Diffraction grating apparatus and method of forming a surface relief pattern in diffraction grating apparatus

A phase diffraction grating apparatus which is usable to generate an array of N spots, where n is an even integer, and the N spots are substantially equally spaced. The generation of an even number of spots is achieved using a translation symmetry in the grating design. Illustratively, the intensities of the N spots are substantially equal. Binary, multi-level, and continuous phase grating embodiments are disclosed. In the case of the multi-level and continuous embodiments, the uniform intensity is obtained using a reflection symmetry in the grating design.

TECHNICAL FIELD 
This invention relates to optics. 
BACKGROUND AND PROBLEM 
Digital optical systems that employ arrays of optical logical devices 
require an array of high intensity light beams in order to set and query 
the logic values of the devices. At the present time, laser diode arrays 
neither provide the beam quality nor have the appropriate configuration to 
generate a beam spot array that could adequately operate such an optical 
system. Binary phase gratings are diffractive optical elements that, in 
combination with imaging optics, produce a set of spots. These binary 
phase gratings (BPG), also referred to as Dammann gratings and described 
in H. Dammann and K. Gortler, "High-Efficiency In-Line Multiple Imaging by 
Means of Multiple Phase Holograms", Opt. Comm., 3, 312-315 (1971), and H. 
Dammann and E. Klotz, "Coherent Optical Generation and Inspection of 
Two-Dimensional Periodic Structures", Optica Acta 24, 505-515 (1977), are 
usable to produce an array of uniform intensity spots from a single 
collimated laser beam in the manner disclosed in U. Killat, G. Rabe, and 
W. Rave, "Binary Phase Gratings for Star Couplers with High Splitting 
Ratio", Fiber and Integrated Optics 4, 159-167 (1982). 
In order for digital optical systems to move from the realm of theory to 
practical prototype, a number of issues related to spot array generation 
must be addressed. The issues associated with spot array generation fall 
primarily into the categories of theoretical design concerns and 
fabrication methods and tolerances and are summarized as follows: 1) 
design--matching the spot array with the associated system function and 
configuration; 2) uniformity--assuring that the relative intensities of 
the desired spots meet system tolerances; and 3) efficiency--optimizing 
usage of the illuminating laser beam power. 
Design issues have primarily been centered on the ability to calculate 
solutions for large arrays of spots. Since the number of transitions 
between the two levels of a binary phase grating for a specific solution 
is proportional to the number of spots, the complexity of designing a 
large array escalates quickly. Because of this complexity the optimization 
programs and associated computational resources can quickly become a 
severe constraint. However, methods have been demonstrated to overcome 
this constraint as disclosed in F. B. McCormick, "Generation of Large Spot 
Arrays from a Single Laser Beam via Multiple Imaging with Binary Phase 
Gratings", Opt. Eng. 28, 299-304 (1989). 
Intensity uniformity is the primary limitation of spot arrays now 
confronting optical system architects. Spot intensity uniformity is 
determined by requirements on the operation of the optical logic devices. 
Critical biasing and/or contrast differences between interconnected logic 
devices will often dictate these tolerances. Non-uniform intensities are 
primarily introduced by limitations of the fabrication process and may be 
impractical to control in many configurations. 
Further, the phase grating must be able to maximally utilize the energy of 
the impinging laser beam by efficiently distributing the light to the 
desired spots. This is critical since the speed of the system is 
influenced by the energy available per optical device and economical high 
power laser sources are not currently available. 
The typical binary phase grating of the prior art, referred to herein as 
the standard grating solution, forms an odd number of spots. An example of 
a spot array generated by a standard solution is shown in FIG. 1. Note 
that the spot array of FIG. 1 is a two-dimensional array of 25 spots 
comprising lines of five spots in one dimension and lines of five spots in 
a second dimension. More generally, a two-dimensional array of N.times.M 
spots comprises one-dimensional arrays (lines) of N spots in one dimension 
and one-dimensional arrays (lines) of M spots in a second dimension. The 
standard spot array design has a high intensity spot central order 
surrounded by uniform intensity sets of both positive and negative order 
spots. Symmetries inherent in the BPG design require that each positive 
and negative order pair have matching intensities. Therefore, only the 
central order spot can be influenced individually. Elimination of the 
central order from the standard solution in the manner disclosed in the 
Killat et al. paper, creates an array with an even number of spots; 
however the spatial regularity of the array is unfortunately destroyed. 
This is particularly significant in applications such as the optical 
crossover network disclosed in the U.S. patent application Ser. No. 
07/349,281 of T. J. Cloonan et al. filed on May 8, 1989, allowed on May 
20, 1991, now U.S. Pat. No. 5,077,483 issued Dec. 31, 1991 and assigned to 
the assignee of the present invention, where the use of each device of an 
array of optical devices is important in achieving the reduced blocking 
characteristics of the network. The elimination of the central order spot 
in the Killat arrangement means that the spacing between the two first 
order spots is twice as large as the spacing between the other spots. 
Therefore, use of the Killat arrangement in the Cloonan optical crossover 
network to illuminate arrays of equally spaced optical devices would 
render the devices corresponding to the suppressed central order unusable. 
SOLUTION 
This problem is solved and a technical advance is achieved in accordance 
with the principles of the invention in an exemplary phase diffraction 
grating apparatus having a surface relief pattern obtained using an 
optimization method such that the grating is usable to generate an array 
of N spots, where N is an even integer, and the N spots are, in a 
departure in the art, substantially equally spaced since all of the even 
order spots are suppressed in addition to the central order spot. The 
surface relief pattern includes a plurality of replicated periods; the 
spot spacing present in the spot array (FIG. 2) produced by an exemplary 
binary phase grating embodiment of the invention is made to be the same as 
the spacing of the spot array (FIG. 1) produced by the standard grating 
solution, by doubling the period within the surface relief pattern and 
thereby dividing the order spacing of potential spots by two. The 
generation of an even number of spots is achieved using a translation 
symmetry in the grating design. Illustratively, the intensities of the N 
spots are substantially equal. In the case of multi-level and continuous 
embodiments disclosed herein, the uniform intensity is obtained using a 
reflection symmetry in the grating design. 
A phase diffraction grating in accordance with the invention has a surface 
relief pattern formed in the grating such that when a monochromatic plane 
wave of light is transmitted to the grating, light is transmitted from the 
surface relief pattern to form an array of N spots, where N is an even 
integer, and the N spots are substantially equally spaced. 
Illustratively, the spots are formed by transmitting the plane wave of 
light through the grating, or alternatively, by reflecting the plane wave 
of light from the grating. For example, in one optical setup (FIG. 4), a 
collimated laser beam 41 is directed to a binary phase grating 42 and the 
diffracted light is focused by a transform lens 43 into a line 44 
(one-dimensional array) in an image plane, one focal length away from lens 
43. Note that if lens 43 is removed, the line of spots would be formed in 
the far field. A magnified view of the one dimensional phase grating 42 is 
shown in FIG. 5. 
