Phase modulation holographic device for generating two different wavefronts

A phase modulating device for spatially modulating a phase of an incident light beam is of a binary type having a plurality of rectangular protrusions on a cross section thereof. A distribution is provided to a diffraction efficiency at each point of the phase modulating device by varying a width of each protrusion.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to a phase modulation holographic device for 
use in a multi focal length holographic lens system and a holographic 
coupler. 
Particularly, the present invention can be employed as a zone plate for 
simultaneously generating a plurality of wave-fronts in a shape measuring 
apparatus as described in Japanese Patent Application H4-168825 of the 
present Applicants. 
2. Description of the Prior Art 
As shown in FIG. 1, a typical holographic device 3 is constituted by a 
phase modulation holographic device 2 and an amplitude modulation 
holographic device 1. As the phase modulation holographic device 2, a 
phase plate called a kinoform has been known. This is a holographic device 
for providing to a plane wave a phase modulation which is necessary for 
reconstituting an arbitrary image. For example, by providing saw-toothed 
convex portions 2a as shown in FIG. 1, an uneven surface is formed with 
which the phase modulation is performed. 
The amplitude modulation holographic device 1 is a plate-like device where 
portions la having a small intensity distribution and through which it is 
difficult for light to pass and portions lb having a large intensity 
distribution and through which it is easy for light to pass are 
alternately arranged. The amplitude modulation holographic device 1 and 
the phase modulation holographic device 2 are joined together to form one 
optical device (i.e. the holographic device 3). 
The amplitude modulation holographic device 1 and the phase modulation 
holographic device 2 can easily be produced. However, since highly 
accurate positioning is required in the process of joining the two 
devices, various problems are presented such as difficulty in the joining 
of the two devices, insufficient precision and high cost. 
SUMMARY OF THE INVENTION 
An object of the present invention is to provide a phase modulation 
holographic device where the above problems have been solved. 
To achieve the above-mentioned object, according to the present invention, 
in a phase modulating device for spatially modulating a phase of an 
incident light beam, a distribution is provided to a diffraction 
efficiency at each point of said phase modulating device. 
According to such a feature, the phase modulation holographic device is 
provided with a diffraction efficiency distribution as well as a phase 
distribution. As a result, the electromagnetic field after the diffraction 
is provided with not only a phase distribution but also an amplitude 
distribution in accordance with the diffraction efficiency distribution, 
whereby a plurality of desired wavefronts can be obtained. Further, with 
the arrangement of the phase modulation holographic device where not only 
a phase distribution but also an amplitude distribution is provided, it is 
unnecessary to attach an amplitude modulation holographic device.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
First Embodiment 
FIG. 2 is a conceptional view of a dual focal length holographic device 
embodying the present invention. FIG. 3 shows an H(r)-r graph showing a 
surface configuration, a phase difference distribution P(r) and a phase 
difference distribution B(r), which is a binarized P(r), of a binary phase 
holographic device embodying the present invention. The H(r)-r graph of 
FIG. 3 represents a surface configuration of a binary phase holographic 
device where a diffraction grating is formed in which the height of convex 
portions is set to be a height h so that the phase difference between 
light passing through the convex portions and light passing through 
concave portions is .pi., and where the pitch and the upper surface width 
of the convex portions are set to be a pitch d and a width a(r), 
respectively, calculated in advance so that the binarized phase difference 
distribution B(r) is realized. The calculation method of the height h and 
the upper surface width a will be described later. 
A method will be described of producing a dual focal length lens system 
constituted by the binary phase holographic device embodying the present 
invention, i.e. a binary phase zone plate 4 having focal lengths Z1 and Z2 
in FIG. 2. 
An optical axis 5 is taken so as to be perpendicular to the zone plate 4. 
It is assumed that positions 7 and 8 at each of which a point light source 
which emits coherent light having a wavelength .lambda. is to be placed 
are located at positions on the optical axis 5 which are distances Z1 and 
Z2 away from the zone plate 4, respectively. When an intersection of the 
optical axis 5 and the zone plate 4 is an original point 0 and a 
coordinate r is taken on the zone plate 4 along the diameter (in a 
direction 6), a complex amplitude distribution H(r) of an electromagnetic 
field formed by superposition of wavefronts of both of the two coherent 
light rays emitted from the point light sources placed at the positions 7 
and 8 is obtained by 
##EQU1## 
Here, A1 and A2 can be given as constants since they are variables which 
are dependent on the light intensities of the point light sources placed 
at the positions 7 and 8. When the intensities of light rays emitted from 
the point light sources placed at the points 7 and 8 located at 
coordinates Z1 and Z2 on the optical axis 5 are approximately the same, A1 
and A2 are set to be A1=A2=A, and the expression (1) can further be 
simplified. 
