Laser beam expanders with plastic and liquid lens elements

Design forms are disclosed for 5X and 10X laser beam expanders whose lens elements consist only of plastic and liquid optical materials, which provide diffraction-limited performance without refocussing over a wavelength range from 0.4 micron to 0.8 micron.

TECHNICAL FIELD 
This invention relates generally to laser beam expanders, and more 
particularly to high-performance laser beam expanders whose lens elements 
are made of plastic and liquid optical materials. 
BACKGROUND ART 
The lens elements of laser beam expanders are ordinarily made of optical 
glasses. In general, optical glasses are considerably more expensive than 
optical-quality plastic and liquid materials. 
A need has been recognized in the optical industry for reducing the cost of 
laser beam expanders. However, until the present invention, design forms 
had not been available for high-performance laser beam expanders whose 
lens elements consist only of plastic and liquid lens elements. 
SUMMARY OF THE INVENTION 
It is an object of the present invention to provide design forms for laser 
beam expanders (i.e., a focal optical systems) whose lens elements consist 
only of plastic and liquid lens elements. 
It is a more particular object of the present invention to provide design 
forms for laser beam expanders that achieve substantially 
diffraction-limited performance over a broad wavelength band using only 
plastic and liquid lens elements. 
It is a specific object of the present invention to provide design forms 
for laser beam expanders that comprise relatively inexpensive plastic and 
liquid lens elements, which can be used to provide substantially 
diffraction-limited performance without refocussing over a wavelength 
range from 0.4 micron to 0.8 micron. 
The invention as disclosed herein includes exemplary embodiments of laser 
beam expanders in which polymethylmethacrylate, polystyrene and 
polycarbonate are used for the plastic lens elements, and in which 
specified optical-quality liquids of proprietary composition marketed by 
R. L. Cargille Laboratories of Cedar Grove, N. J. are used for the liquid 
lens elements.

BEST MODE OF CARRYING OUT THE INVENTION 
In FIG. 1, a 5X laser beam expander according to the present invention is 
illustrated in which four lens elements 11, 12, 13, and 14 are coaxially 
disposed along an optic axis. The first lens element 11 is made of 
optical-quality polymethylmethacrylate (also called "acrylic"), which is 
commercially obtainable from suppliers such as U.S. Precision Lens, Inc. 
of Cincinnati, Ohio. The second lens element 12 is likewise made of 
polymethylmethacrylate. The third lens element 13 consists of an optical 
liquid of proprietary composition, which is marketed by R. L. Cargille 
Laboratories of Cedar Grove, N.J. The fourth lens element 14 is made of 
polymethylmethacrylate. 
The lens elements 11, 12, 13, and 14 can be mounted in a conventional 
manner. An effective technique for containing the liquid lens element 13 
between the rigid lens elements 12 and 14 made of polymethylmethacrylate 
is described in co-pending U.S. patent application Ser. No. 03/014,596 
filed on Feb. 8, 1993. 
The Cargille liquid of which the lens element 13 is made is a siloxane of 
proprietary composition, which is marketed primarily as a laser liquid. 
This particular Cargille liquid can be uniquely identified according to 
the U.S. Mil-Spec system by code number 479370 which indicates a liquid 
whose index of refraction at the wavelength of the sodium d spectral line 
(i.e., 0.58756 micron) at a temperature of 25.degree. C. has the value 
1.479 to the third decimal place, and whose Abbe number has the value 37.0 
to the first decimal place. The optical properties of 
polymethylmethacrylate (i.e., acrylic) are well known. Index-of-refraction 
measurements for polymethylmethacrylate over a broad wavelength band are 
listed in an article by R. M. Altman and J. D. Lytle entitled "Optical 
Design Techniques for Polymer Optics", 1980 International Lens Design 
Conference Proceedings, SPIE, Vol. 237, page 380. 
Cargille 479370 liquid has a density of approximately 1.013 gm/cc; and 
polymethylmethacrylate has a density of approximately 1.19 gm/cc. By way 
of comparison, the densities of most optical glasses are in the range from 
2.5 gm/cc to 5.0 gm/cc. In general, refractive elements made of plastics 
and liquids are significantly lighter in weight than refractive elements 
made of optical glasses. 