In a binary phase grating embodiment, the surface relief pattern has two 
levels and has a set of transitions between the levels, the set including 
a plurality of periods of length P, where each period has a plurality of 
transitions between the two levels. When the plane wave of light is of 
wavelength .lambda., and the N spots are formed by passing the transmitted 
light through an objective lens of focal length f, the substantially equal 
spacing S of the N spots is given by 
##EQU1## 
or twice the spacing of the standard grating solution. The two levels are 
separated by a phase depth substantially equal to .pi. (or equivalently 
3.pi., 5.pi., 7.pi.. . . ). The N spots are formed in a line and, with 
respect to a plurality of orders spaced apart by a spacing 1/2 S and 
comprising a central order and even and odd orders on each side of the 
central order, the N spots correspond to the odd orders. The transitions 
in the second half of each period are obtainable by a translation of the 
first half period transitions and have a phase offset substantially equal 
to .pi. with respect to the first half period transitions. 
In a multi-level phase grating embodiment, the surface relief pattern has L 
levels, L being a positive integer greater than two, and the surface 
relief pattern has a set of transitions between ones of the levels, the 
set including a plurality of periods of length P, where each period has a 
plurality of transitions between ones of the levels. When the plane wave 
of light is of wavelength .lambda., and the N spots are formed by passing 
the transmitted light through an objective lens of focal length f, the 
substantially equal spacing S of the N spots is given by 
##EQU2## 
or twice the spacing of the standard grating solution. The levels are 
separated by a phase depth substantially equal to 
##EQU3## 
The N spots are formed in a line and, with respect to a plurality of 
orders spaced apart by a spacing 1/2 S and comprising a central order and 
even and odd orders on each side of the central order, the N spots 
correspond to the odd orders. The transitions in the second half of each 
period are obtainable by a translation of the first half period 
transitions and have a phase offset substantially equal to .pi. with 
respect to the first half period transitions. The transitions in the 
second quarter of each period are obtainable by reflection of the first 
quarter period transitions about a midpoint of the first half period. 
In a continuous grating embodiment, the surface relief pattern comprises a 
continuous surface comprising a plurality of periods of length P. When the 
plane wave of light is of wavelength .lambda., and the N spots are formed 
by passing the transmitted light through an objective lens of focal length 
f, the substantially equal spacing S of the N spots is given by 
##EQU4## 
or twice the spacing of the standard grating solution. The N spots are 
formed in a line and, with respect to a plurality of orders spaced apart 
by a spacing 1/2 S and comprising a central order and even and odd orders 
on each side of the central order, the N spots correspond to the odd 
orders. The transitions in the second half of each period are obtainable 
by a translation of the first half period transitions and have a phase 
offset substantially equal to .pi. with respect to the first half period 
transitions. The transitions in the second quarter of each period are 
obtainable by reflection of the first quarter period transitions about a 
midpoint of the first half period. 
Optical spot-generating apparatus in accordance with the invention 
comprises a source of a monochromatic plane wave of light of wavelength 
.lambda., an objective lens of focal length f, and a phase diffraction 
grating. The grating is responsive to the plane wave of light and has a 
surface relief pattern comprising a plurality of periods of length P for 
transmitting light from the surface relief pattern through the lens to 
form an array of N spots, where N is an even integer. The N spots are 
substantially equally spaced at a spacing S given by 
##EQU5## 
An optical system in accordance with the invention has a plurality of 
stages, with each stage comprising a source of a monochromatic plane wave 
of light, a phase diffraction grating, and an array of N.times.M optical 
devices, where N and M are even integers. The phase diffraction grating 
has a surface relief pattern formed in the grating. The grating is 
responsive to the plane wave of light for transmitting light from the 
surface relief pattern to form an array of N.times.M spots. The N spots 
are substantially equally spaced, and the M spots are substantially 
equally spaced. The array of N.times.M devices is responsive to the array 
of N.times.M spots. 
A method in accordance with the invention is used for forming a surface 
relief pattern in a phase diffraction grating such that when a 
monochromatic plane wave of light is transmitted to the grating, light is 
transmitted from the surface relief pattern to form an array of N spots, 
where N is an even integer, and the N spots are substantially equally 
spaced. First half periods are formed for a plurality of periods of length 
P which comprise the surface relief pattern. Second half periods are 
formed for the plurality of periods, where the second half periods are 
obtained by a translation of the first half periods and having a phase 
offset substantially equal to .pi. with respect to the first half periods.

DETAILED DESCRIPTION 
FIG. 2 is a diagram of a two-dimensional array of 16 spots produced by a 
diffraction phase grating in accordance with an embodiment of the present 
invention. 
The exemplary grating design producing even numbered spot arrays can be 
compared with the standard design FIG. 1. The exemplary grating design for 
even number spot arrays preserves spot regularity while creating other 
advantages preferable in digital optical system applications. For the case 
of the even number spot array, all even order spots are suppressed as 
shown in FIG. 2. It is this suppression of every other spot in the one 
dimensional array that leads to a regular array. 
The functional dependence of the central or zero order intensity differs 
radically from that of the other orders and is the primary contributor to 
non-uniformity problems. It is thus very advantageous to eliminate the 
zero order spot from the spot array design. Indeed, the zero order 
intensity is so highly sensitive to the optical path difference (also 
referred to as the phase depth or phase difference) between levels that it 
is unlikely that larger arrays will be economically fabricated since 
substrates with a sufficient surface quality will be unduly expensive. 
This situation is illustrated in FIG. 3 where the deviation of the central 
order intensity is plotted for both a standard and even numbered spot 
array. Both arrays contain a similar number of spots, yet the phase depth 
sensitivity for the standard solution is considerably greater than that of 
the new design. As the number of spots increases, the sensitivity becomes 
even more severe. For example, the fabrication of a 1.times.45 
diffraction grating with a maximum 10% intensity deviation would require 
that the fabrication error be no larger than about 1/20 of a wavelength. 
The advantages of the even numbered arrays are considerable. First, the 
system designer is no longer hampered by the sensitivity of the central 
order intensity to phase depth error. The central order intensity will be 
small and, if the system contains an intermediate focus, can easily be 
masked and eliminated. Second, systems based on regular square arrays of 
devices containing on the order of 2.sup.n logic devices will be most 
suitable for prototype systems. Thus the even numbered design can be used 
to precisely match spot arrays with these device arrays. Lastly, since the 
optical path difference of the one dimensional even numbered grating is a 
phase difference of .pi. (or equivalently 3.pi., 5.pi., 7.pi.. . . ), it 
is simple to combine two one-dimensional solutions to obtain a two 
dimensional array and still conform to a two level phase design. 
The production of spot arrays from diffraction gratings is performed in an 
optical Fourier transform setup (FIG. 4) where a collimated laser beam 41 
is directed to a binary phase grating 42 and the diffracted light is 
focused by a transform lens 43 into a line 44 (one-dimensional array) in 
an image plane, one focal length away from lens 43. Note that if lens 43 
is removed, the line of spots would be formed in the far field. A 
magnified view of the one dimensional phase gratings 42 is shown in FIG. 
