Optical scanner using plane linear diffraction gratings on a rotating spinner

An optical scanning system including a spinner containing at least one plane linear diffraction grating. As the spinner is rotated, a plane reconstruction wave of wavelength .lambda..sub.r is directed onto the grating creating a diffracted wavefront which can be focused onto an image plane. The grating is constructed so that the ratio of the reconstruction wavelength .lambda..sub.r to the grating period d is a value lying between the range of 1 and 1.618. The angles of incidence and diffraction of the reconstruction wavefront in a preferred embodiment, approximate 45.degree..

BACKGROUND OF THE INVENTION 
The present invention relates to an optical scanning system and more 
particularly, to a system which uses a rotating spinner, having at least 
one plane linear diffraction grating formed thereon, as the scanning 
element. 
The high brightness characteristics of laser illumination has stimulated 
interest in sequential optical scanning and has resulted in systems 
capable of generating high resolution images at high scan rates. These 
systems have typically used galvanometers, rotating mirrors, acousto-optic 
or electro-optic elements and rotating holograms as the light spot 
deflecting elements. Holographic scanning has come to be preferred for 
many applications due to simplicity of the mechanical geometry, ease and 
economy of fabrication and higher resolution and scanning speed. 
One form of prior art holographic scanner is typically made by 
holographically forming a plurality of zone-type lenses on the surface of 
a rotatable disc. The lenses act to focus normally incident reconstruction 
beams at a locus of focal points which define a desired scan line. 
Examples of such scanners are found in the systems disclosed by McMahon et 
al. "Light Beam Deflection Using Holographic Scanning Techniques," Applied 
Optics, pp. 399-401, Vol. 8, No. 2, Feb. 1969, and U.S. Pat. No. 3,953,105 
(Ih). These prior art spinners are subject to certain inherent problems, a 
principal one being scan line "bow". As the spinner is rotated in the 
reconstruction mode, the locus of the reconstructed point source is a 
circle in space. If the image plane is a flat sheet positioned 
tangentially to the scan circle, the loci of the scan incident on the 
sheet is, in general, a curved or bowed line. Various solutions have been 
used to overcome this problem but each has its own drawbacks. A curved 
image plane can be used in applications where the imaging plane is 
sufficiently flexible but excluded would be the use of an imaging member 
such as a xerographic drum. Additional optics can be utilized as proposed 
by Ih in U.S. Pat. No. 3,953,105 but this type of system is difficult to 
align and is sensitive to spinner decentration errors. 
Another problem is that the spinner is subject to a wobble effect which 
results in formation of colinear multiple scan lines. This problem has 
been addressed by the applicant in copending applications, U.S. Ser. No. 
708,245 (U.S. Pat. No. 4,239,326) and 921,409 (U.S. Pat. No. 4,243,293) 
which disclose solutions based on wobble invariance obtained through 
specific optical geometries. U.S. Pat. No. 4,067,639 and copending 
application U.S. Ser. No. 921,411 (now abandoned), by the same applicant 
disclose spinner mounting techniques for reducing wobble. 
A third problem not addressed by any of the prior zone-type plate systems, 
is that of spinner decentration. If the facets (lenses) of a spinner are 
decentered due to initial fabrication and/or mounting, during the scan 
mode, the focal position of the facet will oscillate causing output scan 
distortion. 
A fourth problem is chromatic aberration resulting from a wavelength shift 
in the reconstruction beam, i.e. the image is reconstructed at a 
wavelength different from the one used for construction. One technique 
known for compensating for this is the simulated computer holographic lens 
design programs disclosed by J. N. Latta in his article "Computer-based 
Analysis of Hologram Imagery and Aberrations", Applied Optics, Vol. 10, 
No. 3, pp. 599-608, March 1971. 
A still further problem encountered when transmission disk spinners are 
used as the scanning element is the influence of spinner "wedge" on 
scanner performance. Variations in the thickness of the substrate 
material, which can occur in the form of localized deviations or of a 
constant wedge, cause perturbations in the direction of the diffracted 
beams resulting in colinear multiple scan lines. 
Another problem is exposure non-uniformities which may occur either within 
a scan line (caused, for example, by noise in the reconstruction beam) or 
from line-to-line (caused by grating-to-grating differences. 
It can thus be observed that prior art holographic scanning systems are 
subject to a multiplicity of problems, many of which can at best be solved 
on an individual basis leaving other problems still present or can be 
solved at the expense of worsening existing problems or introducing new 
ones. In the following description, Applicant discloses a scanning system 
which by its construction, and playback either eliminates these problems 
completely or neutralizes their effects. 