When a function to give a phase component of the complex amplitude is 
represented by Arg[ ], the phase distribution P(r) and the intensity 
distribution I(r) of the complex amplitude to be reconstituted on the zone 
plate 4 is calculated by 
EQU P(r)=Arg[H(r)] (5) 
EQU I(r)=H(r)H*(r) (6) 
where H*(r) is a conjugate complex amplitude distribution function of H(r). 
When, in order to reconstitute holographic images at the positions 7 and 8, 
a luminous flux G is incident from the opposite side of the positions 7 
and 8 as a plane wave advancing perpendicularly to the zone plate 4, P(r) 
can be considered to be a phase difference distribution which is dependent 
on a distance r from the original point 0 along the zone plate 4. Further, 
when Imax is the maximum value of I(r), I(r)/Imax means a transmission 
distribution which is dependent on the distance r from the original point 
0 along the zone plate 4. 
When a threshold value for binarization is given by PTH (0&lt;PTH&lt;2.pi.), B(r) 
which is a binarized P(r) is expressed as 
EQU B(r)=.pi.(PTH.ltoreq.P(r)&lt;2 .pi.) (7)a 
or as 
EQU B(r)=0(0.ltoreq.P(r)&lt;PTH) (7)b 
FIGS. 6A and 6B are cross-sectional views of the binary holographic device 
embodying the present invention. With these figures, the reason why the 
binarized phase difference distribution B(r) and the intensity 
distribution I(r) are simultaneously realized will be described. 
As shown in FIG. 6A, a resist film FI having a refractive index n is 
uniformly applied on a glass plate SB. 
At this time, in order to shift by .pi. the phase difference between 
diffracted light having passed through the convex portions and diffracted 
light having passed through the concave portions, the resist film FI is 
formed to have a thickness h which is calculated in advance by 
##EQU2## 
The uneven surface is easily made by drawing a pattern with an electron 
beam or a laser beam and developing it so that portions .alpha., .beta. 
and .gamma. of the resist film FI where B(r)=0 are removed to form concave 
portions. As a result, the binarized phase difference distribution B(r) 
and the intensity distribution I(r) are realized with the uneven surface 
serving as a diffraction grating. 
Luminous fluxes having an intensity I are incident on the pitch d at 
portions (1), (2) and (3) and transmitted. After diffraction, zero-order 
diffracted light rays having intensities J1, J2 and J3, respectively, and 
first-order diffracted light rays having intensities K1, K2 and K3, 
respectively, are outputted. At this time, it is assumed that at the 
portion (1) 
##EQU3## 
at the portion (2) 
##EQU4## 
and at the portion (3) 
##EQU5## 
where a1, a2 and a3 are upper surface widths of the convex portions formed 
in resist films FI1, FI2 and FI2 applied onto glass plates SB1, SB2 and 
SB3 at the portions (1), (2) and (3), respectively. 
The diffraction efficiencies at the portions (1), (2) and (3) are given by 
K1/I, K2/I and K3/I, respectively. As described later (expression (12)), 
at the portion (2) where the expression (10) holds, the diffraction 
efficiency thereof given by K2/I is the maximum. At the other portions (1) 
and (3) where the expressions (9) and (11) hold, respectively, the larger 
the difference between a and d/2 is, the smaller the diffraction 
efficiency is. Thus, the intensity distribution I(r) of the diffracted 
light ray at the portion (2) is larger than the intensity distributions 
I(r) of the diffracted light rays at the portions (1) and (3). 
That is, the portions (1) and (3) perform the function of the portions la 
of the conventional amplitude modulation holographic device 1 of FIG. 1 
having a small amplitude distribution, while the portion (2) performs the 
function of the portions lb having a large amplitude distribution. Thus, 
with this binary holography, it is possible to omit the high-cost, 
difficult process of joining the amplitude modulation holographic device 1 
and the phase modulation holographic device 2. 
Subsequently, it will be specifically described that in the binary phase 
holography of FIGS. 6A and 6B, a desired binarized phase difference 
distribution B(r) and intensity distribution I(r) can simultaneously be 
realized by forming the convex portions in each pitch d so as to have an 
upper surface width a and a height h which are calculated in advance. 
First, in order to realize a diffraction efficiency distribution .eta.(r), 
the surface configuration of the binary phase holographic device is 
calculated in the following manner. 