5. The condition under which it is assumed that the output spot array is 
described by the Fourier transform of the light that passes through the 
phase grating, is referred to as the Fraunhofer assumption (see for 
instance the book Introduction to Fourier Optics by John Goodman, 
McGraw-Hill Book Company, 1968, p.61, equation (4-12)). Given a incident 
plane wave of maximum radius, R, and wavelength, .lambda., the image is 
formed in the far field at a distance z described by 
##EQU6## 
However the patterns can be observed at distances closer than that implied 
by the equation. In general, it is the property of a converging lens to 
perform a two-dimensional Fourier transformation. Several periods of a one 
or two dimensional replicated pattern are illuminated by a collimated 
laser source. The staggered surface levels of the grating introduce a 
phase difference between light transmitted through the various regions 
resulting in interference in the far field of the optical system. The 
distribution of the light intensity in the resulting spot array is a 
combination of both the Fourier transform of the source distribution and 
the Fourier transform of the grating topology. Neglecting the contribution 
from the source beam distribution, the output consists of a series of 
spots (due to the periodic nature of the pattern) where each order 
intensity is provided by the Fourier transform of one complete period of 
the grating pattern. 
Since the spot arrays of the present embodiments are either square or 
rectangular, the two-dimensional structure is separated and each 
one-dimensional solution is individually obtained. The amplitude for a 
specific order of a one-dimensional grating is expressed by the equation: 
##EQU7## 
where n designates the order of spot, .theta.(x) is the function 
describing the phase relief of the one dimensional diffraction grating 
surface, and the coordinate system has been normalized so that a single 
period of size P is the unit length. Since the intensity of the spot is 
the complex square of the amplitude, only relative coordinate and phase 
shift differences need to be considered, i.e., the solution is shift 
invariant with respect to an arbitrary coordinate and phase offset. The 
spots generated by the array in the image plane are separated by a 
distance 
##EQU8## 
where S is the spot spacing, f is the focal length of the Fourier 
transform lens, P is the period of the grating pattern, and .lambda. is 
the wavelength of the illuminating laser beam. In the case of the even 
numbered arrays every other order is suppressed; therefore, the distance 
between high intensity spots is twice that given by equation [2] or 
##EQU9## 
The grating consists of a series of repeated patterns fabricated on an 
optically transparent substrate. The coordinate system used to describe 
the grating structure is shown in FIG. 6. The location of the transitions 
are given by x.sub.k where x.sub.k, (1&lt;k&lt;N) is the position of a 
transition within the range 0.ltoreq.x.sub.k .ltoreq.1 and N is the total 
number of transitions with N being an even number. The term .theta..sub.k 
represents the relative phase shift experienced by the light in the region 
between x.sub.k-1 and x.sub.k. The case of the two level or binary phase 
grating is limited to two phase levels where .DELTA..theta. is the 
magnitude of the phase difference. The amplitude relationship is 
formulated by solving equation [1] using the configuration shown in FIG. 
6. The solution consists of two equations: 
##EQU10## 
whereas the central order term (n=0) is given by: 
##EQU11## 
These last two equations are adequate to specify the operation of both the 
standard and even numbered spot array gratings. 
It is expected that the number of transitions necessary to produce a given 
array is proportional to the number of restrictions applied to the given 
orders. At first glance this would appear to favor the design of the 
standard spot array grating since it is only necessary to restrict the N 
central uniform intensity spots, while the even numbered design must 
regulate both the N high intensity spots and suppress the N-1 intermediate 
spots. Fortunately, the application of a translational symmetry reduces 
the number of independent transitions to a number consistent with the 
standard design. 
The symmetry condition that is imposed on the even numbered spot array 
solution leads to the suppression of all even order spot intensities. In 
addition, if a phase depth of .pi. is specified, the central order 
intensity also becomes zero. The symmetry is illustrated in FIG. 7. All 
transitions in the first half of the period are reproduced in the second 
half of the period by translating the coordinate by one half period. Next, 
the phase value of the corresponding region in the second half of the 
period is set to the opposite value of that in the first region. This is 
equivalent to adding a value of .pi. to the phase value and then reducing 
it modulus 2.pi.. More formally, 
EQU X.sub.k =X.sub.k-N/2 +0.5, for N/2+1.ltoreq.k.ltoreq.N [6] 
EQU .theta..sub.k =.theta..sub.k-N/2 +.pi.. [7] 
It can be shown that the outcome of this symmetry is the solution: 
##EQU12## 
In addition, the choice of .pi. as the phase difference, .DELTA..theta., 
leads to A(0)=0. 
To summarize, a translational symmetry is utilized to naturally suppress 
the undesired orders and to reduce the number of independent transitions 
necessary to describe the grating structure. Due to this symmetry, the 
complexity of locating a set of transitions for an even numbered design is 
similar to that for a comparable size standard array. Unfortunately, in 
spite of the inherent symmetry, the design does require about twice the 
number of transitions that are required by a standard grating. This 
additional complexity is diminished, though, by the use of a period length 
that is twice that of the conventional design to form high intensity spots 
with the same spacing. Finally, by judiciously choosing the separation of 
the two levels to be equal to a phase difference of .pi., the central 
order spot intensity is eliminated and a solution is obtained that is 
easily implemented as a two-level, two-dimensional spot array. 
Although equation [8] specifies the amplitude and hence the intensity at 
each order, it is the task of the gratings designer to locate the set of 
transitions that adequately satisfy the criteria for the spot array 
intensity distribution, nominally a uniform intensity spot array of a 
specific size. These criteria include the overall efficiency for 
diffracting the impinging beam into the set of desired orders, henceforth 
referred to as .eta., and the standard deviation of the desired order 
intensities from a nominal value, .sigma.. A cost or merit function is 
constructed that maximizes the efficiency and minimizes the intensity 
deviation. The cost function, C, which is a function of the transition and 
phase coordinates sets, was chosen to be 
##EQU13## 
where .eta., the diffraction efficiency, and .sigma..sup.2, which measures 
the intensity nonuniformity, are defined as 
##EQU14## 
In these formulae, M is the extreme order and I(n) is the order intensity 
given by: 
EQU I(n)=A(n).sup.2 [ 12] 
where A(n) is the amplitude given by equation [8]. 
Potential solutions were identified in the following manner. First, a 
random set of transitions were generated and the efficiency and intensity 
deviation were compared against an initial set of criteria. If the 
solution failed to meet a specified criteria, it was discarded and a new 
random solution was generated. A successful solution is passed to the 
simulated annealing routines. 
The simulated annealing algorithm as disclosed in N. Metropolis, A. W. 
Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, "Equation of 
State Calculations by Fast Computing Machines", J. Chem. Phys. 21, p. 
1087-1092 (1953) provides a means of locating a global minimum in the 
merit function. It avoids the pitfall of becoming trapped in local minima 
as can often occur in a simple gradient descent algorithm. Starting with 
the initial transition set, a series of new transitions located in the 
vicinity are sampled. If the new solution's cost is lower than that of the 
current test solution, then it is selected to become the current optimal 
solution. However if the cost value is greater than that of the current 
solution, a decision is made based on the following criteria. A random 
number generated over the range 0 to 1 is compared against a probability 
given by 
##EQU15## 
where the prime denotes the test solution and T is a slowly decreasing 
parameter. If the random number exceeds the probability, then the new 
transition set is retained. Otherwise, the solution is discarded. 