SUMMARY OF THE INVENTION 
Applicant has, in the present application, disclosed a scanning system in 
which a reconstruction beam is directed in non-normal incidence against a 
spinner surface having formed therein at least one plane linear 
diffraction grating. The grating is constructed so as to have a 
.lambda..sub.r d ratio (.lambda..sub.r =wavelength of the reconstruction 
beam; d=grating period) of between 1 and 1.618. As the spinner is rotated 
a diffracted reconstruction wavefront is generated and, in one embodiment, 
focused with the aid of additional optics, onto an imaging plane. 
The scanning system, although lacking the rotational symmetry possessed by 
prior art systems, nevertheless possesses imaging characteristics which 
remain nearly constant with the relative changes in reconstruction beam 
orientation. These characteristics include a scan trajectory which is 
essentially bow-free; invariance to spinner centration errors; 
insensitivity to angular misalignments of the spinner; ready fabrication 
using either holographic or conventional ruling techniques and simple 
wavelength conversion. Additionally, for the transmissive case, spinner 
wobble is eliminated if the incident and diffraction angles are made 
approximately equal. The influence of substrate wedge is reduced for the 
equal angle case. (A reflection-type spinner would be invariant to wedge 
effects). Also for the transmission case, higher than expected diffraction 
efficiencies were obtained.

DESCRIPTION 
Referring to FIG. 1, the holographic formation of a single plane linear 
diffraction grating facet 2 on the surface of transmission type spinner 3 
is shown. The gratings will hereinafter be referred to as PLDGs and the 
term will be understood to mean a grating having a flat surface and a 
constant grating period. 
As shown in FIG. 1a, a single PLDG facet 2 is formed by directing an object 
wavefront 4 and a reference wavefront 5, both of which are plane waves 
lying in the same plane onto a recording medium 6 disposed on the surface 
of spinner 3. It is assumed that these beams have been split upstream and 
individually conditioned (spatial filtered, collimated) to the desired 
wavefronts. The lines of the grating, as shown in FIG. 1b are formed 
perpendicular to the center line of the spinner. 
The choice of medium 6 is determined primarily by the resolution required 
to record the fringes of the interfering wavefronts. As is known in the 
art, spinner 3 can be indexed so that a plurality of PLDG facets 2 can be 
formed on its surface. 
An example of a suitable material for the recording medium is a silver 
halide photographic emulsion; the resulting grating would be of the 
absorption type. If this hologram were bleached, the grating would be a 
phase type. The gratings formed in this material are of the volume type, 
either thick or thin. 
According to the present invention and in a preferred embodiment, the 
grating is formed on the surface of medium 6 and can be characterized as a 
thin surface relief grating. Such a characterization means that a 
sinusoidal phase variation is introduced into the reconstruction wave. 
FIG. 2 shows an embodiment for making the PLDG facets; one which results in 
facets having essentially identical properties. Facet uniformity is 
difficult to achieve due to laser power fluctuations during the sequential 
exposure process and/or due to differential phase fluctuations between the 
interfering beams during the exposures. These fluctuations typically arise 
following the splitting of the laser beam into object and reference beams 
at some upstream point and accumulates during the fairly long separate 
paths travelled by the beams as they interact with loosely coupled optical 
components. Compensation of laser power fluctuations can be achieved by 
monitoring the output of the laser and correcting the individual facet 
exposures for total energy. Compensation of the differential phase 
fluctuation is difficult and in theory is accomplished by monitoring an 
interference pattern established utilizing a portion of each of the two 
beams used to form the hologram and adjusting the phase of one of the 
beams so that the pattern is stationary in time. 
In FIG. 2, preconditioned beam 12 enters prism assembly 11 and is split 
into beams 14 and 15 by beamsplitter 16. These beams are reflected from 
the sides of the prism and directed onto medium 6 of spinner 3 to form a 
facet as previously described in FIG. 1. To enhance high fringe contrast, 
the entrance and exit faces 17, 18 of the prism can be coated with an 
antireflection material to reduce flare light. Other embodiments are 
possible besides the prism assembly shown. For example, a pair of mirrors 
could be positioned downstream of the beamsplitter to provide reflection 
at the desired angle of the two wavefronts. The advantage of using the 
prism assembly is that the beams after the beamsplitter interact only with 
closely coupled optical components. Also, the path length after the 
beamsplitter is minimized. Other prism geometries are also possible. 