With respect to the diffraction efficiency distribution .eta.(r) of a 
binary diffraction grating, when the pitch of the grating at a coordinate 
r is d(r)=d (i.e. the pitch is constant) and a(r) is a distribution of the 
upper surface width a of a concave portions formed in each pitch d of the 
diffraction grating located at a coordinate r which distribution is 
provided to determine the configuration of the convex portions for 
providing a phase difference .pi. to transmitted light, the following 
proportional expression holds: 
##EQU6## 
where C is a constant. That is, the diffraction efficiency .eta.(r) is the 
maximum when the upper surface width a(r) of the convex portion is d/2, 
and the larger the difference between a(r) and d/2 is, the smaller the 
diffraction efficiency .eta.(r) is. Substituting the expression (12) with 
C=1 with respect to a(r)/d, 
##EQU7## 
That is, by forming the upper surface width a(r) of the convex portion in 
each pitch d of the diffraction grating located at a coordinate r so as to 
have a width a(r) fulfilling the expression (13), a desired diffraction 
efficiency distribution .eta.(r) can be realized at the coordinate r. 
If the upper surface width a(r) is given by the expression (13) at a part 
of the binary diffraction grating and the diffraction efficiency 
distribution .eta.(r) is applied to the binary diffraction grating, the 
same function will be achieved as that of the desired transmission 
distribution I(r)/Imax provided to the amplitude modulation holographic 
device 1 of FIG. 1. Hence, in binarizing the phase distribution P(r) to 
obtain B(r), based on the expression (13), the threshold value PTH thereof 
is set as 
##EQU8## 
In producing a binary phase holography with the phase difference 
distribution B(r) binarized as described above in order to pseudo-realize 
of the phase difference distribution P(r), the value of the phase 
difference distribution B(r) is set so as to fulfill the expressions (7)a, 
(7)b and (14). Hence, the phase difference distribution B(r) is obtained 
by 
EQU B(r)=.pi.(PTH.ltoreq.P(r)&lt;2 .pi.) (15)a 
or by 
EQU B(r)=0(0.ltoreq.P(r)&lt;PTH) (15)b 
##EQU9## 
FIG. 3 shows the phase difference distribution B(r) graphically shown based 
on the expression (15)a or (15)b. In order to realize the phase difference 
distribution B(r), a height H(r) of the convex portion at a coordinate r 
is obtained by 
##EQU10## 
Subsequently, the diffraction efficiency distribution .eta.(r) will be 
described. As described above, in this embodiment, the diffraction 
efficiency distribution .eta.(r) of the diffraction grating shown in FIGS. 
6A and 6B performs the function of the intensity distribution I(r) of the 
binary holographic device 1 of FIG. 1. 
For simplicity, the displacement of r is set to 
EQU 0.ltoreq.r&lt;d (17) 
P(r) to be pseudo-realized shown in the P(r)-r graph of FIG. 3 is given 
within the range of the expression (17) by an approximation of the 
following expression: 
##EQU11## 
Assuming that the expression (18) and the expression (14) are equal to 
each other, the coordinate r at which P(r) and PTH intersect is 
##EQU12## 
Since the expression (20) coincides with the previously described 
expression (13), a desired diffraction efficiency distribution .eta.(r) is 
realized. Since the same holds at coordinates other than those fulfilling 
the expression (17), a description thereof will be omitted. 
That is, when PTH is as defined by the expression (14), the upper surface 
width a(r) of a convex portion formed in one pitch of the diffraction 
grating located at a coordinate r which is an intersection of PTH and P(r) 
is a distance from the coordinate r to a coordinate d, and the height of 
the convex portion formed in one pitch of the diffraction Grating is H(r) 
obtained by the expressions (16)a and (16)b, the diffraction efficiency 
distribution .eta.(r) and the binarized phase distribution B(r) are 
precisely realized. 
In the process, the resist film FI having a thickness h (=.lambda./(n-1)) 
is prepared, and to form the surface configuration shown in the H(r)-r 
graph of FIG. 3, the convex portions with a height H(r)=h are formed by 
removing portions where the height H(r) is to be H(r)=0 to form concave 
portions so that the calculated distribution a(r) of the upper surface 
width of one pitch of the diffraction grating located at a coordinate r is 
provided. 
Second Embodiment 
Another embodiment of the present invention employing a blazed holography 
will be described. 
FIG. 4 shows a surface configuration H(r) of a saw-toothed blazed 
holography where a phase difference distribution to be provided to 
incident light waves is formed to Give a relative value between [0, 
2.pi.]. 
In FIG. 4, an original point 0 is an intersection of the zone plate 4 and 
the optical axis 5 shown in FIG. 2, the horizontal axis represents a 
coordinate r on the zone plate 4, and the vertical axis represents a 
height H(r) of saw-toothed portions formed on the surface of the zone 
plate 4. 
At this time, the saw-toothed portions are formed so that a peak height ho 
thereof is 
##EQU13## 
The coordinate interval between each peak is a constant value, i.e. a 
pitch d. 
FIG. 7 shows a cross-sectional view of a blazed holography where the peak 
height of the saw-toothed portions is provided with a distribution. The 
diffraction efficiency thereof will be described. 