Initially the value of T is set at a moderate value so that shallow local 
minima do not bind a particular solution. Over the course of the simulated 
annealing, the value of T is gradually decreased and the search region is 
narrowed so that the solution eventually locates a minimum. 
The simulated annealing is performed in a series of steps. At the end of 
each step the solution must meet threshold values on the cost, efficiency 
and intensity deviation. Whenever a solution fails to meet the criteria 
the program generates a new random solution and begins the simulated 
annealing again. Usually the final conditions require the potential 
solution to have an efficiency greater than 70-75% and an intensity 
deviation of approximately 5%. 
If these conditions are satisfied a gradient descent algorithm is used to 
locate the nearest minimum. The gradient search uses an approximation to 
the first and second derivative in a second order Taylor series expansion 
to predict the coordinates of the minimum. This process typically requires 
only a few steps. If the final solution set meets the overall requirements 
(typically 75-80% efficiency and less than 1% intensity deviation) the 
transitions set is retained. 
The optimization program proceeds until several solutions are collected for 
a specific array of spots. Many unique solutions are often located for the 
larger spot arrays. Ultimately the optimal solution depends on factors 
beyond the efficiency and intensity uniformity criteria. For example, the 
tolerances of the fabrication process often dictate the minimum size of a 
feature and the degree of quantization of the pattern. 
Collected in Table 1 are solutions for the 1.times.4, 1.times.8, 1.times.16 
and 1.times.32 binary phase gratings together with the theoretical 
efficiency. Each solution has two phase levels which differ by .pi. 
radians. FIGS. 8-11 show three periods of the amplitude pattern that would 
be used to produce gratings for 1.times.4, 1.times.6, 1.times.8, and 
1.times.16 spot arrays, respectively. The translational symmetry between 
the first and second half of the period is clearly visible in FIGS. 8-11. 
These theoretical solutions are used in the fabrication and 
characterization of the diffraction gratings. 
TABLE 1 
______________________________________ 
Transitions in first half of period of even 
number spot array designs 
Solution 
1 .times. 4 
1 .times. 6 
1 .times. 8 
1 .times. 16 
1 .times. 32 
Efficiency 
0.705 0.845 0.761 0.805 0.837 
______________________________________ 
1 0.0000 0.0000 0.0000 0.0000 
0.0000 
2 0.2171 0.2910 0.0617 0.0610 
0.0746 
3 0.2716 0.3855 0.2436 0.1152 
0.1515 
4 0.5000 0.5000 0.3537 0.2775 
0.1714 
5 0.3858 0.3141 
0.2002 
6 0.5000 0.3456 
0.2066 
7 0.3925 
0.2406 
8 0.4260 
0.2637 
9 0.4426 
0.3019 
10 0.5000 
0.3308 
11 0.3701 
12 0.3907 
13 0.4068 
14 0.4254 
15 0.4441 
16 0.4612 
17 0.4810 
18 0.5000 
______________________________________ 
The fabrication procedure for creating the even numbered binary phase 
gratings uses many of the techniques commonly used in semiconductor 
processing technology. FIGS. 12-15 illustrate the basic procedure for 
producing the binary phase structure via reactive ion etching. Starting 
with an optically transparent substrate such as a flat fused silica disc 
that has been thoroughly cleaned to remove all residue, a thin layer of 
photoresist (about 1 micron thick) is applied on one side of the disc. 
Next, the coated disc is placed in contact with an amplitude mask, 
produced by e-beam or optical lithography, which holds the theoretical 
pattern. While the two are in contact, they are exposed with a UV light 
source to transfer the pattern into the photoresist (FIG. 12). The optical 
substrate is then immersed in photoresist developer to remove the soluble 
regions of the exposed thin film. The substrate is next placed into an 
evaporation system where a thin layer of chromium metal is deposited. 
After metal deposition, a liftoff process is used to remove areas with 
metal-coated photoresist (FIG. 13). 
Finally the optical path difference is created in the surface by 
reactive-ion etching of the exposed material (FIG. 14). The etch depth, h, 
is related to the phase depth by the relationship 
##EQU16## 
where .DELTA..theta. is the phase depth, .DELTA.n is the difference 
between the index of refraction of the substrate and air, and .lambda. is 
the wavelength of the illuminating laser. For even numbered gratings a 
phase depth of .pi. is desired. When fused silica is used as a substrate 
and an 850 nm laser is used for illumination, the pattern is etched to a 
depth of about 940 nm with a precision of about 1/2 to 1%. Prior to use, 
the remaining metal is removed and the surface cleaned leaving a 
completely transparent phase diffraction grating (FIG. 15). 
One important procedure used in the production of the etch mask is the 
utilization of a compensation procedure to precisely control the 
transition location. Due to the finite thickness of the photoresist and 
the diffraction of light at the edge of a mask feature, some of the 
photoresist that should nominally be shadowed is exposed. Although the 
amount is small, its effect on the intensity uniformity can be substantial 
for gratings patterns with relatively small pitch or high complexity. It 
has been found that decreasing the lateral space between opaque mask 
regions by 0.7 microns allows transitions to be obtained that are 
consistent with design criteria. 
Currently, the primary criteria for evaluating BPG performance in proposed 
optical systems is the uniformity of the spot array intensity. The 
uniformity determines, for example, how effectively the limited operating 
range of the logic device in a cascaded system can be exploited. 
Efficiency is also a concern; however, multi-level phase gratings are 
usable to improve efficiency. 
A computerized data acquisition system was used to measure the intensity of 
each spot in the spot array and thereby determine the uniformity and spot 
array generation efficiency. A schematic of this setup is shown in FIG. 
16. The binary phase grating was illuminated by a collimated light beam 
emanating from an infrared laser diode operating at a power of about 3 mW 
at a wavelength of 850 nm. This beam was spatially filtered by a 30 micron 
pinhole to improve the beam characteristics. The beam spot at the grating 
was typically 7 mm by 10 mm. Next, the diffracted beam was focused via a 
100 mm objective lens onto a pinhole aperture. The pinhole was mounted 
directly to an HP81520A/HP8152A optical power meter that was connected to 
a GPIB data acquisition bus. The 50 micron pinhole and detector was 
translated between spots using a computer controlled 3 axis stage. The 
entire spot intensity measurement process was controlled by a SUN3/160 
workstation. In addition, the system contained a removable beam splitter 
to allow inspection of the spot array when necessary. Since the gratings 
were fabricated according to specifications for a separate optical 
experiment, there was no attempt to tune the laser diode wavelength to 
optimize the central order performance. 
Several 1.times.4, 1.times.6, 1.times.8, 1.times.16, and 1.times.32 phase 
gratings have been fabricated and their performance characterized. 
Although the results presented within this paper are exclusively from 
gratings with a 1000 micron period, several other gratings with periods as 
small as 331 microns have also been fabricated. 
FIGS. 17-18 present measurements for a grating producing a 1.times.16 spot 
array. The histograms of FIGS. 17 and 18 show the desired and suppressed 
intensity distribution, respectively, relative to the average intensity of 
the desired orders. The desired order intensities of the one dimensional 
16 spot array (FIG. 17) all lie within +/-2% of the intensity average, 
while the majority of the suppressed orders (FIG. 18) are less than 0.5% 
of the average intensity. As expected the suppressed central order spot 
exhibits the largest deviation having an intensity of 1.7% of the average 
intensity corresponding to a etch depth error of about 17 nm. 