Although the gratings shown in FIGS. 1 and 2 have been formed by a 
holographic process, they may also be formed by ruling techniques. For 
example, a master grating may be made with the desired grating period. 
Portions of the grating can then be separated into individual facets and 
mounted on the surface of the spinner in proper orientation with the 
reconstruction beam. 
For the transmission grating shown in FIG. 1, the spacing between fringes d 
(shown greatly exaggerated) is given by the diffraction grating equation. 
##EQU1## 
where .lambda..sub.f is the wavelength of the forming wavefront, 
.phi..sub.o and .phi..sub.r are, respectively, the angles that the object 
and the reference waves make with the normal to the recording medium. Both 
beams lie in the plane defined by the spinner normal and the spinner 
diameter. 
FIG. 3 shows spinner 3 of FIGS. 1 and 2 located in an XY plane and rotating 
about a Z axis. A reconstruction wavefront 20 falls upon facet 2 at an 
angle of incidence .theta..sub.i and is diffracted at an angle of 
diffraction .theta..sub.d. Under the condition where the rotation angle is 
zero and the grating lines of facet 2 are parallel to the X axis, the 
incident and diffracted rays satisfy the following general equation. 
EQU sin .theta..sub.ix +sin .theta..sub.dx =(.lambda..sub.r /d) sin 
.theta..sub.R (3a) 
EQU sin .theta..sub.iy +sin .theta..sub.dy =(.lambda..sub.r /d) cos 
.theta..sub.R (3b) 
wherein sin .theta..sub.ix and sin .theta..sub.iy are the components of the 
incident reconstruction wave vector along the X and Y axes, respectively; 
sin .theta..sub.dx and sin .theta..sub.dy are the components of the 
diffracted wave vector along the X and Y axes; .lambda..sub.r is the 
wavelength of the reconstruction beam; d is the grating period and 
.theta..sub.R is the grating rotation angle. 
FIG. 4 schematically illustrates a scanning system utilizing the 
transmission spinner of FIGS. 1-3. Reconstruction wavefront 20 is incident 
on the spinner at angle .theta..sub.i and is diffracted at an angle 
.theta..sub.d. Since wavefront 20 is a plane wave, the diffracted wave 21 
is also a plane wave which is focused by lens 22; mirror 23 directs signal 
beam 24 to image plane 25 which lies on the focal plane of lens 22. 
As spinner 3 is rotated about shaft 40 by a motive means (not shown) which 
can be a conventional motor, facet 2 is rotated through wavefront 20 at 
some angle causing rotation of the diffracted wavefront. The focal 
position of signal beam 24 will be displaced vertically producing a single 
scan line. As additional facets are rotated through wavefront 20, 
additional scan lines are generated. 
With a collimated reconstruction wavefront, lens 22 can be positioned as 
shown between the spinner and the image plane. In this position, the lens 
provides resolution and could provide field flattening and scan 
linearization. The lens can also be located upstream from the spinner in 
which case, beam convergence would be initiated before the spinner and 
continue on to focus at the image plane. A combination of a diverging lens 
upstream of the spinner followed by a converging lens downstream of the 
spinner is also possible. Finally, in certain applications where 
resolution requirements are not stringent, the lens may be completely 
eliminated. 
As will be described in further detail below, the scanning system shown in 
FIG. 4 is capable of producing line scans at plane 25 which are almost 
completely bow free, are invariant with respect to any irregularities 
(wobble) of the surfaces of spinner 3 and are completely free of 
distortion due to decentration. 
These simultaneous beneficial results have been obtained by deriving 
certain useful insights into the forming of the PLDG facets and their 
playback. These insights were obtained through vigorous mathematical 
inquiry into whether optimum scanning conditions could be obtained by 
forming certain relationships between grating period, playback wavelength 
and angles of incidence and diffraction during reconstruction. 
As a first observation, however, applicant realized that the use of 
rotating plane linear diffraction gratings as the scanning elements would 
avoid one of the major causes of scan distortion in spinners with zone 
type lenses, that caused by decentration problems between the lenses. As 
an analogy, if the grooves of a record are not exactly centered, the 
needle moves, or accelerates, back and forth. In the fringe pattern 
comprising a zone lens, it is the focal point which is subject to 
oscillation causing distortion in the diffracted ray. Plane gratings, 
lacking power, are simply not subject to the centration problem. Applicant 
then conducted an analysis to discover whether there were other advantages 
in the use of PLDG as the facets. These results are described below with 
reference to FIGS. 5-12. For convenience, discussion of the various 
problems and their solutions are considered separately below. 
