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^ ii NBS TECHNICAL NOTE 1015

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

Design of a Reflection Apparatus for

Laser Beam Profile Measurements
NATIONAL BUREAU OF STANDARDS

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the Institute for Computer Sciences and Technology.

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OCT 2 2 19/9

Design of a Reflection Apparatus for

^"-"^^-^

Laser Beam Profile Measurements u^<7S3

Eric G. Johnson, Jr.

Electromagnetic Technology Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

lOf^.

m ^OiC

c'-

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued July 1979
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1015

Nat. Bur. Stand. (U.S.), Tech. Note 1015, 124 pages (July 1979)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE WASHINGTON: 1979

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Stock No. 003-003-02103-2 Price $4.00 (Add 25 percent additional for other than U.S. mailing)
TABLE OF CONTENTS

Page

1. INTRODUCTION 1 2. AN OVERVIEW OF THE BASIC APPARATUS 3 3. PRELIMINARY ANALYSIS FOR THE THEORY ACCOUNTING FOR FIRST-ORDER ASTIGMATISM-- 7 3.1 The Local Coordinates for the Eleven Surfaces of the Apparatus 14 3.2 Some Definitions of Terms for Each Surface Used by the Computer Program 16

3.3 Surface Phase and Amplitude Modulation Effects 22

3.4 The Development of the Practical Transfer Functions Between Surfaces

23

3.5 Application of the Formulas in the Apparatus under Design 26

3.5.1 Assume a Gaussian Beam Undulating through the Apparatus 25

3.5.2 What Happens When the Initial Beam Is a Spot of Uniform

I r radiance 29

4. THE COMPUTER PROGRAM 30

4.1 The Logic Flow of the Computer Program 30

4.2 The Detailed Listing of the Computer Program 49

4.3 Sample Output to Illustrate the Program 49

5. THE 10.6 ym WAVELENGTH --- 50

5.1 Selecting Some Initial Control Values 50

5.2 The Convergence Criteria for the Rest of the Control Values 52

5.3 The Spherical Optics Data 55

5.4 The Elliptical Optics Data 58

5.4.1 Effects of Wavelength Change to the Base Field at Surface 11 63

5.4.2 Effects of 1 mm Changes in Positions to Base Field at Surface 11 68

5.5 Alignment of Apparatus Using 0.6328 ym 68

5.6 High Irradiance Constraints in the Design for 10.6 ym Wavelength 59

6. THE 1.06 ym WAVELENGTH 70

6.1 Selecting Some Initial Control Values 71 6.2 The Convergence Criteria for the Rest of the Control Values 71 6.3 The Spherical Optics Data 71 6.4 The Elliptical Optics Data 74

6.4.1 Effect of Wavelength Change to the Base Field at Surface 11 78

6.4.2 Effects of 1 mm Changes in Positions to Base Field at Surface 11 84

6.5 Alignment of Apparatus Using 0.6328 ym 84

5.6 High Irradiance Constraints in the Design for 1.06 ym Wavelength 85

7. PUTTING THE RESULTS OF PREVIOUS AND NEW SIMULATIONS TOGETHER

TO UNDERSTAND THE APPARATUS 85

7.1 Key Features of Sections 5 and 6 86

7.2 Results of New Simulations 86

7.3 Phase Changes by Noise, Amplitude, Beam Direction, and

Various Adjustments in Wavelength and Mirror Positions 87

8. DEVELOPMENT OF THE SURFACE HOLOGRAM 91 8.1 The Concept: Ideal and Practical 92

8.2 The Parameters for the Surface Hologram 99

8.3 The Computer Codes 101

8.4 The Proposed Construction Sequence 105

9. THE EQUIPMENT NECESSARY FOR CONSTRUCTION OF THE REFLECTION APPARATUS 105

9.1 Mirrors 106

9.2 Translation Mounts 107

9.3 Detectors--Types and Construction of the Receiver Array 107

9.4 Prefilter Construction 109

9.5 The Hologram 110

9.6 Support Equipment 110

9.7 Necessary Machine Shop Work 111

10. CONCLUSIONS 111

11. ACKNOWLEDGMENTS 113

12. REFERENCES 114

13. APPENDIX—THE HARTMANN PLATE METHOD 115

1 1

1

DESIGN OF A REFLECTION APPARATOS FOR LASER BEAM PROFILE MEASUREMENTS*

Eric G. Johnson, Jr.

Measurement of both the irradiance and phase front (the beam profile) in real time from the output of a laser has interest for control of that beam

and for efficient energy and economic design of the source and the resulting

optical systems. The National Bureau of Standards (NBS) has begun a program

to build a unit that can measure, at numerous wavelengths from 1.06 pm to 10.6 pin, a selected spatial sample of the beam profile. This device would

have the following features: (1) The different carrier wavelengths use the same apparatus by changing two mirrors. (2) The beam profile is sampled

simultaneously with no time-shift distortions. (3) The output data streams

documenting the sampled beam profile are continuous and are distorted only by

the finite number and the time constants of the detectors. (4) The

phase-front information is generated before the detectors create the data

streams. (5) The apparatus uses mirrors and a reflection hologram that is

computer generated. (6) The unit is calibrated piecewise over the range of

relative phase and irradiances for each pair of neighboring sampling holes

which are 5 mm apart. (7) The resulting calibrated unit can measure profiles

near 10 cm in diameter with phase-front variations of less than

5 wavelengths. (8) The expected response time for measurements as controlled

by the electronics is of the order of several tens of nanoseconds.

The design analysis reported here includes: (1) the theory which uses Fourier optics concepts with off-axis reflections and rough surfaces to provide the basis for accurate computer simulation of laser beams; (2) the

program, BEAM, which generates the expected behavior of the apparatus under

variation of laser wavelength, physical dimensions for curvatures, hologram

structure, and changes in positions of the various components; (3) the

simulation results which demonstrate the expected characteristics for the

apparatus; and (4) the key element in the apparatus, namely the reflection

hologram, which requires discussion of the design, construction, and testing

of this element.

The Hartmann plate method is described briefly so that a comparison between

it and the holographic method can be made. The comparison shows why the

holographic method is best for a standard for irradiance and phase-front

measurements

.

Key words: Beam profile; calibrated system; holography; irradiance; laser

diagnostics; phase front.

INTRODUCTION

A previous publication [1] has documented the background for beam profile

measurements and has indicated the basis of the holographic method. Here we develop

the design details for a reflection system. This technical note presents sequentially

the concepts and results necessary to estimate for selected accuracy the allowed range

of irradiance levels and phase-front variation for an apparatus using the holographic

method. This presentation format has been chosen to give the reader the option either

for scanning each section for a sense of the design or for studying the analysis in

detail in order to construct a similar apparatus.

''Funded in part by the Calibration Coordination Group (CCG) under contract No. 78-109.
The design development, grouped as nine sections, begins in section 2 with a general description of the apparatus and an operation synopsis. The remaining sections

detail, in turn, particular points of the design process. We indicate those points

below--one for each paragraph.

In section 3, we derive the necessary improvements to the scalar theory for

coherent wave propagation under the Fourier optics approximation with due consideration

for the surface character of the mirrors and the hologram and for the effects due to off-axis illumination of these optics.

In section 4, we describe the computer program generated from these corrected

formulas so the reader can copy and use this program. This program allows a reader to

simulate the apparatus before construction. There are numerous adjustment parameters

for a given apparatus. It is impossible to select the correct version; rather, we arbitrarily select certain convenient choices and then adjust the remaining parameters

to make the design as accurate as possible.

In section 5, with the computer program as given, we apply this capability to study one possible configuration for the 10.6 urn wavelength. Here we select the

arbitrary parameters and adjust the remaining parameters to get the optimum system.

The numerous quantitative results show what can happen. These results range from

alignment with an HeNe laser to sensitivity studies from variation of parameters such

as wavelength, equipment dimensions, and curvatures of the mirrors.

Because the apparatus is expected to be used at wavelengths in addition to 10.6 ym

and because 1.06 pm has significant use, in section 6 we repeat the process described

in the previous section for 1.06 um wavelength. These results should give the designer

a clear picture of how to use the holographic method over a range of wavelengths.

Because these simulations have developed large blocks of apparently unrelated

information, in section 7 we extract key results from sections 5 and 6 to establish a pictorial sense for the ideal operations, given these results. Additional simulations

are performed- here to drive home the capabilities and limits of this apparatus.

Normally in construction of a complex apparatus, there is great interest in the

electronics of the device. Here the action of the optics on the laser radiation is more important; therefore, in section 8 we present the details for the surface

hologram. This item is a key optical component for successful operation of the

apparatus. We discuss the ideal concepts and the practical limitations such as allowed

variation of the carrier frequency (wavelength), beam splitting efficiencies, and

idipl ications about the ultimate accuracies of the apparatus for beam profile

measurements.

To complete the design, we indicate the equipment that can be bought and its

approximate cost. Section 9 contains specifications for the equipment. We identify

the necessary custom machining for the apparatus. The section also contains a summary

on the detector unit which must be custom built. In subsection 9.3 we define some options for construction of the detector unit.

In section 10, we conclude the design phase for the reflection unit by summarizing

the estimated cost for construction of the apparatus and by discussing other bottom

line issues such as what is the expected accuracy of the device and why the unit is the

best system for a standard of beam profile measurements compared with other

techniques.
To provide a clear comparison between the holographic and the Hartnann plate

methods, v/e briefly describe in the appendix v/hat the Hartnann plate unit would do for

the same design conditions as already detailed for the holographic method.

2. AN OVERVIEW OF THE BASIC APPARATUS

We define the reflection apparatus using the background terminology and concepts

described in technical note [1] as a basis for the discussion in this section. In figure 2.1, a block diagram indicates the key optical stages for the unit. Ue describe

the actions of each stage going from the prefilter to the array of detectors in the

cross-correlation plane. The construction details and technical limits of the

apparatus are relegated to section 9 and its subsections. The details of the hologram

are in section 8.

The prefilter has an array of holes with a beam sampling pattern as shown in

figure 2.2. The field of view of the apparatus is fixed as a circle 10 cm in diameter.

Each hole spatially samples the incident radiation to get a laser beam, 1 mm in

diameter, which then undulates through the apparatus. To visualize the action by each

stage of the apparatus, wo trace in the paragraphs below what happens to one beam,

exiting from an arbitrary hole in the prefilter.

The prefilter is located at approximately one focal length (mirror 1) in front of

the first Fourier transform mirror. The beam from a single hole in the prefilter

becomes an Airy pattern with a flat phase front at approximately one focal length

(mirror 1) after the reflection off this mirror.

The magnifying telescope has two mirrors which correctly scale this Airy pattern

to the reflection pattern on the hologram. Mirror 2 has its radii of curvature changed

to match the wavelength of the incident radiation. The nominal position of this mirror

is one focal length (mirror 2) after the Fourier plane of mirror 1. The position of

mirror 3 is one focal length (mirror 2) plus one focal length (mirror 3) after the

mirror 2.

The properly matched Airy pattern, again with a flat phase front, at the hologram

is approximately one focal length (mirror 3) after mirror 3. The surface hologram

splits each Airy pattern into a single reflected beam plus eight diffracted beams

exiting in a square array all of which diverge around the reflected beam.

The Fourier transform done by mirror 4, located one focal length (mirror 4) after

the hologram, causes the resulting nine beams to form a three-by-three array of spots

in the Fourier plane of mirror 4, namely one focal length (mirror 4) after this

mi rror.

Because the detector array has physical constraints that will prevent a match to the pattern at this last Fourier transform, mirror 5 magnifies and images the resulting

spot pattern onto a detector array at the cross-correlation plane. Figure 2.3 shows

the expected pattern when all holes in the prefilter are illuminated and the wavelength

of the carrier frequency is properly matched to the curvatures of the five mirrors.

The detector array has a detector at each spot that has only one or two beams

contributing to the irradiance on the detector. If maximum accuracy in real time is

needed, then there are detectors at the spots which have four-beams contributing to the

irradiance at that spot. (At this stage, all phase-front details can be ignored.)

Each detector measures the resulting laser power in its intersected spot to produce

electrical signals. These signals represent the needed information about the original

beam profile at the prefilter plane once the apparatus has been calibrated properly

(see [1]).
Surface*" 1

Prefilter

Plane

Surface* 2

Surface**4

First

Mirror

of

Magnifying

Telescope

Surface*?

Optical

Computer

'Hoiogrann'

.vr>oV\co\

i^

Fourier

Transform

Mirror

Surface*6

Second

Mirror

of

Magnifying

Telescope

Surface* 8

Surface^lO

Magnification

Mirror

\

^

u

Fourier

Transform

Mirror

Surface '^ll

Cross

Correlation

Plane

Figure 2.1. Beam profile measuring apparatus using reflection optics (not to scale)
o o o o o

o o o o o

o o o o o

o o o o o

5 mm

J-O o

5mm

o o o

1 mm

Figure 2.2. Beam sampling pattern in the prefilter plane.
@o@o@o@o@

OXOXOXOXO

@o@o@o@o@

OXOXOXOXO

@o@o@o@o©

O X X O X o X o

@ O @ O @ O @ o @

4 -o X X O X o X o

2.5 mm

+ -@ O @

J m m O @ O @ o ©

X Four Beams contribute to ttiis intensity

O Two Beanns contribute to this intensity

pattern. ( phase here ) O One Beam produces this intensity.

Figure 2.3. Resulting spot pattern in the cross-correlation plane.
3. PRELIMINARY ANALYSIS FUR THE THEURY ACCUUNTING

FUR FIRST-URDER ASTIGMATISM

To deduce the appropriate corrections to the theory of Fourier optics, we must yet

the proper integral equations to start the approximation sequence. Unlike the usual

approximations for Fourier optics, here the source surfaces are not necessarily plane;

they can be curved surfaces such as mirrors and bumpy surfaces such as holograms. To reflect this fact, the starting integral equation for approximation is formulated in

general coordinates. (NUTE: The polarization features of laser radiation are ignored.

See the conclusions in section 10 about this point.)

To make this development specific to the actual apparatus, we follow and define

the surfaces illustrated in figure 3.1. Here they are labeled with £ = 1 being 'che prefilter surface shown in figure 2.1, 1^1 being the surface of mirror 1, etc. Surfaces 3, 5, and 9 are convenient mathematical surfaces. Here the laser radiation

continues without reflection to the next surface. Surfaces 1 and 11 also have no reflections; the rest reflect and may modify both the phase front and the irradiance of

a beam. The b^ quantifies the distance along the optical axis of the center laser

beam between surfaces i and £ + 1. This diagram assumes the optical axis reflects at a constant angle from the normal of each reflecting surface. Obviously, the real

apparatus would not do this; therefore, errors arise in the final apparatus from this

failure. We ignore this situation and accept that the errors can be minimized by

proper adjustments of each mirror. The nonref lecting surfaces are all perpendicular to the optical axis, and the global x, y, and z coordinates for this apparatus have

positive z pointing to the right, positive x pointing toward the top of the page, and

positive y pointing perpendicular out of the page. Finally, the origin of this

coordinate system is at the center of the prefilter. The optical axis of the apparatus

is defined by a laser beam exiting from a hole at the center of the prefilter and

undulating off the mirrors and hologram until it is captured by the detector that is

centered relative to the other detectors at surface 11.

Given the above definitions, we now develop the improved theory accounting for

lowest-order astigmatism. To reduce the details in this derivation, we gather key

notions from different references so that the resulting corrected, but still

approximate, theory is plausible.

For the exact theory for laser radiation, we should use Maxwell's equations. The

publication [1] and papers referenced by it establish that the scalar wave equation is

adequate for many uses of laser radiation, namely almost plane-wave conditions and no significant polarization effects. Unfortunately, these discussions are primarily

transmission analyses where the optical axis is not bent. Because we have reflections,

we select the Fresnel equations from [2] when the electric field has only a y component. The other polarization case, namely the magnetic field has only a y component, is ignored in this discussion. The apparatus can be adjusted and calibrated

for either polarization condition. However, once the unit has been adjusted for one polarization condition, that adjustment will not apply exactly to the other condition.

Small errors in the measured results will occur which ^ro. caused by changes in the

polarization state of the incident laser beam. Thus, we must select only linearly

polarized beams if we ^.tq. to get results of maximum accuracy. Most uses can accept

these small errors.

Figure 3.2 shows the consequence of the reflection to the right from one medium to

a second for a plane wave. The mathematical details for the phase functions ^.tq:
* z

Figure 3.1 The "Surfaces" of the Holographic Apparatus

There are Eleven Such Surfaces
E| e'*^i

Figure 3.2 The case of plane wave reflection to the right

with electric field parallel to reflecting surface
t"^ = k^[-(x - x^) sin e + (z - z^) cos e].

K = k [-(x - X ) sin o - (z - z ) cos o], and

^3 " ^2^'^^^ " ^^ ^''" °' + (^ - ^1^ '^^^ °'^' ^^-^^

where:

k^ = k/e u , k :. k/e p , n^ sin e' = sin 6, n = /e y /e y ,

and X]^ and Z]^ are the coordinates of the origin for the reflection betv^een the

two mediums. The nathenatical details for the strength of the electric fields are:

E = 2 cos e E /(cos e + n cos e'), and

E = -E,(n cos e' - cos e)/ n cos e' + cos 6). (3.2)

If we have a perfect reflector, that implies n„, = oo; hence,

Ej = 9' - and E3 = -E]^. For convenience in this analysis, we assume

Uj^G]^ = 1 for propagation in air and set k]^ = k in our subsequent discussion.

Figure 3.3 shows the consequence of the reflection to the left from one medium to

a second for a plane wave. The mathematical details for the phase functions are:

<j) = k[-(x -X ) sin 8 - (z - z ) cos e],

Si d a

<t), = k[-(x - X ) sin e + (z - z ) cos 6], and

b a a

^ = k^[-(x - X ) sin 6' + (z - z ) cos e']. (3.3)

c i_ a a

The strengths of these electric fields are the same form as before for all values of

n^,. Here Xg and Zg are the corresponding coordinates for the origin of this

reflection.

From the above results for the ideal reflector, we note three points that are used

in the scalar theory analysis.

1. The reflected term has a sign change regardless of the direction of the incident

beam.

2. For incident wave propagation to the right, the derivative of the reflected term

evaluated at z = z^ and normal to that surface is:

-- (E^) = -k cos E^ = k cos e E,

.

dz 3 3 1

(Here this derivative is toward the reflecting surface.)

3. For incident wave propagation to the left, the normal derivative of the reflected

term at z = z^ is:

10
Epe'*^b

Figure 3.3 The case of plane wave reflection to the left

with electric field parallel to reflecting surface

11
— (E, ) = k cos e E^ = -k cos e E, , dz 3 3 1

(Here this derivative is away fron the reflecting surface.)

If we make the convention that the normal derivative is away from the reflecting

surface into the volume surrounded by these surfaces and make slight notational

changes, we have the reflected electric field {^) related to the incident electric

field (ipj) as ij^ = -iii at each surface and the normal derivative as

Vp(j^ = -k cos 9 \pi at each surface.

We are ready now to develop the appropriate scalar theory for off-axis

reflections. We indicate briefly and somewhat cryptically the approximation sequence

in the paragraphs that follow. For more background on this sequence, see [3] for the

generalized coordinates, [4] for the Green's function, [5] for various details on aberrations, and [6] for discussion of the Fourier optics approximations.

We get the integral equation first for the scalar theory assuming harmonic time

dependence. The scalar function ^ is defined by the Helmholtz equation as:

(V^ + k^) i^ = 0. (3.4)

To construct the transfer integral, we use Green's function method, where:

2 2 (V + k ) G = -4Tr6(r - r )

with the solution

G = exp(ikF)/F, (3.5;

where

F = [(x - x )^ + (y - y^) + (z - zfl^^^. (3.6)

The integral equation is:

i>{r) = /dS[-G Vnoi(^(r ) + i(^(r on) V G]/4Tr, (3.7)

where Vp is the operator for normal derivative in Vq coordinates to the surface

pointing into the volume. The surface surrounds the volume, and the origin for Green's

function singularity is within this surface integration. The range of r is entirely

within this volume and this surface. The range of rg is the surface only.

To make the integral equation useful for simple modeling of the apparatus requires

approximations. Basically, the vector features of the electric field is unimportant

either left or right between the surfaces shown in figure 3.1. This condition means

that the r coordinate is far from the Vq coordinates and that the characteristic

phase function is like the plane wave shown in eq (3.1). These facts permit

simplification of eq (3.7), For example, the reflected wave leaving a surface at the

left and the incident wave ipj arriving at that same surface are related as:

12
((;(r) = ik cos e / dS G (j; (r )/2Tr. (3.8)

In the case where the surface, S, has only transmission through the surface that is

normal to the incident beam, the exit beam is related to the incident beam as:

^{r) = -ik / dS G ^ (r )/2tt . (3.9)

The propagation from right to left is the same form for this apparatus. We can rewrite eqs (3.8) and (3.9) to the form for each surface of the apparatus as:

^ , = -C / du dv A ^ exp(ikF )/iAb , (3.10)

where

2 2 2 1/2

F = [(x - X ) + (y - y ) + (z - z ) ]^ . (3.11)

SL SL+l I i+l I £+1 £

Here x^, y^, and z^ ^ro. the coordinates of the £th surface. They are constrained by the surface coordinates u^ and Vjj^ according to the form shown in the

next subsection. The surface orientation factor C^ and the surface aperture function

Aj^ b^ro. defined as:

C = cos 9 for Jl = 2, 4, 6, 7, 8, 10

= -1 for £=1,3, 5, 9, 11, and (3.12)

A = exp[-(u " + V ^)/p ^]. (3.13)

£ £ £ £

The factor h^ is added to original equations to simulate the effects of aperturing

by the finite mirrors. This Gaussian form is chosen for convenience and has no major

significance beyond the fact that it permits explicit integration of eq (3.10) when the

beam is Gaussian and when the F^ has been approximated appropriately. The p^ is

the effective width of this Gaussian aperture.

The b£ factor in eq (3.10) is the propagation distance between surfaces at the

optical axis; this term is very large compared to any distances due to curvatures and

bumps in these two surfaces. Therefore, we can replace the nonphase term F^ by b^

in eq (3.5).

We have completed the initial development of the integral equation for the

apparatus. In subsection 3.1 we put in the explict local coordinates for this analysis

so that we can extract the lowest-order correction for astigmatism as shown in

subsection 3.4. To make the computer program listing convenient, we define a collection of formulas in subsection 3.2. Because we wish to develop various

holograms, we note the modulation process at surface 7 in subsection 3.3. In subsection 3.4 we generate three classes of formulas with the necessary corrections for

astigmatism [7,8]. We derive: first, the Fourier transform with its associated

conditions on curvatures and placement of the mirror; second, the imaging relations by

a single mirror with its associated conditions; and third, the formulas that describe

coefficients for Gaussian beam propagation through the apparatus. As the final

exercise of this section, we apply these results in subsections 3.5.1 and 3.5.2 to the

13
actual alignnent of the apparatus for two bean shapes, nanely Gaussian and an initially

uniforn spot, respectively. This discussion is presented here because the concepts

also apply to the computer simulations for fixing the position of the optics. Using

the same technique documents how close to ideal the system is before it is

constructed.

3.1 The Local Coordinates for the Eleven Surfaces of the Apparatus

In this subsection we develop the local coordinates so that the F^^ can be

expanded in b£ to reflect the Fourier optics approximation.

The mirror surfaces in the apparatus are approximated here by parabolas defined by

curvature functions, hj^^, with different radii of curvature possible for the u^^ and

V£ coordinates, respectively. This means that the actual mirrors with spherical or elliptical surfaces will have some aberrations due to their nonparabolic features.

These effects are minimized by proper adjustment of all b£ and by correct placement

of the detectors in the cross-correlation plane. For convenience, we use CI e cos 9 and SI = sin 9 throughout this analysis and in the computer listing. The functional

forms for the x^, y^, and Z£ in terms of local coordinates are:

x = -SI

i-l

E b, j=l

- + u C ,

y = v , and

z =h +Sgu +C1

£-1

T b.S.

3 = 1 '

' [3.14a)

The curvature functions are:

h = ( u HiVd + v II/e )/2, [3. 14b)

where the radii of curvatures, d^ and e£, are less than zero for those left-side

mirrors and greater than zero for those right-side ones shown in figure 3.1. Those

flat surfaces are simulated by the required curvatures being very large positive

numbers. The expression | C£| means the absolute magnitude of C£ is used. The

remaining parameters S£ and g£ are defined respectively as:

1 for £=1,4, 5, 7, 10, 11

= -1 for £ = 2, 3, 6, 8, 9 (3.15)

and

g

= SI

£ for £=1,3, 5, 9, 11

for £ = 2, 4, 6, 7, 8, 10. ;3.16)

In addition, but not shown in eqs (3.14), surface 7 can have a term in z-, that

represents a bumpy surface. This would be added to the hy term. For convenience in

14
subsequent analysis, this term is ignored until subsection 3=3.

To develop the practical transfer function, we now restructure F^. First note:

X , - X = IJ - SI b ,

where

where

W = u I C I - u C £ i+\ ' £+1' £' S)

y -, - y = V , - V , and

£+1 £ £+1 £

z ,-z =r +Cl(bS;

£+1 £ £ £ £

r = t + h , - h , and

£ £ £+1 £ t=S ,g ,u ,-Squ. £ £+1 £+1 £+1 £ £ £

Thus F^ becomes:

where

2 2 F = b + 2b H + K £ £ £ £ £

H = r s CI - W SI , and

£ £ £ £

2 2 2 K=(v,-v)+r +W

£ £+1 £ £ £

The Fourier optics approximation has F^ changed so that only the first three

terms in a Laurent series in b£ need be kept. In this case:

F - b + H + T /2b , (3.18)

£ £ £ £ £

where H^ is as before and

T = K - H

^ ££££+!££ = (v , - V )^ + (W CI + t SI S )^. (3.19)

£ £

The t£ term is an r^ term; however, the Fourier optics approximation neglects the

hj^ terms in eq (3.19). In contrast, we do not and must not neglect these terms in

Hjj of eq (3.18), which represent the lowest-order action by each mirror in the

apparatus.

15
Sonne Definitions of Terms for Each Surface Used

by the Computer Program

This subsection converts the notation and some formulas of the previous subsection

to derive as briefly as possible new expressions that use notation and symbols in the

computer language BASIC. This exercise simplifies the identification process in the

computer program listed in subsection 4.2 and eliminates the extensive subscripting in

these equations which makes typing and transcription difficult. During this

conversion, there are expressions that mix the two notations. For example, note that

an asterisk symbolizes multiplication. It will be present most places except when the

local coordinates u^ and V£ are shown in the formulas. The expression "f" symbolizes exponentiation. It is used when there is squaring of the number. General

exponential operations are shown as EXP(X). In this discussion, £ = L, and

I + I = LI. The imaginary i = /A is not defined in BASIC; therefore, it is just

inserted as needed, lie define Z(L) = ^^ as the field function.

y(l: EXP d(i,l; iD(2,L) + u^[D(3,L) + iD(4,L)]

+ v^[D(5,L) + iD(6,L)] + u^ [D(7,L) + iD(8,L)]

+ v^ [D(9,L) + iD(10,L)] + u^^^[E(l,L) + iB(2,L)]

+ v^^^[E(3,L) + iE(4,L)] + u^^^[E(5,L) + iE(6,L)]

+ v^^^[E(7,L) + iE(8,L)] + u^ u^^^[E(9,L) + iE(10,L)]

+ v^ v^^^[E{ll,L) + iE(12,L)]

(3.20a)

is the transfer function between surface L and LI. The D and E are arrays of real

numbers fixing this function. Using the array C, we can define:

Z(L)e EXP C(1,L) + iC(2,L) + u^CC(3,L) + iC(4,L)]

+ v^[C(5.L) + iC(6,L)] + u^ [C(7,L) + iC(8,L)]

+ v^ [C(9,L) + iC(10,L)]

[3. 20b:

if the beam profile is Gaussian throughout the apparatus. Otherwise, Z(L) is not simply defined. In any case, eq (3.10) becomes in this notation:

16
Z(L1) = / du^ dv^ Y(L)*Z(L) . (3.20c)

To use eqs (3.20) we nust fix the D, E, and C set of parameters in terms of the

previous definitions. Ue define:

B(1,L) = 1 -1

B(2,L) = 1 -1

B(3,L) = 1

B(4,L) = 1

B(6,l:

for L = 13

2

for L = 4

2

1

for L = 12

for L = 2

1

;5,L) = C = CI for L = 2

i

1 for L = 1

7, 11

98, 10

10, 11

QO

5, 9

5, 6, 7,

8, 10

9, 11

6, 7, 8,

5, 9

10, 11

6, 7, 8, 10

5,9, 11

5, 7, 10 , 11

6, 8, 9.

lith eqs (3.21) to (3.26), v;e can define:

H = C1*[B(1,L) h - B(2,L) h ]

Si l+l Si

-S1*[B(3,L) u^^^ - B(4,L) u^] ,

(3.21)

(3.22)

(3.23)

(3.24)

(3.25)

(3.26)

and

ABS[B(5,L1)] u

_^ - ABS[B(5,L)] u + 2 + (v , -V ) + 2. £+1 £ (3.27)

More definitions are:

A(1,L) = K/(2d ) as curvature factor for x coordinate,

A(2,L) = K/(2e ) as curvature factor for y coordinate,

A(3,L) = -1/p as the Gaussian aperturing factor, and

A(4,L) =- b^ as the distance between surfaces L and LI.