FIG. 19 shows the performance when two 1.times.16 gratings are cascaded 
with their patterns oriented orthogonally to form a 16.times.16 spot 
array. All 256 spots are contained in a region spanning 6% on either side 
of the average intensity value and the standard deviation of the 
intensities is 4.0%. It was found that the measured efficiency of the 
diffraction gratings was consistent with the theoretical value when losses 
due to surface reflection were taken into account. It is expected that 
most of these reflection losses could be eliminated if anti-reflection 
coatings were applied to the surfaces. 
The demonstrated performance of two dimensional structures have also shown 
exceptional performance. Several 8.times.8 BPGs with a 662 micron period 
in each dimension have been fabricated and measured to have an intensity 
standard deviation of from 1.4% to 1.8%. In fact, with current fabrication 
technologies, moderately large spot arrays (up to 16.times.16) with 
periods on the order of 500 microns can be fabricated with spot 
intensities limited to a spread of less than 10%. 
Dammann gratings as disclosed in the above-referenced Dammann et al. 
articles are diffraction gratings that produce replicated images or arrays 
of spots from an incoming beam of monochromatic light. This property makes 
them useful for image duplication applications and spot array generation 
in optical computing. The process of designing an appropriate grating 
pattern typically involves an optimization procedure, therefore, the 
primary limitation in designing gratings that produce complex arrays is 
the computational resources required. In this description the 
incorporation of symmetries into the pattern is exploited to significantly 
reduce the complexity associated with the design process. 
Diffraction gratings that are used in optical computing applications are 
designed to create a finite set of uniform intensity images whose 
separation is inversely proportional to the size of the pattern's period. 
FIG. 4 illustrates the optical setup that achieves this result. The 
intensity distribution in the output array is determined by the structure 
of one period of the grating and is related to the Fourier transform of 
this period. Since the application requires the efficient use of the 
available light, diffraction is chosen to occur from the interference of 
the light passing through separate regions having differing optical path 
lengths, i.e., a phase grating. Due to the nature of the fabrication 
process, these diffraction gratings are designed as a set of discrete 
phase levels associated with regions of various sizes (see for example 
FIG. 20). Most of the current work has centered on the two level or binary 
phase grating where the solution can be described by the set of transition 
points alone. In order to improve operating characteristics of the 
gratings, however, multilevel designs are also under examination as 
disclosed in S. J. Walker, J. Jahns, "Array Generation With Multilevel 
Phase Gratings," OSA Annual Meeting, 1988 Technical Digest Series, Vol. 
11. 
The process of designing the spot array generation gratings begins by 
applying scalar diffraction theory to predict the amplitude relationships 
for the set of spots. Next, these relationships are matched with criteria 
describing the array size, intensity uniformity, and desired efficiency 
and then provided as constraints to an optimization program. As disclosed 
in J. Jahns, M. M. Downs, M. E. Prise, N. Stribl, and S. J. Walker, 
"Dammann Gratings for Laser Beam Shaping," Optical Engineering 28(12), 
1267-1275 (1989), the number of transitions needed to describe a specific 
spot array generated by binary phase gratings grows with the size of the 
array, and the resulting complexity of determining a reasonable solution 
grows exponentially with the array size. Therefore, the ability to locate 
a satisfactory solution for larger array sizes is restricted by the 
computational resources available. Indeed, if the transition regions are 
not constrained to two possible phase levels, the complexity of the 
corresponding solution may become even greater. 
Fortunately, symmetry properties are incorporated into the grating design 
to significantly reduce the number of independent parameters required for 
a solution. In the description which follows, some of the symmetry 
properties that have been used to reduce the complexity of locating an 
optimal solution set are first reviewed. Specific symmetries that were 
introduced to produce an even numbered spot array design are then 
described and extended. The solution for a general functional 
representation of phase surface with reflection symmetry about the period 
midpoint and the application of translational symmetry leading to the 
even-numbered array design are described. The previous results are then 
applied to discrete multi-level designs. Binary phase gratings and their 
inherent properties and several grating designs based on these symmetries 
are then described. 
A general representation of a phase diffraction grating consists of a 
periodic repetition of a transparent surface relief pattern that imposes a 
space variant phase shift on an impinging monochromatic plane wave. The 
diffractive effect of this structure leads to a periodic array of spots in 
the far field whose intensity is given by the Fraunhofer approximation to 
scalar diffraction theory. FIG. 4 illustrates a setup that creates such a 
spot array. The grating is specifically designed to modulate only the 
phase of the plane wave so that no light intensity is absorbed. When the 
intensity distribution of the illuminating beam covers an area that is 
significantly larger than the size of a single period, the amplitude of 
each order is essentially given by the Fourier transformation of the 
transmission function of the diffraction grating. 
The expression t(x,y) exp[i.theta.(x,y)] is used to represent the 
functional relationship describing the grating's modification of the 
incident plane of light. Here t(x,y) represents the amplitude transmission 
and .theta.(x,y) gives the phase modulation of the plane wave. A value of 
t(x,y)=1 represents non-attenuated light energy while a value of zero 
represents a fully attenuated light beam. When gratings are fabricated in 
an optically transparent material, the t(x,y) term may be ignored. The 
exponential term, .theta.(x,y), gives the relative phase delay imparted by 
the grating on the impinging light. A delay of the wave by one wavelength, 
.lambda., corresponds to a phase shift of 2.pi.. Since phase shifts that 
differ by multiples of 2.pi. represent effectively equivalent wave fronts, 
the range of phase shift values is ultimately restricted to lie between 0 
and 2.pi.. The maximum difference between two phase levels thus approaches 
2.pi. and is equivalent to an optical material of thickness, t, given by 
##EQU17## 
where .DELTA.n is the difference between the indices of refraction of the 
grating material and the surrounding environment. If, for example, fused 
silica is used as a substrate surrounded by air, the maximum level 
difference required would be about 2.2 wavelengths. The relative amplitude 
of a spot at a specific order is given by the equation, 
##EQU18## 
where n.sub.x and n.sub.y designate the order of the spots in the x and y 
dimension respectively, and by convention the coordinates are normalized 
to a unit period in both dimensions. If an objective lens having a focal 
length f is used to form the image and the grating is illuminated by a 
monochromatic plane wave of wavelength .lambda., the orders in the output 
spot array in each dimension are spaced by an amount, 
##EQU19## 
where P is the size of one period of the pattern in the corresponding 
dimension. The actual spot size and intensity distribution depends on the 
characteristics of the illuminating beam and aberrations inherent in the 
lens system. 
In this description, only regular rectangular arrays having independent 
performance in each dimension are described. Thus the amplitude and phase 
terms can be separated as t(x,y)=t(x) t(y) and .theta.(x,y)=.theta.(x) 
.theta.(y). Using this simplification, the analysis is separated into two 
single one-dimensional solutions and, if necessary, the solutions combined 
to create a two-dimensional spot array. 