BOW MINIMIZATION 
The first investigation was to determine whether there were any conditions 
under which a minimum bow in a scan line could be achieved for an oblique 
incident reconstruction beam. Referring to FIGS. 5 and 6 it is assumed 
that a PLDG facet is located in the XY plane and that it rotates about the 
Z axis. (In order to keep FIG. 5 as clear as possible, the facet 2 is not 
shown in this figure but is instead shown in FIG. 6.) The following 
notation conventions are observed: subscripts i and d refer to the 
incident and diffracted waves, respectively, subscript o indicates the 
parameter is defined for the case of .theta..sub.R (the grating rotation 
angle) =0; subscript n indicates the parameter is defined for 
.theta..sub.R .noteq.0; .theta..sub.s is the scan angle of the diffracted 
beam measured in the plane defined by the diffracted wave vector and the X 
axis and .theta..sub.s ', is the scan angle of the diffracted beam 
measured in the XY plane. Incident diffracted angles for the grating are 
measured with regard to the grating normal (z axis). It will be assumed 
that the incident wave vector always lies in the YZ plane so: 
EQU .theta..sub.ix =0 and .theta..sub.iy =.theta..sub.i 
With further reference to FIGS. 5 and 6 it is evident that if the 
diffraction angle remains constant with scan angle, the scan line 44 on 
image plane 46 will bow upward as indicated. In order for the scan line to 
become straight, .theta..sub.d must increase with scan angle. From the 
grating equations, 
EQU sin .theta..sub.i +sin .theta..sub.d =.lambda..sub.r /d (4) 
It is apparent that in order for .theta..sub.d to increase, the incidence 
angle .theta..sub.i must decrease. Insight into the relationship between 
apparent incidence angle and grating rotation angle can be achieved with 
the aid of FIG. 7. To illustrate how the apparent incidence angle changes 
as a function of rotation angle, the spinner 50 in FIG. 7 is depicted as 
being kept stationary while the incident beam 49 is rotated about the Z 
axis while keeping the incidence angle .theta..sub.i constant. The 
projection of the rotated incident angle onto the YZ plane is: 
EQU tan .theta..sub.i =y.sub.1 /z.sub.1 cos .theta..sub.R (5) 
The apparent incidence angle decreases from .theta..sub.i to 0 as 
.theta..sub.R increases from 0.degree. to 90.degree.. It will be shown 
that if .theta..sub.d resides within a certain range, there exists a 
.theta..sub.i value that minimizes the scan line bow. 
The first step in deriving the conditions that minimize bow is to define 
parameters which are useful for characterizing the magnitude of the scan 
line bow. It is evident from FIG. 5 that the change in the Z coordinate of 
the scan line is an accurate measure of the bow. Since the change in the 
angle .phi..sub.n is directly related to the change in the Z coordinate, 
the bow can be expressed in terms of this variable. From the geometric 
relationships depicted i FIG. 5 
EQU tan .theta..sub.n =cot .theta..sub.d sec .theta..sub.s ' (6) 
For a bow free line, .phi..sub.n is constant with regard to .theta..sub.s 
'. To determine the conditions under which this occurs Eq. (6) is 
differentiated with respect to .theta..sub.s ' while keeping .phi..sub.n 
constant: 
EQU d .theta..sub.d =sin .theta..sub.d cos .theta..sub.d tan .theta..sub.s 'd 
.theta..sub.s ' (7) 
In order to solve Eq. 7, an expression is first derived to express 
.theta..sub.s in terms of .theta..sub.R. 
By definition: 
EQU tan .theta..sub.s '=x.sub.d /y.sub.d (8) 
where x.sub.d and y.sub.d are the components of the diffracted wave vector 
along the X and Y axis, respectively. Substituting from Eq. (3) gives: 
##EQU2## 
With the aid of FIG. 5, it can be shown that: 
##EQU3## 
Additional equations are derived to approximate the conditions under which 
the PLDG is made. (.theta..sub.R o). It can be demonstrated that the 
general relationship between .theta..sub.i .theta..sub.d, .lambda..sub.r 
and d, for the case of minimized bow in the scan line, is given by 
EQU sin .theta..sub.i =(.lambda..sub.r /d) cos.sup.2 .theta..sub.d sec 
.theta..sub.R (11) 
If .theta..sub.R is set to zero, then E.sub.q (11) reduces to the following 
set of equations for the playback condition: 
EQU sin .theta..sub.i =.lambda..sub.r /d-d/.lambda..sub.r (12a) 
EQU sin .theta..sub.d =d/.lambda..sub.r (12b) 
It is readily apparent that these solutions depend on only the wavelength 
of light and the grating period. 
