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The s and t formulae for holographic lens elements
Volume
21 number
1
April 1977
OPTICS COMMUNICATIONS
THE s AND t FORMULAE
FOR HOLOGRAPHIC
LENS ELEMENTS
R.W. SMITH Physics Department, Received
20 December
Imperial College of Science and Technology,
London SW7 2BZ, UK
1976
By considering the propagation of an astigmatic wavefront through a hologram on a curved surface expressions distances to the principal foci are obtained. In the special case of a system with a plane of symmetry these reduce sions similar to the conventional refraction/reflections and t formulae.
The s and t formulae [l] describe the imagery of and astigmatic pencil of rays by refraction or reflection at a curved surface. In this paper we consider the equivalent formulae for imagery by a hologram lens made on a curved surface S which separates media of refractive index y1and n’ both of which may depend on the wavelength of the light. The hologram on the surface S is constructed using light of vacuum wavelength X by the interference of the object and reference wavefronts x0 and CR respectively in a medium of refractive index no. The reconstruction is made using a wavefront C R’ of wavelength h’ and the problem is to locate the astigmatic foci of the diffracted wavefront Co,, fig. 1. In the most general case the object, reference and reconstruction wavefronts at any point P on the surface S will be astigmatic. We take a rectangular coordinate system (xyz) centred on P with the z axis along the normal to the surface and the x and y axes parallel to the principal directions [2] at P, fig. 2. The rays passing through P are the principal rays of each of the wavefronts. In the region of P the surface S may be represented by the approximation z =+ (c,x2
t c,y2)
+ 0(x3, . ..).
106
Fig. 1. The hologram on the surface S is formed by intcrference between the two wavefronts Z. and CR. Subsequently it is illuminated by the reconstructing wavefront xR~ and the reconstructed wavefront X0, is obtained.
lying in the same plane as this direction and they axis. The coordinates of a point (x,y, z) on the surface S with respect to this new set of axes are 0
(1)
where terms of greater than second order in x and y have been neglected. The quantities cx and cy are the principal curvatures at the point P. Consider any wavefront I: centred on P which has its principal ray travelling in the direction (L,M,N). We take a second set of coordinate (XYZ) with its Z axis along the direction (L,M,N) and with its Y axis
for the to expres-
JYis
X
Y.
(2)
M i: Z An astigmatic wavefront 2 can, in the paraxial approximation, be expressed in the form ZA
=n(+ FX2 +f
GY2 tHXY),
(3)
Volume
21, number
April 1977
OPTICS COMMUNICATIONS
1
Fig. 2. Diagram showing coordinate systems 3~2 surface S at P and the x the principal curvatures matic wavefront whose
the relationship between the two and XYZ. The z axis is normal to the and y axes are along the directions of of the surfaces at P. x is any astigprincipal ray through P is in the direction (L, IV,A’). This direction is taken as the Z asis. The Y axis is chosen to lie in the plane containing they axis and the direction (L, &J, N).
where n is the refractive index. If R, and R, are its principal curvatures and if the first of these is in the principal direction at an angle 0 to the Y axis in the XY plane then F=-++
c0s2e
sin28
RI
RZ3
sin20 G=----+--Rl
cos2t3
(4,536)
R2’
Thus the distribution of phase of the disturbance within the paraxial approximation
at the point x,y, z on the surface S due to this wavefront is,
@(X,Y,Z) = (27rlVn (Z-2,) - [G(I-M2)-NC,,+
=(28ih)n[rx+My-(~+~-=5Vcx);
+(GLM-HN)xy],
(7)
where we have used eq. (2) and substituted for z using eq. (1). The phase at P is taken to be zero. The distribution of phase in the interference pattern between the object and reference waves Co and Z‘, on the surface S is @o - Q~. When the hologram formed from these waves is illuminated by the reconstruction wave CR, the reconstructed wave Cot has a phase distribution over the surface S given by @()’(x, Y, z) = @R’* (@O- @R).