(lie arbitrarily set A(4,ll) = 1.) K is the wave nunber.

In the definitions of D and E, it is convenient to use Tl = K/[2*A(4,L)] during

this generation of each surface coefficient. Thus D and E are:

17
D(1,L) = LN fABS[B(5,L)]*Tl/TT

D(2,L) = (Tr/2)*SM[B(5,L)] ,

D(3,L) = D(5,L) = D(6,L) = ,

D(4,L) = K*S1*B(4,L) ,

D(7,L) = D(9,L) = A(3,L) ,

D(8,L) = T1*[B(5,L) + 2] - C1*B(2 ,L)*A(1 ,L) ,

D(10,L) = Tl - C1*B(2,L)*A(2,L) ,

E(I,L) =0 for I = 1, 3, 4, 5, 7, 9, and 11.

E(2,L) = -K*S1*B(3,L) ,

E(6,L) = T1*[B(5,L1) t 2] + C1*B(1 ,L)*A(1 ,L1) ,

E(8,L) = Tl + C1*A(2,L1)*B(1,L) ,

E(10,L) = -2*T1*ABS[B(5,L1)*B(5,L)] , and

E(12,L) = -2*11 . (3.28)

Above, Sri(X) means the sign of X, ABS(X) means the absolute value of X, and LN(X) means the loge(X).

The D(2,L) should contain +K*A(4,L). This factor is common to all beams going

through the apparatus; therefore, it cannot be measured here. This fact makes it

unimportant. Retaining this term in D(2,L) would generate large numbers which can produce significant and unnecessary round-off errors in subsequent computations.

There remain additional definitions that are used in the computer program BEAM.

It assumes Gaussian beam propagation and is listed in subsection 4.2. When Z(L) is

Gaussian, we define the complex numbers:

18
Al - C(1,L) + D(1,L) + i[C(2,L) + D(2,L)] ,

A2 = C(3,L) + D(3,L) + i[C(4,L) + D(4,L)] ,

A3 = C(5,L) + D(5,L) + i[C(6,L) + D(6,L)] ,

A4 = C(7,L) + D(7,L) + i[C(9,l) + D(8,L)] ,

A5 = C(9,L) + D(9,L) + i[C(10,L) + D(10,L)] ,

A6 = E(1,L) + i E(2,L) ,

A7 = E(3,L) + i E(4,2) ,

A8 = E(5,L) + i E(6,L) ,

A9 = E(7,L) + i E(8,L) ,

AlO = E(9,L) + i E(10,L) , and

All = E(11,L) + i E(12,L) .

With these definitions, the integrand in eq (3.20) becones:

'3.29:

Y(L)*Z(L) = EXP Al + u A2 + V A3 + u A4

Z H Z

+ V A5 + u A6 + V A7

£ £+1 Si+1

2 2 + u AS + V A9

£+1 £+1

+ u , u AlO + V V , All

£+1 £ £ £+1

(3.30)

We nov/ integrate eq (3.20) to find Z(L1) for this case. Here eq (3.30) can be

written in the form:

EXP [A4*(u - 02) + 2 + 03 + A5*(v - 05) + 2] ,

where

02 = (A2 + u A10)/(-2*A4;

05 = (A3 + V , All)/(-2*A5) , and

£+1

03 = Al + u A6 + V A7 + u AS + v A9

£+1 £+1 £+1 £+1

:A2 + u , AlO) + 2/(4*A4) - (A3 + v , All) f 2/(4*A5'

£+1 £+1

(3.3i;

19
The integration over U£ and V£ gives:

Z(L1) = [ti//W^ ) EXP(G3) . (3.32)

For the computer program, we separate the complex numbers and place all terms into

the exponential so the C array can be defined for the LI surface.

We write:

F(l) = C(7,L) + D(7,L) ,

F(2) = C(8,L) + D(8,L) ,

F(3) = C(9,L) + D(9,L) ,

F(4) = C(10,L) + D(10,L) ,

Rl = F(1)*F(3) - F(2)*F(4) ,

R2 = F(1)*F(4) + F(3)*F(2), and

R3 = SQR(R1 ^ 2 + R2 + 2) . (3.33)

(Here SQR(X) is v^.

)

Note that a separation into polar coordinates for the complex numbers gives:

A4*A5 = R3*EXP(i R5)

where

C0S(R6) = R4 = R1/R3 , and

Siri(R6) = R5 = R2/R3 .

Here the exponential form for Tr/SQR(A4*A5) = EXP(R8 + i R9), where:

R9 = -R6/2 , and

The phase term

= LN[7r/SQR(R3)]

R6 = sin" (R5) if R4 >

= TT - Sirf'^ (R5) if R4 <

= SN(R5)*tt/2 if R4 = . (3.34)

The numbers C(I,L1) are related to the Al, etc., by the relationships:

20
C(1,L1) + iC(2,Ll) = R8 + iR9 + Al - A2 + 2/(4*A4) - A3 f 2/(4*A5),

C(3,L1) + iC(4,Ll) = A6 - (A2*A10)/(2*A4)

,

C(5,L1) + iC(6,Ll) = A7 - (A3*A11)/(2*A5 )

,

C(7,L1) = iC(8,Ll) = A8 - AlO + 2/(4*A4), and

C(9,L1) + iC(10,Ll) = A9 - All t 2/(4*A5). (3.35)

Separation of A6 throughA9 into rea] and imaginary is simple. See eq (3.29). The

expression

-A *A /(4*A ) (3.36)

X y z

requires a computer subroutine. With control parameters Jl, J3, J5, and J7 that are shown for each eq (3.36) expression in the computer program (here J2 = Jl+1, J4 = J3+1,

and J6 = J5+1) , we get:

A

X

= F(5) + i F(6) ,

A = y

F(7) + i F(7) , and

A

z

= F(9) + i F(10).

Here:

Here:

F(5) = C(J1,L) + D(J1,L), and

F(6) = C(J2,L) + D(J2,L), if J7 = 1 or 2.

F(5) = E(J1,L), and

F(6) = E(J2,L), if J7 = 3.

F(7) = C(J3,L) + D(J3,L), and

F(8) = C(J4,L) + D(J4,L) if J7 = 1.

F(7) = E(J3,L), and

F(8) = E(J4,L), if J7 = 2 or 3.

Finally,

F(8) = C(J5,L) + D(J5,L), and

F(9) = C(J6,L) + D(J6,L).

The multiplications defined in eq (3.36) are completed by using:

21
Ql = F(5)*F(7) - F(6)*F(8) ,

Q2 = F(5)*F(8) + F(6)*F(7) ,

Q3 = F(9) + 2 + F(10) + 2 ,

Q5 = -F(9)/Q3 ,

QC = F(10)/Q3 ,

Q7 = (Q1*Q5 - Q2*Q6)/4 , and

Q8 = (Q1*Q6 + Q5*q2)/4, (3.37)

where finally we define:

Q7 + i Q8 = -A *A /(4*A ). (3.38)

The above completes for the computer program the definitions that are needed in

the remaining subsections 3.3, 3.4, and 3.5. There will be further definitions used in

the computer program which will be defined in subsections 4.1 and 4.2.

3.3 Surface Phase and Amplitude Modulation Effects

Surface 7 in figure 3.1 is a reflector with a bumpy surface modifying the ideal

plane. This unit acts as a hologram to split each incident beam into eight additional

diffracted beams and into one reflected beam. To represent this bumpy surface, we add

a term to the integral in eq (3.20) for the description of the field at surface 8. Thus:

Z(8) = / du dv F(u,v)*Y(7)*Z(7), (3.39)

where the- surface function correcting for deviation from a plane surface of reflection

is:

F(u,v) = EXP[-i 2*K*Cl*h(u,v)] . (3.40)

Here the local coordinates of surface 7 are abbreviated to:

u H u and v s v .

One simple mathematical example of h is:

h(u,v) = s[sin(u*S7) + sin(v*T7)] , (3.41)

where s is the variation from the plane front for surface 7 in pm units. A possible

value for s is 0.084 ym. Notice that the S7 and T7 can be different. They are proportional to the spatial frequencies for u and v, respectively.

To identify the individual beams generated by this bumpy surface given in eq (3.41), we expand eq (3.40) to show the explicit relative directions for the

diffracted beams from each reflected incident beam. Thus we have:

22
F(u,v) = I 5: J (z) J (z) EXP(iD) (3.42) m n

111= -a, 111= -m

where z = -2*K*Cl*s, and the beam diffraction indicator is:

D = m*u*S7 + n*v*T7 . (3.43)

We will discuss more general examples of h in section 8. Here they are selected

by practical fabrication issues. In many cases, eq (3.42) can be generalized to:

F(u,v) = E EXP[G(n,n)] EXP(iD), (3.44)

m,n=-<i>

where G(m,n) are complex numbers giving the phase shift and the strength of the beam

diffracted from the reflected beam. The order m corresponds to the u coordinate and

the order n corresponds to the v coordinate.

The ideal hologram for the reflection apparatus would have the G(m,n) show the

following form. All C(m,n) would equal -<= except when | m |, | n j < 2. In section 8, we

discuss ways to realize this ideal.

The example shown by eq (3.41) can approach the ideal situation when z = -0.1.

This happens if the wavelength is 10.6 ym and the amplitude of the bumpy surface has

s = 0.084 ym. The Bessel functions become:

Jq(z) = 0.998 .

J (z) = -J (z) - -0.050 , and

0^(1) = J_2(z) = 0.001 .

The rest are less than 0.001 and can be neglected in one percent precision

measurements. This example shows that the:

J (z) % f or

I n

I >_ 2 .

I " I

Therefore, there are operationally nine beans generated by the single incident beam.

The m = n = case represents the pure reflected beam. The four cases, namely m = 0,

n = j^l, or m = +1, n = 0, represent the diffracted beams that can generate the spot

formed by two beams. These are shown in figure 2.3. To get this condition, we scale

S7 and T7 relative to the optics of the apparatus and to the wavelength of the laser

radiation. The remaining diffracted beams contribute to the spots shown in figure 2.3

as the spots with four beams contributing.

3.4 The Development of the Practical Transfer Functions Between Surfaces

In subsection 3.2 we generated in eq (3.20) the formula for the Fourier optics

approximation when only one surface process acts on the laser beam. Uhen a mirror

generates a transform or image, it is a two-surface process. These are shown in

references [1] and [9] for the simpler on-axis case. This section defines the off-axis

case.

23
In the reflection apparatus, we have definite surfaces for each nirror of

interest. In all cases, we assume that these surfaces in front of and after a given

mirror are plane and that their surface normals etre either parallel or at an angle 9 to the surface normal of the mirror. Each circumstance uses appropriate values of

B(5,L) in eq (3.25). Because the actual formulas are complex, we only indicate the

algebraic form with many definitions of convenience. The reader should use a computer

to generate from these equations a code that constrains the parameters of the

apparatus. In the discussion below we assume the arrays D and E are defined in

eq (3.28). lie presume that the values of the arrays are known, and in this section we deduce the constraints on these values. IJe make no attempt to reduce the resulting

formulas. The image formula is defined first, because it is the simplest.

First, we construct the two-surface transfer equation using eq (3.20) twice and

defining L2 = L+2. Thus:

Z(L2: / du dv Y1(l; Hi z(l; :3.45;

where

Y1(L) = / du dv , Y(L) Y(L1) . £+1 £+1

(3.4G:

Integration of eq (3.46) generates the form:

Y1(L) = EXP Kl + u K2 + V K3 + u K4 + v

^ K5

Z i i Si

+u ^K6+v ^K7+u ^K8+v ^K9

i+2 £+2 SL+2 £+2

+ u u ^ KIO + V V ^ Kll

£ £+2 £ £+2

7here the complex numbers Kl to Kll and 84 and 85 are defined:

Kl = Dl + Fl + LN[Tr/SQR(84*85)] - (El + F3) + 2/(4*84)

-(E3 + F5) + 2/(4*B5) ,

K2 = D3 - E9*(E1 + F3)/(2*84) ,

K3 = D5 - E11*(E3 + F5)/(2*B5) ,

K4 = D7 - (E9 + 2)/(4*B4) ,

K5 = 09 - (Ell + 2)/(4*B5) ,

K6 = Gl - G9*(E1 + F3)/(2*B4) ,

K7 = G3 - G11*(E3 + F5)/(2*B5) ,

KB = C5 - (G9 i 2)/(4*B4) ,

K9 = G7 - (Gil + 2)/(4*B5) ,

(3.47)

24
KIO = -E9*G9/(2*B4) ,

Kll = -E11*C11/{2*B5;

B4 = E5 + F7 , and

85 - E7 + F9 . [3.48]

ThG above terns are defined by the following complex numbers that are correlated to the

terns in Y(L) and Y(L1), respectively. Thus:

Dl = D(1,L + iD(2,L) . Fl = D(1,L1 + iD(2,Ll) ,

D3 = D(3,L) + iD(4,L), F3 = D(3,Li; + iD(4,Ll) ,

D5 = D(5,L. + iD(6,L), F5 = D(5,L1 + iD(6,Ll) ,

D7 = d(7,l; + iD(8,L) , F7 = D(7,Li; + iD(8,Ll) ,

D9 = D(9,L + iD(10,L) , F9 = D(9,Li; + iD(10,Ll)

El = E(1,L] + iE(2,L) . Gl = Ed.Li; + iE(2,Ll) ,

E3 = E(3,l: + iE(4,L) , 03 = E(3,Li; + iE(4,Ll) ,

E5 = E(5,L] + iE(6,L) , G5 = E(5,Lr + iE(6,Ll) ,

E7 = E(7,L + iE(8,L), G7 = E(7,Li; + iE(8,Ll) ,

E9 = E(9,L] + iE(10,L) , G9 = E(9,L1] + iE(10,Ll)

Ell - E(11,L) + iE(12,L), and Gil = E(11,L1) + iE(12,Li; (3.49)

To get the inage formula, we want a delta function for the u^ and v^ variables

in Y1(L). To define the image plane for a single mirror from a known object plane, we

first fix the ratio:

R = b /b . £+1 £

This is adjusted subject to the derived constraints in eq (3.50) to generate the

appropriate magnification of the desired image. To locate conveniently the object and

the image plane, we fix the curvature of the mirror's y coordinate, e -, . Now we

set the curvature d imaginary parts of

, and the image distance b , by requiring that t B4 and B5 are zero. This implies:he

E(6,L) + D(8,L1) = 0, and

E(8,L) + D(10,L1) = 0. (3.50)

If these are true, then K4 and K5 in eq (3.47) have been made as large as possible; we

approach the delta function. To do better requires that the aperturing is

insignificant so the real parts of 84 and B5 can be near zero. In this case, the

integral in eq (3.45) can be approximated appropriately, and the image can look like

the object. If the aperturing is significant, there is a convolution of the object

function, Z(L), with a Gaussian-like intensity distribution, Y1(L). The resulting

25
image, Z(L2), is thus smeared and cannot be made sharp. The high-frequency features of

Z(L) are filtered out by the convolution process.

In the reflection apparatus, the R is fixed by the required dimensions for imaging

on the detector array. The curvature of the mirror is fixed by the required distances

between mirrors to allow a clear field of view for the object and the image plane.

The Fourier transform in eq (3.45) has the position, b ., , and the curvature,

d ,, such that the imaginary parts of K4 and K5 are zero in eq (3.47). Here we presume that the y curvature, e ,, is fixed by design constraints. The ideal

transform has the real parts of K4 and K5 zero also. However, the usual aperturing by

the mirror causes eq (3.45) to convolute the Fourier transforms of Z(L) and Y1(L). The

transform of Y1(L) is almost a delta function if the aperturing is small enough;

otherwise, it smears the Fourier transform of Z(L).

After identifying the Fourier tranform plane, we adjust the distance b^ in front

of the mirror to make the phase front in the Fourier plane as flat as possible. This

happens when the imaginary parts of K9 and K8 are both zero. Usually, adjustment of one cannot make both zero simultaneously; therefore, we just make these two numbers as small as possible. The practical situation has their curvatures equal but of opposite

si gns.

This concludes the discussion of the practical formulas, and we proceed to subsection 3.5.

3.5 Application of the Formulas in the Apparatus under Design

As can be seen from the previous subsection, the mathematics for the reflection

apparatus is complex algebraically. However, the discussions in the previous

subsections have made clear the qualitative features of the apparatus so that it is

unnecessary to know its quantitative details. In subsections 3.5.1 and 3.5.2 we use these previous discussions to note what is pertinent to each surface of the reflection

apparatus.- By confirming the presence of these features in the alignment of the

apparatus, we can test its status on a surface-by-surface basis to confirm whether the

apparatus is aligned and constructed properly. Because the computer code in

sections 4, 5, and 5 uses Gaussian beams to propagate through the simulated apparatus,

we discuss in subsection 3.5.1 how to align the apparatus when it has Gaussian beams.

In subsection 3.5.2 we discuss how the results of subsection 3.5.1 are modified because

the actual apparatus has circular sampling holes in the plate at the prefilter plane.

3.5.1 Assume a Gaussian Beam Undulating through the Apparatus

In section 2 we traced the actual apparatus in general terms. Here we get more

technical about each surface, using the fictitious and simplifying condition that the

Hartmann plate at surface 1 has holes which generate an array of beams with Gaussian

profiles when irradiated by an arbitrary, unknown laser beam. Each surface of the

apparatus is labeled in figure 3.1. To construct the apparatus requires understanding

numerous technical details. In the paragraphs below we define these details so that

each component has a proper position. The beams indicated below are those generated

behind the prefilter by illumination with a cw laser of the desired wavelength. The

discussion below requires numerous background points. In the following paragraphs we

address them first, and then we indicate the adjustment sequences.

There are three stages in the adjustment process for each surface. The first

stage uses the results of the theory and the computer simulations to fix the distances

(b^) between surfaces. The second stage uses the expected results at a key surface

26
to make precision adjustments of optical components tied to that surface. The end

result of this adjustment sequence is an apparatus assembled vnth mininun deviation

from the expected. The third stage makes the apparatus a precision unit by calibrating

it v/ith the aid of the mathematical model defined in reference [1, pages 31-4].

We make here a collection of general conditions for alignment of the apparatus:

(1) The precise positions for each surface are documented in sections 5 and G for an apparatus using Gaussian beams. In this subsection, v/e accept that these beams exist

and that they can be used to adjust the optics. (2) A single laser beam going through

the hole at the center of the Hartmann plate operationally defines the optical axis of

the apparatus. (3) The first series of position adjustments can be easily set correctly v/ithin 1 mm. (4) If the room containing the apparatus is large enough, the

corresponding angle adjustments can be adjusted to 0.1 mrad. (5) The initial placement

of each optical component is added as the beams from the prefilter transverse the

apparatus. (6) Because placement errors accumulate, the preliminary positioning and

precision adjustments of each component may have to be performed together. (7) Because

radiation of a nev; wavelength requires substantial shifts in position for all optical

components, the apparatus must be recalibrated to optimize its use.

In addition to the list of general conditions defined above, we must prescribe the

mirrors. Selecting a mirror depends on what can be purchased. Each may have either an elliptical or a spherical surface. In the simulation process in section 4 we ignore

the deviations from the two-curvature parabolic symmetry and pretend that the real

optics behave similarly to these parabolas, lie presume in the alignment process

defined below that the deviations from ideal symmetry cause small changes in the

placement of each mirror. These changes are less than the travel range of the

micropositioning equipment holding each optical component. In the assembly of a particular apparatus, we use the results from the computer code to identify

sensitivities indicated by the deviations from ideal. The positioning devices are tied

to a stable reference table so that they bracket the conditions that make the real

apparatus behave as ideal as possible. In brief, each mirror position must be

insensitive to distortions by curvature errors and surface defects in the mirrors. Of

course, these effects are never zero; therefore, they affect the ultimate accuracy and

dynamic range of the apparatus. To establish precision and dynamic range, v/e made

simulations at two carrier wavelengths which contain the best optics using elliptical

mirrors with two curvatures. For contrast, we simulate the two wavelengths using

spherical mirrors with a single curvature. These simulations identify possible errors during the assembly and design of the apparatus.

Ue discuss beam properties and their ease of measurement before we define the

adjustment process. We can measure the positions for the center of irradiance of each

beam relative to the optical axis. In addition, we can measure the degree to which the

irradiance of each beam deviates from a cylindrical symmetry. These measurements are the practical data available during adjustments of the mirrors and other parts of the

apparatus. If two beams overlap at a particular surface, we can then scan the

resulting irradiance and interference pattern to establish the relative curvature of

the phase front for two beams assuming their separate irradiance patterns are sufficiently similar. Alignment with more than two beams could be used if the resulting

complex irradiance patterns at each surface have the required intuitive character. It is much easier in both the computer simulations and in the actual apparatus to use only

one- or perhaps two-beam structures to adjust all the mirrors to their required

positions. Measurement for the center of irradiance is easily done with a beam-scanning technique. Ue measure the deviation from cylindrical symmetry by using a spinning slit in a mask before an appropriate detector. The slit is rectangular with

100 pm and 1 pm edges. These dimensions are arbitrary but consistent with the design

conditions discussed in sections 4 and 5. To make the cylindrical check, the mask

27
spins parallel to the optical axis. The unit is moved perpendicular to the optical

axis to establish three numbers, namely the x, y, and z coordinates of the maximum dc

signal with a minimum time-varying signal [10].

By selecting individual beams near the optical axis and on the edge for the field

of view in the apparatus, we develop at each surface the true structure for the image

or Fourier transform relative to their ideal. The adjustment sequence in either the

simulation or the real apparatus can use a minimum least squares fit relative to the

ideal. The proper number of necessary test beams depends on the form of the final

apparatus. In our test of a proposed apparatus, we use six beams. During construction

of the final apparatus, I would expect a fairly complete map. Once that simulation has

been done, I would expect the six beams used in this simulation to be adequate for

final adjustments.

We now define the position adjustments for a mirror when it is used either for

imaging or for making a Fourier transform.

When a fixed-curvature mirror creates an image, a critical positioning of this

mirror can minimize the deviations from ideal for all beams at the image plane. This

action can fix the position of the mirror. If we allow x-axis curvature to be adjusted

in addition to the position adjustment for the image, then we can also get a different

position for the mirror. It is also unique. This latter adjustment will reduce the

size of the deviation for all beams from the ideal more than the case where the

curvatures are fixed and the mirror is simply moved.

In like manner, when the mirror acts to generate the Fourier transform of a beam,

its position coordinates can be fixed by some least square criteria. If possible, we

adjust the curvature for the x axis of the mirror to improve the cylindrical symmetry

for the Fourier transform of each beam.

These adjustments do not prescribe uniquely the Fourier transform position. To fix the position coordinate in front of the mirror that makes the Fouier transform

requires- a confirmation that the phase front of each beam at the Fourier plane is flat.

Two tests of this flatness are possible for the real apparatus. One method uses two

beams of equal irradiance to intersect in the Fourier plane and to show a modulation of

dark and straight lines. The second method uses the six beams individually and locates

where the beam size of each has a minimum for both the x and y coordinates. For these

methods, we presume the beams have peak irradiance at the center of each beam. In contrast, the computer simulation allows us to adjust the phase front for a single beam

directly because the mathematics contains those phase details.

One adjustment to these mirrors has been ignored in the computer simulations of

this paper. All rigid bodies have five degrees of freedom which must be adjusted

relative to another rigid body. Four degrees are fixed by the constraints on the

optical axis and by the conditions for proper imaging or Fourier transforms. There

remains the rotation of each mirror so that its two radii of curvature are consistent

with the general x and y coordinates of the apparatus. If a mirror is cylindrically

symmetric, this rotation has no effect. If there is any nonsymmetry, then the rotation

affects the details of each laser beam as well as the location for the image or Fourier

transform plane as defined by that mirror. Therefore, during the adjustment of these

mirrors, it will be necessary to rotate each to test and confirm that the ideal and

actual curvatures are as intended.

28
Given the above discussion, we define our alignnent sequence as follows: (1) Set

each mirror to an accuracy of 1 mn on the basis of the simulation results using the

design parameters for each mirror. (2) Adjust each mirror in sequence using the six test beams and the spinning slit-detector unit to fine-tune the position of each

mirror. These two procedures should get each mirror within 100 ym of the ideal position

and within 1 mrad in the angle adjustments. The final adjustments would employ the

output from the array of detectors and would move slightly each unit in the apparatus

so that the measurements of the phase and irradiance are modified to get best

accuracy.

This paper does not specify the final adjustments. They are unique to a given

apparatus. For example, one mirror may be most sensitive in its final positioning

because it has peculiar surface properties. Further, the position of the hologram at surface 7 will affect significantly the output for phase measurements. Both

ambiguities mean the final adjustment must use a walk-in process where each available

parameter is moved one at a time. We would expect the most sensitive parameters to be

moved after the least sensitive ones are positioned for their best conditions. In these final adjustments, some minimum-maximum criteria must be uniformly applied.

Examples of the criteria are: (1) a maximum dynamic range for each detector in the

array; (2) a maximum range of allowed wavelengths for a given dynamic range; or (3) a minimum distortion by apparatus on the phase fronts of a plane wave.

All the general discussion on the alignment process is finally complete;

therefore, we pass to the specific adjustment conditions for each mirror in

figure 3.1.

The mirrors at surfaces 4, 8, and 10 each generate a periodic array of spots of

minimum size, similar in size and relative position to those in the Hartmann plate at surfaces 5, 9, and 11, respectively. The scale size for these arrays of spots can be

different for each laser wavelength. The adjustments for the ideal imaging of these

spots will minimize barrel distortion in these image planes.

Mirrors 2 and 6 generate Fourier transforms of the original spot patterns at surfaces 3 and 7, respectively. Here we have one spot centered around the optical

axis. The sequential illumination of each aperture generates the required test beams

from the Hartmann plate. A stationary center of irradiance of the spot at the Fourier

plane exists when these two mirrors are properly adjusted. To verify that the phase

front of each spot in this Fourier plane is flat, we either inspect the resulting

interference pattern produced by two beams simultaneously intersecting at the Fourier

plane or check that the spot size in each transverse coordinate for a single beam is a minimum at the Fourier plane. In the computer simulation, the test is simple: we loo^

at the phase front for each beam. In the actual apparatus, the test is more difficult:

we must make a numerous measurements using the spinning slit-detector unit and a cw laser source of the desired wavelength. The signal out of the detector allows

deduction of the optimum adjustments. This paper does not detail what should be done;

reference [10] develops the necessary concepts.

3.5.2 What Happens When the Initial Beam Is a Spot of Uniform Irradiance

Because the actual apparatus cannot realize Gaussian apertures in the Hartmann

plate, we note the change in the alignment process for real beams. All the discussion

in subsection 3.5.1 applies except that the reader must know these beams are not simple

structures. Nevertheless, the Airy patterns at the Fourier transform planes and spots

at the imaging planes do not change the basic alignment procedures at either plane.

k

29
The signal v^aveforn from the spinning slit-detector unit will change; therefore, this

output nust be properly understood to allov/ proper adjustnent of the apparatus.

4. THE COMPUTER PROGRAM

Mornally, computer programs are relegated to appendices. We include one because

this program provides a detailed understanding of the simulations in sections 5 and 6. The summary of these simulations appears in section 7. In the next three subsections

of section 4 we explain the structure of the computer program and define the input and

the output variables so the program can be used. In subsection 4.1 we show the logic

flow and define the input parameters. In subsection 4.2 we list the appropriate blocks

either for function or for logical flow. The identification also ties the listing to

those equations in section 3 and in particular to those in subsections 3.1, 3.2, and

3.3. In subsection 4.3, we describe the output from a sample run so that the data from

such output can be understood.

4.1 The Logic Flow of the Computer Program

Figure 4.1 shows, in three pages, the structure for the possible flows in logic

within the program BEAfl. This figure has four columns of descriptions. The first

column shows the entry statement number for a block of statements. The second column

defines the function of the block and how the block is correlated to other blocks in

the program. The third column gives various temporary jumps to subroutines and other

such blocks. Finally, the fourth column defines the range of statments in the listing

containing the function block. To understand completely the potential logic flows

requires four items: the actual computer system; the definitions for the control

parameters as shown in figure 4.3; the descriptions for the output statments on the

operator console as shown in figure 4.2; and finally, a detailed listing of the

program, BEAM, as shown in figure 4.4.

Because there are numerous possibilities for operation of this program, it is

impossible to give a simple sequence showing this program, nevertheless, the program

can be executed after appropriate study of these four figures which collectively

provide the needed explanations. The discussion in this subsection only indicates the

basic possibilities. This program has a general design purpose; therefore, the

decisions for a computing sequence are numerous.

BEAM traces a beam with a Gaussian profile from surface 1 to 11. Each trace

corresponds uniquely to a chosen aperture in the prefilter. This trace remains a single beam until it reaches the hologram at surface 7. The tracing from surface 1 to

7 is called stage I. At surface 7, the operator selects one of the diffraction orders

so a single beam trace can continue. This continued trace between surface 7 to 11 defines stage II in the program.

Stages I and II represent two logic flows in BEAM. First, program use centers on stage I for beam traces up to surface 7. Second, assuming fixed values from stage I results, the program now centers on stage II with a series of beam traces after

surface 7.