As shown in FIG. 7, saw-toothed portions FI4 and FI5 are formed on glass 
plates SB4 and SB5. The saw-toothed portions FI4 and FI5 perform the 
function of a diffraction grating by diffracting light rays incident on 
the pitch d with an intensity I at a predetermined diffraction angle and 
outputting diffracted rays K4 and K5. Thus, a phase difference 
distribution and the intensity distribution I(r) are realized with these 
saw-toothed portions serving as a diffraction grating. 
That is, luminous fluxes having an intensity I are incident on the pitch d 
at portions (4) and (5) and transmitted. After diffraction, transmitted 
light rays having intensities J4 and J5, respectively, and diffracted 
light rays having intensities K4 and K5, respectively, are outputted. At 
this time, it is assumed that at the portion (4) 
EQU h5&lt;h4.ltoreq.ho (22) 
and at the portion (5) 
EQU 0.ltoreq.h5&lt;h4 (23) 
where h4 and h5 respectively represent peak heights of the saw-toothed 
portions FI4 and FI5 formed on the glass plates SB4 and SB5 at the 
portions (4) and (5). 
The diffraction efficiencies at the portions (4) and (5) are given by K4/I 
and K5/I, respectively. As described later (expression (24)), at the 
portion (4) where the expression (22) holds, the diffraction efficiency 
thereof given by K4/I is larger than the diffraction efficiency given by 
K5/I at the portion (5) where the expression (23) holds. Thus, the 
intensity distribution I(r) of the diffracted light ray at the portion (4) 
is larger than the intensity distribution I(r) of the diffracted light ray 
at the portion (5). 
That is, the portion (5) performs the function of the portions la of the 
conventional amplitude modulation holographic device 1 of FIG. 1 having a 
small amplitude distribution, while the portion (4) performs the function 
of the portions lb having a large amplitude distribution. Thus, with this 
blazed holography, it is also possible to omit the high-cost, difficult 
process of joining the amplitude modulation holographic device 1 and the 
phase modulation holographic device 2. 
Subsequently, it will be specifically described that in the blazed 
holography shown in FIG. 7, the phase difference distribution and the 
intensity distribution I(r) can simultaneously be realized by forming the 
saw-toothed portion located at a coordinate r so as to have a peak height 
h(r) which is precisely calculated. 
First, in order to realize a diffraction efficiency distribution .eta.(r), 
the surface configuration of the blazed holographic device is calculated 
in the following manner. 
The diffraction efficiency .eta.(r) when the peak height h(r) (0&lt;h(r)&lt;ho) 
of the saw-toothed portions is provided is obtained by 
##EQU14## 
where C is a constant. That is, the larger and the closer to ho the peak 
height h(r) of the saw-toothed portions is, the larger the diffraction 
efficiency distribution .eta.(r) is, and the diffraction efficiency 
distribution .eta.(r) is a maximum value .eta.max when h(r)=ho. 
Hence, the following expression holds: 
##EQU15## 
Substituting the expression (25) with respect to the peak height h(r) of 
the saw-toothed portions in order to obtain a desired diffraction 
efficiency distribution .eta.(r), 
##EQU16## 
By modulating the surface configuration H(r) of the saw-toothed portions as 
shown in FIG. 5 so as to have the peak height h(r) provided by the 
expression (26) and processing the surface in accordance therewith, a 
blazed holography having the diffraction efficiency .eta.(r) can be 
produced. 
In FIG. 5, the original point 0 is an intersection of the zone plate 4 and 
the optical axis 5 shown in FIG. 2, the horizontal axis represents the 
coordinate r on the zone plate 4, and the vertical axis represents the 
height H(r) of the saw-toothed portions formed on the zone plate 4. 
In the present invention, the diffraction efficiency distribution .eta.(r) 
provided to the blazed holography having the phase distribution performs 
the function of the intensity distribution I(r) of the binary phase 
holographic device 1 of FIG. 1 for providing amplitude distribution 
modulation. 
Moreover, while a transmission holography is used in the above embodiments, 
when a reflection holography is used, with a similar structure, it is also 
possible to provide to an incident wavefront a phase distribution and an 
amplitude distribution in accordance with a diffraction efficiency 
distribution by use of one holographic device. 
Further, in the case of the binary holographic device, while in the 
above-described embodiments, a distribution is provided to the diffraction 
efficiency by varying the widths of the convex portions on its cross 
section, a distribution may be provided to the diffraction efficiency by 
varying the heights of the convex portions. Moreover, in the case of the 
blazed holographic device, while in the above-described embodiments, a 
distribution is provided to the diffraction efficiency by varying the peak 
heights of the saw-toothed portions, a distribution may be provided to the 
diffraction efficiency by varying the widths of the saw-toothed portions. 
Obviously, many modifications and variations of the present invention are 
possible in light of the above teachings. It is therefore to be understood 
that within the scope of the appended claims, the invention may be 
practiced other than as specifically described.