Unfortunately, the determination of a suitable phase profile that satisfies 
specific spot array intensity distribution criteria can not be approached 
via analytic means. Optimization techniques are necessary to locate the 
solution that adequately represents the surface. Since the 
parameterization of the surface is described by sets of coefficients 
associated with some set of basis functions (for the case of continuous 
function descriptions) or a set of transitions and phase levels (for 
discrete representations) it is advantageous to select topologies that 
either reduce the complexity of the optimization process. Reflection 
symmetry and translational symmetry are used here to substantially reduce 
the optimization complexity. 
One of the primary objectives in designing a regular rectangular array is 
to ensure that all the intensities, in particular a positive and negative 
order pair, have the same value. It is first shown how the reflection of 
the transmission function about the center point of the pattern leads to 
matching order intensities. Since many of the simplification steps for the 
various symmetry conditions are similar, the first proofs are examined in 
detail to show the nature of the process. 
First, one period of the structure is analyzed to determine the spot 
amplitude, A(n), given at each order, n, using the Fourier transform of 
one period, 
##EQU20## 
The transmission, t(x), is limited to the range 0&lt;t(x)&lt;1. The intensity of 
each order, I(n), is calculated by taking the complex square of this 
amplitude. 
In order to ensure that the intensities of a specific negative and positive 
order are equivalent, the intensities must satisfy the relationship 
EQU I(n)=A(n).multidot.A*(n)=A(-n).multidot.A*(-n)=I(-n), [2-5] 
where A*(n) is the complex conjugate of the amplitude. This implies that 
either A(n)=exp[i.phi.] A(-n) or A(n)=exp[i.phi.]A*(-n), where .phi. is an 
arbitrary angle. As discussed later herein, the second condition is 
automatically satisfied for the case of a binary phase grating, while the 
first condition is met by using a reflection symmetry in the grating 
pattern. 
The equation that describes the negative order amplitude is given by 
##EQU21## 
By making the change of variable, x'=1-x, the integral becomes 
##EQU22## 
The symmetry condition that will ensure that order pairs given by 
equations [2-4] and [2-7] are the same intensity is the requirement that 
the first half of the period be a mirror image of the second half 
differing at most by a phase offset given by .phi.. This is expressed by 
EQU t(1-x)=t(x), 
EQU and 
EQU .theta.(1-x)=.theta.(x)+.phi.. [2-8] 
FIG. 21 shows how one period of the phase shift structure would appear if 
the midpoint of the period is used as the reflection point and a phase 
offset of .phi.=0 is used. If these symmetry conditions are inserted into 
the amplitude solution, the general solution is simplified in the 
following manner. First, the integral is divided into two halves, 
##EQU23## 
Substituting the reflection symmetries expressed by equation [2-8] into 
the second integral, the formula becomes 
##EQU24## 
Making the change variables x'=1-x in the second integral results in, 
##EQU25## 
Thus it has been shown that the design of a diffraction grating requires 
information on only one half of the period. If the choice of .phi. is 
restricted to either 0 or .pi. and it is assumed that there is no 
attenuation (i.e., t(x)=1), equation [2-11] becomes 
##EQU26## 
Thus, application of this reflection symmetry has simplified the 
optimization process. If the surface is to be represented by a continuous 
function, the choice of basis functions can be limited to a set of either 
even or odd symmetric functions. When the solution is parameterized by a 
set of transitions and phase levels, only one half of the transition 
points of each solution set are independent and need to be optimized. 
Since the complexity of the optimization problem grows exponentially with 
the number of independent transitions, this reduction is critical. 
The previous description illustrated that the reflection of half of the 
pattern about the midpoint of the period leads to the condition that both 
the negative and corresponding positive order intensities are equal. In 
the following description, it is shown that the translation of a section 
of a period with an additional phase offset leads to a property that is 
desirable in spot array generation, namely even-numbered spot arrays. 
The analysis begins with the general solution expressed in equation [2-4]. 
As before, the integral is split into two regions and the effects of 
adding a translational shift are examined. The translational symmetry is 
expressed by 
EQU t(x)=t(x-1/2), for 1/2.ltoreq.x.ltoreq.1, 
EQU .theta.(x)=.theta.(x-1/2)+.phi., [3-1] 
where .phi. is an arbitrary phase angle. If .phi. is equal to 0 or 2.pi., 
the pattern is replicated twice as often per period and the result is 
trivial. Substituting equation [3-1]into the integral of equation 
[2-4]yields 
##EQU27## 
After making a change of variables x'=x-1/2 in the second integral, the 
terms can be combined to form 
##EQU28## 
An interesting result occurs when .phi. is chosen so that particular 
orders have zero amplitude. When .phi. is chosen to be .pi., then, 
EQU A(n)=0, 
for n even, and 
##EQU29## 
Thus applying the translation of the pattern and setting the phase offset 
of the resultant shifted region to .pi. leads to a natural suppression of 
all even orders including the zero order. Next, the condition that 
I(n)=I(-n) is enforced. The analysis proceeds in the same manner as in the 
previous section, that is, reflection symmetry of a quarter section of the 
pattern about the midpoint of the half period unit is required. In this 
case, the midpoint 1/4 is used, with the conditions represented as 
EQU t(x)=t(1/2-x), for 1/4.ltoreq.x.ltoreq.1/2, 
EQU .theta.(x)=.theta.(1/2-x)+.phi.', [3-5] 
where .phi.' is an arbitrary phase offset. One period of the phase relief 
is shown in FIG. 22 where the offset, .phi.' is chosen to be 0. The center 
arrow in FIG. 22 denotes the point about which the translational symmetry 
is maintained and the other two arrows mark the points of reflection 
symmetry of the smaller half periods. Simplification of [3-4] is made by 
splitting the integral into two halves, 0&lt;x&lt;1/4 and 1/4&lt;x&lt;1/2, applying 
the symmetry condition, and combining the two results. This leads us to 
the formula 
##EQU30## 
The same result could also be arrived at in a different manner by applying 
two sets of symmetric reflections about the mid- and quarter-points, where 
one reflection has a phase offset of .pi.. Again, simplification will 
result if .phi.' is chosen to be either 0 or .pi.. Finally, it is noted 
that the design of this even number grating requires knowledge of only one 
quarter of the full period. 
Even numbered designs are useful for a variety of applications. FIGS. 1 and 
2 illustrate the difference between the standard spot array, e.g., a 
5.times.5 array (FIG. 1), and a spot array generated by the even numbered 
design, e.g., a 4.times.4 array (FIG. 2). The even numbered spot array 
contains an even number of spots along one dimension, where high intensity 
orders alternate with suppressed orders. This design is advantageous since 
it eliminates the central order. Normally, the central order amplitude, 
when evaluated for discrete grating patterns, leads to a functional 
dependence on .theta.(x) that is different from the other orders. This 
difference leads to a dramatic sensitivity of the zero order intensity on 
the phase depth. With the even numbered design, this sensitivity becomes 
less critical since the spot is no longer an important constituent of the 
regular array. 