There are numerous uses for the program such as determining the proper control

parameters in a given apparatus before purchase of expensive equipment. Sections 5 and

6 show such examples. A second use of BEAM would test the sensitivity for adjustment

of these parameters at all surfaces during a beam trace. These results permit choice

of adjustment strategies and estimates for the allowed dynamic ranges of beam profile

parameters.

30
The progran, v/ritten in BASIC, uses the input and output codes appropriate to an Interdata computer system. It is necessary to use 14 significant figures to prevent

serious round-off errors in the simulations. The program assumes there are six modules

for real-time use--the CPU with 16- or 32-bit words, about 40 kilobytes of memory, a disk for file storage, a printer (PR), an operator terminal with cathode ray tube (CRT)

for read out, a keyboard for input, and finally the appropriate software for the

operating system.

During a beam trace, the operator would use the information in figure 4.2 to choose appropriate control parameters. Mo further discussion of this figure should be

necessary.

Also during this same beam trace, the operator would use the explanations from

figure 4.3. Here some comments should expedite its use. The number on the left is the

statement number that put the quoted statement on the CRT. The following explanation

defines what it means and what the operator should do, if anything. If questions

remain, the operator must use figure 4.4 and puzzle out the answer. I have tried to construct a program that works correctly; however, the many options can cause inconsistencies in status. If the reader gets such a case, the best course is to start with RUN and reset the program.

There remains one logic flow that requires further discussion, namely how the

least-squares-fit sequence works. To execute it properly, the operator repeats the

beam trace one more time than the number of varied parameters. For example, an adjustment sequence at a given surface requires the four values defined in figure 4.3

to be zero for surface 3 and three adjustment parameterS'-b^ , b2, and do. Each

parameter changes one at a time to generate the necessary data. In this example, four

beam traces result. The first trace uses the base parameters--bi , b2, and d2.

The second changes b^ to b]^ ' . The third returns b]^ ' to bj and changes b2 to

b2'. Finally, the last trace in this LSQ sequence returns b2' to b2 and changes

d2 to d2'. Each change is about 1 percent or less of the original parameter. The

LSQ part of the program then generates a new set of b^ , b2, and d2. These latter

values should produce a better working apparatus. The LSQ process is converged when

all differences between the old and new sets of parameters are less than 1 ym for a given LSQ. The simulations in sections 5 and 6 tabulate such results for the surface

of interest and for the chosen beam.

Figure 4.2 refers to an equation defining the initial parameters of the traced

beam as it exits from the prefilter in surface 1. We give that equation here.

31
riAiri PROGRAM LOniC FLOW

Entry Function Jump to Outside Block Range

RUM To run progran BEAM --- 10-34

(Initializes progran arrays)

Flow to 36

36 Control data defined 840 36-36

Flow to 38

38 Read and change control data

Options: blocks--134

individual--162 134 or

set initial conditions and 162 or 38-58

flow to 60 194

60 Propagates Gaussian from surface

L = 1 to L = 7 390 60-64

Flow to 66

66 Set up LSQ fit controls 912 66-66

Flow to 68

68 Decision to print Stage I data

Options: IF YES, go to 566

If no, continue 566 68-74

Flow to 76

76 Decision to generate new Stage I data

Options: YES go to 40

no continue

Calls 796 for S7 and

T7 values

Flow to 94

40 or

796

76-94

94 Selected exit beam from surface 7 (Propagates Gaussian beam from

surface L = 7 to L = 11)

Flow to 114

390

662

94-112

114 Decision to generate new beam from

surface 7 Options: IF YES, go to 94

IF no, continue

Flow to 120

94 114-118

120 Decision to generate new beam from

surface 1 Options: IF YES, go to 40

IF no, continue

40 120-124

126 Preparing main program for exit

Flow to 130 (STOP) 830 126-130

Figure 4.1. Logic flow of BEAM (page 1).

32

^
VARIOUS SUBFUnCTION LOGIC BLOCKS

Entry Function Jump to Outside Block Range

840 Read control data from 840-362

floppy disk (Subroutine)

Options are: Do it or return

8C4 Print control data to floppy 134 854-886

disk (Subroutine)

Options: Do it or return

134 Print to CRT block of control 132-152

data (Subroutine)

Options are: Range of each

block and nunber of such

blocks

155 Prints to CRT one control value

at a time (Subroutine)

Options are: To select a value

or to quit

Flow to 162

162 Decide to get new control data

IF YES, go to 156 155 152-165

IF no, continue

Flow to 168

158 Display range of control data 158-192

Options: Select new ranges

or return

194 Set up initial data using control 390 194-384

data for each surface (Subroutine)

390 Propagating the Gaussian beam from 514 386-508

surface L to L + 1 (Subroutine) (many times)

514 Computes form: AX*AY/Y*AZ

Given input Jl, J3, J5, J7 510-562

(Subroutine)

564 Decision tree

3 options: bulk print —612 578 or individual print--612 612 or 564-576

skips everythi ng--652 652

578 Print on printer (PR) initial

data in proper form 524 578-610

Figure 4.1. Logic flow of BEAM (page 2).

33
ADDITIONAL SUBFUMCTION LOGIC BLOCKS

Entry Function Junp to Outside Block Range

612 Print individual control — 612-622

values

Options: Select value or flow to 624

624 Selects surfaces to be printed 624-646

Flow to 648

648 Print initial data for -— 648-660

surface L = 1 Options: Do it or skip it

Flow to 662

662 Complicated sequence

Options are: Print on CRT or PR and print out selected 888 662-790

surfaces as queried, or print

according to control data

796 Computes expected S7 , T7 to compare — 792-828

with actual S7, T7 (Subroutine)

830 Starts exiting from program and commences the filing of data in 864 830-838

storage (Subroutine)

840 Read in control data from floppy

disk (Subroutine)

Options: Do it or skip it --- 840-862

864 Print control data to floppy disk

(Subroutine) -— 864-886

Options: Do it or skip it

888 Store control data for least squares

fit (LSO) process (Subroutine) 682 or

Options: Do it--returns 966 888-964

Skip it--returns to 632

data for LSQ fit —goes to 966

966 LSO fit analysis 12 966-1082

Figure 4.1. Logic flow of BEAM (page 3).

34
1) = X in jjm: wavelength of laser beam

[2) = 9 in rad: angle cf reflection for all elenents of apparatus

[3) = LA: hole coordinate for real X (an integer) [see eq (4.1)]

[4) = LB: hole corrdinate for real Y (an integer) [see eq (4.1)]

[5) = SI in mm: the X distance between holes in prefilter [see eq (4.1)]

'6) = Tl in mm: the Y distance between holes in prefilter [see eq (4.1)]

'7) = LC: hole coordinate for imaginary X (an integer) [see eq (4.1)]

(8) = LD: hole coordinate for imaginary Y (an integer) [see eq (4.1)]

I = S7 in l/(mm): the spatial frequency for the X coordinate of the hologram

[see eq (3.41)]

[10) = T7 in l/(mm): the spatial frequency for the Y coordinate of the hologram

[see eq (3.41)]

111) = d]^ in m: curvature of X coordinate for surface 1 (positive)

112) = d2 in m: curvature for surface 2 (negative)

|13) = d3 (positive)

[14) = d4 (positive)

[15) = d5 (positive)

[16) = d5 (negative) [see eqs (3.14)]

[17) = dy (positive)

[18) = dg (negative)

[19) = dg (positive)

[20) = dio (positive) X(21) = d^l (positive)

[21) = dxi (positive)

[22) = ei in m: curvature of Y coordinate for surface 1 (positive)

[23) = 62 (negative)

[24) = 63 (positive)

[25) = 64 (positive)

[26) = 85 (positive)

[27) = 95 (negative) [see eqs (3.14)]

[28) = ey (positive)

[29) = eg (negative)

[30) = eg (positive)

[31) = eio (positive)

[32) = e^i (positive)

[33) = pi in mm: aperture for surface 1 [see eq (3.13)]

[34) = P2

[35) = P3

[36) = P4

[37) = P5

[38) = P6

[39) = P7

Figure 4.2. Control parameters for BEAM (page 1).

35
X(40 ) = P8 X(41 = P9 X(42 = PIO X(43 I = Pll

X(44 = bi i X(45 > = b2

X(46 = b3

X(47 ) = b4

X(48 ) = b5

X(49 ) = bg

X(50 = by

X(51 = bg

X(52 = bg

X(53 = bio

X(54 = -30:

X(55 = Wx i X(56 = Wy i X(57 = Rx i X(58 = Ry i X(59 = A in

x(6o; = $ in x(6i; : prin

X(62;

X(63]

X(64;

X(65;

X(66]

X(67;

X(68]

X(69

x(7o;

X(71)

n m: distance for surface 1 and 2 [see figure 3.1)]

this parameter has no meaning; it is present to satisfy array

constraints

n mm: the width of the X coordinate [see eq (4.1)]

n mm: the width of the Y coordinate [see eq (4.1)]

n m: the curvature of the beam in X [see eq (4.1)]

n m: the curvature of the beam in Y [see eq (4.1)]

(H/m2)l/2: amplitude of beam [see eq (4.1)]

rad: phase of beam [see eq (4.1)]

t control for surface 1 (if 1 print; if no print)

234

56

78

910

11 Beam deflection control below.

These are changed during the computer runs at Stage II.

X(72) = m = LI: the order of X coordinate beam [see eq (3.43)]

X(73) = n = L2: the order of Y coordinate beam [see eq (3.43)]

X(74) = real part of G(m,n) [see eq (3.44)]

X(75) = imaginary part of G(m,n) [see eq (3.44)

Remaining control variables are generated by operator during a simulation run,

Figure 4.2. Control parameters for BEAM (page 2).

36
40 "BLOCK DATA YES = 1 or flO = 0" Here operator decides if a range of control data is to be typed by operator

at teletype.

Use 1 i f yes, i f no.

56 "ALL DATA STAGE I INITIALIZED"

58 "now GEriERATE SURFACES: C(,L) WHERE L = 1 to 7" Hero program informs operator of the status of the program. Purpose is to

detect program traps and give flov/ information to operator. Stage I generates results for surfaces 1 to 7.

rio response needed.

68 "DEAL WITH STAGE I DATA FOR PRINT?, YES = 1, MO = 0" Here operator decides if Stage I data should be printed at all. When using LSQ

features of program and a surface adjustment has L less than 8, operator must say

yes.

Use 1 if yes, if no.

76 "riElJ VALUE STAGE I YES = 1, NO = 0" Here operator decides if new control values are needed to compute stage I results vjithout going to stage II. Use 1 if yes, if no.

90 "BEGIN STAGE II" Program informs operator it is in stage II, namely generating the details of

surfaces 8 to 11.

94 "LI, L2, AND G(L1, L2) RE AND IMAC PAIR VALUES ARE"

LI is order on surface 7 in X coordinate; L2 is order on surface 7 in Y coordinate. G{L1,L2) is the complex number in eq (3.44). Type LI, then L2,

then real part of G(L1,L2), and finally imaginary part of G(L1,L2).

114 "NEW VALUES OF LI, L2, G(L1, L2) RE AND IMAG YES= 1, NO = 0" Here operator decides if new stage II beam trace is wanted.

Use 1 if yes, if no.

126 "EXITING FROM PROGRAM"

Status information on logic flow.

No response needed.

134 "BLOCK DATA GROUP GIVE LOU 12, HIGH 13 RANGE, FIRST"

To read from teletype the block of control data, operator must give

inclusive range of that data.

Give the low and then the high.

140 "HERE I = ", 14, "X(I) =

" To aid in typing the block of control data, each subscript is noted. Type

the appropriate X(I).

146 "A NEll INITIAL DATA BLOCK YES = 1, NO = 0" Operator has chance to transmit new block of control data.

Use 1 if yes, if no.

Figure 4.3. Statements at CRT generated by BEAM (page 1).

37
156 "INITIAL X(I) DATA, GIVE I FIRST, THEfl X" Operator can give a single control value.

First give subscript, then the value for the subscript.

162 "flElf X DATA YES = 1, r!0 = 0" Operator has chance to transnit a single value of control data,

lise 1 if yes, if no.

168 "DISPLAY RAriOE OF X VALUES, 14 LOWER, 15 UPPER, 14 = 15 = SKIPS"

Operator selects inclusive range of control values for display on CRT.

Select 14 first, then 15. If operator wishes to skip display type for 14 and 15.

176 SHOWS I, X(I)

Operator can decide if control value is correct or not.

If correct, type 0; if not, type -1, and then type correct value.

186 "OOriE WAflT TO DISPLAY flEW X RANOE YES = 1, riO = 0" Operator can choose to display new range of control data.

Use 1 if yes, if no.

202 "OEriERATinG lUITIAL DATA"

Program informs operator where it is.

f!o response needed.

566 "BULK irilTIAL DATA PRT = 1, NO = 0, SKIP TO PRINT CONTROL = -1"

Operator has three options: first, generate the details of the initial control

data on the printer; second, to select individual control values for print on CRT;

finally, skip control data, print details, and continue automatic control of

printing for each surface.

First option, use 1; second, use 0; third, use -1.

612 "DESIRED X(L) VALUE, L =

" Operator selects the control value to be printed on CRT.

Choose L.

618 "NEW X VALUE YES = 1, NO = 0" Operator has option to print new control values.

Use 1 if yes, if no.

624 "PRINT OUTPUT CONTROL"

Here the control values for printing data on each surface are shown.

No response needed here.

634 "A 1 r^EANS PRINT, MEANS NO PRINT IN THIS CONTROL"

636 "I = SURFACE TO BE CHANGED MEANS NO CHANCE"

Operator decides which surfaces should print automatically to CRT or PR. If some surfaces to be printed show incorrect control, then that surface

control must be changed.

Type surface number if change is desired. Then type new control value

either at or 1. Type is no change is desired.

Figure 4.3. Statements at CRT generated by BEAM (page 2).

38
648 "irilTIAL DATA L = 1 SURFACE PRIflT YES = 1, flO = 0" Operator has option to print on PR initial conditions of the bean at surface

1 beyond that already recorded in statements 578-608.

Use 1 if wanted, if not.

662 "COriT = 1, SIflCLE SURFACE = 0: PRITIT 01! 1 = CRT, 3 = PR" Operator has option to use or not use the control data to select surfaces to

be printed on either the PR or CRT.

If control is used, type 1; if not, type 0; to select output nediun, type 1 or 3.

804 "SiriPLE THEORY EXPECT S7 = ", S7 , "ACTUAL = ", X(9)

806 "CHAflGE ACTUAL VALUE YES - 1, FIO = 0" Operator looks at expected S7 compared with the control S7

,

If value to be changed, type 1; otherwise, type 0.

812 "S7 = ?" Asks what should S7 be.

Type it in.

816 SIMPLE THEORY EXPECT T7 = ", T7 , "Actual =", X(10)

818 "CHAriGE ACTUAL VALUE YES = 1, flO = Operator looks at expected T7 compared with the control T7.

If value to be changed type 1; othen-zise, type 0.

824 "T7 = ?" Asks what should T7 be.

Type it in.

830 "DOriE WITH PROGRAM AND riOL' EXITIflC"

Program telling operator whore it is. No response needed.

840 "READ IN FROM FILE X(l-75) DATA"

842 "YES = 1, NO = 0" Operator has option to read control data from a disk.

Use 1 if yes, if no.

864 "PRINT TO FILE X(l-75) DATA"

866 YES = 1, NO = 0"

Operator has option to print on disk the current control data for subsequent

use in a new simulation session.

Type 1 if yes, if no.

888 "STORE LSQFIT DATA, 1 = YES, = NO"

Operator has option to store desired data from a selected surface to deduce

the proper values of control parameters. Both control parameters and test values are saved in LSQ file.

If you want to save data, type 1; if not, type 0.

Figure 4.3. Statements at CRT generated by BEAM (page 3).

39
912" "RESET AND OPEN LSQ FILE 1 = YES, 0' = NO" Operator has option to clear the LSQ file for a least squares fit sequence.

If the sequence is to be started, type 1; otherwise, type O.

924 "NUMBERS OF PARAMETERS = ?" Operator supplies number of control parameters to be varied in the least

square fit sequence.

Type a number less than four. Up to three can be varied.

928 "LOCATION OF PARAMETERS"

930' "FIRST L3 =" Give subscript of X(L3), the first parameter of variation.

942 SECOND L4 =" Give subscript of X(L4), the second parameter of variation.

952 "THIRD L5 =" Give subscript of X(L5), the third parameter of variation.

974 "NUMBER OF VALUES TO BE DRIVEN TO REFERENCE"

A given surface has up to four values that are required to be fixed so a surface

can have the desired form. The first value is XI, the center of irradiance in the

X coordinate. The second value is Yl, the center of irradiance in the Y coordinate. The third value for all surfaces except 11 is proportional to the

inverse of the curvature in the X coordiante. If this term and the following are

zero, then plane waves are present. The fourth value for all surfaces except 11

is proportional to the inverse of the curvature in the Y coordinate. Surface 11 has its third value as the ratio of the beam width in the X and Y irradiances,

namely, WX/WY. Surface 11 's fourth value is proportional to the inverse of the

curvature for the X coordinate. Normal least squares fit sequence uses the first

three values for surface 11 and all four values for the remaining surfaces.

982 "REFERENCES VECTOR C(L2) ="

986 "I =", I, "C(I) = ?" Here operator supplies the desired XI, Yl, inverse curvature in X, and

inverse curvature in Y values, if not, surface 11; otherwise, the set is XI,

Yl, WX/WY, etc. If less than four values are requested, then the desired

terms are the same order up to the number of requested values. For example,

two values would only give XI and Yl, respectively. Note: the order and

types of values are fixed. Only the number can be varied.

1070. "I, OLD, NEW, INCREMENT RESPECTIVELY"

Here the least squares fit sequence tabulates the subscript for the

parameter, the extrapolated value of the parameter, and finally the change

between the old and new state of each parameter. In an adjustment sequence,

convergence of this LSQ process occurs when the increments are less than

adjustment accuracy of the apparatus, namely about 1 \im.

Figure 4.3. Statement at CRT generated by BEAM.

40
10 DIM AS(60) ,B3(60)

12 DIH A(2G0) ,B(200) ,C(2G0) ,D(20O) .E(200) ,F(200) ,G(2O0) ,X(200)

14 DIM Y(200) ,S(20)

16 EtH NOTE THE VERSION miNT DATE

18 riEn "BEAM" IS WliEIlE THIS PROGRAM IS STORED"

20 CLOSE 1 22 CLOSE 3 24 OPEN "I'R: ",3.1 26 OPEN "LO: ", 1,

1

20 DIM A(4, 11) ,B(5, 11) ,CC 10, 11) ,D( 10, 10) 30 DIM E( 12, 10) ,F( 10) ,X(75) 32 MAT C=(0)

34 CS= "BEAM TRACE **;|:*:c^:*;i:** 36 GOSUB 840

38 lUCM MAIN PROGRAM FOR LOGIC FLOW. 40 PRINT "BLOCK DATA YES= 1 OR K0=0"

42 INPUT II 44 IF 11=0 GOTO 50

46 GOSUB 134 40 REM ABOVE HAS INPUT 1 DATA BLOCK FORM

50 GOSUB 162 52 I'EM ABOVE HAS INPUT DATA ONE AT AT I ME. 54 GOSUB 194 56 PRINT "ALL DATA STAGE I INITIALIZED"

50 PRINT "NOW GENERATE SURFACES: C( , L) TvTffiRE L= 1 TO 7" 00 FOR L= 1 TO 6 62 GOSUB 390

64 NEXT L 66 GOSUB 9 12 68 PRINT "DEAL WITH STAGE I DATA FOR PRINT?, YES= 1 , NO=0"

70 INPUT II 72 IF 11=0 GOTO 76

74 GOSUB 566

76 PRINT "NEW VALUES STAGE I YES= 1 , N0=0"

78 INPUT II 80 IF 11=1 GOTO 40

82 P8=D( 1,7) 84 T8=D(4,7)

86 P9=D(2,7)

88 T9=D(6,7)

90 PRINT "BEGIN STAGE II" 92 GOSUB 796

94 PRINT "L1,L2, AND G(L1,L2)RE AflD INAG PAIR VALUES ABE"

96 INPUT XC72) ,X(73) ,X(74) .X(75)

98 D(4,7)= r8+X(72)*X(9)

100 D( 1,7)=X(74)+P8

102 D(6,7)=T9+X<73)*X( 10) 104 D(2,7)=X(75)+P9

106 FOR L=7 TO 10 108 GOSUB 390

1 10 NEXT L

1 12 GOSUB 662

114 PRINT "NEW VALUES OF LI , L2, G( LI ,L2) RE AND IMAG YES=1,RO=0"

116 INPUT 11

J 1 IF 11=1 GOTO 94

120 PrTnT 'NEW INITIAL DATA YES= 1 , NO=0"

VZ2. U4PUT II 124 IF 11=1 GOTO 40

126 PRINT "EXITING FROM PROGRAM"

123 GOSUB 830

130 STOP

132 REM THIS SUBROUTINE INPUTS THE INTITIAL DATA X( 1 ) TO X(75)

134 PRINT "BLOCK DATA GROUP GIVE LOW 12, HIGH 13 RAHGE FIRST"

136 INPUT 12, 13 138 FOR 14=12 TO 13 140 PRINT "HERE 1= " , 14, "X(I) =

142 INPUT X( 14) 144 NEXT 14 146 PRINT "A HEW INITIAL DATA BLOCK YES= 1 , NO=0"

148 INPUT 15 150 IF 15=1 GOTO 134

152 RETURN

154 HEM THIS SUBROUTINE INPUTS ONE INITIAL VALUE AT ATIME.

156 PRINT "INITIAL X( I ) DATA, GIVE I FIRST, THEN X " 158 INPUT 12 160 INPUT X( 12) 162 PRINT "NEW X DATA YES= 1 , NO=0"

164 INPUT 15 166 IF 15=1 GOTO 156 168 PRINT "DISPLAY RANGE OF X VALUES, 14 LOWER, 15 UPPER, 14=15=0 SKIPS"

170 INPUT 14, 15 172 IF 14=0 GOTO 192

174 FOR 1= 14 TO 15 176 PRINT "1=", I, "X( I)= ",X( I) , "TYPE TO COW, -1 TO GET NEW VALUE"

178 INPUT 18 180 IF 18=0 GOTO 184

102 INPUT X( I) 184 NEXT I 186 PRINT "DONE WANT TO DISPLAY NEW X RANGE YES= 1 , NO=0"

188 INPUT IB 190 IF 18=1 GOTO 168

192 IlETUnn 194 C1=C0S(X(2))

Figure 4.4. List of computer program BEAM (page 1).

41
196 S1 = SIN(X(2))

198 1.7= 1 200 L0= 1 I 202 PIUNT "GENERATING INITIAL DATA" 204 REM ALL DATA INITIALIZED HERE EXCEPT L1.L2. 206 REM ALSO INCLUDE G(L1,L2) IN EXCEPTION. 208 Pl=3. 14159 210 K=P1>r20OO/X( 1) 212 FOR 1=1 TO 11 214 11=1+10

216 12=1+21 218 13=1+32

220 14= 1 + 43 222 A( 1 , I ) = ( KXX( ID) /2O00

224 A(2. I)=(K/X( 12) )/2000

226 A(3, I)=-l/X( I3)-^2 228 A(4, I)=X( I4)*IOO0

230 NEXT I 232 REM ALL UNITS IN PROGRAM IN MM SCALE. 234 REM INITIAL PROFILE INPUT GENERATED HERE. 236 C(7, l)=-l/X(55)'-2

238 C(9, l)=-l/X(56)'-2

240 C(8, 1)=K/X(57)/'2OO0 242 C( 10. l)=KyX(5O)/20eO

244 C( 3 , 1 ) =-2:kC( 7,1) s-X( 3) :f-X( 5) 246 C(5, 1)=-2*C(9, I ) s:X( 4) *X( 6) 248 C(4, 1)=-2W:C(8, l)*X(7)*X(o)

250 C(6, 1)=-2*C( 10, 1)*X(8)*X(6)

252 C( 1 , 1)=L0G(X(59) )+C(7, 1 ) *( X( 3) >t:X( 5) ) '~2+C( 9 , 1 ) *( X( 4) *X( 6) )'"2 254 C(2, 1)=X(60)+G(8, 1 ) *( X( 7) :i:X( 5) )-"2+C( 10, 1 ) =«( X( 8) *X( 6) ) ^"2 256 REM FIX THE B(1-5,L) VALUES FOR CONTROL

258 FOR 11=1 TO 11 260 FOR 12=1 TO 5 262 B( 12, 1 1)=

1

264 NEXr 12 266 NEXT II 268 B( 1 ,2)=0

270 B( 1,4)=0 272 B( 1 ,6)=-l 274 B( 1 ,8)=0

276 B( 1 , 10) =0 278 B(2, 1)=0 280 B(2,3)=0

282 B(2,5)=0

284 B(2,6)=-l

286 B(2.9)=0

288 B(3,2)=0

290 B(3,4)=0

292 B(3, 1O)=0

294 U(3,8)=0

296 B(4, 1)=0

298 B( 4,3)=0

300 B(4,5)=0

302 B(4,9)=0

304 B( 1,3)=-1 306 B( 1 ,9)=-l

308 B(2,2)=-l

310 B(2,8)=-l

312 FOR I 1 = 2 TO 10 314 B(5, I1)=C1 316 NEXT II 318 B(5, 1)=-1

320 B(5,3)=-l

322 B(5,5)=-l

324 B(5,9)=-l

326 REM NOW X'fE FIX THE D AND E SET OF INITIAL DATA

328 FOR L= 1 TO 10 330 TI = I</A(4,L)^.5

332 T2=B(5,L)*T1/'P1

334 IF T2<0 GOTO 342

336 DC 1 ,L)=L0G(T2)

338 D(2,L)=Pl/'2

340 GOTO 346 342 D( 1,L)=L0G(-T2)

344 D(2,L)=-Pl/2

346 L9=L+1

348 FOR 12=3 TO 10 350 D( I2,L)=0

352 E( I2,L)=0

354 NEXT 12 356 E( 1 1 ,L)=0

358 E( I ,L)=0

360 D(7,L)=A(3,L)

362 D(9,L)=A(3,L)

364 RJ'M irmciNARY SET BELOW REAL SET ABOVE

366 D(4,L)=S1*IC*B(4,L)

368 E(2,L)=-S1*10:B(3,L)

370 D(a,L)=TI*B(5,L)''2-Cl*B(2,L)*A( 1,L)

372 E(6,L)=T1*B(5,L9)^2+C1*B( 1,L)*A( 1,L9)

374 D( 10,L)=T1-C1:;:B(2,L)*A(2,L) 376 E(8,L)=T1+C1*B{ 1,L)*A(2,L9)

378 E( 10,L)=-2«ABS(B(5.L)3:B(5,L9))*T1

380 E( 12,L)=-2*T1

Figure 4.4. List of computer program BEAM (page 2).

42
382 NEXT L 384 RETURN

386 REM SUBROUTINE NEED L, C( L) , D( L) , E( L) SET

388 REM GENERATE C( , L+ 1 ) FROM D( . L) , E( L) , ADD C( ,L) 390 FOR 1= 1 TO 4 392 11=6+1 394 F( I)=C( I1,L)+D( Il.L) 396 NEXT I 398 R1 = F( 1)*F(3)-F(2)*F(4)

400 R2=F( 1)*F(4)+F(3)*F(2)

402 R3=SQR(R1^2+R2^2)

404 R4=R1/R3

406 R0 = R2/113 408 IF R4<0 GOTO 410 ELSE 416

410 R7=ATN(-R5/R4)

412 116 = P1-R7

414 GOTO 424

416 IF R4>0 GOTO 422

418 R6=(P1/2)«SGN(R5)

420 GOTO 424

422 R6 = ATff(R5/R4) 424 R8=L0G(P1/SQR(R3))

426 R9=-R6/2

428 J 1 = 3 430 J7=

1

432 J3=3

434 J5=7

436 GOSUB 514

438 Ra=RB+Q7

440 R9=R9+Qe

442 Jl=5 444 J3=5

446 J5=9

448 GOSUB 514

450 L1=L+1

452 C( 1 ,L1)=R8+Q7+C( 1,L)+D( 1,L)

454 CC2,L1)=R9+QS+C(2,L)+D(2,L)

456 J7=2

458 Jl=3

460 J3=9

462 J5=7

464 GOSUB 514

466 C(3,L1)=E( 1,L)+Q7*2

468 C(4,L1)=E(2,L)+Q8*2

470 .11 = 5 472 J3= 1

1

474 J5=9

476 GOSUB 514 478 C(5.L1)=E(3,L)+Q7*2

480 C(6,L1)=E(4,L)+Q8*2

482 Jl=9

484 J7=3

486 J3=9 488 J5=7

490 GOSUB 514 492 C(7,L1)=E(5.L)+Q7

494 G(U,L1)=E(6,L)+Q8

496 Jl= 1 1 498 J3=ll 500 J5=9

502 GOSUB 514

504 C(9,L1)=E(7,L)+Q7

506 C( 10,L1)=E<B,L)+Qa

508 RETURN

510 nF.H SUBROUTINE FOR -AX*AY/( 4*AZ) 512 REM NEED J1.J3,J5,J7

514 J2=J1+1

516 J4=J3+1

518 J6=J5+1

520 F(9)=C( J5,L)+D( J5,L)

522 F( 10)=C( J6,L)+D( J6,L)

524 IF J7> 1 GOTO 532

526 F(7)=C( J3,L)+D( J3,L)

528 F(8)=C( J4,L)+D( J4,L) 530 GOTO 536 532 F(7)=E(J3,L)

534 F(8)=E(J4,L)

536 IF J7>2 GOTO 544

538 F(5)=C( Jl ,L)+D( Jl.L) 540 F(6)=C( J2,L)+D( J2,L) 542 GOTO 548

544 F(5)=E(J1,L)

546 F(6)=E(J2,L)

548 Q1=F(5)*F(7)-F(6)«F(8)

550 Q2=F(5)*F(8)+F(6)*F(7)

552 Q3=F(9)-^2+F( 10)-"2 554 Q5=-F(9)/Q3

556 Q6 = F( 10)/a3

558 Q7=(Q1*Q5-Q2*Q6)*.25

560 08= (01*06+05*02)*. 25 562 RETURN

564 REM SELECTED INITIAL .INTERMEDIATE, AND FINAL DATA TO BE PRINTED.

566 PRINT "BULK INITIAL DATA PRT= 1 , HO=0. SKIPTO PRINT C0NTR0L=-1"

Figure 4.4. List of computer program BEAM (page 3).