The laser beam expander illustrated in FIG. I was specifically designed to 
expand an input laser beam from a diameter of 2 mm to a diameter of 10 mm 
(i.e., to produce a 5X expansion) without requiring refocussing anywhere 
in the spectral range from 0.4 micron to 0.8 micron. The design form of 
the laser beam expander of FIG. 1 provides diffraction-limited performance 
over that spectral range, and chromatic aberration is virtually absent in 
the expanded beam. 
The 5X laser beam expander of FIG. 1 has an optical prescription, which is 
specified in tabular format as follows: 
TABLE I 
______________________________________ 
Surface 
Radius Thickness 
No. (mm) (mm) N.sub.d 
V.sub.d 
Material 
______________________________________ 
1 -6.8628 3.0000 1.491757 
57.455 
Acrylic 
2 -14.1495 114.7343 Air 
3 -226.9391 3.0000 1.491757 
57.455 
Acrylic 
4 -55.4824 1.0000 1.480110 
37.080 
479370 
5 392.1598 3.0000 1.491757 
57.455 
Acrylic 
6 -59.7645 10.0000 Air 
______________________________________ 
where the surfaces of the lens elements are numbered consecutively from 
left to right along an optic axis in accordance with optical design 
convention. Thus, surfaces No. 1 and No. 2 are the left and right 
surfaces, respectively, of the polymethylmethacrylate (i.e., acrylic) lens 
element 11. Surface No. 3 is the left surface of the 
polymethylmethacrylate lens element 12. Surface No. 4 is both the right 
surface of the polymethylmethacrylate lens element 12, and the left 
surface of the liquid lens element 13. Surface No. 5 is both the right 
surface of the liquid lens element 13, and the left surface of the 
polymethylmethacrylate lens element 14. Surface No. 6 is the right surface 
of the polymethylmethacrylate lens element 14. 
The radius listed for each lens surface in Table I is the radius of 
curvature expressed in millimeters. In accordance with optical design 
convention, the radius of curvature of a lens surface is positive if the 
center of curvature of the surface lies to the right of the surface, and 
negative if the center of curvature of the surface lies to the left of the 
surface. The thickness listed for each lens surface is the thickness 
expressed in millimeters of the lens element bounded on the left by the 
surface. The thickness of each lens element of the laser beam expander 
shown in FIG. 1 is measured along the optic axis. 
The column headed N.sub.d in Table I refers to the index of refraction of 
the lens element bounded on the left by the indicated surface at the 
wavelength of the sodium d spectral line (i.e., 0.58756 micron). The 
column headed V.sub.d refers to the Abbe number for the lens element 
bounded on the left by the indicated surface. The material listed for each 
surface in Table I refers to the type of material bounded on the left by 
the indicated surface. 
The index of refraction of an optical material varies with wavelength. The 
indices of refraction for the two different materials comprising the lens 
elements of the laser beam expander of FIG. 1 at five representative 
wavelengths in the range from 0.435 micron to 0.706 micron (i.e., N.sub.1 
at 0.58756 micron; N.sub.2 at 0.48613 micron; N.sub.3 at 0.65627 micron; 
N.sub.4 at 0.43584 micron; and N.sub.5 at 0.70652 micron) are tabulated as 
follows: 
TABLE II 
______________________________________ 
Material 
N.sub.1 N.sub.2 N.sub.3 
N.sub.4 
N.sub.5 
______________________________________ 
Acrylic 
1.491757 1.497760 1.489201 
1.502557 
1.487781 
479370 1.480110 1.489225 1.476227 
1.496483 
1.474177 
______________________________________ 
It is instructive to evaluate the performance of the 5X laser beam expander 
shown in FIG. 1 at each of the above-specified wavelengths. A graphical 
indication of performance of a lens system at a particular wavelength is 
provided by a plot of normalized aperture height as a function of optical 
path difference for that wavelength. In FIG. 2, plots of normalized 
aperture height as a function of optical path difference are shown for the 
five wavelengths for which the indices of refraction are specified in 
Table II. It is apparent from FIG. 2 that the maximum wavefront error for 
any one of the five plotted wavelengths is less than .lambda./65 for the 
5X laser beam expander shown in FIG. 1. 
The optical path difference (OPD) error of an optical system at a 
particular wavelength is defined as the difference between the optical 
path length of a ray traced through a particular location in the entrance 
pupil of the system at a specified field angle and the optical path length 
of a reference ray, where the reference ray is usually taken to be the 
"chief ray" or "principal ray" traced through the center of the pupil at 
the specified field angle. The wavefront error of an optical system at a 
particular field angle is calculated as the statistical root-mean-square 
(RMS) of the optical path differences of a number of rays traced through 
the system at a single wavelength. A grid of equally spaced rays is 
generally used; and statistics are accumulated with respect to the point 
in the image plane where the RMS wavefront error is a minimum for the 
particular field angle and wavelength. 