The results presented so far have been derived for the case of a general 
functional representation of the grating surface. During the actual 
fabrication of diffraction gratings, one relies on lithographic methods 
that have been developed for the semiconductor industry. These methods 
result in a surface profile that is composed of a set of discrete phase 
levels. The number, L, of phase levels is finite (often 2.sup.m levels) 
and is determined by the number, m, of repeated fabrication steps. 
Therefore it is necessary to derive the formulae for patterns that are 
modeled as sets of discrete phase levels. 
First, the general solution for the amplitude of the one-dimensional 
structure shown in FIG. 20 is derived. The structure is periodic with N 
phase transitions [x.sub.1, x.sub.2, x.sub.3, . . . , x.sub.N ]. The light 
traveling through the area between x.sub.i-1 and x.sub.i differs in phase 
by an amount .theta..sub.i from a reference phase. In addition, the 
assignment t(x)=1 is made, that is, no absorption occurs. The coordinate 
system is normalized so that unit length is equal to one period, 
therefore, all transitions exist in the range [0,1]. Thus the periodic 
nature of the N transitions and phases is expressed by x.sub.i 
+1=x.sub.i+N and .theta..sub.i =.theta..sub.i+N. The form for the 
intensity equation is shift invariant in both position and phase, thus all 
transitions and phase levels are relative. 
As the case of discrete phase levels is solved, it is necessary to analyze 
A(n) for two cases: n not equal to zero and n equal to zero. In the 
following analysis, the non-zero terms are analyzed first followed by the 
central order result. 
First consider the general solution before symmetric properties have been 
incorporated. First equation [2-4] is expanded with order n not equal to 
zero. In performing the integral, the terms containing the initial and 
final boundary are isolated. 
##EQU31## 
The second term becomes the series 
##EQU32## 
Evaluating the integrals of the first and third terms of equation [4-1] 
and using the periodic relatinships of the solution set (x.sub.o =x.sub.N) 
yields 
##EQU33## 
It can be seen that this combination is the k=1 term of the series in 
equation [4-2], thus the full amplitude equation becomes 
##EQU34## 
The terms of the series may be rearranged as a difference of phase shift 
terms as given by 
##EQU35## 
This form can be computationally more efficient when the relative phases 
for each region are held static and only the transitions are varied during 
optimization. It is necessary to derive the zero order amplitude 
separately from that above. Equation [2-4] with n=0 gives, 
##EQU36## 
Using the periodic relationships for the transitions and phases this 
becomes 
##EQU37## 
or equivalently, 
##EQU38## 
Together equations [4-5] and [4-8] describe the amplitudes of a 
generalized discrete level structure. 
Except for the case of binary gratings, which we shall cover in the next 
section, the positive and negative order amplitudes given by the general 
solution are not guaranteed to be equal. In order to assure this 
condition, reflection symmetry is imposed about the midpoint of the 
period. Two possible configurations exist. In one case, a transition point 
exists at the location of the midpoint, and the phase offset is nonzero. 
For this case the symmetry conditions are (from equation [2-8]): 
EQU x.sub.k =1-x.sub.N-k, for N/2+1.ltoreq.k.ltoreq.N, 
EQU .theta..sub.k =.theta..sub.N+1-k +.phi., [4-9] 
For the second case, the phase offset is zero and thus a transition point 
is not required at the midpoint. In this case the conditions are, 
EQU x.sub.k =1-x.sub.N+1-k, for N/2+1.ltoreq.k.ltoreq.N, 
EQU .theta..sub.k =.theta..sub.N+2-k +.phi., [4-10] 
where N is the total number of transitions. In either case, the result of 
evaluating equation [2-11] is, 
##EQU39## 
If the phase offset is assigned the value 0, the amplitude equation 
reduces to 
##EQU40## 
If the zero order amplitude is examined, the result is 
##EQU41## 
When the phase offset, .phi., is chosen to be zero this reduces to 
##EQU42## 
while for .phi. equal to .pi. A(0)=0 is obtained. FIG. 23, which is phase 
plot for one period of a discrete multi-level phase grating with 
reflection symmetry about the period midpoint, illustrates a pattern with 
a value of 0 used for .phi.. 
It is now appropriate to examine the even numbered spot array solutions 
that are produced by applying both a translational shift with phase offset 
about the midpoint and a reflection about the half period midpoint. Using 
equation [3-1] it is required that 
EQU x.sub.k =x.sub.k-N/2 +1/2, for N/2+1.ltoreq.k.ltoreq.N 
EQU .theta..sub.k =.theta..sub.k-N/2 +.pi.. [4-15] 
Following this reflection symmetry is required about the midpoint to assure 
that the positive and negative orders have matching intensities. This 
symmetry (equation [3-5]) is given by, 
EQU x.sub.k =1/2-x.sub.N/2-k, for N/4+1.ltoreq.k.ltoreq.N/2, 
EQU .theta..sub.k =.theta..sub.N/4-k +.phi.. [4-16] 
FIG. 24 is a phase plot of one period for an even numbered multi-level 
phase grating pattern with translation about the midpoint, 0.5, and 
reflection symmetry about the points 0.25 and 0.75. (The pattern has had 
its phases reduced modulus 2.pi..) If the reflection symmetry point is not 
a member of the transition set then .phi. must be chosen to be zero. Using 
the symmetries of equations [4-15] and [4-16], the value of the amplitudes 
are given by 
##EQU43## 
for n odd. Again, by choosing .phi. equal to either 0 or .pi. the n odd 
amplitudes reduce to 
##EQU44## 
respectively. 
It is evident that the application of symmetry properties has simplified 
the optimization problem. In the case of the standard design where only 
the reflection symmetry is incorporated, the number of independent 
transitions has been cut to half the total number. When an even numbered 
design using both translational and reflectional symmetry is optimized, 
only one fourth of the transitions are independent. 
The simplest fabrication scheme for manufacturing diffraction gratings is 
to either etch or deposit a pattern on an optical substrate leaving two 
levels that differ by a predetermined phase. Such binary phase gratings 
are constructed using a single step procedure that eliminates the critical 
alignment process required at each step of multi-level fabrication. 
Because of an inherent symmetry property in the binary phase model, 
reflection symmetry is not necessary to guarantee that positive and 
negative orders are equivalent. Although this inherent symmetry was not 
initially exploited in the design of Dammann gratings, others have removed 
the reflection symmetry and located solutions with better efficiencies as 
disclosed in U. Killat, G. Rabe, and W. Rave, "Binary Phase Gratings for 
Star Couplers with High Splitting Ratios," Fiber and Integrated Optics 4, 
159-164 (1982). In the following analysis, this natural property of 
standard binary phase gratings is investigated as well as the use of 
translational symmetry to create even numbered gratings. 
For the case of a binary phase grating, the phase delays for the two levels 
are described by 
EQU .theta..sub.k =.theta..sub.0 +(-1).sup.k .DELTA..theta./2, [5-1] 
where .theta..sub.0 represents an arbitrary reference level midway between 
the two levels and .DELTA..theta. is the phase difference between the two. 