43
568 L7=

1

570 INPUT n

572 L8=7

574 IF Il=(-1) GOTO 662

576 \V 11=0 GOTO 6 12 578 PRINT ON (3)"VEHSI0N 3/9/78"

580 PRINT ON (3) PjS , GETDATES( 0) , GE'ITIWESC 0) 582 PRINT ON ( 3) "WAVLGN= " ; X( 1 ) , "rilCROMETEKS "

, "AWG= " ; X(2) ; "RAD" 584 PRINT ON ( 3) "COS( ANG) =

" , CI 586 PRINT ON ( 3) "LA= " ; X( 3) , "LB= " ; X( 4) , "SI . Tl( ml) = " . X(5) , X( 6) 508 PRINT ON ( 3) "LC, LD= " ; X( 7) , X( 8) 590 AG="XCUR(N) YCUR( M) APER( M) RAD I US L TO L+ 1 DIS(M)"

592 PRINT ON (3) "< 10> " 594 PRINT ON (3)"L=",AS

596 FOR 1= 1 TO 11 598 13=1+10

600 14=1+21 602 15=1+32

604 16=1+43

606 PRINT ON (3) I , X( 13) . X( 14) , X( 15) , X( 16) 608 NF.Xr I 610 GOTO 624

612 PRINT "DESIRED X( L) VALUE, L=

614 INPUT 16

6 16 PRINT "X(L)= ",X( 16) 618 PRINT "NEW X VALUE YES =1. N0=0"

620 INPUT 16 622 IF 16=1 GOTO 6 12 624 PRINT "PRINT OUTPUT CONTROL"

626 FOR 1= 1 TO 11 628 16=60+1 630 PRINT "I=", I,X( 16) 632 NEXr I 634 PRINT " A 1 MEANS PRINT, MEANS NO PRINT IK THIS CONTROL"

636 PRINT "I=SURFACE TO BE CHANGED, MEANS NO CHANGE" 638 INPUT 17 640 IF 17=0 GOTO 648

642 16=17+60

644 INPUT X( 16) 646 GOTO 636 648 PRINT "INITIAL DATA L= 1 SURFACE PRINT YES=1,NO=0"

650 INPUT 17 652 IF 17=0 GOTO 662 654 PRINT ON ( 3) "\/X( MM) = "

j X( 55) , "WY( MM) = "; X(56) , "RJ« M) = "; X( 57) 656 PRINT ON ( 3) "AMPL= " ; X( 59) , "PHASE( RAD) =

" ; X( 60) . "RY( M) = " ; X( 58)

658 PRINT ON ( 3) "*'**"*«'''#*#*# END OF INITIAL DATA #########*##*"

660 PRINT ON (3) "< 10> " 662 PRINT "CONr= 1, SINGLE SURFACE=0 : PRINT ON 1=CRT,3=PR"

664 INPUT J5,J6

666 IF J5=0 GOTO 682

668 FOR J5=L7 TO L8

670 I7=J5+60

672 GOSUB 702 674 NEXr J5 676 L7=a

678 L8= 1

1

680 GOTO 698

682 PRINT "J5=0 SKIPS SURFACE"

684 PRINT "SURFACE TO BE PRINTED J5=, HERE J6= 1 FOR CRT, 3 FOR PR" 686 INPUT J5,J6

688 I7=J5+60

690 IF J5=0 GOTO 676

692 GOSUB 702

694 GOTO 888

696 GOTO 682

698 RETURN

700 REM OUTPUT PRINT CONTROL J5 SURFACE, J6 PRINT MEDIUM

702 IF X(I7)=0 GOTO 790

704 IF J5=ll GOTO 706 ELSE 722

706 PRINT ON (J6)"<10> ***«'**ORDER OF BEAM REFRACTED BY HOLOGRAM < 10> 708 PRINT ON ( J6) "L1,L2= ".X(72) ,X(73) , "RG(L1,L2)= ";X(74)

i

710 PRINT ON ( J6) "IMG(L1,L2) = ";X(75) , "S7,T7 ( l/MM) =

" ; X( 9) ,X( 10) 712 PRINT ON ( J6) "C( 1-10, 11)

714 FOR 1=1 TO 10 716 PRINT ON (J6) C(I,11),

718 NEXT I 720 PRINT ON ( J6) "*•<'*''"'#•<'#*<'##*»«'#**'«'«'«'*'*##< 10> " 722 Xa= . IE- 19 724 IF C(7,J5)<0 GOTO 730

726 PRINT "IMPROPER WX FOR L= " , J5 728 GOTO 126 730 W1=SQR(-1/C(7, J5))

732 IF C(9,J5)<0 GOTO 740

734 PRINT "IMPROPER \r^ FOR L=",.J5 736 PRINT "C(9,L)= ",C(9, J5)

738 GOTO 126 740 W2=SQR(-1/C(9, J5)) 742 T5=C(8,J5)

744 IF ABS(T5)>X8 GOTO 750

746 T5=T5+. 1E-29*SGN(T5)

748 T5=X8*SGN(T5)*K*.5

750 R1=K/T5*.5

752 T6=C( 10, J5)

Figure 4.4. List of computer program BEAM (page 4).

44
754 IF ADS(T6)>X0 GOTO 760

756 'r6 = T6+. 1K-29=.:SGN(T6)

7511 •16 = >;U:l-t?<;N( r6)WK;i:.5 760 Il2=]C/r6*.5 762 X1 = C(3,.)5)/C(7. J5):i;(-.5) 764 ><;i = -C(4. J5)/T5M;.5 ' 766 Y1=C(5, J5)/C(9,J5)«(-.5)

768 Ya=-C(6,,I5)/-T6*.5 77G r3=C(2, J5)-C(8,J5)*K3'"2-C( 10, J5)*Y3-"2 772 PO=C( 1, J5)+(X1/W1)'^2+(Y1/W2)'^2 774 r2=KXy(P0)

776 Q9=P2*P2*Pl*Wl*W2-i;( . 5) *ABS( B( 5 , J5) ) 778 PRINT ON ( J6) "SUKi'ACE PARAME'reilS PniNTED NEXT HAS L= " , J5 780 PRINT ON (J6)"ALL IN NN, WX, )A', RX, RY= " , Wl , W2, Rl , R2 702 PRINT ON (J6)"RATfO WX/^fi'= , 1/RX, I/RY PROP TO" , W1/W2 , C( 8, J5) , C( 10, J5) 784 PRINT ON (J6)"ALL IN PIfl, XI , X3, Yl , Y3= " , XI , X3 , Yl , Y3 786 PRINT ON ( J6) "MIPLrHJDE= " , P2, "PHASE( RAD) = "

, P3, "POh'ER= ", Q9

788 PRINT ON ( J6) "< I0>< 1G> " 790 RETURN

792 REH THIS SURROUTINE COriPARES EXPECTED S7 , T7 Wrra ACTUAL S7,T7

794 REM USE ( A*2>::P I/LAriDA/E) « . 5m:( C 1 OR 1.0) TO GET S7 , T7 RESPECTIVELY

796 IY=Pl:i;4-.:X(25)/(X( 1 ) «X( 23) *X( 27) ) 798 T7 = T7:i:X(6)>^.5a:Cl 800 S7=Pl:i:4*XC 14)/(X( 1 ) s:X( 12):bX( 10) ) 802 S7 = S7.';X(5):i:.5 804 PRINT "SIMPLE THEORY EXPECT S7= " , S7 , "ACTUAL= " , X( 9) 806 PRINT "CHANGE ACTUAL VALUE YES= I , HO=0"

808 INPUT II 810 IF I 1 = GOTO 816 812 PRINT "S7=?" 814 INPUT X(9) 816 PRINT "SIMPLE THEORY EXI'ECT T7= " , IT, "ACTUAL= "

, X( 10) 813 PRINT "CHANGE ACTUAL VALUE YES=1,HO=0"

820 INPUT II 822 IF I 1-0 GOTO 828

824 PRINT "T7=?" 826 INPUT X( 10) 828 RETUIW

8:t() I HINT "DONE WITH PROGRAM AND NOW EXITING"

ll;i2 CUSUD 064

834 CLOSE 1 836 CLOSE 3 838 RETURT)

840 PRINT "READ IN FROM FILE X( 1-73) DATA" 842 PRINT "YES=1, N0=0"

844 INPUT 11 846 IF 11=0 GOTO 862

848 CLOSE 2 850 OPEN "DATA ",2,2 852 REW 2 054 FOR 18= 1 TO 75 856 INFILE OK (2) X( IB) 858 NEXT IB 860 CLOSE 2 862 RETURH

864 PRINT "PRINT TO FILE X( 1-75) DATA"

066 PRINT "YES=1, H0=0"

068 INPUT II 870 IF I 1 = GOTO 886 072 CLOSE 2 874 OPEN "DATA", 2,

1

076 REW 2 878 FOR 18=1 TO 75 800 OUTFILE ON (2) X( IB) 082 NEXr 18 884 CLOSE 2

805- RETURN

888 PRINT "STORE LSQF IT DATA, 1=YES. = WO" 890 INPUT 18 892 IF 18=0 GOTO 602 894 OUTFILE OK (4) X( L3) .X( L4) , X( L5) , L6 096 IF J5= II GOrO 904

898 OUTFILE ON (4) XI , Yl ,C( 8, J5) , C( 10, J5)

900 GOTO 906 902 PRINT "CASES LEFT L6=",L6

904 OUTFILE OK (4) XI , Yl , WI/W2, C( 8, 1 1) 906 PRINT "CASES LEFT L6=",L6

903 L6=L6-1

910 GOTO 960

912 PRINT "RESET AND OPEN LSQ FILE 1=YES, 0=NO°

914 INPUT 18

9 16 IF 10=0 GOTO 958

918 CLOSE 4 920 OPEN °ERIC:LSa",4,

1

922 REW 4 924 PRINT "NUMBERS OF PARAMETERS=7"

926 INPUT L6 920 PRINT "LOCATION OF PARAMETERS?"

930 PRINT "FIRST L3=

Figure 4.4. List of computer program BEAM (page 5]

45
932 INl'UT 1,3 934 IF L0> 1 GOTO 942

936 1,4= 1 93a u;= 1 940 <;0T() 956 942 PRIWT "SECOND L4=

944 INPUT L4 946 IK L6>2 GOTO 952

940 I,a= 1 950 GOrO 956 952 riUNT "THIHD L5= " 954 IWl'UT L5 956 OUTFILE ON (4) L6,L3,L4,L5

95y lil'nUItH 960 IF L6X-1) GOTO 698

902 GOTO 966 964 ItFlUnH 960 GLOSE 4 968 Ol'EH "LSO",4,2

970 HEW 4 972 INFILE ON (4) L1,L3,L4,L5

974 Fill NT "NUrOJER OF VALUES TO BE DRIVEN TO REFERENCE"

970 inruT L2 97« DIfl E(L2) ,C(L2) ,D(Ln ,H(LI) ,B(L2,L1) ,F(L1,LI) ,G(L1,L1)

9U0 DIM A(4)

902 riUHT "lUOFERENCE VECTOR C(L2)="

9a4 FOil 1=1 TO L2

9U\. VUINT "1= ", I, "C( I)=?" oaa H'Ji'UT c( I) 990 NEXT I 992 INFIEE ON (4) S( 1 )

, S( 2) , S( 3) ,S( 4) 994 INKILE ON (4) A( 1 ) , A( 2) , A( 3) , A( 4) 990 FOR I,!i=l TO EI 990 INFILE ON (4) Y( 1 ) , Y( 2) , Y( 3) , Y( 4) 1000 A1=Y(L8)-S(L8)

1002 INFILE ON (4) S(5) ,S( 6) ,S( 7) ,S( 8) 1004 FOR 1=1 TO L2

1006 B( I ,LO)=(S( I+4)-A( I))/A1

1000 NEXT I 1010 NEXr LO

1012 FOR 1= 1 TO L2

1014 E( I)=A( I)-C( I) 1016 NEXT I 1018 FOR 11= 1 TO LI 1020 D( I1)=G

1022 FOR 12= 1 TO LI 1024 F( U, 12) =0

1026 FOR 13= 1 TO L2 1028 IF I2> 1 GOTO 1032

1030 0( I 1)=D( Il)+B( 13;, I1)*E( 13) 1032 F( I I, I2)=F( 1 1, I2)+B( 13, I 1 ) *B( 13, 12) 1034 NEXT 13 1036 NEXT 12 1030 NEXT II 1040 FOR 11=1 TO LI 1042 FOR 12= 1 TO LI 1044 IF 12=11 GOTO 1048

1040 F( I 1, I2)=F( II, I2)/F( U, ID

1048 NEXT 12 1050 D( I1)=D( Il)/F( II, ID

1052 F( II, ID = 1 1054 NEXT II 1056 riAT G= INV(F) 1058 FOR 11= 1 TO LI 1060 H( I1)=0

1062 FOR 12= 1 TO LI 1064 H( I D=H( Il)-G( II, I2)*D( 12) 1066 NEXT 12 1068 NEXT II loro l-RIHT "I, OLD, NEW, INCREIIEHT RESPECTIVELY"

1072 FOR I 1= 1 TO LI 1074 PRINT "1= "; I1,S( ID ,S( ID+H( ID ,H( ID

1076 ne;;t ii 1078 CLOSE 4 1000 GOTO 12 1082 END

Figure 4.4. List of conputer progran BEAM (page 6).

46
VERS 1 OH 6/29/78

BEAM TRACE

WAVLGN= 1 . 06

COS( ANG) = LA= r LC,LD= 7

4:*4:af:^^^**4f MICROHETERS

.990216

LB= 7

7 XCUR(M)

. 1E2I -2.0397183

. 1E21 .2039718

. 1E21 -2.0396975

. 1E21 -2.0397384

. 1E21 .20397546

. 1E21

WY( MM) = .5 PHASE(RAD)=

ANG= 14 RAD

S1,T1(MM) =

YCUR( M)

. 1E21 -2

. 1E21

.2

. 1E21 -2

. 1E21 -2

. 1E21

.2

. 1E21 RX(M) = RY( M) =

APER(

1E21

1E21

12

3

4

5678

910

11 TO( MM) » .5 AMPL= 1 *#»#<•«#<*** EBD OF INITIAL DATA tttt/iesitt/aittfrt SURFACE PARAMETERS PRINTED NEXT HAS L= ALL IN MM. WX,WY,RX,RY= .5 RATIO VX/Wy=, 1/RX, 1/RY PROP TO 1 ALL IN MM, X1,X3,Y1,Y3= 35

AMPLITtn)E= 1 PHASE(RAD) = SURFACE PARAMETERS PRINTED NEXT HAS L= ALL IN MM, WX,WY,RX,RY= .85358927

RATIO WX/WY=, 1/RX, 1/RY PROP TO 1.0098807

ALL IN MM, X1.X3,YI,Y3= 35.345824

AMPLITUDE= .59153504 PHASE( RAD)

SURFACE PARAMETERS PRINTED NEXT HAS L= ALL IN MM, WX,WY,RX,RY= .68148381 RATIO WX/WY=, 1/RX, 1/RY PROP TO 1.0000001

ALL IN MM, X1.X3,Y1,Y3= .39192519E-6

AMPLITUDE= .69836857 PHASE( RAD)

SURFACE PARAMETERS PRINTED NEXT HAS L= ALL IN MM, WX.WY,RX,RY= .69006626

RATIO WX/WY= , 1/RX, 1/RY PROP TO 1.0098808

ALL IN MM. X1,X3,Y1,Y3= -3.5345964

AMPLITUDE= .6964964 PHASE( RAD)

SURFACE PARAMETERS PRINTED NEXT HAS L= ALL IN MM. WX,WY,RX,RY= .500O4885E-1

RATIO WX/VY=, 1/RX, 1/RY PROP TO 1.0000017

ALL IN MM, X1,X3,Y1,Y3= -3.4999781

AMPLITUDE= 9.4229956 PHASE( RAD) =

SURFACE PARAMETERS PRINTED NEXT HAS L= ALL IN MM, TO{,VY,RX,RY= 6.8816677

RATIO IOC/ WY=, 1/RX, 1/RY PROP TO 1.0098789

ALL IN MM, X1.X3,Y1,Y3= -3.5409303

AMPLITUDE= .69137E-1 PHASE( RAD)

SURFACE PARAMETERS PRINTED NEXT HAS L= ALL IK MM, VX.WY.BX,RY= 6.8782011

RATIO WX/WY= , 1/RX. 1/RY PROP TO 1.0098595

ALL IN MM, X1,X3,Y1,Y3= -. 12029812E-4

AMPLITUDES .691047e3E-l PHASE( RAD) = SURFACE PARAMETERS PRINTED NEXT HAS L= ALL IN MM, VX,WY,RX,RY= 6.504488

RATIO WX/WY= , 1/RX, 1/RY PROP TO 1.0088113

ALL IN MM, X1,X3,YI,Y3= 3.5316034

AMPLITUDE= .6910239E-1 PHASE( RAD) = SURFACE PARAMETERS PRINTED NEXT HAS L= ALL IN MM, WX,WY,RX,RY= .529277 IE1

RATIO WX/WY= , 1/RX, 1/RY PROP TO 1.0010675

ALL IN MM, X1,X3,Y1.Y3= 3.5000479

AMPLITUDE= 8.4104591 PHASE(RAD)=

SURFACE PARAMETERS PRINTED NEXT HAS L= ALL IN MM. WX.VY,RX,RY= .71716338

RATIO WX/Vri'=, 1/RX. 1/RY PROP TO 1.0088158

ALL IN MM. X1,X3.Y1.Y3= 3.5352909

AMPLITUDE= .62607663 PHASE( RAD)

06/29/78

UN) RADIUS

10000 224 400

50400

224

20

224

400

50

400

II: 17: 19

L TO L+1 DIS(M)

1.0098859

1.0098713

. 10098901

. 10098787

1.0098759

1.0107332

1.0090289

1.0098854

. 11108752

1 . 1 10876

1

-30

1.5.29637642E-19

35

. 1E24 .29637642E-19

35

. 431 4083 lE-31 POWER=

.84523775

.43183256

1110.8293

-538306.88

6863.2254

. 44040807

35 POWER=

.68148373

.20267644E-6

.50679937E9

-.345684e9Ell

4.68331455

-14.234022

-36. 162624

. 14623131Eli

-.60330258E-6

. 10080396E-5

POWER= -208.21691

-14.516697

-3.5000142

18613.48 POWER=

.50004798E-1 80789106

. 36685 196E-4 . 73553 177E-4 501.11065 -3.4999791

16.900377 POWER=

6.8143492

1.4386543

280 . 4025

1

113119.63

2060.0948

1 . 4672392

-3.5061631

P01ffi:R=

6.8110474 -.19980748E9

-. 14833099E-4 . 143027E-4

-27196046 . 12082362E-4

.10963553E1I POVER= 8

6.4476755

-1.4386165

-280.40989

113113.34

-2060. 1489 -1.467172

3.4970375

POWER=

.5287 127 IE- 1 -4032425.7

- . 73498294E-3 - . 12239038E-2

27.233733 3.4999786

-2.4272773 P0WER=

10.71089624

11.626878

43.479015

-2206 1 . 484

254.90627

11.858143

3.5006205

P0WER=

. 1E24

35 .39269875

6729.5864

151.61552

.39267951

-.49125667E10

-. 17025672E9

.35579506

-204. 16244 -7.0378505

.35579403

40294169

243 . 24743

.34875593

2019.9598

-7.0003012

.34864914

.2072171 1E9 -718159.61

.34797931

-2020.0522

7.0004621

.31149557

-2421566.2

17.494229

.31092939

249.93493

7.8310115

.31083417

###*#*ORDER OF BEAM REFRACTED BY HOLOGRAM

L1,L2= e

C( 1-10. 11) -8753.6118 -3.7435917 -249.83561 -.52222223E-1

-3.5765753 2.9349362 <**•*#*#*#**###*#*•###«•***# SURFACE PARAMETERS PRINTED NEXT HAS L= 11 ALL IN MM, WX.VY,RX,RY=: .52932434 .5287694

RATIO WX/WY=, 1/RX. 1/RY PROP TO 1.0010495 2.9339998

ALL IN MM, X1,X3,Y1,Y3= -35.000001 . 88994933E-2

AMPLITtn)E= .83246726 PHASE( RAD)

-3.74384

RC(L1,L2)= • IMC(LI.L2>> » -200.36028

1010. 1446 2.9349362

-35.000001

POWER=

S7,T7 (1/MM>= 1.4530308 1.4673733

.13694027E-1 -3.56908 2.9339998

1009 . 8223

-.23329344E-2

. 30467864

Figure 4.5. Sample print of BEAM (page 1),

47
SURFACE PARAMETERS PRINTED HEXT HAS L= ALL IN MM, WX, WY.RX,RY= 6.504488

RATIO VX/WY= . 1/RX, l/RY PROP TO 1.0088113 ALL IN MM. X1,X3,Y1,Y3= 3.5316034

AMPLITUDE= .6910239E-1 PHASE( RAD) = SURFACE PARAMETERS PRINTED NEXT HAS L= ALL IN MM. WX,WY,RX,RY= .5292771E-1 RATIO WX/WY=. l/RX, 1/RY PROP TO 1.0010675 ALL IN MM, X1.X3,Y1,Y3= 3.5000479

AMPLITUDE= 8.4107413 PHASE( RAD) =

8

6.4476755

-1.4386165

-280.40989

113108.38

-2060. 1489 -1.467172

3.2472501

POWER=

.5287 127 IE1

-4032425.7

-.73498294E-3 -. 12239038E-2

27.233733 3.2499804

-2.4645316 POWER=

-2020.0522

6.5004295

.31149557

-2421566.2

16.24279

.31095025

SURFACE PARAMETERS PRINTED NEXT HAS L= 10 ALL IN MM, VX,VY,RX,RY= .71716338 .71089624 254.90627

RATIO WX/WY=, 1/RX, l/'RY PROP TO 1.0088158 11.626878 11.858143 ALL IN MM. X1,X3,Y1,Y3= 3.5352909 43.479015 3.2505763

AMPLITUDE= .62610424 PHASE(RAD)= -22006.085 POWER=

249.93493

7.271654

.31086158

#***##OHDER OF BEAM REFRACTED BY HOLOGRAM

LI.L2= -1

C( 1-10, 11) -8130.0644 -3.7815591 -249.83561 -.52222223E1

-3.5765753 2.9349362 ###*###«'######**#*#*<'*«## SURFACE PARAMETERS PRINTED NEXT HAS L= 11 ALL IN MM, WX,¥Y,RX,RY= .52932434 .5287694 RATIO WX/WY=, 1/RX, 1/RY PROP TO 1.0010495 2.9339998

ALL IN MM, X1,X3,Y1,Y3= -35.000001 . 88994933E-2

AMPLITUDE= .83306618 PHASE(RAD)= -3.7818052

RG(L1,L2)= IMC(L1,L2)=

-232.47742

1010. 1446 2.9349362

-32.500003

FOWER=

S7,T7 (1/I1M)= 1.4539308 1.4673733

.127I6337E-1 -3.06998 2.9339998

1009 . 8223

-.21663736E-2

.3051172

Figure 4.5. Sanple print of BEAM (page

48
Z(l) = A*EXP ( -[(u - Sl*LA)/lJX] + 2

[(v - Tl*LB)/WY]+2

+ il<[(u - Sl*LC)+2/(2*RX)]

+ ik[(v - Tl*LD)+2/(2*RY)]

|. + 10 ( . (4.i:

4.2 The Detailed Listing of the Computer Program

The logic flow gives the boundaries for each function in the program. In most

cases, knowing these functions and using the list of the program gives sufficient

information. In subsection 3.3 and in other parts of section 3, we define the

mathematics. All that remains to be described about the program's logic flow is the

fact that there is temporary storage of numbers which allows appropriate interface

between function blocks. Examples are those statements at 100 to 104 and those at 456

and 462.

There are many system-sensitive statements in this listing such as the opening and

closing of files. The program will have to be adjusted to reflect the system used so the program can function as intended.

For convenience in analysis, the program computes RX and RY and X3 and Y3. In some cases, RX and RY can be so large that the computation becomes meaningless;

therefore, statements around 744 and 754 will cause the value of RX and RY to be set to

a maximum. The prime result of this constraint is an absurd value for X3 and Y3.

Since this condition occurs when the phase front is almost plane wave, X3 and Y3 are poorly defined anyway. If a different range is wanted, X8 in statement 722 can be

changed to some other value.

4.3 Sample Output to Illustrate the Program

Figure 4.5 shows a print for two different beam traces (a zero and a -1 diffraction order) from surfaces 1 to 11 to illustrate what output can appear on the

pri nter.

Given all previous discussions and definitions, most terms should be clear except

the POWER term. This represents the total power in watts in the Gaussian beam incident

on the designated surface. This term helps the user to select appropriate aperturing

by each mirror so that there is no serious attenuation for off -axis beams, namely those

with large LA, LB, LC , and LD.

notice that this print has the same stage I results for these beam traces. In contrast, the stage II results are printed twice to reflect changes.

Also note that the control parameters for stage II are printed next to surface 11 print rather than at the beginning of stage II. This is done because many traces will

print only the initial conditions and the surface 11 results. The intermediate data

usually have no interest and are therefore suppressed. To reproduce results, control

49
data cannot be suppressed and are recorded in the output.

5. THE 10.6 un UAVELEriCTH

Here we select the control values for an apparatus that would be optimum for

10.6 pn wavelength. As already stated, some of these values must be arbitrary. Once

they have been selected, then other values can optimize the desired output of the

apparatus. In subsection 5.1 we select the arbitrary values, and in subsection 5.2 we define the criteria that optimize the remaining values.

lie examine two configurations, namely a version that uses only off-the-shelf

mirrors with spherical or single curvature and a more expensive version that uses custom-made mirrors with two (elliptical) curvatures.

In subsection 5.3 we develop the expected output when all mirrors have fixed

spherical curvatures.

In subsections 5.4, 5.4.1, and 5.4.2, we deal with the elliptical optics. In subsection 5.4.1 we show what happens for changing wavelength, and In subsection 5.4.2 we show how the positions of the mirrors change the output.

Because 10.6 ym is not a convenient wavelength for alignment, we show In subsection 5.5 what to expect for selected beams at 0.6328 ym wavelength going through

the apparatus using the elliptical optics.

This section Is concluded in subsection 5.6 by a discussion of high Irradlance

effects on the surfaces of the mirrors and hologram. This consideration Is important

to establish the limits to the Incident irradlance at the prefllter.

5.1 Selecting Some Initial Control Values

To design this reflection apparatus, we must select some arbitrary values. We

discuss them In this subsection. We note at this point that there are numerous other

choices, and some may prove to be better than those described here.

The first arbitrary value is the focal length for each focusing mirror. For an apparatus to be of a moderate size when using 10.6 ym wavelength, the unit should not

be much larger than a meter. Therefore, we set the radius of curvature In the

Y coordinate at 2 meters for all focusing mirrors. This curvature Implies a focal

length of 1 meter.

The second arbitrary parameter Is the width of a beam. I selected 10 cm as a convenient size for the diameter of the beam because most reflection optics that are

not custom made are restricted to 20 cm (8 inches) In diameter or less and because we

need adequate clearance for each beam to account for diffraction effects created at the

Hartmann plate.

The third parameter is the size of the sampling holes; they should be as small as possible. To drill a hole less than 1 mm In diameter requires extraordinary

procedures; therefore, we set the sampling hole size to 1 mm.

The fourth parameter is the sampling frequency for the square array In the

Hartmann plate. The key constraints here are the size of the detectors In the array at the cross-correlation plane and the size of each cross-correlation spot. Aperturing

50
and cross-correlation effects cause each spot to be 2 nrn or less in diameter. For a 10 ns speed of response, each detector should be less than this diameter. Since the

distance between detectors should allow clearance for adjustment, say 0.5 mm, to account for various astigmatic effects, we assume each detector in the array is separated by 2.5 mm to capture the center of these cross-correlation spots. Given this

form, we infer the separation at the prefilter is 5 mm between centers of sampling

holes.

In the test beam analysis, our nominal sampling values for (LA, LB) are: (10,0)

for a deflection in the x axis; (0,10) for a deflection in the y axis; (7,7) for a deflection along the 45° diagonal; and (0,0) for the zero deflection or optical axis

beam. These four principal values are arbitrarily chosen to define all simulation

studies. Additional test beams may be used; however, each is keyed to one of the

principal beams.