Real solutions for .theta..sub.i and .theta..sub.d exist only for the range 
where: 
EQU -1.ltoreq.(.lambda..sub.r /d-d/.lambda..sub.r).ltoreq.1 (13) 
and 
EQU 0.ltoreq.d/.lambda..sub.r .ltoreq.1 (14) 
Since d can have only positive values, the maximum and minimum values that 
d can have are: 
EQU 0.618.ltoreq.d.ltoreq..lambda..sub.r (15) 
or 
EQU 1.ltoreq..lambda..sub.r /d.ltoreq.1.618 (16) 
The maximum and minimum corresponding values of .theta..sub.i and 
.theta..sub.d are: 
EQU 0.degree..ltoreq..theta..sub.i .ltoreq.89.445.degree. 
EQU 90.degree..gtoreq..theta..sub.d .gtoreq.38.17.degree. (17) 
The above analysis demonstrates that there is a wide range of incident and 
diffracted angles over which this scan line bow minimization technique can 
be used, although as will be shown below, certain angles are preferred. 
As is apparent from an investigation of FIG. 8, the proposed technique does 
not completely eliminate bow. FIG. 8 illustrates how the shape of the scan 
line changes as the incident angle is varied about the value predicted by 
Equation (12). Curve 60 indicates what the scan line would look like when 
the incident and diffracted angles satisfy Equation (12). The bow is 
essentially zero for small rotation angles but increased monotonically 
with rotation angle. For incident angles less than .theta..sub.i the 
monotonic increase in bow (curve 62) is more pronounced; for much larger 
incidence angles than .theta..sub.i the scan line begins to develop a 
fairly large negative bow as indicated by curve 64. It is, however, 
apparent that a minimum bow can be achieved if curve 66 can be produced. 
This curve first goes negative and then monotonically increases thereby 
developing three separate inflection points. Its deviation from an ideal 
straight line can be made less than curve 60 for a given rotation angle. 
To achieve the minimum bow in the scan line, the scanner would be set up to 
produce curve 66 of FIG. 8. Since the position of the inflection points of 
the scan line determine the maximum bow in the line, it is important to be 
able to predict the location of these points. Due to the differential 
approach utilized to solve this problem, information about the inflection 
points is contained in Eq. (11). By inspection of Eqs. (11) and (12) the 
following more generalized equations for the incident and diffraction 
angles can be obtained. 
EQU sin .theta..sub.i =(.lambda..sub.r /d-d/.lambda..sub.r) sec .theta..sub.R ( 
18a) 
EQU sin .theta..sub.d =(1-sec .theta..sub.R).lambda..sub.r /d+d/.lambda..sub.r 
(18b) 
When the incident and diffracted angles satisfy Eq. (18) the three 
inflection points occur at .theta..sub.R =0 and .+-..theta..sub.R. In a 
typical scanner arrangement the inflection points are chosen to occur at 
50-65% of the maximum rotation angle per scan. The following approximation 
is very useful for relating the grating rotation angle to the beam 
scanning angle: 
EQU .theta..sub.s =.theta..sub.R /sin .theta..sub.d (19) 
The above analysis will enable one skilled in the art to design a system 
which will minimize bow regardless of whether a transmission or reflection 
grating scanner is used. Further investigation into the transmission 
grating scanner characteristics, however, has led to the discovery of a 
geometry where the system is invariant with regard to spinner wobble. It 
has been found that invariance occurs when .theta..sub.i 
.apprxeq..theta..sub.d ; by Eq. (12), .theta..sub.i =.theta..sub.d 
=45.degree.. The general relationship between the change produced in the 
diffraction angle, d.theta..sub.d. by tilting the spinner by the angle 
d.theta. is: 
EQU d.theta..sub.d =.+-.[1.-+.(cos .theta..sub.i /cos 
.theta..sub.d)]d.theta.(19.1) 
where the upper signs apply to transmission gratings and the lower to 
reflection gratings. 
