ( r , r ) ] { n ̂ × [ H s * r ̂ ( r ̂ H s * ) ] } + j k Γ f ( r , r ) [ n ̂ × ( r ̂ × H s * ) ] ) d S ,
H b ( r ) = S h { [ × Γ f ( r , r ) ] ( n ̂ × H s * ) + Γ f ( r , r ) [ n ̂ × ( × H s * ) ] } d S ,
H e ( r ) = S h Γ f ( r , r ) [ n ̂ × ( × H s * j k r ̂ × H s * ) ] d S .
× H s * = j k r ̂ × H s *
H t ( r ) = S h ( [ × Γ f ( r , r ) ] { n ̂ × [ H s r ̂ ( r H s ) ] } + j k Γ f ( r , r ) [ n ̂ × ( r ̂ × H s ) ] ) exp ( 2 j k r ) d S .
Π t ( r , r ) = S h ( [ × Γ f ( r , r ) ] [ n ̂ × Γ f ( r , r ) ] + j k Γ f ( r , r ) { n ̂ × [ r ̂ × Γ f ( r , r ) ] } ) × exp ( 2 j k r ) d S ,
Π t ( r , r ) = j k S h ( Γ f ( r , r ) { n ̂ × [ r ̂ × Γ f ( r , r ) ] } + Γ f ( r , r ) { n ̂ × [ r ̂ × Γ f ( r ̂ , r ) ] } ) exp ( 2 j k r ) d S ,
Π t ( r , r ) = 2 j k D S h n ̂ r ̂ G f ( r , r ) × G f ( r , r ) exp ( 2 j k r ) d S
n ̂ × ( r ̂ × Γ f ) = r ̂ ( n ̂ Γ f ) Γ f ( n ̂ r ̂ ) , r ̂ Γ f = 0 ,
Γ f ( r , r ) Γ f ( r , r ) = D G f ( r , r ) G f ( r , r ) .
K t r ( r , r ) = 1 4 ( 2 j / π k r ) 1 2 exp { j k [ r + r cos ( ϕ ϕ ) ( r ) 2 sin 2 ( ϕ ϕ ) / r ( ϕ ) ] } + 1 4 ( 2 j / π k r ) 1 2 × exp { j k [ r + r cos ( ϕ ϕ ) ( r ) 2 sin 2 ( ϕ ϕ ) / r ( ϕ + π ) ] } .
[ a × ( × B ) + ( × a ) × B ] = ( × × a ) B a ( × × B )
S h n ̂ { H ( r ) × [ × Γ f ( r , r ) ] + [ × H ( r ) ] × Γ f ( r , r ) } d S = V { [ × J e ( r ) ] Γ f ( r , r ) j ω J m ( r ) Γ f ( r , r ) } d V { H ( r ) , r inside , 0 , r outside .
{ H r , υ ( r ) , r , υ space 0 , υ , r space } = S h ( [ n ̂ r , υ × H r , υ ( r ) ] [ × Γ f ( r , r ) ] { n ̂ r , υ × [ × H r , υ ( r ) ] } Γ f ( r , r ) ) d S ,
lim r ( r { r ̂ × [ × H ( r ) j k H ( r ) ] } ) = 0
H ( r ) = { H r ( r ) , r space H υ ( r ) , υ space } = s h { J e ( r ) [ × Γ f ( r , r ) ] j ω J m Γ f ( r , r ) } d S .
Γ f ( r , r ) = Γ f T ( r , r )
× Γ f ( r , r ) = [ × Γ f ( r , r ) ] ,
S [ ( × A ) T ( n ̂ × B ) ( n ̂ × A ) T ( × B ) ] d S = V [ ( × × A ) T B A T ( × × B ) ] d V ,
S { [ × Γ f ( r , r ) ] T [ n ̂ × Γ f * ( r , r ) ] [ n ̂ × Γ f ( r , r ) ] T [ × Γ f * ( r , r ) ] } d S = 0 ,
S ( [ × Γ f ( r , r ) ] [ n ̂ × Γ f * ( r , r ) ] + Γ f ( r , r ) { n ̂ × [ × Γ f * ( r , r ) ] } d S = 0 ,