As a matter of convenience, the simulation studies in this paper set LC = LD = 7 for all runs because the initial curvatures imply a plane wave which means the

positions of X3 and Y3 are irrelevant.

The plane surfaces, namely 1, 3, 5, 7, 9, and 11 all have a radius of curvature

equal to 10^0 n for both the X and Y coordinates.

The 1 mm sampling holes imply at a distance of 1 m an Airy disk with a 26 mm

diameter. To avoid significant aperturing by a sample beam 5 mm from the optical axis, we increase the radius of our optics by two Airy diameters. This means 102 mm (4 inch)

radius optics. To simulate the aperture properly for the Gaussian beams, we arbitrarily require that 80% of the Gaussian beam power pass when the beam radius is

100 mm. This implies: —P 2 2 2 2

= exp(-x /a ) or (x /a) = 0.2 .

With x = 100, then o = 225 mm.

We arbitrarily set those flat surface apertures at 400 mm which means no significant absorption takes place at these surfaces. Since surface 7 has an Airy disk

of approximately 26 mm in diameter (the second ring is at 48 mm), we choose an aperture

that will pass this basic beam with room to spare. A convenient size for the Gaussian

beam and the design sample is 40 mm in diameter. This dimension will comfortably fit

the 50 mm optical flats that can be plated with available equipment during metal

evaporation. The remaining 10 mm can be used for fiducial marks which are needed

during the multiple evaporation cycles on these flats. In the final construction of

the apparatus, it may be necessary to use a larger diameter aperture such as 100 mm to

avoid a significant distortion to the final spots in the cross-correlation plane.

The remaining aperture at surface 1 is arbitrarily set to 10 m which obviously is

a meaningless number. It makes the aperturing by surface 1 unimportant. All aperturing by the prefilter shows in the initial beam parameters.

We fix the initial beam parameters by four assumptions: (1) lie assume the

curvatures exiting from each sampling hole are very large, say 10^0 m. (2) We set the phase of a single beam to zero and its amplitude to 1. This implies no loss in

generality, but a gain in convenience for comparison of various simulations. (3) We

51
assume circular sampling holes; hence, we set WY = WX to fix the y width of the

Gaussian beam. (4) We select the nominal value for WX by the following ad hoc

reasoning. At the Fourier plane, the true pattern produced by the sampling holes at the Hartmann plate is an Airy disk with a diameter of 26 mm. This disk contains

84 percent of the power in the beam. If we set the width of a Gaussian beam to this

diameter, we would have a Gaussian beam with 8o percent of the power within this area. This beam size implies a Gaussian beam of 0.5 mm in diameter at the prefilter. Because

we wish less variation in beam size, we double this value to equal the sample hole. In this case, the beam at surface 7 is 13 mm in diameter. We use WX = 0.5 mm for the

simulations.

This choice for beam width affects critically the necessary size for the aperture

at surface 7 that will minimize distortion in the cross-correlation plane. A builder

of the apparatus should run additional simulations to test the effect of aperturing on the beam profile measurements. For now, we dispense with these detailed studies. Our

goal in this paper is to identify what must be done. This paper does a first-order

pass at the design so the preliminary costs for construction of the apparatus can be

estimated as realistically as possible.

There remains one control parameter to be fixed within this subsection, namely the

angle of reflection, 9, along the optical axis. Given the 204 mm optics and the 1 m distance between surfaces 1 and 2, we require this angle to exceed 0.102 radians. We

clearly need significant clearance to allow proper baffling of stray beams in the

apparatus and appropriate room for mechanical fixtures. A reasonable margin would make

a 50 percent increase in angle. For example, when cos (9) = 0.99, the angle is

0.14 rad. We choose this value for our simulations. In the final construction of the

apparatus, the reader should expect this angle to change because some mechanical or optical part will bend some beam in the unit. A smaller angle implies less elliptical

curvatures. Unfortunately, this reduction of the angle implies mirrors with much

larger focal lengths and a correspondingly longer apparatus.

5.2 The Convergence Criteria for the Rest of the Control Values

There remain seventeen control values to be fixed in these simulations: the five

x-curvature values for the mirrors; the ten position values for placement of these

mirrors; and the hologram parameters S7 and T7. In this subsection we set them for our

design. The selection process is slightly different for spherical mirrors than for

elliptical mirrors. This procedure is not unique because there are arbitrary choices.

The best method depends on the allowed parameters for adjustment.

Given the above, we fix these control values as follows: First, we guess initial

values for each type of mirror. Second, some arbitrary formulas constrain others.

Third, figures 5.1 and 5.2 show the required convergence conditions for other control

values. Finally, the values of S7 and T7 are adjusted to get the desired diffraction

of the incident beams at surface 7.

As already discussed, this optimization uses one beam, namely that with

LA = LB = 7 for the simulation. Here we use a nominal focal length for our optics,

namely f = 1 m, and use the angle for astigmatism effects as 9 = 0.14 rad. Three

constants are useful for fixing initial data, namely Cx = (Cy)2 = 1.019858981,

Cy = f/cos 9 = 1.009880676, and M = A/10.6. Here X is the wavelength of the laser

beam in micrometers. Thus, M = 1 for x = 10.6 pm and 0.1 for X = 1.06 ym.

Given these constants, the initial values for the elliptical mirrors are:

52
X(12) = X(16) = X(18) = -2*Cx/f, and

X(14) = X(2C) = 2*ri*Cx/f . (5.1)

The distance values are started at:

X(44) = X(45) = X(48) = X(49) = X(50) = X(51) = Cy ,

X(46) = X(47) = Cy*M ,

X(52) = Cy*M/(l+n) , and

X(53) = X(52)/M . (5.2)

If the simulations uses spherical mirrors only, then exactly the same distance

values are used, but the mirror curvatures are set to:

X(12) = X(16) = X(18) = -2*f , and

X(14) = X(20) = 2*M*f . (5.3)

S7 and T7 use the formula in the program and their values are printed during a computer run. This fixes an initial guess for starting the simulation process.

Because the apparatus has discrete beams progressing through it and because the

number of those beams are restricted during the adjustment sequence in the interest of

keeping the process simple, we must constrain somewhat arbitrarily several distance

values between the mirrors. These values are insensitive to the critical features of

the beams; thus, the constraints remove simply unneeded degrees of freedom. If a different alignment or simulation technique is chosen which uses many beams in addition

to the LA = LB = 7 case, then the adjustment sequence changes and these arbitrary

constraints should be dropped for a different criteria such as a least squares fit

technique. The constraints are:

X(46) = M*Cx/X(45) ,

X(50) = Cx/X(49), and

X(52) remains constant. (5.4)

Thus, two distances are constrained relative to other distances. These

constraints arise by requiring the mirrors to be separated by their respective focal

lengths with a modification due to the fact there is off -axis illumination. The choice

for X(52) arises by using the image equation, again with the adjustment for the

off-axis illumination.

In the case of elliptical mirrors there remain twelve control values to adjust for

the desired beam behavior in the apparatus. These are shown in figure 5.1 along with

the desired target values for the beam parameters at each surface. In this simulation,

we track four terms, namely the center of the beam, XI and Yl, and two terms

proportional to inverse of the curvature, ir/(Rx*A) and (ir/Ry*x).

53
In the real apparatus, these curvature parameters are not accurately or easily

neasured; therefore, two beams can be used to create an interference pattern at each

surface exposing the details of the phase front. Wo have not established which pair of

beans is best for this adjustment; therefore, we only suggest that an appropriate bean

pair nay be either (LA, LB) = (7,7) and (-7,-7) or (7,7) and (0,0).

In considering other adjustnent strategies, we suggest that the reader consider

the changes in the X3 and Y3 values for each bean to help infer appropriate positions

for the mirrors.

In the case of only spherical mirrors, there remain five control values to be

adjusted. The mirror curvatures are fixed, and the distances X(45) and X(49) are set

to the initial values as defined in eqs (5.1)-(5.3). The results in figure 5.2,

similar to those in figure 5.1, show the desired converged values for the spherical

mirrors. When a physical apparatus is adjusted, we have fixed-curvature mirrors;

therefore, the strategy for its adjustment would follow the sequence defined for these

spherical mirrors. The difference between the two types of mirrors shows how

accurately the beams remain astigmatic at each surface. For the ideal situation with

no astignatism, surfaces 3, 5, 9, and 11 would each show a square array of spots when

the Hartmann plate is illuminated by a uniform irradiance and flat, phase front beam.

To get accurate control values during these simulations, we use the LSQ part of

the program. The simulation process adjusts the parameters of each surface as shown in

figures 5.1 or 5.2. These parameters are adjusted completely to get as close as possible the desired final values for each surface before any adjustments are made at the next surface. This process causes the accumulative errors to transfer to the

remaining surfaces. IJe do not prove that this sequence is best for adjusting an apparatus. Ue assume that it is best both for finding the ideal control parameters and

for minimizing the astigmatism effects for the actual apparatus. As already stated,

the preliminary adjustments in the physical apparatus would use these results.

Surface Final Values of:

XI Yl Tr/(RX*X) Tr/(Ry*A;

Adjust

Only

X(44), X(45)

X(47), X(14)

X(48), X(49)

X(51), X(18)

X(53), X(20)

X(12)

X(16)

3

5 -35*f1 -35*M

79 35*M 35*M

11 -35 -35 ** **

** Means these values are not used for convergence.

Figure 5.1. Desired final values for beam parameters in elliptical case. Here we show

the allowed control variables that are changed for each surface. The adjustment

sequence is surface 3 first, then 5, etc.

54
Surface Fi nal Values of:

XI Yl Tr/(RX*A) TT/(Ry*

3 ** **

5 -35*M -35*M ** **

7 ** **

9 35*M 35*ri ** **

Adjust

Only

X(44)

X(47)

X(48)

X(51)

X(53) 11 -35 -35  

**  Means these values are not used for convergence.

Figure 5.2. Desired final values for beam parameters in spherical case. Here we show

the allowed control variables that are changed for each surface. The adjustment

sequence is surface 3 first, then 5, etc.

As already stated the final adjustments would use the desired form of the spots at the cross-correlation plane as the reference base and then use all available

adjustments in distances as well as angles between mirrors to fine tune the actions of

the apparatus. If possible, the positions of detectors may also be adjusted during

this adjustment sequence.

After the curvatures and distances are fixed by the computer runs, then the ideal

hologram can be identified. Using LSQ and LI = L2 = 1, we adjust S7 and T7 so the

diffracted beams form spots for surface 11 at XI = Yl = -37.5 mm. Because the real

apparatus has a hologram with fixed S7 and D7, the adjustment strategy changes. Here

adjustments use the desired structure of all spots in the cross-correlation plane.

This structure controls the final adjustment for the distances between mirrors, the

various angle coordinates of the mirrors, and finally the spatial distribution of the

array of detectors at the cross-correlation plane. The form of this final adjustment

process critically depends on the allowed dynamical range for incident laser

wavelengths at a given setup as well as on the allowed field of view for the incident

beam at the Hartmann plate.

5.3 The Spherical Optics Data

We list here a summary of the data generated in the computer simulation runs using

spherical optics. These data are shown in figures 5.3 to 5.8. This subsection briefly

explains each figure and expands where necessary on the results to explain what each

figure means.

Figure 5.3 contains a summary of the initial data for the simulation. These data

and the convergence criteria of subsection 5.2 imply the final data shown in

figure 5.4. To sense what these results imply, we print selected numbers in

figures 5.5 through 5.8 for some surfaces and some beams. Figures 5.5 and 5G show

results for the initial configuration case, and figures 5.7 and 5.8 show those same

results after the mirrors have been adjusted according to the convergence criteria.

The principal conclusion drawn from these results is that beam profile work of

high precision with spherical mirrors is not possible. To use these mirrors requires

substantial reduction of the reflection angle with corresponding substantial increases

in the distances between mirrors and in the curvatures of these mirrors. In addition,

we need significant reduction of the field of view for the apparatus.

55
VERSION 6/::9/78

BEAM TRACE ********* 07/14/78 14:42:43

WAVLGN 10 .6 MICROMETERS ANG. .14 RAD COS(ANG) .990216

LA 7 LB 7 S1,T1(MM) 5 5 LC.LD 7 7

L XCUR(M) YCUR(M) APER(MM)RADIUS L TO L+1 DIS(M

1 .1E21 .1E21 10000 1.0098807

2 -2 -2 224 1.0098807

3 .1E21 .1E21 400 1.0098807

4 2 2 224 1.0098807

5 .1E21 .1E21 400 1.0098807

6 -2 -2 224 1.0098807

7 .1E21 .1E21 20 1.0098807

8 -2 -2 224 1.0098807

9 .1E21 .1E21 400 2.0197613

10 2 2 224 2.0197613

11 .1E21 .1E21 400 -30

WX(MM) .5 UY(MM) 5 RX(M) .1E21

AMPL 1 PHASE(RAD)

END OF iniTIAL DAT^

RY(M) .1E21

S7, T7 (1/MMD) = 1.4530308 1.4673733

Figure 5.3. Initial guess of control data for spherical mirrors.

VERSION 6/29/78

BEAM TRACE

WAVLGN 10.6

COS(ANG)

LA 7 LC.LD 7

L

12

3

4

56

78

910

11 UX(MM) .5 AMPL 1

********* 07/17/78 09:29:17

MICROMETERS ANG. .14 RAD .990216

LB 7

7 S1,T1(MM) 5 5

XCUR(M) YCUR(M) APER(MM)RADIUS L TO L+1 DIS(M)

.1E21 .1E21 10000 .9997152

-2 -2 224 1.0098807

.1E21 .1E21 400 1.0098807

2 2 224 .98995757

.1E21 .1E21 400 1.0100388

-2 -2 224 1.0098807

.1E21 .1E21 20 1.0098807

-2 -2 224 .98169863

.1E21 .1E21 400 2.0197613

2 2 224 2.0027554

.1E21 .1E21 400 -30

WY(MM) 5 RX(M) .1E21

PHASE(RAD) RY(M) .1E21 END OF INITIAL DATA

S7, T7 (1/MM) = 1.4530308 1.4673733

Figure 5.4. Final values of control data for spherical mirrors.

56
Bean LA = LB = Surface XI (nm) Yl(mm) tt/(RX * X)[(nim)

1 0.000 0.000 .296E-20

3 -.344E-18 -.353E-18 -.591E-2

5 -.115E-19 .208E-25 -1.64

7 .339E-18 .352E-18 -.174E-1

LI = 9 -.214E-19 -.401E-24 1.78

11 .493E-19 -.314E-23 1.66

LI = 1 9 2.450 -.401E-24 1.78

11 -2.543 -.314E-23 1.66

LI =

9 -.214E-19 2.4 50 1.78

11 .493E-19 -2.450 1.66

2] 7t/(RY * A)[(nm)-2]

.296E-20

.292E-5

.319E-5

.291E-5

L2 = .553E-4

.293

L2 = .553E-4

.293

LI = 1 .553E-4

.293

Fiquro 5.5. Bean trace at selected surfaces with control data from figure 5.3,

Bean LA = 7 LB = 7 Surface XI (mm) Yl(mm) tt/(RX * X)[(mm)

1 35.000 35.000 .296E-20

3 -.727 -.324E-1 -.591E-2

5 -34.972 -35.000 1.64

7 2.137 .323E-1 -.174E-1

LI =

9 -34.906 34.999 1.78

11 -36.075 -34.999 1.66

LI = 1 9 37.356 34.999 1.78

11 -38.619 -34.999 1.66

LI =

9 34.906 37.499 1.78

11 -36.075 -37.499 1.66

.296E-20

.292E-5

.320E-5

.292E-5

L2 - .541E-4

.293

L2 = .541E-4

.293

L2 = 1 .541E-4

.293

Figure 5.6. Beam trace at selected surfaces with control data from figure 5.3,

57
Beam LA = LB = Surfdce XI (mm) Yl (mm) Tr/(RX * A)[(mm)--2] tt/(RY * A)[(mm)

1 .296E-20 .296E-20

3 -.344E-18 -.353E-18 -.284E-2 .296E-2

5 .116E-20 -.105E-19 .471 -1.39

7 .339E-18 .352E-18 -.852E-2

LI = .870E-2

L2 =

9 -.390E-20 .182E-19 .816 -1.68

11 .259E-19 -.231E-19 1.99

LI = 1

-1.53

L2 =

9 -.390E-20 2.500 .816 -1.68

11 .259E-19 -2.4 58 1.99

LI = -1.53

L2 = 1 9 2.452 .182E-19 .816 -1.68

11 -2.504 -.231E-19 1.99 -1.53

Figure 5.7. Beam trace at selected surfaces with control data from figure 5.4.

Beam LA = 7 LB = 7 Surface XI (mm) Yl(mm) tt/(RX * x)[(mm)-2] Tr/(RY * X)[(mm)

1 35.000 35.000 .295E-20 .296E-20

3 -.727 -.320E-1 -.283E-2 .296E-2

5 -35.000 -35.000 .477 -1.39

7 2.136 .304E-1 -.852E-2

LI = .870E-2

L2 =

9 35.000 34.999 .816 -1.68

11 -35.623 -34.413 1.99

LI = 1

-1.53

L2 = 9 37.452 34.999 .816 -1.68

11 -38.126 -34.413 1.99

LI = -1.53L2 = 1 9 35.000 37.499 .816 -1.68

11 -35.623 -36.871 1.99 -1.53

Figure 5.8. Beam trace at selected surfaces with control data from figure 5.4.

5.4 The Elliptical Optics Data

Here the mirrors with elliptical surfaces are used in the computer simulation

runs. Figure 5.9 shows the initial data for the simulation, and figure 5.10 shows the

final results for best fit mirrors for the 10.6 ym wavelength. These data were derived

by using the initial conditions of figure 5.9 and the criteria of subsection 5.2.

Figures 5.11 through 5.15 show selected data for some beams and surfaces.

The conclusions drawn from these figures are:

1. The change in curvature for best elliptical mirrors is less than 0.002% from the

nominal identical mirrors shown in figure 5.9. Clearly, this result is beyond

fabrication accuracies; therefore, identical mirrors can be used.

2. It is possible to have a significant field of view and plane phase fronts at surfaces 1, 3, 5, 7, and 9 of the apparatus.

58
The power and phase distribution at surface 11 as shown in figure 5.15 shows

significant changes over the field of view. These changes are small for each bean

pair; therefore, the device can neasure the bean profile at 10.6 pn under proper

calibration.

VERSION 6/29/78

BEAM TRACE

UAVLGfl 10.6

COS(AriG) LA 7 LC.LD 7

*********** MICROMETERS AHG. .14 RAD .990216

LB 7 S1,T1(MM)

7

07/17/78 10:26:11

12

34

56

78

910

11 IIX(MM) AMPL 1

.5

XCUR(M)

.1E21

-2.039718

.1E21

2.039718

.1E21

-2.039718

.1E21

-2.039718

.1E21

2.039718

.1E21

WY(MM)

PHASE(RAD)

ycur(m; APER(MM)RADIUS L TO

.5

END OF INITIAL DATA

.1E21 10000

-2 224

.1E21 400

2 224

.1E21 400

-2 224

.1E21 20

-2 224

.1E21 400

o

(- 224

.1E21 400

RX(M) .1E21

RY(M) .1E21

L+1 dis(m;

1.0098807

1.0098807

1.0098807

1.0098807

1.0098891

1.0098807

1.0098807

1.0098807

2.0197613

2.0197613

30

S7, T7 (1/MM)= 1.4530264 1.4673833

Figure 5.9. Initial guess of control data on elliptical mirrors.

59
VERSION 6/29/78

BEAM TRACE ********* 08/04/78 15:03:36

WAVLGM= 10.6 MICROMETERS AriC= .14 RAD COS(AMG) = .990216

LA= 7 LB= 7 S1,T1(MM) = 5 5 LC,LD= 7 7

L= XCUR(M) YCUR(M) APER(MM)RADIUS L TO L+1 DIS(M)

1 .1E21 .1E21 10000 1.009895

2 -2.0397555 -2 224 1.0089468

3 .1E21 .1E21 400 1.0108155

4 2.0397242 2 224 1.0098771

5 .1E21 .1E21 400 1.0098841

6 -2.0397115 -2 224 1.0089475

7 .1E21 .1E21 20 1.0108147

8 -2.0397422 -2 224 1.0098816

9 .1E21 .1E21 400 2.0197614

10 2.0397172 2 224 2.0197912 1

11 .1E21 .1E21 400 -30 1

WX(MM)= .5 UY(Mri)= .5 RX(M)= .1E21 1

AMPL= 1 PHASE(RAD)=

END OF INITIAL

RY(M)=

DATA

.1E21

1

37, T7 (1/MM)= 1.4530308 1.4673733 1

Figure 5.10. Final values of control data on elliptical mirrors,

Beam LA = LB = urface XI (mm) Yl(mm) tt/(RX * X)[(mm)- 2] it/(RY * A)[(mm)

1 0.000 0.000 .296E-20 .296E-20

3 -.351E-18 -.353E-18 .298E-5 .292E-5

5 -.188E-20 .208E-25 .115E-5 .319E-5

7 .346E-18 .352E-18 .292E-5

LI - .291E-5

L2 = 9 .206E-20 -.401E-24 .538E-4 .553E-4

11 .102E-20 -.314E-23 .293

LI = 1

.293

L2 = 9 2.500 -.401E-24 .538E-4 .553E-4

11 -2.500 -.314E-23 .293

LI = .293

LI = 1 9 .206E-20 2.500 .538E-4 .535E-4

11 .102E-20 -2.500 .293 .293

Figu re 5.11. Beam trace at selected surfaces with control data f

60
Beam LA = 7 LB = 7 Surface XT Yl tt/(RX * A)[(mm)

1 35.000 35.000 .296E-20

3 -.330E-1 -.324E-1 .298E-5

5 -35.000 -35.000 .115E-5

7 .333E- 1 .323E-1 .292E-5

LI = 9 34.999 34.999 .538E-4

11 -34.999 -34.999 .293

LI = 1 9 37.499 34.999 .538E-4

11 -37.499 -34.999 .293

LI =

9 34.999 37.499 .538E-4

11 -34.999 -37.4 99 .293

Figui-e 5.12. Bean trace at se lected surfaces

^ A)L(TMj'

.296E-20

.292E-5

.319E-5

.291E-5

L2 = .553E-4

.293

L2 = .553E-4

.293

L2 = 1 .553E-4

.293

2]

Bean LA = LB = Surface XI Yl Tr/(RX * X)[(mm)- 2] tt/(RY * A)[(mm)

1 0.000 0.000 .296E-20 .295E-20

3 -.351B-18 -.353E-18 .273E-5 -.270E-5

5 -.188B-20 .376E-23 -.581E-3 +.578E-3

7 .346E-18 .352E-18 .269E-5

LI = -.266E-5L2 =

9 .207E-20 -.491E-23 -.653E-3 .64 7E-3

11 .103E-20 .107E-22 .293

LI = 1

.294

L2 =

9 2.500 -0.491E-23 -.6538-3 .647E-3

11 -2.500 .107E-22 .293

LI =

.294

L2 = 1 9 .207E-20 2.500 -.653E-3 .64 7E-3

11 .103E-20 -2.500 .293 .294

Figure 5.13. Beam Trace at Selected Surfaces wi th Conti"ol Data f

61
Bean LA = 7 LB = 7 jurfaco XI Yl ^/(RX * X)[(mn)- 2] 7r/(RY * X)[(mtii)

1 35.000 35.000 .296E-20 .296E-20

3 .639E-6 -.137E-6 .273E-5 -.270E-5

5 -34.999 -35.000 -.581E-3 .578E-3

7 -.147E-5 -.149E--5 .269E-5

LI = -.266E-5

L2 - 9 34.999 34.999 -.G53E-3 .64 7E-3

11 -35.000 -35.000 .293

LI = 1

.294

L2 = 9 37.499 34.999 -.653E-3 .637E-3

11 -37.500 -35.000 .293

LI =

.294

L2 = 1 9 34.999 37.499 -.653E-3 .64 7E-3

11 -35.000 -37.500 .293 .294

Figure 5.14. Beam trace eit selected surfaces wi th contt-ol data f

Bean Parameters

Phase(Rad)

LA LB LI L2

10 10 -1 .9 1 9 10

"0

.0

10 -1

9 - 1 9

1 '1 -1

.0 1

"0

.0

1

1 -1

1 7 G "7.7

6 1

7 -1

7 7 7

.6

7 -1

7 1 6 7

Position Uidth Power(watts]

Xl(inni) Yl(iiim) Ux(mm) Wy(mn)

-50.000 .107E-22 .531 .530 0.197

-47.500 .107E-22 .531 .530 0.202

-4 7.500 .107E-22 .531 .530 0.215

-45.000 .107E-22 .531 .530 0.220

.102E-20 -50.000 .531 .530 0.199

.103E-20 -47.500 .531 .530 0.204

.103E-20 -47.500 .531 .530 0.217

.103E-20 -45.000 .531 .530 0.222

-5.000 .107E-22 .531 .530 0.347

-2.500 .107E-22 .531 .530 0.348

-2.500 .107E-22 .531 .530 0.349

.103E-20 .107E-22 .531 .530 0.349

.103E-20 -2.500 .531 .530 0.349

.103E-20 -2.500 .531 .530 0.348

.103E-20 -5.000 .531 .530 0.348

-35.000 -30.000 .531 .530 0.216

-35.000 -32.500 .531 .530 0.213

-35.000 -32.500 .531 .530 0.204

-35.000 -35.000 .531 .530 0.201

-32.500 -35.000 .531 .530 0.204

-32.500 -35.000 .531 .530 0.213

-30.000 -35.000 .531 .530 0.216

-1.294

-1.407]

-1.538J

-1.645

1.787

1.372

1.243

0.850

-3.123

-3.132

-3.139

-3.142

-3

-3

-3.093

-0.462

-0.196

-0.108

0.179

]

]

i.l31 1

i.l24 J

:]

.lOO"! .011 J -.062

[•••]' means overlapping beams.

Figure 5.15. Here is spot position, beam width, power, and phase at surface 11 with

base control data given in figure 5.10.

62
5.4.1 Effects of Wavelength Change to the Case Field at Surface 11

In the design of an apparatus, it soeins reasonable to change various paraneters to

test how they influence the final output at the detectors in surface 11. Figures 5.1G

to 5.21 contain examples of such changes. In this subsection we discuss what happens

when there is a change in wavelength x in these figures. Basically, this change of 10%

implies a shift of position for the detracted beans (LI or L2 not zero) by 0.2359 mn. Since the bean width at surface 11 is 0.53 mm, this shift causes significant change in

functional form. This form change affects significantly the accuracy for determining

the relative phase at the neighboring sampling holes of the Hartmann plate.

This sensitivity to the change in the wavelength of the laser radiation is a principal limit to this holographic technique. To increase the dynamic range for

allov/ed changes in wavelength requires: (1) substantial adjustments to the mirror

curvatures to reduce the path differences for the diffracted beams; (2) some means to apply the correct equivalent of eq (5.4.5) shown in reference [1] to the measured

output of the detectors at surface 11; or (3) sufficient aperturing of the beams

striking the hologram at surface 7 to generate at surface 11 beams with widths near

1 mm.

BEAM LA = LB = LI = L2 =

Variable of

Ch ange

Base Case

AX = 1 pm

Abj = -1 mm

Ab^ = -1 mm

'^3 = 1 mm "^ = -1 mm

"•5 = -1 mm "6 = -1 mm '^ = 1 mm

'"8 = -1 mm

">, = -1 mm

'^0 = -1 mm

Position of Spot

Xl(mm) Yl(mm)

.1029E-20 .10C9E-22

.1219E-2G .1091E-22

.7301E-21 -.3008E-21

.1030E-20 .1048E-22

.1027E-20 .1090E-22

.7301E-21 -.3008E-21

.7301E-21 -.3008E-21

.1030E-20 .1049E-22

.1027E-22 .1070E-22

.7301E-21 -.3008E-21

.7301E-21 -.3008E-21

.7286E-21 -.3008E-21

Figure 5.16. Spot position change at surface 11 due to one control parameter

change. Base control data given in figure 5.10.

63
BEAM LA = LB = LI = 1 L2 =

Variable of

Change

Base Case

AA = -1 ym

Ab = -1 mm

Ab = -1 mm

Ab = 1 mm

Ab, = -1 mm

4 Ab = -1 mn

5 Ab = -1 mm

6 Ab = 1 mm

Ab = -1 mm

8

Ab = -1 mm

Ab = -1 mm

Figure 5.17. Spot position change at surface 11 due to one control parameter

change. Base control data given in figure 5.10.

«

Position of Spot

Xl(mm) Yl(mn)

2.5001 .1069E-22

2.2642 .1091E-22

2.5001 -.3008E-21

2.5001 .1048E-22

2.5001 .1090E-22

2.5001 -.3008E-21

2.5001 -.3008E-21

2.5001 .1049E-22

2.5001 .1070E-22

2.5001 -.3008E-21

2.5001 -.3008E-21

2.4976 -.3008E-21

64
BEAM LA =

Varia 3le of

Change

Base (lase AX = -1 pn

Abj = -1 nrn

'^2

= -1 mn

'"3

= 1 nn ^ = -1 mm

ib^ = -1 nn '^ = -1 nn

'"7

= 1 nn %- -1 nm %- -1 mm

'So

= - L nn

LB - LI = 1 L2 = 1

Position of Spot

Xl(mm) Yl(mm)

1029E-20 -2.5000

1219E-20 -2.2642

7301E-21 -2.5000

1030E-20 -2.5000

1027E-20 -2.5000

7301E-21 -2.5000

7301E-21 -2.5000

1030E-20 -2.5000

1027E-20 -2.5000

7301E-21 -2.5000

7301E-21 -2.5000

7286E-21 -2.4976

Figure 5.18. Spot position change at surface 11 due to one control parameter

change. Base control data given in figure 5.10.