(8)
The f signs correspond to the normal and conjugate images, respectively. We find the location and orientation of the astigmatic foci in the image space by substituting eq. (7) into eq. (8). By considering the coefficients of the terms in x, y, x2, y2 and xy we obtain
n0
+-h
FoN;+GoL;M;-2HoLoMoNo
K
1-M;
FRN; +GRL~M&~HRLRMRNR -NRC~
)I(11) ,
107
Volume 21, number 1
;
OPTICS COMMUNICATIONS
[G,(l-I@)
-No,cY]
=;[G,l(l-M;,)-A+,]
+
April 177
{[Go(l-M&VocY]
-[GR(l-M&N,+]}, (12)
;
[GoLo~Mo~ -ZZo,No,] =T [GRLRAfR~-ZZR~NR~] &q
Fol, Go, and Hoc may be found from these equations tions. tan 28,~ = HOMO, -
1
[(GoLoMo-HoNo)-(G&,M,-ZZ,N,)].
(13)
and Oo,, RIO, and R20 f calculated using the following equa-
-~ Go,)>
(14)
= Fez cos2001 t GoI sin28ol t Ho, sin O,, cos tIo,,
RIO’
1
= Fo! sin2Bof t Go, cos2tIo, -HOI sin eOrcos2f30c. R201 (15, 16)
The equations given above correspond to the most general case in which the object, reference and reconstruction wavefronts arriving at the point P on the general surface S are astigmatic. Similar equations for a flat hologram were given by Miles [3,4]. When the system, within the paraxial approximation, has a plane of symmetry equations corresponding to the conventional refraction or reflection s and t formulae are obtained. Thus in this instance the object, reference and reconstruction principal rays lie in the same plane and hence L,, L o, L R’ and Lo, are all zero and N2 = l- M2. This symmetry is completed by taking all the coefficients H to the zero, that is that one of the principal directions of each of the wavefronts is along the x axis at P. The quantities N are the cosines of the angles between the principal ray through P and the normal to the surface S. Thus NR, is the cosine of the angle of incidence Z and Not is the cosine of the angle of refractiomdiffraction Z’. We can define similar angles Z. and ZR for the object and reference wavefronts at P. The resulting equations with F, = l/s and G = l/t where s and t refer to the customary saggital and tangential radii of curvature are n’ --;T;no(;-;)=cx(
n’cos I’ - n cos Z T:
s’
n2’cos2Z’ ___tr
n cos2Z _ A’ t -+-in0
cos2zo __-___ (\ to
, COS2ZR n’cosZ’-ncosZT~nO(cosZo-cosZR) =cr tR ) (
where so, to and sR, tR are the radii of curvature of the object and reference wavefronts at P and
4-2
n’sinZr=nsinZfhn
nrcos21’
;
O
(sinfo-sinZR).
(19)
Similar equations for diffraction gratings have been given recently by Velzel [5]. These s and t equations include refraction due to the difference in index between the two media as well as the diffraction due to the hologram fringe structure. The conventional s and t formula for refraction or reflection [6] which in eq. (8) correspond to Qof = GR,, are obtained by omitting all the terms associated with the hologram from eqs. (17), (18) and (19). Thus these are
108
(17)
nO(cos Z. - cos ZR) , J
s’
s
t’
cx
1
,
(n’cos I’ - n cos I),
n cos2Z - ____ = cv (n’cosl’ 6
(18)
(20)
- n cos I),
(21)
and Snell’s law n’sinZ’ = n sin Z.
(22)
The results obtained in this paper can be used to locate the two principal foci of the astigmatic image wavefront produced by a holographic lens element. These formulae can also be used to find holographic lens geometries which produce images free from astigmatism [7].
Volume
21, number
1
OPTICS COMMUNICATIONS
Discussions with Professor W.T. Welford are gratefully acknowledged.
References [l] W.T. Welford, Aberrations System
(Academic
of the Symmetrical Press, 1974) 165.
April 1977
[2] C.E. Weatherburn, Differential Geometry of Three Dimensions (Cambridge, 1927) 66. [3] J.F. Miles, Optica Acta 19 (1972) 16.5. [4] J.F. Miles, Optics Acta 20 (1973) 19. [S] C.H.F. Velzel, J. Opt. Sot. Am. 66 (1976) 346. [6] W.T. Welford, lot. cit., p. 168, 170. [7] R.W. Smith, Opt. Commun. 19 (1976) 245.
