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Recording an efficient holographic optical element from computer generated holograms
Volume 80, number 2
OPTICS COMMUNICATIONS
15 December 1990
Recording an efficient holographic optical element from computer generated holograms Y a a k o v A m i t a i a n d J o s e p h W. G o o d m a n Information Systems Laboratory, Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA Received 25 July 1990; revised manuscript received 10 September 1990
A method for recording an efficienttransmissionholographic optical element from computer generated hologramsis presented. It is based on transferring the two-dimensionalgratingfunction of the computer generatedhologramsinto a thick emulsion layer, where the volume grating distribution of the final hologram satisfiesthe Braggcondition over the entire hologram surface. The method is applicable even when the recording wavelengthis different from the readout wavelength.
1. Introduction Recently, there has been a significant advance in the technology of computer generated holograms ( C G H s ) [ 1 ]. The increasing capabilities of computers and new advanced plotting techniques enable the production of complicated CGHs with excellent optical characteristics even for elements with high space-bandwidth product. However, the main drawback of the present techniques is the limited diffraction efficiencies of the CGHs. One of the main methods to overcome this problem is to fabricate the C G H as a blazed surface relief element [2], where each fringe in the grating is shaped during the writing process in order to give optimum diffraction efficiency into the required order. Unfortunately, the fabrication of the desired continuous surface relief pattern is impossible with any existing technology. Instead, the grating is shaped as a multi-level phase structure. The problem is that the minimum necessary number of levels for achieving high efficiency is 8-12 [3 ] which multiplies the space-bandwidth product of the C G H by a factor of 100, and hence makes this technique very complicated, especially for the visible and the near infrared domains. A different approach is to copy a binary amplitude C G H into a thick phase emulsion layer to produce a high efficiency volume holographic optical element ( H O E ) [4]. In this paper we present a method to
copy the exact desired grating function into the hologram surface, where the Bragg condition is satisfied over the entire hologram surface. Since in a lot of cases the recording wavelength differs from the readout wavelength, we generalized our method to allow such a wavelength shift; we assume that 2¢/2o---/z>I 1, where 2¢ and 2o are the readout and the recording wavelengths, respectively.
2. Design procedure The desired grating function, ~H, of a HOE is in general ¢~H = "[- (~i -- ~c),
(1)
where ¢¢ and ~i are the reconstruction and the desired image phases of the element, respectively. It was shown previously [ 5 ] that when ~¢, 0i and/z are given, the necessary recording phases for an efficient transmission HOE are: o~off=a¢i+b¢¢,
0reff-- a~b¢+ b~bi,
(2)
where ogff and Or~ff are the desired phases of the object and the reference waves of the element, respectively, and a and b are defined as a_-__(/z+ l)/2/z,
b-= (/Z- 1)/2#.
(3)
Note that if/z= 1, eq. (2) degenerates into
0030-4018/90/$03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
107
Volume80, number2
OPTICSCOMMUNICATIONS
15 December1990
Hint We assume that we have the facilities to fabricate two CGHs, ~0 and ( ~ r , with the following grating functions, respectively:
H
int
Lcns~ff Filter
~ _ ~
~
~ H
L
e
n
~
~0~
Fig. 3. Reconstructingthe intermediatehologramand recording the finalhologram. O~i ~- o~¢~ -~- ~~ o,
/
/
]
Fig. 1. Reconstructingthe first CGH and recordingpart of the intermediatehologram.
H int
O h _~ r~¢fr -~_ - ~~ r,
where O~ and O~ are phases of spherical waves, which may be chosen from various considerations such as minimizing the space-bandwidth product or separating the different diffracted orders of ~o and c~r. The transfer of the image wavefronts from ego and ~r into the recording plane of the final hologram ~ , can be achieved with the help of an intermediate hologram [6] ~int. The transfer procedure includes three parts: (a) ~go is reconstructed with the spherical wavefront 0~ (fig. 1 ). The image wave 0 °, using the first negative order, is --V'o--'H--V'n--~'o
4:' Filter
j
C r i
i
Fig. 2. Reconstructingthe second CGH and recordinganother part ofthe intermediatehologram. 108
(4)
--V'o----
•
(5)
The image wave - 0 g ff is used to record the intermediate hologram ~ t with a plane wave 0~' as a reference wave. If it is necessary to separate between the reconstruction and the image waves of ~go,a converging lens and a spatial filter may be inserted between the planes of ~fo and o~int. (b) Similarly, ~g° is reconstructed with the spherical wavefront ~ (fig. 2), and the image wave - 0~ff is used to record o~ nt with 0i~,. To avoid an overlap between the two exposures, different parts of 0 in' are used to expose ~ ' with "0gff and -0~ff, respectively. (c) o~i~t is reconstructed with the conjugate of the reference wave _0~nt (fig. 3). The output phases from o~¢i~' will be precisely the conjugates of -0~off and _0~fr on the recording plane of the final hologram ~ , namely,
Volume 80, number 2
OPTICS COMMUNICATIONS
is recorded with the desired object and reference wavefronts, 0~off and Oreff, respectively.