65
BEAri LA = 7 LI = L2

Variable of

Change

Position of Spot

Xl(mni) Yl(mm)

Base Case

AA = -1 yn

Ab = -1 mm

Ab, 1 mm

Ab = 1 mm

Ab^ = -1 mm

4 Ab^ = -1 mm

5 Ab = -1 mm

5 Ab = 1 mm

Ab, -1 mm

Ab = -1 mm

9 Ab = -1 mm

-35.0000

-35.0000

35.0001

-35.0000

-35.0000

-35.0001

-35.0001

-35.0000

-35.0000

-35.0001

-35.0001

-34.9654

-35.0000

-35.0000

-35.0000

-35.0000

-35.0000

-35.0001

-35.0001

-35.0000

-35.0000

-35.0001

-35.0001

-34.9654

Figure 5.19. Spot position change at surface 11 due to one control parameter

change. Base control data given in figure 5.10.

66
BEAM LA = 7

Variable of

Change

Base Case

AA = -1 pm

•>! = -1 mm Abj- -1 mm

Ab3 = 1 mm

Ab, = -1 mm ^

= -1 mm Abg. -1 mm

Ab^ = 1 mm

Abg = -1 mm

*bg = -1 mm

'^10'= - . mm

LB = 7 LI

Position of Spot

Xl(mm) YKmml

-32.4999 -35.0000

-32.7358 -35.0000

-32.5000 -35.0001

-32.4999 -35.0000

-32.4999 -35.0000

-32.5000 -35.0001

-32.5000 -35.0001

-32.4999 -35.0000

-32.4999 -35.0000

-32.5000 -35.0001

-32.5000 -35.0001

-32.4678 -34.9654

Figure 5.20. Spot position change at surface 11 due to one control parameter

change. Base control data given in figure 5.10.

67
BEAM LA LB LI = L2 = -1

Variable of

Change

Position of Spot

Xl(mm) Yl(mm)

Base Case

AA = -1 yn

Ab.

Ab,

Ab.

Ab,

-1 nin

-1 nrn

Ab = 1 rnn

Ab, = -1 mm

4 Ab = -1 mm

5

-1 mm

Ab = 1 mm

-1 mm

-1 mm

Ab = -1 mm

-35.0000

-35.0000

35.0001

-35.0000

-35.0000

-35.0001

-35.0001

-35.0000

35.0000

35.0001

35.0001

34.9654

-32.5000

-32.7358

-32.5000

-32.5000

-32.5000

-32.5000

-32,5000

-32.5000

-32.5000

-32.5000

-32.5000

-32.4679

Figure 5.21. Spot position change at surface 11 due to one control parameter

change. Base control data given in figure 5.10.

5.4.2 Effects of 1 mm Changes in Positions to Base Field at Surface 11

Figures 5.16 to 5.21 show, in addition to the wavelength effects, the influence of

a 1 mm shift of each mirror on the output of the detectors at surface 11. In all cases except changes in the imaging mirror at surface 10, these movements are completely

negligible. Moving the imaging mirror causes primarily the obvious change in

magnification of the pattern at surface 9 to the pattern at surface 11. Clearly this

adjustment requires sensitive movement of the mirror.

These results imply that the final imaging mirror can cause adjustment problems.

Using fine adjustment of the detector array to find the best position of surface 11 improves the situation.

5.5 Alignment of Apparatus Using 0.6328 pm

Figures 5.22 and 5.23 show the position of two beams at selected surfaces of the

apparatus. These beams permit accurate placement within 100 um of the mirrors, the

hologram, and the detector array. This final adjustment of the apparatus will require

using a cw laser beam with 10.6 pm wavelength for maximum precision and dynamic range

of the apparatus.

58
Bean LA = 7 LB = 7 Surface XI (mm) Y1(mm) ^/(RX * A)[(mm)-2] w/(RY * X)[(mm)

1 35.000 35.000 .49CE-19 .496E-19

3 .329E-1 .323E-1 -.G93E-2 .689E-2

5 -34.999 -35.000 .657E-4 .300E-4

7 -.332E-1 -.322E-1 -.678E-2

LI = -.687E-2

L2 =

9 34.999 34.999 .8G2E-4 .361E-4

11 -35.000 -35.000

Ll= -1 L2 = 9 34 .850 34.999 .862E-4 .361E-4

11 -34.851 -35.000 4.916

LI = 4.916L2= -1

9 34.999 34.850 .852E-4 .361E-4

11 -35.000 -34.851 4.91G 4.916

Figure 5.22. Beam trace at selected surfaces with control data from figure 5.10,

X = 0.6328 pm.

tt/(RY * X)[(mm)-2]

.496E-19

-.689E-2

.300E-4

-.687E-2

L2 = .361E-4

4.916

L2 = .361E-4

4.916

L2 = 1 .361E-4

4.916

5.6 High Irradiance Constraints in the Design for the 10.6 pm Wavelength

Reference [1] established that the peak irradiance allowed at the Hartnann plate

is 200 W/cm2. The surface hologram fixes this limit because the coherence effects in

the Fourier plane at surface 7 can increase the peak irradiance by a factor of r|2,

where N is the number of sampling holes. This sampling process also reduces the peak

irradiance by the factor [7ra2/(4Af )]2, where a is the diameter of the sampling

holes, f is the effective distance to the Fourier plane from the Hartmann plate, and x is the wavelength of the radiation. For the current design, this factor becomes

5.5 X 10"3. Given that N = 300, we find the peak irradiance at the hologram is

99 kW/cm2 which approaches the cw limit for mirrors with copper substrates.

This 200 W/cm2 at the Hartmann plate implies a 15 kW bean, which is a low range

in high-power laser beams. This constraint implies the beam profile apparatus must use

69

Bean LA = LB - Surface XI (nm) Yl(mm) w/(RX * A)[(mm)

1 0.000 0.000 .496E-19

3 -.141E-18 -.353E-18 -.693E-2

5 .432E-19 .376E-23 .657E-4

7 .588E-20 .353E-18 -.678E-2

LI = 9 .350E-20 -.508E-23 .862E-4

11 .310E-21 .120E-22 4.916

LI = 1 9 .149 -.508E-23 .862E-4

11 -.149 .120E-22 4.916

LI =

9 .350E-20 .149 .862E-4

11 .310E-21 -.149 4.916

Figure 5.23. Beam trace at selec ted surfaces wi
a beam splitter or grating in front of it to look at 20 kW and higher-power beams.

This apparatus captures from the splitter one of the lower peak irradiance beams that

can satisfy this 200 W/cm2 limit. In the conclusion (section 10), I mention a way to increase the allowed flux significantly. Consequently, we must test if this

lower-power beam copies the primary beam.

There are at least three test strategies depending on the primary beam.

First, if the original high-power beam has known range of irradiances and its

highest value does not exceed the damage threshold of the beam splitter or grating,

then the one-time test using a laser beam of much smaller cross section but with peak

irradiance equal to the peak of original laser beam can be used. Here we would

determine the splitting ratio as a function of position and verify that this peak

irradiance does not change the splitting ratio [11,12] from the value measured at much

lower irradiances.

Second, if the potential peak irradiance has significant power compared with the

power in the rest of the beam, then a pair of calorimeters can check for potential

power dependence in the splitter ratio.

Finally, if the primary beam has unknown peak irradiances, then real-time

measurements are needed. Pulse systems or situations that require control or extreme

precision are examples for real-time measurements. Two methods to obtain these

measurements are: (1) an ideal technique that is s/ery expensive, uses two beam profile

apparatuses to sample simultaneously several of the lower-power diffraction orders and

compares their beam profiles in real time. This process tests for significant

distortion by the grating or splitter; (2) scattered radiation from the splitter or grating can show changes in the splitter ratio. Implementation of this second

technique depends on the expected scattering. Again, ratio measurements of several

beams are necessary for both methods.

In conclusion, measuring high-power laser beams requires boostrap techniques that

extrapolate the results from direct measurements in low-power beams. Appropriate tests

must be made to confirm that the extrapolation process is valid.

6. THE 1.06 ym WAVELENGTH

Here we change the scale of two mirrors in the apparatus to make it work

accurately for 1.06 ym. We reduce their curvatures by approximately 10 for surfaces 4 and 10. This section is briefer than section 5 because most concepts and criteria for

adjustment of the control variables are defined in section 5 and its subsections. In section 6 and its subsections we report only changed values because we wish to use the

same apparatus as much as possible for a range of wavelengths. In this way, the unit can justify its cost. If a unit with multiple ranges of wavelength is not needed, then

three mirrors could be eliminated if the resulting physical size of the apparatus

proves to be convenient.

Because the design of the apparatus is directed first toward the 10.6 ym and

second toward the 1.06 ym, the choice of arbitrary values is controlled by the

decisions in subsections 5.1 and 5.2. In subsection 6.1 we note the new changes in some values and in subsection 6.2 we define the new adjustment criteria given these

changes.

We again examine the spherical and elliptical cases. In subsection 6.3 we document the data for spherical mirrors, and in the remaining subsections we document

the data for the elliptical mirrors in direct correspondence to that done for 10.6 ym.

70
In subsection 6.4 we generate the base data for the apparatus using elliptical

mirrors. Here we accept that all mirrors can change their x curvatures. These results measure how far the ideal apparatus deviates from the practical that is optimized for

10.6 urn.

In subsection 6.4.1 we show what happens when the wavelength is changed to 0.96 ym, and in subsection 6.4.2 we show how the base data change when each distance

variable is moved by 1 mm.

Because 1.06 ym is inconvenient for initial alignment, in subsection 6.5 we trace

0.6328 ym beams through the apparatus containing the elliptical mirrors.

This section is concluded with subsection 6.6 in which the allowed peak irradiance

on the hologram is identified.

6.1 Selecting Some Initial Control Values

Subsection 5.1 contains all the selected arbitrary control values for the

apparatus. In this subsection we only comment on one situation that becomes different.

The Hartmann plate has 1 mm holes for sampling the laser beam. These sampled beams

become Airy disks with 2.6 mm diameters. This significant reduction in beam size

implies reduction in the absorption rate of the laser beam as it progresses through the

apparatus to one tenth of the 10.6 pn absorption rate.

6.2 The Convergence Criteria for the Rest of the Control Values

The discussion in subsection 5.2 defines the remaining control values for all

wavelengths. It is only necessary to adjust M in each formula.

There remains one discrepancy implied by the discussion in subsection 6 compared

to the data contained in later subsections. Here adjustments for the spherical and the

elliptical mirror cases allow all relevant curvatures and distances to change as if the

previous constraints generated for the 10.6 ym were not present. It becomes obvious

from data in these later subsections that this discrepancy makes no difference.

Nevertheless, I suggest the builder of the final apparatus use the computer program

with the true constraints to achieve a proper adjustment of the apparatus. Because the

possibilities for all types of design are very large, this paper does not try to document a final design.

6.3 The Spherical Optics Data

The discussion of subsection 5.3 applies here. The figures relevant to this

subsection are 6.1 to 6.5. Again, the simulations show spherical symmetries preclude

high precision measurements of beam profile unless significant changes in the design

take place.

71
VERSION 6/29/78

BEAM TRACE

WAVLGN= 1.06

COS(ANG)=

LA = 7 LC,LD = 7

L=

12

34

56

78

910

11 WX(MM) = .5 AMPL = 1

********* 08/08/78 08:39:43

MICROMETERS ANG. .14 RAD

.990216

LB = 7

7 S1,T1(MM) =: 5 5

XCUR(M) YCUR(M) APER(MM)RADIUS L TO L+1 DIS(M)

.1E21 .1E21 10000 1.0098807

-2 -2 224 1.0098807

.1E21 .1E21 400 .10098807

.2 .2 224 .10098807

.1E21 .1E21 400 1.0098807

-2 -2 224 1.0098807

.1E21 .1E21 20 1.0098807

-2 -2 224 1.0098807

.1E21 .1E21 400 .11108688

.2 .2 224 1.1108688

.1E21 .1E21 400 -30

UY(MM) = .5 RX(M) = .1E21

PHASE(RAD) = RY(M) = .1E21

Figure 6.1.

END OF INITIAL DATA

S7, T7 (1/MM)= 1.4818821 1.4673833

Initial guess of control data for spherical mirrors.

VERSION 6/29/78

BEAM TRACE

WAVLCN = 1.06

COS(ANG) =

- LA = 7 LC,LD = 7

L=

12

34

56

78

910

11 WX(MM) = .5 AMPL = 1

********* 08/08/78 09:32:32

MICROMETERS ANG = .14 RAD

.990216

LB - 7

7 S1,T1(MM) ~- 5 5

XCUR(M) YCUR(M) APER( MM) RAD I US L TO L+1 DIS(M)S

.1E21 .1E21 10000 1.0053952

-2 -2 224 1.0098807

.1E21 .1E21 400 .10098807

.2 .2 224 .99019205E-1

.1E21 .1E21 400 1.0008226

-2 -2 224 1.0098807

.1E21 .1E21 20 1.0098807

-2 -2 224 .97981908

.1E21 .1E21 400 .11108688

.2 .2 224 1.1022951

.1E21 .1E21 400 -30

WY(MM) = .5 RX(M) = .1E21

PHASE(RAD) = RY(M) = .1E21 END OF INITIAL DATA

S7, T7 (1/MM)- 1.453029 1.4673735

Figure 6.2. Final values of control data for spherical mirrors

72
Bean LA = LB = Surface XI (mm) Yl(mn) Tr/(RX * A)[(nm)-2] Tr/(RY * X)[(mm)

1 0.0000 0.000 .296E-19 .296E-19

3 -.223E-18 -.353E-18 -.261E-1 .292E-4

5 .313E-20 .149E-24 102.33 -.476E-2

7 .210E-17 .353E-17 -.644E-1

LI = .480E-6

L2 = 9 -.779E-19 -.680E-25 68.851 -.126E-2

11 .849E-18 .679E-24 3.555

LI = 1 2.935

L2 =

9 .250 -.680E-25 68.851 -.126E-2

11 -2.554 .679E-24 3.555

LI = 2.935

L2 = 1 9 -.779E-19 .250 68.851 -.126E-2

11 .849E-18 -2.500 3.555 2.935

Figure 6.3. Bean trace at selected surfaces vn th control data from

Bean LA = 7 LB = 7 Surface XI (mm) Y1(mm) Tr/(RX * X)[(mm)-2] Tr/(RY * X)[(mm)

1 35.000 35.000 .296E-19 .296E-19

3 -.695 -.325E-3 -.261E-1 .292E-4

5 -3.485 -3.500 102.33 -.476E-2

7 7.787 .616E-2 -.644E-1

LI = .480E-6

L2 = 9 3.212 3.500 68.851 -.126E-2

11 -32.648 -35.000 3.555

LI = 1

2.935

L2 = 9 3.462 3.5000 68.851 -.126E-2

11 -35.201 -35.000 3.555

LI = 2.935L2 = 1 9 3.212 3.750 68.851 -.126E-2

11 -32.648 -37.500 3.555 2.935

Figure 6.4. Beam trace at selected surfaces with control data from figure 6.1.

73
Beam LA = LB = Surface XI (mm) Yl(mm) V /(RX * A)[(mm) -2] 7r/(RY * A)[(mm)

1 0.000 0.000 .296E-19 .296E-19

3 -.222E-ia -.353E-18 -.126E-1 .131E-1

5 .lllE-20 .705E-17 1.519 -101.235

7 .210E-17 .353E-17 -.315E-1

LI = .322E-1

L2 = 9 -.320E-20 .129E-18 10.372 -69.244

11 .596E-19 -.128E-17 3.930

LI = 1 2.257

L2 = 9 .245 .129E-18 10.372 -69.244

11 -2.484 -.128E-17 3.930

LI = 2.257

L2 = 1 9 -.320E-21 .250 10.372 -69.244

11 .596E-19 -2.479 3.930 2.257

Figure 6.5. Beam trace at sel ected surfaces with control data fr

Beam LA = 7 LB = 7 Surface XI (mm) Yl(mm) Tr/(RX * A)[(mm)--2] ^/(RY * X)[(mm)

1 35.000 35.000 .296E-19 .296E-19

3 -.695 -.323E- 3 -.126E-1 .131E-1

5 -3.500 -3.500 1.519 -101.235

7 7.787 .616E-2 -.315E-1

LI = .322E-1L2 = 9 3.500 3.500 10.372 -69.244

11 -35.298 -34.704 3.930

LI = 1 2.257L2 = 9 3.745 3.500 10.372 -69.244

11 -37.782 -34.704 3.930

LI = 2.257

L2 = 1 9 3.500 3.750 10.372 -69.244

11 -35.298 -37.182 3.930 2.257

Figurei 6.6. Beam trace at se 1 ected surfaces wi th control data from

6.4 The Elliptical Optics Data

Parallel to the discussion of subsection 5.4, this subsection generates comparable

simulations for elliptical mirrors. Figures 6.7 through 6.13 apply here. Most

conclusions are identical to those in subsection 5.4. We now conclude that the

apparatus can work for both wavelengths. Only the mirrors at surfaces 4 and 10 need to

be changed from those used by the 10.6 pm configuration. Because the focal lengths of

these mirrors change various parts of the apparatus by a factor ten, it is very easy to accommodate the change in wavelength.

74
VERSION 6/29/1^

BEAM TRACE

IJAVLGN = 1.06

COS(ANC) = LA = 7 LC,LD = 7

L=

1234

56

78

910

11 WX(MM) = .5 AMPL = 1

********* 08/08/78 08:25:53

MICROMETERS ANG = .14 RAD .990216

LB = 7 7S1,T1(MM) ^ 5 5

XCUR(M) YCUR(M) APER(MM)RADIUS L TO L+1 DIS(M)

.1E21 .1E21 10000 1.0098807

-2.0397178 -2 224 1.0098807

.1E21 .1E21 400 .10098807

.20397178 .2 224 .10098807

.1E21 .1E21 400 1.0098807

-2.0397179 -2 224 1.0098807

.1E21 .1E21 20 1.0098807

-2.0397178 -2 224 1.0098807

.1E21 .1E21 400 .11108688

.20397178 .2 224 1.1108688

.1E21 .1E21 400 -30

UY(MM) = .5 RX(M) = .1E21

PHASE(RAD) = RY(f1) = .1E21 END OF iniTIAL DATA

S7, T7 (1/MM ) = 1.4530265 1.4673833

Figure 6.7. Initial guess of control data on elliptical mirrors.

VERSION 6/29/78

BEAM TRACE

WAVLGN = 1.06

COS(ANG) = LA = 7 LC,LD = 7

L=

12

34

56

78

910

11 WX(MM) = .5 AMPL = 1

********* 07/17/78 11:17:50

MICROMETERS ANG = .14 RAD .990215

LB - 7 7S1,T1(MM) = 5 5

XCUR(M) YCUR(M) APER(MM)RADIUS L TO L+1 DIS(M

.1E21 .1E21 10000 1.0098859

-2.0397183 -2 224 1.0098713

.1E21 .1E21 400 .10098901

.2039718 .2 224 .10098787

.1E21 .1E21 400 1.0098862

-2.0397399 -2 224 1.0089452

.1E21 .1E21 20 1.0090289

-2.039696 -2 224 1.0098752

.1E21 .1E21 400 .11108687

.20396776 .2 224 1.110874

.1E21 .1E21 400 -30

WY(MM) = .5 RX(M) = .1E21

PHASE RAD) = RY M = .1E21

Figure 6.8.

END OF INITIAL DATA

S7, T7 (1/MM)= 1.453029 1.4673735

Final values of control data on ellipitical mirrors.

75
Bean LA = 7 LB = 7 Surface XI (mm) Y1(mm) tt/(RX * A)[(mm)-2] Tr/(RY * x)[(mm)-2]

1 35.000 35.000 .296E-19 .296E-19

3 -.334E-3 -.325E-3 .297E-4 .292E-4

5 -3.500 -3.500 -.441E-2 -.476E-2

7 .637E-2 .616E-2 .344E-6 .480E-6

LI = L2 =

9 3.500 3.500 .420E-2 -.126E-2

11 -35.000 -35.000 2.93 2.935

LI = 1 L2 =

9 3.750 3.500 .420E-2 -.126E-2

11 -37.500 -35.000 2.93 2.935

LI = L2 = 1 9 3.500 3.750 .420E-2 -.126E-2

11 -35.000 -37.500 2.93 2.935

Figure 6.9. Beam trace at selected surfaces with control data from figure 6.1.

Beam LA = LB = Surface XI (mm) Yl(mm) tt/(RX * x)[(mm)-2] Tr/(RY * A)[(mm)

1 0.000 0.000 .296E-19 .296E-19

3 -.230E-18 -.353E-18 .297E-4 .292E-4

5 -.116E-20 -.149E-24 -.441E-2 -.476E-2

7 .218E-17 .353E-17 .344E-6

LI = .480E-6L2 =

9 .L30E-21 -.680E-25 .420E-2 -.126E-2

11 .118E-19 .679E-24 2.93

LI = 1

2.935

L2 = 9 .250 -.680E-25 .420E-2 -.126E-2

11 -2.500 .679E-24 2.93

LI = 2.935L2 = 1 9 .130E-21 .250 + .420E-2 -.126E-2

11 .118E-19 -2.500 2.93 2.935

Figure 6.10. Beam trace at selected surfaces with control data from figure 6.7.

76
Bean LA = LB = Surface XI (mm) Yl(mm) tt/(RX * X)[(mm)-2] Tr/(RY * A)[(mm)

1 0.000 0.000 .296E-19 .296E-19

3 -.230E-18 -.353E-18 .203E-6 -.603E-6

5 -.116E-20 -.G67E-24 -.105E-1 -.103E-1

7 .218E-17 .353E-17 .159E-4

LI = -.154E-4

L2 =

9 .130E-21 .431E-24 -.717E-3 -.103E-2

11 .118E-19 -.299E-23 2.936

LI = 1 2.935

L2 =

9 .250 .431E-24 -.717E-3 -.103E-2

11 -2.500 -.299E-23 2.936

LI = 2.935L2 = 1 9 .130E-21 .250 -.717E-3 -.103E-2

11 .118E-19 -2.500 2.936 2.935

Figure 6.11. Bean trace at selected surfaces with control data from figure 6.8.

Bean LA = 7 LB = 7 Surface XI (nm) Yl(mm) Tr/(RX * A)[(mm)-2] ^/(RY * X)[(mm)

1 35.000 35.000 .296E-19 .296E-19

3 .392E-6 .lOlE-5 .203E-6 -.603E-6

5 -3.500 -3.500 -.105E-1 -.103E-1

7 .128E-4 -.130E-4 .159E-4

LI = .

-.154E-4

L2 =

9 3.500 3.500 -.717E-3 -.103E-2

11 -35.000 -35.000 2.936

LI = 1

2.935

L2 = 9 3.750 3.500 -.717E-3 -.103E-2

11 -37.500 -35.000 2.936

LI = 2.935

L2 = 1 9 3.500 3.750 -.717E-3 -.103E-2

11 -35.000 -37.500 2.936 2.935

-2]

Figure 6.12. Beam trace at selected surfaces with control data from figure 6.8.

77
Bean Pa rameters Posi tion 111 dth Power(watts)

Phase(Rad)

LA LB LI L2 Xl(m) Yl(m) Wx(nn) Uy(mm)

10 -50.000 -.299E-23 .529 .529 0.315 -0.562

plO -1 -47.500 -.299E-23 .529 .529 0.316 -0.800-1

-0.825-' L

9 1 -4 7.500 -.299E-23 .529 .529 0.322

9 -45.000 -.299E-23 .529 .529 0.322 -1.052

10 .118E-19 -50.000 .529 .529 0.316 -3.170

r

° 10 -1 .118E-19 -47.500 .529 .529 0.316 -3.154-.

-3.179-' L

9 1 .118E-19 -4 7.500 .529 .529 0.322

9 .118E-19 -45.000 .529 .529 0.322 -3.164

1 -5.000 -.299E-23 .529 .529 0.351 -3.116

r

1 -1 -2.500 -.299E-23 .529 .529 0.351 -3.134-.

•-0

1 -2.500 -.299E-23 .529 .529 0.351 -3. 136-' .118E-19 -.299E-23 .529 .529 0.351 -3.142

r° 1 .118E-19 -2.500 .529 .529 0.351 -3.142-,

•-0 1 -1 .118E-19 -2.500 .529 .529 0.351 -3.141^

1 .118E-19 -5.000 .529 .529 0.351 -3.142

7 6 -35.000 -30.000 .529 .529 0.321 -1.888

[7 5 1 -35.000 -32.500 .529 .529 0.321 -1.897-,

-1.881 -7 -' 7 -1 -35.000 -32.500 .529 .529 0.317

7 7 -35.000 -35.000 .529 .529 0.316 -1.891

r

7 7 -1 -32.500 -35.000 .529 .529 0.317 -2.057-1

-2.074 •-6 -

7 1 -32.500 -35.000 .529 .529 0.321

6 7 -30.000 -35.000 .529 .529 0.321 -2.227

"[...]" neans overlapping beans.

Figure 6.13. Here is spot position, bean width, power, and phase at surface 11 with

base control data given in figure 6.8.

6.4.1 Effects of Wavelength Change to the Base Field at Surface 11

The 10 percent change in the wavelength at 1.06 pn inplies exactly the sane shift

as a 10 percent change in the wavelength at 10.6 yn. This fact is shown by the data

in figures 6.14 to 6.19. The conclusions of this subsection are those of

subsection 5.4.1.

78
BEAM LA = LB = LI = L2 =

Variable of Position of Spot

Change Xl(mrn) Yl(rnm)

Base Case .1182E-19 -.2994E-23

AX = -0.1 uf^ .1292E-19 -.3146E-23

Ab = -1 mm .1162E-19 -.3161E-23

Ab = -1 mm .1193E-19 -.2288E-22

Ab = 1 mm .1171E-19 .1689E-22

Ab„ = 1 mm .3070E-19 .3130E-19

4 Ab^ = -1 mm -.7054E-20 -.3131E-19

5 Ab^ = -1 mm .1182E-19 -.3191E-23

6 Ab = -1 mm .1182E-19 -.2995E-23

Ab^ = -1 mm -.7055E-20 -.3131E-19

8 Ab = -1 mm -.7055E-20 -.3131E-19

y Ab = 1 mm .1202E-19 .3101E-21

Figure 6.14. Spot position change at surface 11 due to one control parameter

change. Base control data given in figure 6.8.

79
BEAM LA = LB = LI = 1 L2 =

Variable of Position of Spot

Change Xl(mm) Yl(inm)

Base Case

AX = -0.1 pn

Ab = -1 mm

Ab = -1 mm

Ab = 1 mm

Ab = 1 mm

Ab = -1 mm

5 Ab^ = -1 mm

5 Ab = -1 mm

Ab = -1 mm

Ab = -1 mm

Ab = 1 mm

Figure 6.15. Spot position change at surface 11 due to one control parameter

change. Base control data given in figure 6.8.

2.5000 -.2994E-23

2.2642 -.3146E.23

2.5000 -.3161E-21

2.5000 -.2288E-22

2.5000 .1689E-22

2.5000 .3130E-19

2.5000 -.3131E-19

2.5000 -.3191E-23

2.5000 -.2995E-23

2.5000 -.3131E-19

2.5000 -.3131E-19

2.5025 .3101E-21

80
BEAM LA = LB = LI = L2 = 1

Variable of Position of Spot

Change XI(ciiti) Yl(mm;

Base Case

AA = -0.1 pn

Ab = -1 mm

Ab^ = -1 mm

Ab = 1 mm

Ab = 1 mm

Ab = -1 mm

5 Ab = -1 mm

6 Ab = -1 mm

Ab = -1 mm

8 Ab = -1 mm

Ab = 1 mm

Figure 6.16. Spot position change at surface 11 due to one control parameter

change. Base control data given in figure 6.8.

1182E-19 -2.5000

1292E-19 -2.2641

1162E-19 -2.5000

1193E-19 -2.5000

1171E-19 -2.5000

3070E-19 -2.5000

7054E-20 -2.5000

1182E-19 -2.5000

1182E-19 -2.5000

7055E-20 -2.5000

7055E-20 -2.5000

1202E-19 -2.5025

81
BEAM LA = 7 LB = 7 LI = L2 =

Variable of Position of Spot

Change Xl(m) Yl(m)

Base Case -35.0000 -35.0000

AA = -0.1 ym -35.0000 -35.0000

Ab = -1 mm -35.0000 -35.0000

Ab = -1 mm -35.0000 -35.0000

Ab = 1 mm -35.0000 -35.0000

Ab = 1 mm -35.0000 -35.0000

Ab^ = -1 mm -35.0000 -35.0000

Ab^ = -1 mm -35.0000 -35.0000

Ab = -1 mm -35.0000 -35.0000

Ab„ = -1 mm -34.9999 -34.9999

o Ab = -1 mm -34.9999 -34.9999

Ab = 1 mm -35.03476 -35.0347

Figure 6.17. Spot position change at surface 11 due to one control parameter

change. Base control data given in figure 6.8.