Since the .theta..sub.0 term cancels in the intensity calculation, i.e. 
the intensity is invariant to the absolute phase, it is dropped from 
further analysis. Thus, the term which is dependent on the relative phases 
can be simplified as, 
EQU exp[i.theta..sub.k ]-exp[i.theta..sub.k+1 ]=exp[i(-1).sup.k 
.DELTA..theta./2]-exp[-i(-1).sup.k .DELTA..theta./2]=2i(-1).sup.k sin 
(.DELTA..theta./2). [5-2] 
When this is applied to equations [4-8] and [4-8] the nth order amplitude 
becomes 
##EQU45## 
and the central order amplitude is given by 
##EQU46## 
At this point a determination is made as to whether the positive and 
negative orders are naturally equal. By examining equation [5-3], it is 
seen that A(n)=-A*(-n); therefore positive and negative orders have 
matching intensities. Accordingly, the only reason for applying any form 
of symmetry to binary phase gratings is to create the novel even numbered 
solutions. 
Although it is by no means necessary, it is often convenient to choose 
.DELTA..theta.=.pi.. By doing so one can form a two dimensional spot array 
by using two one-dimensional solutions, and, when the phase delays are 
reduced to the range [0,2.pi.], still have a binary phase surface. If the 
phase difference is set equal to .pi., equation [5-3] becomes 
##EQU47## 
and the central order amplitude becomes, 
##EQU48## 
As has been seen in the previous sections, the design of an even numbered 
solution requires that a translational symmetry exist in the pattern. It 
can be shown that the resulting amplitudes are 
##EQU49## 
where the number of transitions in the first half of the period, N/2, must 
be be odd. In general, for each solution of a binary phase grating 
pattern, the number of independent transitions will be about one half of 
the number of spots. FIGS. 25 and 26, respectively, show standard and even 
numbered designs using a binary phase format. 
In the above description, it has been shown that the incorporation of 
symmetries into the design of spot array diffraction gratings can simplify 
the surface parametrization. The solutions associated with FIGS. 21-22 and 
23-26 are now described, proceeding in the reverse order from the earlier 
description, i.e., first illustrating the binary phase gratings that are 
relatively simple to fabricate, then describing the multi-level gratings, 
and then describing the continuous surface profile solutions. 
The design in FIG. 25 shows a binary level pattern that produces a one 
dimensional line of 13 spots. The phase level difference is equal to a 
.pi. phase delay. The transition set is given by [0.0000, 0.1289, 0.3233, 
0.5862, 0.6139, 0.7290, 0.7919, 0.9074] in units of the period. The 
pattern has a theoretical efficiency of diffracting 78.0% of the incoming 
energy into the desired 13 spots. Note that since matching positive and 
negative order intensities naturally occurs in a binary phase grating, 
there is no observable symmetry present in the pattern. 
An example of translational symmetry in a binary phase grating is shown in 
FIG. 26. This design produces an 8 spot array with 75.9% of the energy 
distributed to the desired orders. The data set for the transitions is 
[0.0000, 0.1812, 0.2956, 0.3282, 0.4392, 0.5000, 0.6812, 0.7956, 0.8282, 
0.9392]. It can be easily verified that the transitions in the second half 
of the period are translated by an amount 0.5 from the first half. 
The multi-level 1.times.13 design shown in FIG. 23 was obtained as an 
8-level pattern although only 7 levels were used in this design. The 
vertical axis labels the levels from 0 through 6 with each increment equal 
to an additional phase shift of .pi./4. The transition set solution is 
given by [0.0747, 0.1164, 0.1257, 0.2196, 0.3358, 0.4113, 0.4477, 0.5523, 
0.5887, 0.6642, 0.7804, 0.8743, 0.8836, 0.9253]. Note that in this case 
the points 0. and 1.0 are not transition points in the solution set. The 
corresponding phase values for this solution are [0, 1, 0, 6, 2, 4, 2, 0, 
2, 4, 2, 6, 0, 1, 0]. This solution has an efficiency of 82.8% and would 
require three etch/deposition steps to fabricate. Only the reflection 
symmetry about the point 0.5 is required in this design. 
FIG. 24 shows a 4 level pattern for creating a 1.times.8 spot array. The 
transition solution set is [0.0000, 0.0632, 0.0931, 0.1548, 0.3452, 
0.4069, 0.4368, 0.5000, 0.5632, 0.5931, 0.6548, 0.8452, 0.9069, 0.9368] 
and the corresponding phases are [2, 0, 1, 2, 3, 2, 1, 0, 2, 3, 0, 1, 0, 
3, 2]. Each level corresponds to an additional .pi./2 phase delay. The 
transitions of the first period are translated into the second half and 
the phase levels in the second half are increased by .pi. and then reduced 
modulus 2.pi. to the illustrated values. This multi-level solution 
demonstrates the translational and reflection symmetry required for an 
even numbered solution. The efficiency of this solution is 85.3% and the 
fabrication would require only two etch or deposition steps. 
Finally, two continuously varying solutions are shown in FIGS. 21 and 22. 
The profiles of each were parametrized using Legendre polynomials. Neither 
of these solutions have been folded to fit within the range [0.2.pi.]. 
FIG. 21 shows a surface profile that produces a 1.times.13 spot array with 
an efficiency of 93%. Six parameters are needed to specify the surface. 
The pattern shows a reflection symmetry about the point 0.5. FIG. 22 shows 
a surface profile that would produce a 1.times.8 spot array with a 
theoretical 95% efficiency. Four parameters are needed to specify this 
profile. The translation symmetry and the reflection about the points 0.25 
and 0.75 in each half period can be seen as well as the .pi. phase offset. 
In order to fabricate a discrete level grating based on this design, the 
profile is converted to a series of discrete levels and further 
optimization of the discrete solution is performed to eliminate potential 
nonuniformities that were introduced. Preliminary studies indicate that 
the efficiency loss resulting from the transformation from continuous to 
discrete levels could be limited to about one percent. 
An optical system illustrating the present invention is shown in FIG. 27. 
The system comprises a plurality of stages and implements a multi-stage 
switching network, illustratively of the type disclosed in U.S. patent 
application Ser. No. 07/349,281 of T. J. Cloonan et al. filed on May 8, 
1989, allowed on May 20, 1991, now U.S. Pat. No. 5,077,483 issued Dec. 31, 
1991 and assigned to the assignee of the present invention. The optical 
system comprises a plurality of stages each including means (e.g., 120, 
130, or 140) for generating a monochromatic plane wave of light, a phase 
diffraction grating (e.g., 121, 131, 141), and an array (e.g., 150) of 
N.times.M optical devices, e.g., symmetric self electro-optic effect 
devices or S-SEEDS. Grating 131, for example, has a surface relief pattern 
formed therein such that the grating is responsive to a plane wave of 
light for transmitting light from the surface relief pattern to form an 
array of N.times.M spots, where N and M are even integers, the N spots are 
substantially equally spaced, and the M spots are substantially equally 
spaced. The array of N.times.M optical devices is responsive to the array 
of N.times.M spots conveyed to the device array via an optical hardware 
module 110. 
It is to be understood that the above-described embodiments are merely 
illustrative of the principles of the invention and that many variations 
may be devised by those skilled in the art without departing from the 
spirit and scope of the invention. It is therefore intended that such 
variations be included within the scope of the claims.