15 December 1990
ciency o f the element was calculated to be > 96%.
4. Conclnding remarks 3. Design illustration The method is illustrated with a focusing HOE. The reconstruction wave is a plane wave with an off-axis angle o f 45 °, the image wave is an on-axis converging wave with focal length o f 30 mm, the recording and readout wavelengths are 514.5 n m and 1.06 gm, respectively, and the hologram diameter is 30 m m (i.e. the f-number is 1 ). To separate the different orders o f the CGHs, 0~ and Or~ were chosen to be offaxis converging spherical waves with the following paraxial parameters, respectively: R o - 4 0 mm, S
__
and
flg=14 °
R~=120mm,
We propose a method for copying binary C G H s into thick phase transmission holograms with high diffraction efficiency. During the copying process we do not have to resort to complicated equipment, and the process itself is simple and can be appropriate for mass production o f the final element, There are cases when o~ff or 0~a, are far from being spherical waves, and then the spatial frequencies o f ~o or ~r will be much larger than in the above example. But usually, the spatial frequencies will not exceed 1/2c, which is much better than the requirement for relief surface C G H s with high efficiencies.
fl~=35 °,
where R~ ( q = o , r) is the distance between the respective point source and the center o f the hologram, and fl~ is the off-axis angle o f the wave. The maxim u m spatial frequencies o f ~o and ~r were calculated to be 220 ram-~ and 180 m m - 1 , respectively. Based on the above parameters, a ray tracing analysis was performed in order to calculate the diffraction efficiency o f the final H O E from the coupledwave theory [ 7 ]. For the calculation we assumed that the emulsion thickness is 15 g m and the refraction index modulation is 0.032. The diffraction effi-
References [ 1] K.S. Urquhart, S.H. Lee, C.C. Guest, M.R. Feldman and H. Farhoosh, Appl. Optics 28 (1989) 3387. [21 H. Dammann, Optik 53 ( 1978) 409. [ 3 ] G.J. Swanson, MIT Lincoln Lab. Tech. Report 854 (1989). [4] R.G. Canas, R.W. Smith and A.A. West, Proc. SPIE 1136, on HolographicOptics II: Principles and Applications (1989) p. 208. [5] K. Winiek, J. Opt. Soc. Am. 72 (1982) 143. [6] Y. Amitai and A.A. Friesem, J. Opt. Soc. Am. A 5 (1988) 702. [ 7 ] H. Kogelnik, Bell Syst. Tech. J. 48 (1969) 2909.
109
OPTICS COMMUNICATIONS
15 December 1990
Recording an efficient holographic optical element from computer generated holograms Y a a k o v A m i t a i a n d J o s e p h W. G o o d m a n Information Systems Laboratory, Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA Received 25 July 1990; revised manuscript received 10 September 1990
A method for recording an efficienttransmissionholographic optical element from computer generated hologramsis presented. It is based on transferring the two-dimensionalgratingfunction of the computer generatedhologramsinto a thick emulsion layer, where the volume grating distribution of the final hologram satisfiesthe Braggcondition over the entire hologram surface. The method is applicable even when the recording wavelengthis different from the readout wavelength.
1. Introduction Recently, there has been a significant advance in the technology of computer generated holograms ( C G H s ) [ 1 ]. The increasing capabilities of computers and new advanced plotting techniques enable the production of complicated CGHs with excellent optical characteristics even for elements with high space-bandwidth product. However, the main drawback of the present techniques is the limited diffraction efficiencies of the CGHs. One of the main methods to overcome this problem is to fabricate the C G H as a blazed surface relief element [2], where each fringe in the grating is shaped during the writing process in order to give optimum diffraction efficiency into the required order. Unfortunately, the fabrication of the desired continuous surface relief pattern is impossible with any existing technology. Instead, the grating is shaped as a multi-level phase structure. The problem is that the minimum necessary number of levels for achieving high efficiency is 8-12 [3 ] which multiplies the space-bandwidth product of the C G H by a factor of 100, and hence makes this technique very complicated, especially for the visible and the near infrared domains. A different approach is to copy a binary amplitude C G H into a thick phase emulsion layer to produce a high efficiency volume holographic optical element ( H O E ) [4]. In this paper we present a method to
copy the exact desired grating function into the hologram surface, where the Bragg condition is satisfied over the entire hologram surface. Since in a lot of cases the recording wavelength differs from the readout wavelength, we generalized our method to allow such a wavelength shift; we assume that 2¢/2o---/z>I 1, where 2¢ and 2o are the readout and the recording wavelengths, respectively.
2. Design procedure The desired grating function, ~H, of a HOE is in general ¢~H = "[- (~i -- ~c),
(1)
where ¢¢ and ~i are the reconstruction and the desired image phases of the element, respectively. It was shown previously [ 5 ] that when ~¢, 0i and/z are given, the necessary recording phases for an efficient transmission HOE are: o~off=a¢i+b¢¢,
0reff-- a~b¢+ b~bi,
(2)
where ogff and Or~ff are the desired phases of the object and the reference waves of the element, respectively, and a and b are defined as a_-__(/z+ l)/2/z,
b-= (/Z- 1)/2#.