82
BEAM LA LB = 7 LI

Variable of

Change

Position of Spot

Xl(mni) Yl(mm)

Base Case

AX -0.1 pm

Ab = -1 nm

Ab, -1 mm

Ab = 1 mm

Ab = 1 mm

Ab,

Ab

Abj

Ab,

7

-1 mm

-1 mm

-1 mm

-1 mm

-1 mm

Ab = 1 mm

-32.5000

-32.7258

-32.5000

-32.5000

-32.4999

-32.4999

-32.5000

-32.5000

-32.5000

-32.4 999

-32.4 999

-32.5321

-35.0000

-35.0000

-35.0000

-35.0000

-35.0000

-35.0000

-35.0000

-35.0000

-35.0000

-34.9999

-34.9999

-35.0347

Figure 6.18. Spot position change at surface 11 due to one control parameter

change. Base control data given in figure 6.8.

83
BEAM LA = 7 LB = 7 LI = L2 = -1

Variable of Position of Spot

Change Xl(mm) Yl(mm)

Base Case -35.0000 -32.5000

AX = -0.1 pn -35.0000 -32.7359

Ab = -1 m -35.0000 -32.5000

Ab = -1 nm -35.0000 -32.5000

Ab = 1 nn -35.0000 -32.5000

Ab^ = 1 nm -35.0000 -32.5000

Ab^ = -1 trim -35.0000 -32.5000

b Ab^ = -1 mm -35.0000 -32.5000

Ab - -1 mm -35.0000 -32.5000

Ab^ = -1 mm -34.9999 -32.5000

o Ab = -1 mm -34.9999 -32.5000

Ab = 1 mm -35.0347 -32.5322

Figure 6.19. Spot position change at surface 11 due to one control parameter

change. Base control data given in figure 6.8.

6,4.2 Effects of 1 mm Changes in Positions to Base Field at Surface 11

Again, figures 6.14 to 6.19 show the effects due to 1 mm shifts. The conclusions

are identical to those of subsection 5.4.2.

6.5 Alignment of Apparatus Using 0.6328 ym

Figures 6.20 and 6.21 show the alignment results for the 1.06 ym configuration.

These results parallel those of subsection 5.5.

84
Bean LA = 7 LB = 7 Surface XI (mn) YHm) ^/(RX * A)[(mni)-2] Tr/(RY * X)[(mm)

1 35.000 35.000 .496E-19 .496E-19

3 .213E-3 .210E-3 -.448E-4 -.4 54E-4

5 -3.500 -3.500 -.359E-2 -.350E-2

7 -.406E-2 -.397E-1 .258E-4

LI = -.265E-4

L2 =

9 3.500 3.500 .689E-2 .665E-2

11 -35.000 -35.000 4.917

Ll= -1 4.916

L2 =

9 3.351 3.500 .689E-2 .665E-2

11 -33.508 -35.000 4.917

LI = 4.917

L2= -1

9 3.500 3.351 .C89E-2 .665E-2

11 -35.000 -33.508 4.917 4.916

Figure 6.20. Beam trace at selected surfaces with control data from figure 6.8,

X = 0.6328 pm.

2]

Bean LA = LB = Surface XI (mm) Yl(mm) n /(RX * A)[(mm)-2] 7r/(RY * X)[(nm)

1 0.000 0.000 .496E-19 .496E-19

3 -.141E-18 -.353E-18 -.448E-4 -.454E-4

5 -.178E-20 -.667E-24 -.359E-2 -.350E-2

7 .119E-17 .353E-17 .258E-4

LI = -.256E-4

L2 =

9 .185E-21 .499E-24 .689E-2 .665E-2

11 .166E-19 -.357E-23 4.917

Ll= 1 4.916

L2 =

9 .149 .499E-24 19.132 19.513

11 -1.492 -.357E.23 4.917

LI = 4.916L2= 1 9 .185E-21 .149 .689E-2 .665E-2

11 .166E-19 -1.492 4.916 4.916

Figure 6.21. Bean trace at selected surfaces with control data from figure 6.8.

6.6 High Irradiance Constraints in the Design for 1.06 ym Wavelength

The discussion in subsection 5.6 applies to this configuration. The only changes

are the wavelength and the effective focal length. Their product does not change;

therefore, the allowed irradiance at the Hartnann plate is unchanged. It is still

200 lJ/cm2, and the potential peak irradiance on the hologram is still 99 kW/cm2.

7. PUTTING THE RESULTS OF PREVIOUS AND NEW SIMULATIONS TOGETHER

TO UNDERSTAND THE APPARATUS

This section fixes some additional conditions for proper use of the apparatus in

three separate discussions. The first lists the key features deduced from sections 5 and 6. The second discussion examines how a phase curvature in a hole at the Hartmann

plate affects the measured output at a detector. These new simulations as summarized

85
in figure 7.1 help define the allov;ed special frequencies for bean profiles. Finally,

the section examines phase sensitivity.

7.1 Key Features of Sections 5 and 6

The key features deduced from the simulations of sections 5 and 6 are:

1. The peak irradiance allowed for the apparatus is 200 IJ/cm^ regardless of wavelength if the hologram is on a metal substrate.

2. Appropriate change in two mirrors allows beam profile measurements at any wave- length between 1.06 and 10.6 urn.

3. To make precision measurements for details in beam profile, the wavelength from a laser source must be within 1 percent of the wavelength used to set up the

apparatus.

4. Given the nominal dimensions and angles in these simulations, the apparatus can be

packaged in a volume 2x2x0.5 m^ if the hologram at surface 7 reflects the beam.

5. The power and phase distortions by the apparatus for each beam tracing through the

apparatus are less than 5 percent in power and less than 0.2 rad in phase between

interfering beam pairs. These simulations shown in figures 5.15 and 6.13 do not reflect defraction details as shown by nonzero G(m,n) at the hologram. Direct

measurements will be necessary to fix the correct values for G(m,n). For our purposes, these details do not change the basic results from these simulations.

They mainly reduce the power received at the detector for each beam pair by one order of magnitude (see section 8 and subsection 3.3 for further details.

6. The power variation is less than 50 percent from the sampled beam at the center of

the Hartmann plate compared with those sampled beams on the edge of the plate.

This variation is sensitive to the apertures of each mirror and the hologram. If

this variation is a problem, then some mirrors will need larger diameters than

those chosen in the simulation. Of course, the cost of the apparatus increases

significantly.

7.2 Results of New Simulations

In all the previous simulations, the phase front at each sampling hole is presumed

to be a plane wave normal to the Hartmann plate. Ue can change this constraint so that

each hole has some curvature of the phase front at the hole and can be structured so that the direction of this front is no longer normal to the plate. To accomplish these

changes, we would adjust LC and LD to allow nonnormal phase fronts in the x and y

directions, respectively. For our purposes, it is unnecessary to make variations for

both directions because the differences between the two directions are not significant

at this time. For more precise simulations, details in direction could become

important. Similar notions apply to the two curvatures, RX and RY, at each sampling

hole. Therefore, we only vary the LC and RX in the simulations shown in figure 7.1.

86
A(un) Power(Watts) RX(m) $ (rad)++ a (LC * c/RX;

1.06 0.316 1E20 -1.891

1.06 0.205 1E3 2937.204 lE-3

1.05 0.316 1E3 25.5328 lE-4

1.06 0.198 1E6 2,963,737.5 lE-3

1.06 0.315 1E6 29,633.5 lE-4

10.6 0.201 IE 20 0.179

10.6 0.201 1E6 296,374.02 lE-3

10.6 0.201 lEG 2963.707 lE-4

10.6 0.201 1E3 293.985 lE-3

10.6 0.201 1E3 2.911 lE-4

+ Here c = 5 mm.

++ Crude model hds ^ = ^q + RX * a'^ * -n * D/X and implies

D

~ 1E3 if a < lE-4 for 1.06 pm

a < lE-3 for 10.6 ym

Conclusionl !1

If RX * a2 * D/x << 1, then

pov/er and phase are unambiguous measurements for either wavelength.

If LC < 10, then we require

RX > 2^5 E6 m for 1.01 ym, and

> 2.5 E5 m for 10.6 pm for unambiguous measurements.

Figure 7.1. Summary of simulation changing the parameters RX and LC for beams

LA = LB = 7 and LI = L2 = LD = 0.

The simulations in figure 7.1 make it clear that there are limits to the allowed

curvature at each sampling hole and that there are severe limits to the angle as defined by a relative to the surface normal of the Hartmann plate. In brief, this

angle must be less than 1 mrad for 10.6 pm and 100 mrad for 1.06 pm if we are to have

insignificant power attenuation of the sampled beams. Further, to avoid ambiguity in

the phase measurements, the curvature must be greater than 2.5 E +5 m for 1.06 pm and

2.5 E +5 m for 10.6 pm. Here we presume that LC cannot exceed 10. This latter result

implies very stringent constraints to the apparent curvature at each sampling hole and

to the degree the phase front deviates from the Hartmann plate normal. To relax these

conditions requires significant change in the focal lengths of the mirrors and the

allowed beam size. For the simulations in this paper, we presume that the local

curvature at each hole is a plane wave with a phase front parallel to the Hartmann

plate.

7.3 Phase Changes by Noise, Amplitude, Beam Direction,

and Various Adjustments in Wavelength and Mirror Positions

We have examined many properties and variables of the reflection apparatus. We now look at the allowed variation in power and the phase between neighboring beams from

the prefilter so there are unambiguous and reasonable accuracy measurements of phase at each sampled point. To be sure you know the phase at each point uniquely requires the

direction variation over the entire beam profile of 10 cm diameter has the total phase

87
variation, a$, less than 2it from one edge of the beam to the other edge along any

diameter through the beam center. This constraint implies a strong constraint to the

average beam direction, a„.

Thus with:

A$ = k a aX,

where

aX = 100 mm, and A9>j< Zir , then

a. _< 10 prad for 1.06 ym and

<_ 100 prad for 10.6 ym.

If this average beam direction varies beyond this range given above, there can be

(2n + 1)tt ambiguities in the phase front. If the beam phase front is sufficiently

understood from previous information, then moderate changes in phase beyond this Zir restriction across the front can be inferred. In this case, the allowed range of

ap^ may increase by a factor of 10 to equal the constraint on beam direction, a, at each hole.

So far, the allowed variation in amplitude to get accurate phase measurements

between neighboring sampling holes has not been fixed. Briefly, each pair of sampling

holes in the prefilter generates three beams at surface 11, two with amplitude-only

information labeled as Ii and I3 and a third labeled as I2. The latter has

relative phase information. If we pretend for this discussion that all calibration

effects by each detector cancel and set the relative phase between the two beams equal to 0, the cos Q is given as:

cos Q . (I^ - l^ - I^)/2^/I^ . (7.1)

The errors in measuring the relative phase, namely DQ, are primarily due to electronic noise in the three detectors of interest at surface 11. We simulate that

noise by:

(DI^)^ = (gl^)^ .

(DI^)^ = (912)^ ' and

(013)^ = (913)^ . (7.2)

Here we model the dominant noise as due to the amplication process in each detector;

therefore, the noise is proportional to the appropriate input signal. If the final

apparatus has different noise sources, this discussion must be changed accordingly.

These results define the procedure.

88
In our analysis, we presune that the noises are independent between each detector;

therefore, the standard deviation of (DQ)^ is given as:

8(Dq)^ sin^ Q = g^ESd^)^ + (I^ - ^^)^^/\ly (7.3)

If we accept for this discussion that I^ is constrained by eq (7.1), that I^ is nore intense than I3, and that I3 = u I^, where u _< 1, eq (7.3) becomes:

8(DQ)^ sin^ Q - g''[3(l + u + Z/il cos Q)'" + (1 - u)^]/u. (7.4)

There are three situations of primary interest in this equation, namely Q = 0,

ir/2, and tt.

If Q = Tr/2, then:

(DQ)^ = g^[3(l + u)^ + (1 - u)^/8u. (7.5)

Here minimum error has u = 1. When u^O, the error increases without bound. Since the error is bounded to be less than ir, this means the phase becomes undefined once u is small enough. How small depends on the noise value indicated by g.

If Q is near tt or 0, the error is unbounded; therefore, the discussion must

change. Here we set DQ = it and estimate how close we can get Q = or ti for a given u. Here we let Q approach 0, then:

Q^ = g^[3(l + u 2 u)^ + (1 - u)'^]/(8w^ u). (7.6)

Here the smallest Q happens when u = 1, and the value of Q increases without bound

when u approaches zero.

When Q is near ir, it is possible for it to become arbitrarily close to it when u becomes close to 1. In this case we set DO = it and equate Q = it - e where:

e^ = g'^[3(l + u - 2 u)^ + (1 - u)^/(8tt^u). (7.7)

Note that e = when u = 1. Again, as u goes to zero, the phase Q becomes

undefined.

In summary, we have seen that the errors in measurement are complex functions of

the actual beam profile. Therefore, a simple picture of the error profile for the

phase front is not possible without significant constraints in the allowed beam

profile. The results of sections 5 and 5 along with this section should permit the

inference of beam profiles that allow precision measurements of phase. When the

measurements require less precision, then the constraints on the profile can be

reduced. One final question remains. How do the phase and power for the selected test

beams change when the wavelengths and positions of the mirrors change. Figures 7.2 and

7.3 show the phase shifts caused by these changes. These results show that there is

significant sensitivity to some position changes. This can allow fine positioning

adjustments of the apparatus. The power variation for each beam is completely

negligible under these conditions and therefore is not tabulated.

89
lie make a brief overview based on these simulations. They show that there can be

accurate phase and amplitude measurements provided: (1) there is less than 1 percent

change in wavelength relative to the setup wavelength; (2) there are significant

restrictions on the local beam curvatures, the average beam direction, and the bounds

for the variation of the phase front; (3) there is a range of wavelengths that can be

used if certain mirrors are changed; and finally, as is always the case, there is an ultimate accuracy of this system for measurements which are controlled by the noise in

the detectors.

Beam LA = 7 7 7 LB = 7 7 7 LI = 1 -1 L2 = 1 C -1

Base -3.142 -3.139 -3.131 .179 .100 -.108

AX -3.142 -3.139 -3.133 -.034 -.100 -.279

VAb -3.130 -3.438 -3.429 -121.338 -113.028 -113.271

'^2 -3.142 -3.137 -3.130 .188 .116 -.093

""3 -3.142 -3.140 -3.133 .170 .085 -.124 '^ -3.130 -3.438 -3.429 -121.340 -113.030 -113.273

*^5 -3.130 -3.4 38 -3.4 29 -121.340 -113.030 -113.273

'^c

-3.142 -3.137 -3.130 .187 .116 -.0930

.b^ -3.142 -3.139 -3.131 .179 .100 -.108 '^ - -3.130 -3.4 38 -3.429 -121.338 -113.028 -113.272

'N

-3.130 -3.4 38 -3.4 29 -121.338 -113.028 -113.272

'^0 -3.130 -3.438 -3.429 -121.4 90 -113.169 -113.413

Figure 7.2. The phase at surface 11 for test beams when the wavelength and mirror

positions are changed (the 10.6 ym case).

90
Beam LA = 7 7 7 LB = 7 7 7 LI = 1 -1 L2 = 1 -1

Base

AX

%

'^0

-3.142

-3.142

-3.141

-3.142

-3.142

-3.262

-3.022

-3.142

-3.142

-3.022

-3.022

-3.143

-3.136

-3.137

-3.161

-3.121

-3.150

-.942

-6.111

-3.135

-3.136

-6.111

-6.111

-3.110

-3.142 -1.891 -2.057 1.881

-3.142 -1.968 -2.110 -1.959

-3.168 -12.462 -11.901 -11.721

-3.128 -1.810 -1.912 -1.737

-3.157 -1.972 -2.201 -2.024

-.938 906.900 844.270 844.171

-5.134 -1218.431 -1135.024 -1134.391

-3.142 -1.890 -2.055 -1.879

-3.142 -1.891 -2.057 -1.881

-6.134 -1218.464 -1135.052 -1134.423

-6.134 -1218.464 -1135.052 -1134.423

-3.116 8.638 7.749 7.921

Figure 7.3. The phase at surface 1 for test beans v;hen the wavelength and mirror

positions are changed (the 1.06 ym case).

8. DEVELOPriENT OF THE SURFACE HOLOGRAM

Subsection 3.3 defines the basic action of the surface hologram in eq (3.39).

Section 8 and its subsections step from this basic formula toward the realization of

such holograms. I define "surface hologram" at surface 7 functionally. Some readers

may consider this hologram to be a filter, a Kinoform, or a complex

grating [13,14,15]. Surface 7 must have a reflection surface with features shown in

eq (3.44). This structure must generate the desired nine beams from each incident beam

out of the prefilter. In addition, because this surface hologram is nonideal, it

generates unwanted background radiation which must not cause significant overlap with

desired beams. This latter fact influences the allowed form for this surface hologram.

In subsection 3.3, we indicated one way to realize the ideal nine-beam output with

minimum extraneous background radiation. In this section, we examine two approaches

that realize the desired nine beams. Each has background radiation whose effects on the final cross-correlation pattern at surface 11 must be known. This section examines

these two methods and develops their limits. The two methods are: (1) the binary type

of hologram (see references [13,14,15]) with both reflecting and transmitting beams;

and (2) a bumpy and essentially 100% reflecting binary type of hologram on a metal

substrate [14].

In subsection 8.1 we discuss why the two methods are examined and develop the

simplifications for modeling these two methods. In subsection 8.2 we apply these

results to fit the constraints of the reflection apparatus, and we define the necessary

work. In subsection 8.3 we list some computer codes that generate the masks needed for

each surface hologram. Finally, in subsection 8.4 we indicate a suggested construction

sequence for each method using the tools of the previous analysis and appropriate

91
facilities for thin-filtn deposition.

Subsection 9.5 contains the conclusions about the surface hologram drawn from the

analysis in section 8, from various inquiries of commercial and technical sources, and

from some preliminary work at NBS.

8.1 The Concept: Ideal and Practical

There are numerous ways to make holograms. Some transmission and reflection

holograms are shown in reference [1] and in selected pages of [6,13,14, and 15]. By

proper scaling as shown in reference [1], it is possible to get the desired hologram

where a single beam illuminates it and nine similar beams exit from it either as a reflection or transmission process. These volume holograms can absorb significant

laser energy; therefore, the substrate which holds the holographic materical can be

heated differentially. For laser pulses of high power or energy, this implies serious

errors in the measurement of the phase front. We avoid this problem by using holograms

which have insignificant heating of the substrate. The bumpy and binary surface

holograms can be constructed to have this condition.

The bumpy surface hologram has a substrate which is a good heat conductor, such as copper. This bumpy surface causes phase modulation of each incident beam. The

modulation generates nine similar beams with almost 100 percent reflection as well as some extraneous radiation. We examine how this hologram could be realized in

subsequent subsections. Reference [14] shows one example for such a 100% reflection

hologram. That paper as well as subsection 3.3 shows that the depth of modulation is

strongly wavelength dependent. We discuss the consequence of this dependence as we analyze the bumpy surface hologram. For example, the dependence makes each bumpy

surface hologram useful only for a narrow range of wavelengths.

The second method or binary type for generating the desired nine-beam pattern uses

a substrate that can transmit the laser beam with high power or energy through the

substrate with little absorption. For example, 10.6 pm radiation could use a beam

splitter wedge of ZnSe, NaCl, or KCl. Here the transmission hologram would be

constructed by etching through a film of gold a complex but periodic pattern of

nonoverlapping transmission ports on the exit surface of this wedge. These ports

sample the incident beams from the prefilter and radiate a pattern in the transmitted

direction that can look like nine beams of each original incident beam plus some background radiation of high spatial frequency. The reflected laser power returns to

some energy dump for absorption. Because the wedge has low absorption, there is little

distortion in this hologram; therefore, the reflection apparatus can operate

accurately. (If anti reflection types of coatings can be used, it may be possible to reverse the roles of the reflected and transmitted beams. We ignore that possibility

here.)

If the use of the apparatus for beam profile measurements implies no distortion by

the hologram, then we can use a film of some appropriate plastic on a flat mirror

surface for this hologram. This film could create a reflection phase hologram. The

plastic could record the needed pattern on a film of variable thickness. Because we

expect to use the apparatus for high-power beams, we do not consider a plastic film;

rather, we examine the two metal film techniques. When the apparatus has been

constructed, then the plastic film approach should be considered. It may prove better

than the techniques using metal films.

92
Figure 8.2 The Example of the Mask Output

93
Base Fiducial Mark Other Fiducial Mark

D

Code Mark

Figure 8.3 The Expansion of the Fiducial Marks and Code Mark

To See Details

94
To nake the analysis of the netal filns as sinple as possible, we eliminate

details that can be considered extraneous.

Under reasonable conditions, we can reduce the two-dimensional features of the

surface hologram to a one-dimensional discussion. IJe do this by observing that the

surface hologram is defined by the function F(u,v) in eq (3.39). This function can be

separated into:

F(u,v) = F(u) C(v), (8.1)

where F(u) and G(v) are similar functions. If there were no off-axis illumination,

these functions could be identical. Equations (3.40) and (3.41) show how the F(u) and

G(v) can be different. First, the S7 and T7 need not be the same because there is off-axis illumination in the x plane. Second, there can be a weak dependence on the

angle of illumination as shown by the CI in eq (3.40). The simulations from sections 5 and 6 established that S7 = C1*T7 to high accuracy. This means that we can analyze the

function F(u) only and accept that the only essential change to get G(v) for

experimental realization of F(u,v) has C(v) = F(v/Cl). The remaining CI term in

eq (3.40) is buried in the form of F(u).

flote that eq (3.40) implies F(u) has unit amplitude. This arises during the

derivation of eq (3.39) where the surface reflection at surface 7 was allowed only a phase shift. Letting F(u) cause amplitude as well as phase modulation does not affect

the form of eq (3.35). This flexibility in F(u) means holograms can cause either pure

phase modulation, pure amplitude modulation, or mixed phase and amplitude modulation.

The coefficients, G(m,n), in eq (3.44) are limited by the possible fabrication

methods.

Ue simplify our discussion by studying F(u) only and ignoring the G(v). This means the two-dimensional array of detectors at surface 11 has collapsed into a one-dimensional array of detectors where there are alternate spots with a single-beam

contribution and others with a two-beam contribution. Instead of nine beams, three

beams now exit from surface 7. These propagate in the directions !Lg^*S7 where the

incident beam is designated with a -Zq and the three exiting beams are labeled as

Zq + I, Zq - 1, and ig.. The last beam is the direct reflected beam, and the

former pair are the defracted beams.

To further simplify this analysis and to eventually make maximum economic use of

equipment and techniques available from integrated circuit technology, we presume that

the F(u) can be completely defined by a periodic function. Thus, F(u + D*n) = F(u),

where n is an arbitrary integer and D is the fundamental spatial period of the pattern.

Here D = 1/f and f = S7/2 . Here f is the fundamental spatial frequency for the

hologram. We can represent F(u) and its Fourier transform as:

F(u) = Z A. exp(i j u Zirf) , (8.2;

and

J=-

A. = f/° du F(u) exp(-i j 2TTfu). (8.3;

J

95
To realize the desired form of Aj , we must look at possible realistic forms for

F(u) that can be made by lift-off techniques and other such processes from the

integrated circuit technology. The ideal hologram for our case would have Aj

nonzero, for j = 0, j^l and the remaining Aj zero. If we accept that this ideal

cannot be made, we must relax this requirement and use the structure of the apparatus

to help achieve the desired pattern at surface 11 even though the hologram does not behave as wished.

The discussion in subsection 3.3 shows that we can approach the ideal provided we allow small but nonzero values for Aj with | j| >_ 2. This case is one example of the

bumpy surface. Subsection 8.2 contains a discussion of how to realize this hologram.

Additional complex patterns using the bumpy surface concept are discussed in this paper

after we explain the binary hologram.

If Aj f for

I j| j< 22 in addition to the j = +1,0, then the radiation from

the terms with | j| >_ 22 will miss the detector ^vx-d.)/; hence, we will operationally have

the ideal hologram plus a significant loss of laser power. Reference [14] shows one example that could use this approach. Here there is almost 100 percent reflection of

the incident beam. We skip this example until we have examined in some detail the

binary transmission case. Once it is understood, then the binary reflection example

for a bumpy surface can be properly discussed.

We represent the transmission structure of the binary type by a grid of segments

(see reference [16]). Each has a transmission hole. Thus:

M-1

E m=0

Here:

F(u) = {9 [u - u^(m)] - e[u - u^Cm)]} . (8.4)

;u) = 1 if u > 0,

= if u < 0, and

= 1/2 if u = 0.

Also:

u (m) = [m + a(m)]g, and

^^{^) = u^(m) + ge .

Here each segment size is g = D/M. The width of each hole in the segment, gp, is constrained by <^ a _< 1. Finally, the position of each hole as indicated by ga(m) is

constrained by:

<_ a(m) <_ (1 - e).

If we define

96
and

B. = [exp(-i 2TT6J/M) - l]/(-2Trij),

C = exp {-i[2TT(m + a(n))/M]} , m

the Fourier coefficients are:

M-1

A. = B. E (C )-^. (8.5)

rn=0

lie discuss this equation system briefly. Note that real F(u) implies

Aj = A_j. Also note that Aq = 3. To satisfy the conditions of the

previous paragraphs, v/e must adjust 2fl-l coefficients. This implies that M is equal to

or greater than 2N-1, so v/e have sufficient a(n) for adjustment. These a(m) and 3 must

be found by an appropriate computer program so that we have required spectra.

Our spectra have | Aj | j< 0.001 for all 2 _<

| j | _< 21, and a maximum

I A |/e and 3 . (8.6)

For example p = 1 causes A^ = 0, so the 3 must be a value between zero and 1.

For brevity, we make no attempt to find the correct set of a(m) and 3. That

exercise is reserved for the persons who construct the final apparatus.

To illustrate the expected form of the results, we look at a simple case where

3 = 1/2 and where a(m) = h[l-sin(2iTm/M)]. Equation (8.5) can be expanded in terms of

Bessel functions to give the results:

A. = MB. Z J. „ (2 hj/M) exp(-ij 27rh/M). (8.7)

J J J+mM

ni=-a>

To have | J2(4Trh/M) |< 0.001 requires h <_ M(0.1)/4Tr. The constraint on a(m)

requires h _< 0.5. For example, we could choose M = 42 to illustrate a possible

condition on h, namely h = 0.33. This case implies 0^(0.05) = 0.025 which is a very

small amount of defracted beam. In this case:

A = 0.5, and (8.8)

A = 0.0125.

I recommend that the reader try new a(m) to see what spectra are possible. For

example, two sine waves with phase factors give three parameters of adjustment that can

be used to eliminate exactly the A2 term and allow the strength for these modulations

to be increased beyond the value allowed for a single sine wave. This discussion

should have explained a(m); therefore, we now pass to the reflection hologram.

97
To construct the 100 percent reflection (binary type) case, we nust make two

adjustments.

First we modify the F(u) representing the surface to a sum of two terms. Namely,

we add a second surface term to F(u) which is the complement of F(u) for transmission

(see reference [13]). This complement is multiplied by a phase shift given as:

Q = exp(-i2ks).

Here s represents the displacement of the complement surface relative to the

original surface.

Second, to make the surface a reflection hologram, we cover this two-level

structure with a metal film. Our reflection hologram now has the following functional

form for F(u), namely:

F(u)^ = F(u)^ + [1 - F(u)^]*Q. (8.9)

The F(u)^ is the transmission function used in eq (8.8). The Fourier coefficients

for the reflection system can be given in terms of those for the transmission system

as:

a"^ = (1 - Q)*A^ + Q*5.Q . (8.10)

We note two cases of interest:

1. First, set s = 0, which is a flat mirror. Here we have pure reflection and no diffraction, namely Aq = 1 and Aj = where j f 0.

2. Second, set s = +X/4 ; thus, Q = -1 and AJ = ZAJ <5jo. Note that

Ag = 2B - 1.

A computer program is necessary to maximize the Ag and (AJ^/Ag)

values and to minimize the remaining AJ values. The optimum for the transmission

case will be different from the reflection case. For example, e = 1/2 implies no Ag term.

We now have three realizations of surface holograms: a simple bumpy surface and

two binary types with either transmission or reflection conditions. At this point, a few comments on the construction of a more complex bumpy surface hologram remain.

The binary type of hologram causes large amounts of laser power to be diffracted

into the | j | > 2 modes. The ideal situation would be to construct an F(u) with surface

variations so that the Aq, A^, and A_i are maximum and so that the remaining

terms are minimized appropriately. This situation implies the following formulation.

To prevent surface heating and to maximize the available pov;er, we write the surface

variation as a pure phase function. Thus:

F(u) = exp i 0(u), (8.11)

where i?5(u) is represented as a linear spline with appropriate support constants. The

integral equations.