Optical
109
21 number
1
April 1977
OPTICS COMMUNICATIONS
THE s AND t FORMULAE
FOR HOLOGRAPHIC
LENS ELEMENTS
R.W. SMITH Physics Department, Received
20 December
Imperial College of Science and Technology,
London SW7 2BZ, UK
1976
By considering the propagation of an astigmatic wavefront through a hologram on a curved surface expressions distances to the principal foci are obtained. In the special case of a system with a plane of symmetry these reduce sions similar to the conventional refraction/reflections and t formulae.
The s and t formulae [l] describe the imagery of and astigmatic pencil of rays by refraction or reflection at a curved surface. In this paper we consider the equivalent formulae for imagery by a hologram lens made on a curved surface S which separates media of refractive index y1and n’ both of which may depend on the wavelength of the light. The hologram on the surface S is constructed using light of vacuum wavelength X by the interference of the object and reference wavefronts x0 and CR respectively in a medium of refractive index no. The reconstruction is made using a wavefront C R’ of wavelength h’ and the problem is to locate the astigmatic foci of the diffracted wavefront Co,, fig. 1. In the most general case the object, reference and reconstruction wavefronts at any point P on the surface S will be astigmatic. We take a rectangular coordinate system (xyz) centred on P with the z axis along the normal to the surface and the x and y axes parallel to the principal directions [2] at P, fig. 2. The rays passing through P are the principal rays of each of the wavefronts. In the region of P the surface S may be represented by the approximation z =+ (c,x2
t c,y2)
+ 0(x3, . ..).
106
Fig. 1. The hologram on the surface S is formed by intcrference between the two wavefronts Z. and CR. Subsequently it is illuminated by the reconstructing wavefront xR~ and the reconstructed wavefront X0, is obtained.
lying in the same plane as this direction and they axis. The coordinates of a point (x,y, z) on the surface S with respect to this new set of axes are 0
(1)
where terms of greater than second order in x and y have been neglected. The quantities cx and cy are the principal curvatures at the point P. Consider any wavefront I: centred on P which has its principal ray travelling in the direction (L,M,N). We take a second set of coordinate (XYZ) with its Z axis along the direction (L,M,N) and with its Y axis
for the to expres-
JYis
X
Y.
(2)
M i: Z An astigmatic wavefront 2 can, in the paraxial approximation, be expressed in the form ZA
=n(+ FX2 +f
GY2 tHXY),
(3)
Volume
21, number
April 1977
OPTICS COMMUNICATIONS
1
Fig. 2. Diagram showing coordinate systems 3~2 surface S at P and the x the principal curvatures matic wavefront whose
the relationship between the two and XYZ. The z axis is normal to the and y axes are along the directions of of the surfaces at P. x is any astigprincipal ray through P is in the direction (L, IV,A’). This direction is taken as the Z asis. The Y axis is chosen to lie in the plane containing they axis and the direction (L, &J, N).
where n is the refractive index. If R, and R, are its principal curvatures and if the first of these is in the principal direction at an angle 0 to the Y axis in the XY plane then F=-++
c0s2e
sin28
RI
RZ3
sin20 G=----+--Rl
cos2t3
(4,536)
R2’
Thus the distribution of phase of the disturbance within the paraxial approximation
at the point x,y, z on the surface S due to this wavefront is,
@(X,Y,Z) = (27rlVn (Z-2,) - [G(I-M2)-NC,,+
=(28ih)n[rx+My-(~+~-=5Vcx);
+(GLM-HN)xy],
(7)
where we have used eq. (2) and substituted for z using eq. (1). The phase at P is taken to be zero. The distribution of phase in the interference pattern between the object and reference waves Co and Z‘, on the surface S is @o - Q~. When the hologram formed from these waves is illuminated by the reconstruction wave CR, the reconstructed wave Cot has a phase distribution over the surface S given by @()’(x, Y, z) = @R’* (@O- @R).