FIG. 3 is a plot of root-mean-square (RMS) wavefront error versus 
wavelength for rays entering the 5X laser beam expander of FIG. 1 parallel 
to the optic axis (i.e., at a field angle of zero) calculated over the 
wavelength range from 0.4 micron to 0.8 micron. It is apparent from FIG. 3 
that the RMS wavefront error of the 5X laser beam expander of FIG. 1 is 
less than .lambda./50 (i.e., less than 0.002 of a wave) between the 
wavelengths of 0.4 micron and 0.8 micron. The average RMS wavefront error 
over this wavelength range for the laser beam expander of FIG. 1 is 
.lambda./222. 
In FIG. 4, an alternative embodiment of a laser beam expander according to 
the present invention is illustrated in which the lens elements are 
likewise made of polymethylmethacrylate and Cargille 479370 liquid. The 
laser beam expander of FIG. 4 was specifically designed to expand an input 
laser beam from a diameter of 2 mm to a diameter of 20 mm (i.e., to 
provide a 10X expansion) with diffraction-limited performance without 
requiring refocussing anywhere in the spectral range from 0.4 micron to 
0.8 micron. 
The 10X laser beam expander of FIG. 4 comprises four lens elements 21,22, 
23, and 24, which are coaxially disposed along an optic axis. The first 
lens element 21 is made of polymethylmethacrylate (i.e., acrylic). The 
second lens element 22 is likewise made of polymethylmethacrylate. The 
third lens element 23 consists of Cargille 479370 liquid; and the fourth 
lens element 24 is made of polymethylmethacrylate. 
The 10X laser beam expander of FIG. 4 has an optical prescription, which is 
specified in tabular format as follows: 
TABLE III 
______________________________________ 
Surface 
Radius Thickness 
No. (mm) (mm) N.sub.d 
V.sub.d 
Material 
______________________________________ 
1 -2.8390 3.0000 1.491757 
57.455 
Acrylic 
2 -6.3701 120.0000 Air 
3 -1175.9144 
3.0000 1.491757 
57.455 
Acrylic 
4 -43.4991 1.0000 1.480110 
37.080 
479370 
5 481.8914 3.0000 1.491757 
57.455 
Acrylic 
6 -69.7898 10.0000 Air 
______________________________________ 
where the surfaces of the lens elements are numbered consecutively from 
left to right along the optic axis according to the same convention as 
described above in connection with Table I. The radii of curvature, axial 
thicknesses, and the values of N.sub.d and V.sub.d, are specified in the 
same manner as in Table I. 
In FIG. 5, performance of the 10X laser beam expander of FIG. 4 is 
illustrated by plots of normalized aperture height as a function of 
optical path difference for the same five wavelengths for which the 
indices of refraction are specified in Table II. It is apparent from FIG. 
5 that the maximum wavefront error of the 10X laser beam expander of FIG. 
4 for any one of the five plotted wavelengths is less than .lambda./10.3. 
FIG. 6 is a plot of RMS wavefront error vs. wavelength for rays entering 
the 10X laser beam expander of FIG. 4 parallel to the optic axis (i.e., at 
a field angle of zero) calculated over the wavelength range from 0.4 
micron to 0.8 micron. It is apparent from FIG. 6 that the wavefront error 
of the 10X laser beam expander of FIG. 4 is less than .lambda./10.9 (i.e., 
less than 0.092 of a wave) between the wavelengths of 0.4 micron and 0.8 
micron. The average RMS wavefront error over this wavelength range for the 
laser beam expander of FIG. 4 is .lambda./37.9. 
In FIG. 7, a 5X laser beam expander according to the present invention is 
illustrated whose lens elements consist of polystyrene and Cargille 480437 
liquid. Optical-quality polystyrene is obtainable from suppliers such as 
U.S. Precision Lens, Inc. of Cincinnati, Ohio. Index-of-refraction 
measurements for polystyrene over a broad wavelength band are listed in 
the aforementioned article by R. M. Altman and J. D. Lytle entitled 
"Optical Design Techniques for Polymer Optics". Polystyrene has a density 
of approximately 1.06 gm/cc; and Cargille 480437 liquid has a density of 
approximately 1.013 gm/cc. Thus, refractive elements made of polystyrene 
and a Cargille liquid are significantly lighter in weight than refractive 
elements made of optical glasses. 