(3)
Note that if/z= 1, eq. (2) degenerates into
0030-4018/90/$03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
107
Volume80, number2
OPTICSCOMMUNICATIONS
15 December1990
Hint We assume that we have the facilities to fabricate two CGHs, ~0 and ( ~ r , with the following grating functions, respectively:
H
int
Lcns~ff Filter
~ _ ~
~
~ H
L
e
n
~
~0~
Fig. 3. Reconstructingthe intermediatehologramand recording the finalhologram. O~i ~- o~¢~ -~- ~~ o,
/
/
]
Fig. 1. Reconstructingthe first CGH and recordingpart of the intermediatehologram.
H int
O h _~ r~¢fr -~_ - ~~ r,
where O~ and O~ are phases of spherical waves, which may be chosen from various considerations such as minimizing the space-bandwidth product or separating the different diffracted orders of ~o and c~r. The transfer of the image wavefronts from ego and ~r into the recording plane of the final hologram ~ , can be achieved with the help of an intermediate hologram [6] ~int. The transfer procedure includes three parts: (a) ~go is reconstructed with the spherical wavefront 0~ (fig. 1 ). The image wave 0 °, using the first negative order, is --V'o--'H--V'n--~'o
4:' Filter
j
C r i
i
Fig. 2. Reconstructingthe second CGH and recordinganother part ofthe intermediatehologram. 108
(4)
--V'o----
•
(5)
The image wave - 0 g ff is used to record the intermediate hologram ~ t with a plane wave 0~' as a reference wave. If it is necessary to separate between the reconstruction and the image waves of ~go,a converging lens and a spatial filter may be inserted between the planes of ~fo and o~int. (b) Similarly, ~g° is reconstructed with the spherical wavefront ~ (fig. 2), and the image wave - 0~ff is used to record o~ nt with 0i~,. To avoid an overlap between the two exposures, different parts of 0 in' are used to expose ~ ' with "0gff and -0~ff, respectively. (c) o~i~t is reconstructed with the conjugate of the reference wave _0~nt (fig. 3). The output phases from o~¢i~' will be precisely the conjugates of -0~off and _0~fr on the recording plane of the final hologram ~ , namely,
Volume 80, number 2
OPTICS COMMUNICATIONS
is recorded with the desired object and reference wavefronts, 0~off and Oreff, respectively.
15 December 1990
ciency o f the element was calculated to be > 96%.
4. Conclnding remarks 3. Design illustration The method is illustrated with a focusing HOE. The reconstruction wave is a plane wave with an off-axis angle o f 45 °, the image wave is an on-axis converging wave with focal length o f 30 mm, the recording and readout wavelengths are 514.5 n m and 1.06 gm, respectively, and the hologram diameter is 30 m m (i.e. the f-number is 1 ). To separate the different orders o f the CGHs, 0~ and Or~ were chosen to be offaxis converging spherical waves with the following paraxial parameters, respectively: R o - 4 0 mm, S
__
and
flg=14 °
R~=120mm,
We propose a method for copying binary C G H s into thick phase transmission holograms with high diffraction efficiency. During the copying process we do not have to resort to complicated equipment, and the process itself is simple and can be appropriate for mass production o f the final element, There are cases when o~ff or 0~a, are far from being spherical waves, and then the spatial frequencies o f ~o or ~r will be much larger than in the above example. But usually, the spatial frequencies will not exceed 1/2c, which is much better than the requirement for relief surface C G H s with high efficiencies.
fl~=35 °,
where R~ ( q = o , r) is the distance between the respective point source and the center o f the hologram, and fl~ is the off-axis angle o f the wave. The maxim u m spatial frequencies o f ~o and ~r were calculated to be 220 ram-~ and 180 m m - 1 , respectively. Based on the above parameters, a ray tracing analysis was performed in order to calculate the diffraction efficiency o f the final H O E from the coupledwave theory [ 7 ]. For the calculation we assumed that the emulsion thickness is 15 g m and the refraction index modulation is 0.032. The diffraction effi-
References [ 1] K.S. Urquhart, S.H. Lee, C.C. Guest, M.R. Feldman and H. Farhoosh, Appl. Optics 28 (1989) 3387. [21 H. Dammann, Optik 53 ( 1978) 409. [ 3 ] G.J. Swanson, MIT Lincoln Lab. Tech. Report 854 (1989). [4] R.G. Canas, R.W. Smith and A.A. West, Proc. SPIE 1136, on HolographicOptics II: Principles and Applications (1989) p. 208. [5] K. Winiek, J. Opt. Soc. Am. 72 (1982) 143. [6] Y. Amitai and A.A. Friesem, J. Opt. Soc. Am. A 5 (1988) 702. [ 7 ] H. Kogelnik, Bell Syst. Tech. J. 48 (1969) 2909.
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