System characteristics were theoretically investigated for the following 
parameters: .lambda..sub.r /d=1.414; .theta..sub.i =45.6.degree.; 
.theta..sub.d =44.41.degree.. Scan line bow was calculated as a function 
of the distance of the spinner from the image plane and as a function of 
scan line length. As shown in FIG. 9, three curves 70, 72, , and 74 were 
plotted for three different spinner-to-image plane distances of 20", 25" 
and 30", respectively, with the image plane oriented so that it was 
perpendicular to the on-axis diffracted beam. 
The various calculated bow values for the different throw distances in FIG. 
9 are related to the rotation angle that the spinner would have to undergo 
to generate the scan line length. The shorter the throw distance the 
larger the rotation angle for a given scan length. FIG. 10 illustrates how 
the bow for a 25" throw distance is minimized for a 14" scan length by 
optimum choice of incidence angle. In this case, .lambda..sub.r /d=1.41; 
.theta..sub.i =45.45.degree.; .theta..sub.d =44.55.degree.. The 1.6 mils 
of bow in this figure is more than acceptable for most high resolution 
scanning applications. If a post-scan focusing lens is utilized in 
conjunction with the spinner, the magnitude of the bow in the image plane 
will be influenced by the angular magnification properties of the lens. If 
the lens is of a telephoto design, there will be an increase in bow, 
whereas, the reverse is true for a wide angle design. 
WEDGE 
As a first comment, a spinner functioning in the reflectance mode would be 
invariant to wedge effects so the following discussion concerns only 
transmission systems. In such a system, variations in spinner thickness 
can occur in many forms, from localized deviations to constant wedge. The 
change in the diffraction angle, d.theta..sub.d, due to spinner substrate 
wedge is: 
##EQU4## 
where .theta..sub.i and .theta..sub.d are the incident and diffraction 
angles, respectively, measured in air; .theta..sub.1 and .theta..sub.2 are 
the incident and diffraction angles, respectively measured in the spinner 
substrate medium; n is the index of refraction of the spinner medium, and 
dx.sub.1 and dx.sub.2 are the angles by which the first and second 
interfaces of the spinner deviate from their ideal parallel position. 
Since it is difficult to measure the individual surface deviations from 
parallelism, wedge is usually measured and specified as the sum of these 
deviations. For the previously disclosed geometry where .theta..sub.i 
.theta..sub.d ; .theta..sub.1 .apprxeq..theta..sub.2 and, therefore, the 
portion of the wedge associated with the second interface of the spinner 
introduces no variation into the diffracted ray angle. The physical reason 
for this is that wedge associated with the same interface on which the 
grating is deposited has the same effect on the beam that wobble would 
and, therefore, the system is invariant with regard to it. Wedge effects 
would still be present however, with the other interface. 
EXAMPLE 
Referring to FIGS. 1 and 4, a scanner system was constructed using a glass 
spinner 3, 0.125" (3.125 mm) thick and 3.75" (93.75 mm) in diameter and 
containing 10 PLDG facets located 1.55" (38.75 mm) from the center of the 
spinner. Each facet was 1 cm wide in the sagittal direction and 2.47 cm 
long in the tangential direction. Lens 22 has a focal length of 50.8 cm. 
The image plane was positioned in the focal of the lens; and .theta..sub.i 
=48.27.degree.; .theta..sub.d =43.37.degree.. The reconstruction beam was 
generated by red H.sub.e -N.sub.e laser with a wavelength of 0.6328 .mu.m 
and the grating period was 0.4416 .mu.m. The .lambda..sub.r /d ratio in 
this case was 1.433. The forming wavelength is discussed in the 
immediately following paragraph. The system functioned in the under-filled 
mode and had an 84% duty cycle over a 14" write position of the scan line. 
FIG. 11 shows that the correlation between the measured bow and the 
predicted bow for this system is excellent. Since lens 22 is of the 
telephoto type, the magnitude of the bow has been increased by 2.4 times 
the nominal value it would have if the lens were not present. It is 
estimated that the bow depicted in FIG. 11 is marginally acceptable for 
scanning applications requiring about 200 lines/in. 
WAVELENGTH CONVERSION 
Due to the fact that a diffraction limited image is formed only when the 
reconstruction wavefront meets certain pre-encoded conditions required to 
form the holographic facet and because of the monochromatic nature of the 
holographic imaging process, it is difficult to achieve diffraction 
limited performance at wavelengths other than the forming wavelength. The 
previously referenced Latta article describes a complex computer designed 
holographic lens program capable of compensating for a wavelength shift. 