The 5X laser beam expander of FIG. 7 comprises four lens elements 31,32, 
33, and 34, which are coaxially disposed with respect to each other along 
an optic axis. The first lens element 31 is made of polystyrene. The 
second lens element 32 is likewise made of polystyrene. The third lens 
element 33 consists of Cargille 480437 liquid; and the fourth lens element 
34 is made of polystyrene. 
The 5X laser beam expander of FIG. 7 has an optical prescription, which is 
specified in tabular format as follows: 
TABLE IV 
______________________________________ 
Surface 
Radius Thickness 
No. (mm) (mm) N.sub.d 
V.sub.d 
Material 
______________________________________ 
1 -8.6405 3.0000 1.590481 
30.866 
Poly- 
styrene 
2 -15.3841 145.6490 Air 
3 -245.7646 3.0000 1.590481 
30.866 
Poly- 
styrene 
4 82.1563 2.0000 1.480097 
43.770 
480437 
5 -54.4036 3.0000 1.590481 
30.866 
Poly- 
styrene 
6 -55.7738 10.0000 Air 
______________________________________ 
where the surfaces of the lens elements are numbered consecutively from 
left to right along the optic axis according to the same convention as 
described above in connection with Table I. The radii of curvature, axial 
thicknesses, and the values of N.sub.d and V.sub.d, are specified in the 
same manner as in Table I. 
The indices of refraction for the two different materials comprising the 
lens elements of the 5X laser beam expander of FIG. 7 at the same five 
wavelengths for which the indices of refraction are specified in Table II 
are tabulated as follows: 
TABLE V 
______________________________________ 
Material 
N.sub.1 N.sub.2 N.sub.3 
N.sub.4 
N.sub.5 
______________________________________ 
Polystyrene 
1.590481 1.604079 1.584949 
1.615446 
1.581954 
480437 1.480097 1.487885 1.476917 
1.494297 
1.475199 
______________________________________ 
In FIG. 8, performance of the 5X laser beam expander of FIG. 7 is 
illustrated by plots of normalized aperture height as a function of 
optical path difference for the five wavelengths for which the indices of 
refraction are specified in Table V. It is apparent from FIG. 8 that the 
maximum wavefront error of the 5X laser beam expander shown in FIG. 7 for 
any one of the five plotted wavelengths is less than .lambda./37.6. 
FIG. 9 is a plot of RMS wavefront error vs. wavelength for rays entering 
the 5X laser beam expander of FIG. 7 parallel to the optic axis (i.e., at 
a field angle of zero) calculated over the wavelength range from 0.4 
micron to 0.8 micron. It is apparent from FIG. 9 that the wavefront error 
of the 5X laser beam expander of FIG. 7 is less than .lambda./18 (i.e., 
less than 0.055 of a wave) between the wavelengths of 0.4 micron and 0.8 
micron. The average RMS wavefront error over this wavelength range for the 
laser beam expander of FIG. 7 is .lambda./191. 
In FIG. 10, an alternative embodiment of a laser beam expander according to 
the present invention, whose lens elements consist of polystyrene and 
Cargille 480437 liquid, is illustrated. The laser beam expander of FIG. 10 
provides a 10X expansion with substantially diffraction-limited 
performance without requiring refocussing anywhere in the spectral range 
from 0.4 micron to 0.8 micron. 
The 10X laser beam expander of FIG. 10 comprises four lens elements 41, 42, 
43, and 44, which are coaxially disposed with respect to each other along 
an optic axis. The first lens element 41 is made of polystyrene. The 
second lens element 42 is likewise made of polystyrene. The third lens 
element 43 consists of Cargille 480437 liquid; and the fourth lens element 
44 is made of polystyrene. 
In FIG. 11, portions of the lens elements 42, 43, and 44 are shown in 
expanded view to illustrate the liquid lens element 43 contained between 
the plastic (i.e., polystyrene) lens elements 42 and 44. 