(8)
The f signs correspond to the normal and conjugate images, respectively. We find the location and orientation of the astigmatic foci in the image space by substituting eq. (7) into eq. (8). By considering the coefficients of the terms in x, y, x2, y2 and xy we obtain
n0
+-h
FoN;+GoL;M;-2HoLoMoNo
K
1-M;
FRN; +GRL~M&~HRLRMRNR -NRC~
)I(11) ,
107
Volume 21, number 1
;
OPTICS COMMUNICATIONS
[G,(l-I@)
-No,cY]
=;[G,l(l-M;,)-A+,]
+
April 177
{[Go(l-M&VocY]
-[GR(l-M&N,+]}, (12)
;
[GoLo~Mo~ -ZZo,No,] =T [GRLRAfR~-ZZR~NR~] &q
Fol, Go, and Hoc may be found from these equations tions. tan 28,~ = HOMO, -
1
[(GoLoMo-HoNo)-(G&,M,-ZZ,N,)].
(13)
and Oo,, RIO, and R20 f calculated using the following equa-
-~ Go,)>
(14)
= Fez cos2001 t GoI sin28ol t Ho, sin O,, cos tIo,,
RIO’
1
= Fo! sin2Bof t Go, cos2tIo, -HOI sin eOrcos2f30c. R201 (15, 16)
The equations given above correspond to the most general case in which the object, reference and reconstruction wavefronts arriving at the point P on the general surface S are astigmatic. Similar equations for a flat hologram were given by Miles [3,4]. When the system, within the paraxial approximation, has a plane of symmetry equations corresponding to the conventional refraction or reflection s and t formulae are obtained. Thus in this instance the object, reference and reconstruction principal rays lie in the same plane and hence L,, L o, L R’ and Lo, are all zero and N2 = l- M2. This symmetry is completed by taking all the coefficients H to the zero, that is that one of the principal directions of each of the wavefronts is along the x axis at P. The quantities N are the cosines of the angles between the principal ray through P and the normal to the surface S. Thus NR, is the cosine of the angle of incidence Z and Not is the cosine of the angle of refractiomdiffraction Z’. We can define similar angles Z. and ZR for the object and reference wavefronts at P. The resulting equations with F, = l/s and G = l/t where s and t refer to the customary saggital and tangential radii of curvature are n’ --;T;no(;-;)=cx(
n’cos I’ - n cos Z T:
s’
n2’cos2Z’ ___tr
n cos2Z _ A’ t -+-in0
cos2zo __-___ (\ to
, COS2ZR n’cosZ’-ncosZT~nO(cosZo-cosZR) =cr tR ) (
where so, to and sR, tR are the radii of curvature of the object and reference wavefronts at P and
4-2
n’sinZr=nsinZfhn
nrcos21’
;
O
(sinfo-sinZR).
(19)
Similar equations for diffraction gratings have been given recently by Velzel [5]. These s and t equations include refraction due to the difference in index between the two media as well as the diffraction due to the hologram fringe structure. The conventional s and t formula for refraction or reflection [6] which in eq. (8) correspond to Qof = GR,, are obtained by omitting all the terms associated with the hologram from eqs. (17), (18) and (19). Thus these are
108
(17)
nO(cos Z. - cos ZR) , J
s’
s
t’
cx
1
,
(n’cos I’ - n cos I),
n cos2Z - ____ = cv (n’cosl’ 6
(18)
(20)
- n cos I),
(21)
and Snell’s law n’sinZ’ = n sin Z.
(22)
The results obtained in this paper can be used to locate the two principal foci of the astigmatic image wavefront produced by a holographic lens element. These formulae can also be used to find holographic lens geometries which produce images free from astigmatism [7].
Volume
21, number
1
OPTICS COMMUNICATIONS
Discussions with Professor W.T. Welford are gratefully acknowledged.
References [l] W.T. Welford, Aberrations System
(Academic
of the Symmetrical Press, 1974) 165.
April 1977
[2] C.E. Weatherburn, Differential Geometry of Three Dimensions (Cambridge, 1927) 66. [3] J.F. Miles, Optica Acta 19 (1972) 16.5. [4] J.F. Miles, Optics Acta 20 (1973) 19. [S] C.H.F. Velzel, J. Opt. Sot. Am. 66 (1976) 346. [6] W.T. Welford, lot. cit., p. 168, 170. [7] R.W. Smith, Opt. Commun. 19 (1976) 245.
Optical
109