According to the present invention, and utilizing equations (1) and (12), 
this conversion process is greatly simplified. Referring to the above 
example, the forming beam was a H.sub.e -C.sub.d laser with a wavelength 
of 0.4416 .mu.m. Solving Eq. (12) for d yields an above-mentioned value of 
0.4416 .mu.m. Inserting this value and the value of .lambda..sub.f (0.4416 
.mu.m) into Eq. (1) yields the making condition requirements by indicating 
that .phi..sub.o and .phi..sub.r must be set at 30.degree. to provide the 
required conversion. 
To summarize the invention up to this point, applicant has demonstrated 
that in an optical system utilizing a spinner having plane linear 
diffraction gratings the scanning elements can produce an essentially bow 
free scanning beam trace in which cross scan motion due to spinner 
decentration, wobble, and wedge could be minimized by selection of a 
.lambda..sub.r /d ratio of between 1 and 1.618, values for .theta..sub.i 
between 0.degree. and 89.445.degree. and for .theta..sub.d between 
38.17.degree. and 90.degree.. In a preferred embodiment in a transmission 
mode .theta..sub.i .apprxeq..theta..sub.d .apprxeq.45.degree.; the system 
is invariant to wobble. Two remaining features remain to be discussed: 
diffraction efficiency and exposure nonuniformities. 
DIFFRACTION EFFICIENCY 
Scalar diffraction theory predicts that the maximum diffraction efficiency 
that can be achieved with thin sinusoidal surface relief gratings is 34%. 
It has been known theoretically that, with metalized thin sinusoidal 
surface relief gratings, diffraction efficiencies of greater than 80% can 
be obtained for certain polarizations when .lambda./d.gtoreq.0.6. 
According to the present invention, it has been discovered that the 
above-described making conditions of an unmetalized thin sinusoidal 
surface relief transmission type grating results in relatively high 
efficiencies. FIG. 12 shows using experimental data for grating 
diffraction efficiency as a function of .lambda..sub.r /d and incident 
light polarization and with .theta..sub.i =.theta..sub.d. Plot 80 is for P 
polarization and plot 82 is for S polarization. 
DIFFRACTION NON-UNIFORMITIES 
In the scanning system shown in FIG. 4, only the diffracted first order ray 
is shown but a zero-order diffracted component (not shown) is also 
present. This zero-order component is spatially stationary and has a power 
P.sub.o which may be compared with the power P.sub.m of reconstruction 
wavefront 20 incident on the grating. Applicant has devised a feedback 
technique which uses a correction signal derived from P.sub.o and P.sub.m 
to remove intensity errors in the deflected first order beams (of power 
P.sub.l). These errors would normally arise from grating-to-grating 
diffraction efficiency differences or as intra-grating diffraction 
efficiency variations. FIG. 13 shows a correction circuit for the 
transmission spinner mode but the method described below is equally 
applicable to reflection holograms. Referring to FIG. 13, a correction 
signal is derived as a function of the rotational angle, .theta..sub.R of 
the holographic spinner 100. Laser modulator 102 is assumed to put out a 
beam power (P.sub.m) which is proportional to the incident laser beam 
power and the modulator driver input voltage E(t), where E(t) is 
maintained within a reasonable range for linear operation. A small 
fraction of the beam power, P.sub.m from the modulator is diverted by 
splitter mirror 104 to light detector 106 whose output (.alpha.P.sub.m) is 
proportional to P.sub.m. Likewise, a portion of the zero order beam power, 
P.sub.o, from a PLDG 108 on spinner 100 is detected by light detector 110 
to provide a signal .beta.P.sub.o proportional to P.sub.o (.alpha. and 
.beta. are proportionality constants). The signals P.sub.m and 
.beta.P.sub.o are amplified by amplifiers 112, 114 which are both adjusted 
so that the output of the following ratio circuit 115 is equal to P.sub.o 
/P.sub.m. The ratio P.sub.o /P.sub.m will be a function of the angular 
position because of inter-facet and intra-facet diffraction efficiency 
variations. This ratio is related to the local diffraction efficiency 
(D.sub.E) of the holographic grating by 
##EQU5## 
If it is desired to simulate an arbitrary but constant diffraction 
efficiency, D.sub.o, so that the scanning beam power is not affected by 
small local variations in D then a correction factor, M(.theta.), equal to 
##EQU6## 
is electronically derived in computation, and storage circuit 118. The 
correction factor, M.theta.' is multiplied in multiplier 120 by the 
incoming "video" signal, E.sub.s to provide a scanning beam power, 
P.sub.l, which is proportional to the video signal, E.sub.s, independent 
of local scanner diffraction efficiency. The symbol, M(.theta.) is used to 
indicate that a time base shift relative to M(.theta.) is often required 
to ensure that the correction factor is applied for the correct 
instantaneous position of the scanner, accounting for the several time 
delays which may occur in the electronics and modulator. This time base 
shift may vary from zero to many line scan times, depending on the mode of 
correction to be adopted. 