The 10X laser beam expander of FIG. 10 has an optical prescription, which 
is specified in tabular format as follows: 
TABLE VI 
______________________________________ 
Surface 
Radius Thickness 
No. (mm) (mm) N.sub.d 
V.sub.d 
Material 
______________________________________ 
1 -7.2143 3.0000 1.590481 
30.886 
Poly- 
styrene 
2 -13.5442 272.3082 Air 
3 -765.9110 3.0000 1.590481 
30.886 
Poly- 
styrene 
4 135.8858 2.0000 1.480097 
43.770 
480437 
5 -75.0698 3.0000 1.590481 
30.886 
Poly- 
styrene 
6 -95.8958 10.0000 Air 
______________________________________ 
where the surfaces of the lens elements are numbered consecutively from 
left to right along the optic axis according to the same convention as 
described above in connection with Table I. The radii of curvature, axial 
thicknesses, and the values of N.sub.d and V.sub.d, are specified in the 
same manner as in Table I. 
In FIG. 12, performance of the 10X laser beam expander of FIG. 10 is 
illustrated by plots of normalized aperture height as a function of 
optical path difference for the same five wavelengths for which the 
indices of refraction are specified in Tables II and V. It is apparent 
from FIG. 12 that the maximum wavefront error of the 10X laser beam 
expander of FIG. 10 for any one of the five plotted wavelengths is less 
than .lambda./11.6. 
FIG. I 3 is a plot of RMS wavefront error vs. wavelength for rays entering 
the 10X laser beam expander of FIG. 10 parallel to the optic axis (i.e., 
at a field angle of zero) calculated over the wavelength range from 0.4 
micron to 0.8 micron. It is apparent from FIG. 13 that the wavefront error 
of the 10X laser beam expander of FIG. 10 is less than .lambda./5.7 (i.e., 
less than 0.176 of a wave) between the wavelengths of 0.4 micron and 0.8 
micron. The average RMS wavefront error over this wavelength range for the 
laser beam expander of FIG. 10 is .lambda./56. 
In FIG. 14, a 5X laser beam expander according to the present invention is 
illustrated whose lens elements consist of polycarbonate and Cargille 
676196 liquid. Optical-quality polycarbonate is obtainable from suppliers 
such as U.S. Precision Lens, Inc. of Cincinnati, Ohio. Index-of-refraction 
measurements for polycarbonate over a broad wavelength band are listed in 
the aforementioned article by R. M. Altman and J. D. Lytle entitled 
"Optical Design Techniques for Polymer Optics". Polycarbonate has a 
density of approximately 1.20 gm/cc; and Cargille 676196 liquid has a 
density of approximately 1.593 gm/cc. Thus, refractive elements made of 
polycarbonate and a Cargille liquid are significantly lighter in weight 
than refractive elements made of optical glasses. 
The 5X laser beam expander of FIG. 14 comprises four lens elements 51,52, 
53, and 54, which are coaxially disposed with respect to each other along 
an optic axis. The first lens element 51 is made of polycarbonate. The 
second lens element 52 is likewise made of polycarbonate. The third lens 
element 53 consists of Cargille 676196 liquid; and the fourth lens element 
54 is made of polycarbonate. 
The 5X laser beam expander of FIG. 14 has an optical prescription that is 
specified in tabular format as follows: 
TABLE VII 
______________________________________ 
Sur- 
face Radius Thickness 
No. (mm) (mm) N.sub.d 
V.sub.d 
Material 
______________________________________ 
1 -11.4624 3.0000 1.585470 
29.909 
Polycar- 
bonate 
2 -25.0349 145.0000 Air 
3 -195.4749 3.0000 1.585470 
29.909 
Polycar- 
bonate 
4 -4643.6103 1.0000 1.676295 
19.644 
676196 
5 85.3339 3.0000 1.585470 
29.909 
Polycar- 
bonate 
6 -64.5436 10.0000 Air 
______________________________________ 
where the surfaces of the lens elements are numbered consecutively from 
left to right along the optic axis according to the same convention as 
described above in connection with Table I. The radii of curvature, axial 
thicknesses, and the values of N.sub.d and V.sub.d, are specified in the 
same manner as in Table I. 
The indices of refraction for the two different materials comprising the 
lens elements of the 5X laser beam expander of FIG. 14 at the same five 
wavelengths for which the indices of refraction are specified in Tables II 
and V are tabulated as follows: 
TABLE VIII 
______________________________________ 
Material 
N.sub.1 N.sub.2 N.sub.3 
N.sub.4 
N.sub.5 
______________________________________ 
Poly- 1.585470 1.599439 1.579864 
1.611519 
1.576832 
carbonate 
676196 1.676295 1.701134 1.666706 
1.722817 
1.661679 
______________________________________ 
In FIG. 15, performance of the 5X laser beam expander of FIG. 14 is 
illustrated by plots of normalized aperture height as a function of 
optical path difference for the five wavelengths for which the indices of 
refraction are specified in Table VIII. It is apparent from FIG. 15 that 
the maximum wavefront error of the 5X laser beam expander of FIG. 14 for 
any one of the five plotted wavelengths is less than .lambda./58. 