It is obvious that the ratio P.sub.o (.theta.)/P.sub.m must be derived at a 
time when P.sub.m is sufficiently greater than zero so that P.sub.o 
/P.sub.m may be computed with sufficient accuracy. This fact places 
certain restrictions on the correction modes. Likewise, since the 
modulator and driver electronics have a limited linear range, there are 
limitations on the selection of the constant, D.sub.c, and the permissible 
variations in grafting diffraction efficiency, D.sub.E. If the range of 
input voltages to the modulator driver electronics varies from zero to 
E.sub.max for linear operation, then the video signal E.sub.s must satisfy 
the inequality 
##EQU7## 
to maintain linear operation of the modulator. 
The signals required for correction may be derived by several other 
techniques. For example, if a self-modulated laser) e. g. a diode laser is 
used, the compensation signal is used to alter the output of the laser. If 
an acousto-optic laser modulator is used, the zeroth order beam from the 
modulator (which is usually discarded) may be used to compute P.sub.m, 
since this zeroth order power and P.sub.m are linearly related. Also, a 
portion of the unmodulated laser beam may be diverted around the 
modulator, measured to generate an equivalent P.sub.m (signal) and 
optically directed to be incident on a facet. The P.sub.m signal derived 
before the modulator could be used to correct the combined modulator and 
spinner signal. 
FIG. 14 illustrates the use of a reflective type scanning system 
constructed according to the invention. In this embodiment, reconstruction 
beam 120 emanating from a point source is collimated by lens 122 before 
reaching PLDG facet 124 formed on spinner 126. The diffracted beam 128 is 
focused by lens 122 onto image plane 150. This embodiment has the 
following advantages associated with it; the number of beam conditioning 
optics before the scanner is reduced: a smaller angular difference between 
incident and diffracted beam is allowed which is advantageous in many 
instances, and the focusing lens can be located closer to the spinner and, 
thereby, be of smaller size. 
Although the disclosure has emphasized the use of plane linear diffraction 
gratings, there may be systems in which the facet possesses some optical 
power either as a result of having a varying grating period and/or because 
it is formed on a curved rather than plane surface. By incorporating power 
into the facets, it is possible in some applications to compensate the 
system for deficiencies. As an example, compensation could be implemented 
for the slight power change produced in a facet as a result of the grating 
period being distorted by the centrifugal force associated with high 
rotational velocities. Through the action of centrifugal force, a spinner 
undergoes slight elongation in the radial direction with the greatest 
elongation per unit cell of the spinner occurring near the center of the 
spinner. A PLDG facet residing on a high speed spinner will be altered 
very slightly in the following manner as a result of the substrate 
elongation; the grating lines of the facet will acquire a concave shape 
when viewed from the center of the spinner and the grating period will 
vary with the largest increase in period occurring toward the center of 
the spinner. Under the dynamic load of the centrifugal force, the PLDG is 
changed into a weak focusing lens whose center is located at a negative 
radial distance from the spinner. This lens effect can be eliminated by 
introducing the opposite lens power into the facet during the fabrication 
process, that is by utilizing a construction wavefront which appears to 
originate from a point source located at some corresponding positive 
radial distance from the spinner. 
Although the present invention has been described as applicable to either 
thin surface gratings or thick (volume) gratings, thin surface gratings 
have a preferred advantage in that they may be conveniently replicated by 
non-optical means. It is well known that optical surfaces can be 
replicated by the technique in which the surface of a master is 
transferred to a substrate by a thin film of epoxy. 
As a final point, and applying a slightly different perspective to what has 
gone before, the PLDG scanner is, in effect, a "flat polygon". The grating 
facets function, for all practical purposes, as plane mirrors. Almost all 
of the techniques developed to improve the performance of a polygon 
scanner can be also applied to the present device.