FIG. 16 is a plot of RMS wavefront error vs. wavelength for rays entering 
the 5X laser beam expander of FIG. 14 parallel to the optic axis (i.e., at 
a field angle of zero) calculated over the wavelength range from 0.4 
micron to 0.8 micron. It is apparent from FIG. 16 that the wavefront error 
of the 5X laser beam expander of FIG. 14 is less than .lambda./21 (i.e., 
less than 0.048 of a wave) between the wavelengths of 0.4 micron and 0.8 
micron. The average RMS wavefront error over this wavelength range for the 
laser beam expander of FIG. 14 is .lambda./152.6. 
In FIG. 17, an alternative embodiment of a laser beam expander according to 
the present invention, whose lens elements consist of polycarbonate and 
Cargille 676196 liquid, is illustrated. The laser beam expander of FIG. 17 
expands an input laser beam from a diameter of 2 mm to a diameter of 20 mm 
(i.e., provides a 10X expansion) with substantially diffraction-limited 
performance without requiring refocussing anywhere in the spectral range 
from 0.4 micron to 0.8 micron. 
The 10X laser beam expander of FIG. 17 comprises four lens elements 61, 62, 
63, and 64, which are coaxially disposed with respect to each other along 
an optic axis. The first lens element 61 is made of polycarbonate. The 
second lens element 62 is likewise made of polycarbonate. The third lens 
element 63 consists of Cargille 676196 liquid; and the fourth lens element 
64 is made of polycarbonate. 
In FIG. 18, portions of the lens elements 62, 63, and 64 are shown in 
expanded view to illustrate the liquid lens element 63 contained between 
the plastic (i.e., polycarbonate) lens elements 62 and 64. 
The 10X laser beam expander of FIG. 17 has an optical prescription, which 
is specified in tabular format as follows: 
TABLE IX 
______________________________________ 
Sur- 
face Radius Thickness 
No. (mm) (mm) N.sub.d 
V.sub.d 
Material 
______________________________________ 
1 -10.9087 3.0000 1.585470 
29.909 
Polycar- 
carbonate 
2 -28.9858 272.3082 Air 
3 -504.7294 3.0000 1.585470 
29.909 
Poly- 
carbonate 
4 -1032.8888 2.0000 1.676295 
19.644 
676196 
5 140.0307 3.0000 1.585470 
29.909 
Poly- 
carbonate 
6 -116.0882 10.0000 Air 
______________________________________ 
where the surfaces of the lens elements are numbered consecutively from 
left to right along the optic axis according to the same convention as 
described above in connection with Table I. The radii of curvature, axial 
thicknesses, and the values of N.sub.d and V.sub.d, are specified in the 
same manner as in Table I. 
In FIG. 19, performance of the 10X laser beam expander of FIG. 17 is 
illustrated by plots of normalized aperture height as a function of 
optical path difference for the same five wavelengths for which the 
indices of refraction are specified in Tables II, V and VIII. It is 
apparent from FIG. 19 that the maximum wavefront error of the 10X laser 
beam expander of FIG. I 7 for any one of the five plotted wavelengths is 
less than .lambda./4.7. 
FIG. 20 is a plot of RMS wavefront error vs. wavelength for rays entering 
the 10X laser beam expander of FIG. 17 parallel to the optic axis (i.e., 
at a field angle of zero) calculated over the wavelength range from 0.4 
micron to 0.8 micron. It is apparent from FIG. 20 that the wavefront error 
of the 10X laser beam expander of FIG. 17 is less than .lambda./7.5 (i.e., 
less than 0.133 of a wave) between the wavelengths of 0.4 micron and 0.8 
micron. The average RMS wavefront error over this wavelength range for the 
laser beam expander of FIG. 17 is .lambda./18.3. 
The present invention has been described above in terms of certain 
exemplary design forms, which were developed for particular applications. 
However, practitioners skilled in the art of optical design could readily 
develop different design forms for laser beam expanders intended for other 
applications by changing parametric values of the exemplary design forms 
and still be within the scope of the invention. Therefore, the present 
invention is defined more generally by the following claims and their 
equivalents.