98
A. = f/° du exp{i[0(u) - j ?TTfu]} . (8.12)

are nonlinear functions of the support constants. Ue nust adjust each support constant so that the diffraction spectra are as ideal as possible. If we can realize

this hologram in both the computer model and in the technical details of thin-film

operations, v/e would have the best hologram for two reasons: the laser power is not diffracted needlessly, and the surface structure has no sharp edges which would

increase the chance for field breakdown at high irradiances. One suggestion by Dr. A. J. DeMaria, United Technologies Research Center, was to use ion beams to charge

the reflectivity of the minor surfaces as a function of position. If a facility has this capability, then this may be a good way to get reflection holograms with the

necessary smoothness. To realize this hologram requires significant development which

has not been done; therefore, we settle for the simpler but less efficient bumpy

surface hologram. The technical approach for it is detailed in subsections 8.2, 8.3,

and 8.4. The technical realization of the binary types of holograms is already

adequately discussed in references [13,16]. However, there remains some development

work for these types, namely: (1) the computer codes to optimize the a(m) and g; (2) the computer codes to generate the masks for the pattern; and finally, (3) some measurements to document the influence of surface defects on the actual output at each

appropriate wavelength.

8.2 The Parameters for the Surface Hologram

The remaining subsections for section 8 become more criptic. We indicate key

details for construction of the surface hologram implied by eq (3.41) but do not step

through each development sequence. This surface hologram can be constructed using this

information as well as available thin-film technology and computer simulation

techniques.

Equation (3.41) shows a sine wave for the 0(u). Thus:

F(u) = exp[i 0(u)] , (8.13)

and

^(u) = -2k*Cl*S*sin(u*S7) . (8.14)

How do we construct such a 0(u)? To do this we need to understand the necessary

mathematical representations and to give the parameters appropriate to the reflection

apparatus under design in this paper. This subsection gives that information. In subsection 8.3 we list the computer codes used to generate the necessary bar-pattern

masks, and finally, in subsection 8.4 we list a suggested construction procedure for

the ^(u).

Here we generate a bumpy surface by starting with a flat substrate of copper. The

ideal final surface should look like a sum of two sine v;aves--one for the x direction

and one for the y.

To create the bumpy surface, we apply a series of uniform films of copper to the

substrate. Each film is placed over a bar pattern of organic material. Each film has

a different thickness. Because the film of copper sticks well to the substrate and

because the organic materical can be dissolved, well-defined ribbons of copper can be

added to the substrate. This deposition process is binary because each application

adds a single uniform thickness of metal film at a time.

99
To represent a sine wave surface using these ribbons requires many applications

and widths of ribbons. The precise number of films depends on the wavelength of the

incident radiation and on how small those unwanted Fourier coefficients in eq (8.3)

need be.

The mathematical representation of this operation shows the sine wave as expanded

into a series of square waves of different heights and frequencies. Thus:

sin p = (it/4^ S(p) + S(3p + 7r/3)/3 + S(5p + Tr/5)/5

+ S(7p + Tr/7)/7 + S(llp + tt/11)/11

+ neglected terms

where

S(p) = 1 if < P < TT ,

= -1 if TT < p < 2tt, and

= if p = or TT.

We control the sign of a square wave, S(p), by shifting it by one-half cycle, S(p + ir) = -S(p).

IJe now specify parameters for the reflection apparatus. For the 10.6 pm

wavelength, we set s = 0.084 ym. This implies the amplitude of each square wave as:

0.0660 \im for the fundamental frequency, f,

0.0220 ufTi for the third harmonic,

0.0132 ym for the fifth harmonic,

0.0094 pm for the seventh harmonic, and

0.0060 pm for the eleventh harmonic.

Each amplitude is doubled to give the thickness of the film because we need

peak-to-peak variation. The half period or bar width of the fundamental square wave is

2.162 mm for the x direction and 2.141 mm for the y direction. The substrate area for

these sine waves is a circle 40 mm in diameter.

If we note that the diameter of an atom is about 0.1 nm, we see that the

eleven-order frequency has about 60 atomic distances for its thickness. This fact

indicates that we can build a high quality sine wave for 10.6 pm. Our technical

constraint for this wavelength comes from the practical number of film deposition

cycles.

If we shift the wavelength to 1.06 pm, then the cut-off frequency for construction

of the sine wave at this wavelength is fixed by the requirement that the final film

should have at least 10 to 100 layers of atoms to get a well-defined, "uniform" film.

It is unclear if we can approach the sine wave with adequate accuracy for this and the

visible wavelengths.

100
8.3 The Computer Codes

Figures 8.1 and 8.3 show the computer codes and an example of a computer run. Since the program is fairly straightforward, we do not discuss the codes in figure 8.1

in any detail. Briefly, the control data at statement 30 are read in for each mask as defined by the comment cards just below PROGRAM. These codes generate a file, TAPEll,

which contains a series of commands to be used by another code [17] to generate the

mask shown in figure 8.2

101
PROGRAM ERIC (INPUT, OUTPUT, TAPES=INPUT, TAPE6=0UTPUT, TAPE 11)

C D(1)=UIDTH OF DARK BAR IN MM.

C D(2)=SHIFT OF DARK BAR IN MM.

C D(3)=RADIUS OF CENTRAL MASK CUT OFF IN MM.

C D(4)=RADIUS OF CROSS MARKS CENTER POSITION IN MM.

C D(5)=X POSITION OF CODE STRUCTURE IN MM.

C D(6)=Y POSITION OF CODE STRUCTURE IN MM.

C D(7)=C0DEa

C D(8)=FRACTI0N OF PI, ANGLE RELATIVE TO X AXIS.

DIMENSION D(8), ID(6)

5 J=5

PRINT 6 6 FORMAT( "NUMBER OF MASKS")

C REWIND 11 8 READ(J,*)I

1 = 1 + 1 10 1=1-1

20 IF(I .LE. 0) STOP

28 PRINT 29

29 FORMAT("NEW DATA 8 VALUES, FREE FORM USE COMMAS")

30 READ(J,*)D

31 FORMAT("DATA RECEIVED")

PRINT 31 URITE(11,32)D(7)

32 FORMAT("START,",F10.5,".")

D(8)=D(8)*3. 141592654

DO 40 11=1,6

ID(Il)=D(Il)*10000/25.4+.5

40 CONTINUE

50 F0RMAT("B0X",4(","I5)".")

PRINT 45,D,ID

45 - FORMAT("DATA=",/4F10.5/4F10.5617)

14=0(7)

IDY=(ID(l)/2)*2

I0=ID(2)

I1=ID(3)/(2*IDY)

IY=I0-2*(I1+1)*IDY

F1=ID(3)

95 F0=IY

100 X=1.-(F0/F1)**2

IF(X)150,160,160

160 X1=SQRT(X)

IX=-ID(3)*X1

IDX=-IX*2

URITE(11,50)IX,IY,IDX,IDY

150 IY=IY+IDY*2

IF(IY-ID(3))95,95,200

200 IDX=50

Figure 8.1. The conputer listing for the mask generation--

bumpy surface case (page 1).

102
HERE WE CODE THE MASK

IDY=50

IX0=ID(5)

IY0=ID(6)

11=100

12=100

DO 240 15=1,3

IX=IX0+(I5-1)*I1

DO 230 16=1,3

IY=IY0+(I6-1)*I2

DA=C0DE(I5,I5,I4)

IF(DA)230,230,220

220 IIRITE(11,50)IX,IY,IDX,IDY

230 CONTINUE

240 CONTINUE

TH=D(8)

DTH=1. 570796327

DO 300 IA=1,4

C1=C0S(TH)

S1=SIN(TH)

IAY=ID(4)*Sl+.5

IAX=ID(4)*Cl+.5

CALL CROSS (lAX.IAY)

TH=TH+DTH

IF(IA-2)290,300,300

290 IX=IAX+120

292 IY=IAY-100

294 IDX=40

296 IDY=200

298 WRITE(11,50)IX,IY,IDX,IDY

300 CONTINUE

WRITE(11,329)

329 FORMATC'END.")

ENDFILE 11 GO TO 10 END SUBROUTINE CROSS(I,J)

10 F0RMAT("B0X",4(","I5),".")

DIMENSION ILL1(6),IL2(6),IL3(6),IL4(6)

DATA(ILl=-100,-100,-100,-2,20,20)

DATA(IL2=-2,20,-100,-100,-100,20)

DATA( IL3=200 ,80 ,80 ,4 ,80 ,80)

DATA(IL4=4,80,80,200,80,80)

DO 30 11=1,6

IX=IL1(I1)+I

IY=IL2(I1)+J

IDX=IL3(I1)

IDY=IL4(I1)

URITE(11,10)IX,IY,IDX,IDY

30 CONTINUE

Figure 8.1. The computer listing for the mask generation —bumpy surface case (page 2.)

103
40 RETURN

50 END FUNCTION CODE(I,J,L)

INTEGER II DIMENSION 11(9)

DATA(I1 = 1,0, 1.0, 1,0, 1,0,1)

K=3*(I-1)+J

IF(I1(K))1,1,30

1 IF(I-2)2,5,3

2 IF(L .LE. 8)22,30

3 Ll=L-(L/2)*2

4 ID(L1 .LE. 0)30,22

5 IF(J-2)8,50,6

6 Ll=(L-l)/4

L2=Ll-(Ll/2)*2

IF(L2)50,22,30

8 Ll=(L-l)/4

L2=L-L1*4

IF(L2-2)22,22,30

22 CODE=0

25 GO TO 40

30 C0DE=1

40 RETURN

50 PRINT 60

70 STOP

60 F0RMAT( "ERROR IN CODE SUBROUTINE'

80 END

Figure 8.1. The computer listing for the mask generation

--bumpy surface case (page 3).

Figure 8.2 has three features: (1) a series of rectangular boxes that correspond

to the dark areas of a bar mask; (2) one basic fiducial mark for the angular position

of a mask and three additional fiducial marks, all of which allow each mask to overlay

properly the previous mask patterns; and (3) a code structure for 16 possible unique

labels of these masks.

For convenience, figure 8.3 shows magnified versions of the fiducial marks. This

sample case used the following values:

D(l) = 2.162 mm bar width D(4) = 24 mm cross marks D(7) = 1 Code No.

D(2) = mm phase shift D(5) = 24 mm x position D(8) = fraction of ir

D(3) = 20 mm radius D(6) = 24 mm y position

104
8.4 The Proposed Construction Sequence

At least 4 masks are necessary for construction of the sin(S7*u) function with the

correct amplitudes for 10.6 ym wavelength. In addition, at least 4 masks are necessary

for the construction of the sin(T7*v) function again with appropriate amplitudes.

The sequence becomes:

1. Start with flat-copper substrate.

2. Follow this basic sequence for each addition of metal film:

a. Use photoresist

b. Use appropriate mask to expose photoresist

c. Remove exposed photoresist

d. Deposit appropriate thickness of build-up material such as copper

e. Remove unexposed photoresist to lift off areas of metal.

There remain ribbons of copper of desired thickness and width.

3. The ideal construction sequence is to repeat the above process eight times--once

for each mask. The suggested order is:

First, the S7 frequency.

Second, the T7 frequency,

Third, the 3*S7 frequency,

Fourth, the 3*T7 frequency,

Fifth, the 5*S7 frequency,

Sixth, the 5*T7 frequency.

Etc.

Once the bumpy surface has been built up to the desired accuracy, then the entire

surface has a gold film added to maximize the reflectivity at 10.6 ym.

9. THE EQUIPMENT NECESSARY FOR CONSTRUCTION OF THE REFLECTION APPARATUS

In previous sections we described a complex theoretical analysis of the reflection

apparatus. The discussions were directed toward an ideal model. In this section we begin the task of identifying necessary hardware. Because we did not actually build

the device, the resulting lists and cost estimates are tentative and may even be

speculative. When actually building the device, the designer should note the

suggestions here and generate more complete lists, specifications, and strategies for

construction of the various parts. For example, the optics part and the detector

package are likely to be separate proposals because the detector package has too many

unknowns that need to be defined. Because we expect the constructed device not to match the nominal design specifications in this paper, this section can serve only to suggest what must be considered and what may be possible. The discussion in these

subsections does not exhaust the construction details. Only key components with their

approximate costs are discussed, and many mechanical and electronic specifications are

ignored. We have chosen a flexible design which is naturally very expensive. To reduce the costs of the device, significant simplifications in the system must be made.

Each simplification may reduce the capability of the final unit.

Despite the above discussion, this paper makes a substantial first step in the

design of a reflection system with potential for high accuracy.

105
The subsections are grouped according to a particular class of items. The choice

of mirrors along with their costs is discussed in subsection 9.1. We need translation

and rotation mounts to hold the mirrors, the prefilter, and the hologram as well as the

arrdiy of detectors. This discussion is summarized in subsection 9.2. The most

expensive part of the reflection apparatus is the array of detectors; in subsection 9.3 we discuss some possibilities. The prefilter is defined in subsection 9.4. Remaining

disucssions beyond that in section 8 about the hologram are contained within

subsection 9.5. Here we estimate costs from a commercial source. To make the

apparatus insensitive to temperature changes as well as to vibration, we need to design

the support of the various components to respond to or to reduce the effects of

environmental changes. In subsection 9.6 we discuss what could be done. Finally,

because it is not possible to just buy parts and then assemble the apparatus, we discuss the needed machine shop work.

9.1 Mirrors

Five mirrors, 200 mm in diameter, are needed for the reflection apparatus under

design. These mirrors define the base apparatus at 10.6 ym wavelength. The nominal

focal lengths of these mirrors are 1.000 m for the y axis and 1.020 m for the x axis.

When we change the wavelength to something like 1.06 ym, we must substitute two

mirrors. For 1.06 um these mirrors are 50 mm in diameter and have nominal focal

lengths of 0.1000 m for the y axis and 0.1020 m for the x axis.

The above mirrors require a certain level of optical quality. There are three

measures of this quality, namely the surface roughness, the optical figure, and the

scratch and dig values. We ignore scratch and dig issues for these mirrors. The

apparatus is too complex to estimate these effects in advance.

If we allow the shortest wavelength to be 1.06 ym and if we require that the

reflections off each mirror distort the phase and amplitude less than 0.5 percent, then

the surface roughness should be less than 5 nm rms for each mirror [18,19]. It is not clear at this-time that we need this level of accuracy. The basic instrument would

have problems induced by surface roughness if the background radiation from each mirror

reaches the detectors and if each pair of beams forming the interference spots comes from significantly different scattering conditions. The quality of mirrors for the

apparatus does not need to be as good as that for laser cavities or for beam forming

and turning apparatuses. Probably more critical for this apparatus is the accuracy of

the surface to the desired curvatures. The prime negative effect of surface roughness

will be to induce unwanted signals at the spots in surface 11. These unwanted signals

will cause a decrease in the signal to noise at each spot. It is difficult to estimate

how this effect influences the accuracy of measurements.

Technically, the surfaces of these mirrors in this mathematical model are the

product of two parabolas of different curvatures for each axis. For convenience, I have labeled the surfaces as "elliptical to reflect shape of the contours." The actual

mirror surface would be almost parabolas with less than A./10 deviation from the ideal.

Since A ranges from 1.06 to 10.6 ym, the allowed deviation for the apparatus under

design is 0.1 ym. This appears to be within the capabilities of industry.

There are three types of mirrors that can be used in this device. The first type

is a solid copper mirror which has been constructed by machining and polishing a copper

blank. The second type uses electroforming to generate the mirror. This structure is

either copper or nickel sheet with a copper or gold film as the mirror surface. The

third type has a copper or gold film on a glass or plastic substrate. Because the

capabilities of production in the industry are changing rapidly, each type should be

investigated at the time of purchase of the mirrors. To specify these mirrors, the

106
surface qualities already mentioned should be fixed. In addition, the potential

distortion in the mirrors by the heat from the laser beams must be considered. To have

high precision measurements of the phase front, the errors in the value of the

curvatures for these mirrors must be kept to less than 10 x. How much less, must be

tested with the simulation programs by varying each curvature. That simulation has not been done. The procedure is similar to the other simulations already performed and

must be done before the mirrors are ordered. Remember, surface 11 output is the key

control for all sensitivity tests.

If solid copper mirrors [20] are used because there can be significant thermal

loading by the laser beams, we estimate the cost to be S6,000 for each ZOO mm mirror [19]. They each weigh 10 kg. The solid copper, mirrors of 50 mm in diameter

would cost about SIOOO each and would each weigh 0.5 kg.

If we can use metal mirrors made of 6 mm thick copper plate by an electrofonning

technique, both the cost and weight of these mirrors are reduced substantially. We estimate [21,22] the cost of the two mandrels to be about S6000 each. After they

have been made, the cost for construction of each mirror would be about £200. Some

polishing may be necessary. The mirror surfaces made by this method may be distorted

by the high-power loading. If appropriate water cooling can be done, then this

distortion should not be a problem. My understanding [22] is that the surface figure

of these electroformed mirrors can be adequate for this apparatus, namely x/10.

I have no comments on the glass substrate.

In summary, the range of costs are 532,000 for solid mirrors and 514,400 for

electroformed mirrors.

9.2 Translation Mounts

We have five degrees of freedom to position each mirror, the prefilter, the

hologram, and the array of detectors. Thus, we use three translation mounts, namely

one for x, y, and z motion, as well as two rotation mounts. When we use the 200 mm diameter solid copper mirrors, these mounts have to take significant weight. For units

that have approximately 50 mm and 50 kg load capacity, the approximate price of each

unit is S500 [23]. If the range is reduced and the loading is made smaller, then the

cost is closer to £100. The two-angle rotation and mirror holder can cost about

S500 for the 200 mm diameter mirror and about £100 for the 50 mm diameter mirror.

In summary, we need about £2000 for the heavy-duty positioning system and about

$400 for the light-duty system. I have not included cost of nuts and bolts or the

possibilities of high precision types of mounts.

We require seven heavy-duty systems for the five mirrors, the prefilter, and the

array. In addition, we use three light-duty systems for the two substitution mirrors

and the hologram. The net cost is £15,200. If we use only light-duty systems, then

the cost is £4000.

9.3 Detectors--Types and Construction of the Receiver Array

In this subsection we indicate one method for recording the information presented

at surface 11. The correct method to use depends on the desired time response for

measurement of the beam profile and on the available technology. The design goal of

this reflection apparatus is to have a system capable of measuring at the 20 ns resolution so that the beam profile of laser pulses of 20 to 1000 ns in duration can

be measured in real time. To make this design goal as realistic as possible in terms

107
of available capability, we will presume that the measurement process by a method would

accomplish the following:

1. Each of the beams at surface 11 should be sampled in an analog manner using a time

period of 20 ns. The time interval between these samplings would be variable from

20 ns to 1 ms. The maximum number of these samples that would be recorded for a data reduction cycle is 10.

2. To be sure that the measurement of beam profile is a faithful snapshot at selected

time points, these samples dre synchronized.

3. To get maximum accuracy from the reflection apparatus, each analog sample will be

digitized for subsequent processing by a computer. This computer may be

electronic or optical [24,25].

4. The minimum number of spots that must be measured at surface 11 are: (1) those

with one beam contributing from the prefilter; and (2) those with two beams

contributing from the prefilter. This implies about 600 spots for this

apparatus.

For best accuracy, redundancy, and real-time control of the errors in the beam

profile of the incident laser beam, it is also necessary to measure the power in those

beams at surface 11 that are formed by four beams from the prefilter. If there are 300

holes in the prefilter, then there are approximately 600 beams that would be measured

if the error signal beams are not used. If they are used, then there are approximately

1200 such beams.

In summary, for a single data reduction cycle we need to store the digitized value

of the energy in each of the 1200 beams during the ten sampling periods. If we can construct such a unit, then the reflection apparatus can be of high value for beam

profile control of the inital laser source as well as of high value as a primary method

for absolute measurement of laser profile. The latter capability will be of use for

checking less costly units such as the Hartmann plate unit.

From the above discussion, we will presume that a single cycle of data processing

responds to the 12,000 words of data before a new block of data is needed. If the

cycle cannot respond that fast, then the measurements must stop until the next data

cycle can be processed properly. This rate of processing limits the throughput rate of

the apparatus.

We now define one system. Remember that it has not been built and therefore some development may be needed.

The system would use individual pyroelectric detectors with diameters of 0.5 mm to convert the laser energy to an electron current. These currents would each flow in an appropriately impedance-matched coaxial delay line. Ten spatially separated sample and

hold circuit units are attached to each delay line. The final readout sequence has a serial interrogation of these 12,000 sample and hold units for a sequential A/D

conversion and storage in a computer memory. This system can be called a brute force

technique. We estimate that it would cost £100,000 to buy the hardware. There are

no new technologies needed to construct this device; just manpower and money. I have

no cost estimate for the manpower.

108
9.4 Prefilter Construction

In this subsection we describe one possible prefilter design. There are obviously

others. We selected the easiest, and hopefully the least expensive, version to construct. Its influence on the final accuracy is unknown but is presumed to imply a calibration correction near 1 to 5 percent.

Here we start with a square plate of copper showing volume of 12 by 12 by

0.6 cm3. A square array of identical holes spaced 5 mm apart are drilled in this

plate. This array is contained in a circle 10 cm in diameter. This implies

approximately 300 holes.

There are three basic points about this prefilter that must be considered in the

design. First, the plate of copper will be heated from the incident laser beam. This

will induce stresses in the prefilter and thereby cause errors in the measurement of

the beam profile. Second, the reflecting surface of the prefilter must have low

absorption for all wavelengths of interest and must be structured so that the reflected

laser beam does not return to the source. The former is necessary to minimize the

impact of absorption on the prefilter as well as on the beam profile measurements. The

latter is necessary to avoid feedback in the laser. Third, the structure of these

holes must use the diffraction process properly so that there is accurate detection of

the phase and amplitude at the center of each hole.

Surrounding the prefilter by a water-cooled mount and using various periodic beam

blocking techniques such as a chopper will reduce the heating to a single correction.

To account for the heating requires calibration of the apparatus. The measured beam

profile is then used to correct for these heating effects and is thereby in turn

modified to infer the corrected beam profile.

The reflecting surface is polished so that the reflectivity is better than

98 percent for the wavelength range of 1.06 to 10.6 ym. The holes are drilled at an angle of 0.5 degrees off normal so that the reflecting surface prevents the reflected

beam from returning to the laser. Copper has less than 2 percent absorption for these

wavelengths. To do better would require plating the reflection surface of the

prefilter with other metals and/or various layers of dielectric coatings. This action

is most useful when a particular wavelength is under study. For the beam profile

apparatus with multiple wavelengths, we cannot use the dielectric coatings; rather, we could reduce the total power loading of the laser beam by some beam splitters. For the

allowed irradiance of 200 W/cm^, we find that a 1 percent absorption level implies

about a 200 W load for a cw beam of 10 cm in diameter. This is easily handled by water

cooling a mounting plate holding the prefilter. Pulsed lasers have different limits

which need to be carefully considered. For example, we need to define the class of

pulse shapes that will strike the device so we can determine the likelihood of surface

damage at each optical component.

To keep the calibration process simple, the structure of each hole should be

alike. This means the diameter of the hole should expand rapidly in a cone structure

so that the surface of the hole has little effect on the diffracted beam within the

hole. Reference [27] gives a criteria for the rate of expansion of this hole.

Ideally, the holes should not shadow the exiting laser beam. Of course, that is only

possible if the depth of the hole is zero. We can use a finite thickness by requiring

that the influence from the surface of the hole on the diffraction pattern be less than

10 percent. Because the variation between each hole is the important cause of errors

and because it is very likely that each hole can be made alike within a 10 percent

accuracy, we can expect these differences to imply around 1 percent changes in the

phases. These changes are reasonable for correction by calibration. The ideal hole is

109
an expand! ny cone with an angle of 5° or more. This means a 1 mm diameter initial

hole at the reflection surface would expand to a 2 mm diameter hole on the exit side

for a distance of 6 mm. Construction of such a cone would be costly, so the simplest

approach is to drill a 2 mm diameter hole for the 5 mm depth on the exit side and

finish the hole with a 1 mm diameter hole in the reflection surface. This last hole

would then be reamed for the 5° cone. This configuration should easily satisfy the

constraint for calibrations to eliminate 1 percent differences between holes.

Constructing such a prefilter is estimated to cost about $4000 for a solid plate

unit. For the electroforming approach, we estimate the cost to be about $10,000 for

the mandrel with about $500 cost per copy. The electroform units could be

constructed so that they are water-cooled throughout the prefilter.

9.5 The Hologram

Most discussion on the holograms was completed in section 8. Here we make

tentative conclusions on construction costs.

Using our preliminary construction attempts of the bumpy surface holograms, we conclude that they can be built [22]. The exact number of cycles for applying thin

films is still an unknown. Our present information shows that at least four can be

applied. This is insufficient for a high accuracy hologram. We need at least 6 cycles

for each sine wave. The two sine wave systems described in section 8 would need 12

such cycles. We conclude that a sine wave hologram cannot be constructed properly at this time. Rather, the binary surface hologram should be constructed for the

reflection apparatus. This method requires only two thin-film applications on the

polished mirror substrate. Although the discussion of section 8 shows that the binary

methods have significant loss in efficiency, this is unimportant for measurement of

high-power laser beams.

Using present metal turning and polishing capabilities [18,19], we can develop an optical quality flat of stainless steel with a surface roughness of 50 nm rms. By

using chromium as the evaporation metal for creating either the binary or some complex

bumpy surface, we can create a mandrel which allows electroforming of the copper

hologram with appropriate bumpy surfaces. This electroforming process permits mass production of the holograms. Using the estimates for the mirrors in subsection 9.1,

this mandrel would cost $4000 and the copies would cost $200 each [22]. Custom

construction of a single binary surface hologram on a polished mirror substrate is

estimated to cost $500 once the binary masks have been developed. The cost of

development of the binary mask is a manpower issue. It is estimated to cost $4000

using the codes shown in section 8.

9.6 Support Equipment

This category refers to two types of equipment, namely that used to insure the

structural integrity of the apparatus and that used in feedback loops to adjust

dynamically the positions of parts.

The former represents the support plate of honeycomb material and the rigid box to hold all the mirrors, mounts, etc. This equipment should fix the distances between

mirrors as well as other parts even if the apparatus is moved, the temperature of the

apparatus changes, or the apparatus is stressed. Moving parts can change the

calibration of the unit. If recal ibration is possible between moves, then the rigidity

can be reduced. We suggest that this is the appropriate design mode for the reflection

apparatus. The rigidity needed for the other approach would add much weight to the

unit which would affect the unit's mobility.

110
Dynamic tuning of the apparatus would dampen the effects of temperature changes

and vibration but would retain calibration. This can be done with numerous solid-state

lasers and detector arrays to measure relative movements of parts. By appropriate

adjustments of the mirrors with stepping motors and a computer system to define the

appropriate adjustments, it will be possible to keep the apparatus calibrated for a selected changes in temperature, vibration, and other such strains.

It is difficult to estimate the cost of this support equipment. An estimate is

S5000 for the nondynamic part and $150,000 for the design, development, and

implementation of the dynamic part. Most likely, the apparatus would be built without

dynamic feedback. Then when the need for more accuracy and flexibility increased, the

feedback system would be added.

Copies of the dynamic support are estimated to cost $15,000. This will be

mainly for parts and assembly.

9.7 Necessary Machine Shop Work

We include this subsection because first-time assembly of an apparatus requires

custom work. It is impossible to specify this work before purchased parts dre

received. As a rough rule of thumb, we would expect a 20 percent cost for such custom

machining added to the cost of the basic parts. Thus, if the parts cost $100,000, we would expect an additional $20,000 cost for the fitting process. This estimate does

not include unique or complicated elements such as the electronics. These costs

require different estimates as already indicated in subsection 9.3.

10. CONCLUSIONS

Costs :

1. Using heavy-duty equipment, solid mirror, and other cost estimates, we estimate

the cost for construction of the reflection apparatus without the electronics to

be $200,000 (see figure 10.1 for a summary).

2. If electroforming is used wherever possible, the comparable apparatus without the

electronics would cost $150,000. I assume this unit would be built after the

heavy-duty unit so most learning would be made there.

3. Once the mandrels have been made, then the cost of the additional copies would be

about $40,000. This number is probably even less firm than the previous

estimates. We need data that are not available at this time to establish a better

number.

4. The final completion of this apparatus requires substantial research and

development of the detector and electronics package. Cost estimates at this time

are guesses.

Error Estimates :

1. The first source of error arises from the quality of the prefilter. I estimate

this prefilter will create errors around 1 percent. This can be reduced by

calibration and by not overheating the surface of the prefilter.

2. The second source of error in the apparatus is the effects from the failure of the

optics to have the ideal surfaces. Calibration can reduce these errors.

Ill
3. A third source of error can come from the scattering off the mirrors and the

hologram due to the scratches and the rms types of surface fluctuations. This error cannot be removed by calibration and will limit the final accuracy of the

measurements. I cannot estimate the size of this error at this time. It is

presumed not to be significant for beam profiles with amplitude changes of less

than 20 percent between sampled spots.

4. A fourth source of error comes from mechanical instabilities. Appropriate control

can reduce this error. The largest error would come from vibration of the mirror

at surface 10.

5. The fifth and final source of error arises from the detection and amplification

processes. The amplitude part of the beam profile will have the least error since

it is directly measured. The phase measurement error depends on the value of the

phase. Briefly, individual measurements at each sampled signal should have

precision better than 0.1 percent to be sure the phase precision is better than

1 percent. The actual structure for these phase errors is complex and requires

careful analysis.

6. If extensive processing of the signals at surface 11 is available, the error

signals from those signals with four beams contributing from the prefilter can improve the accuracy of the phase measurements to better than 1 percent.

Copies Discussed

Heavy Electroform of the in Item Duty Initial Electroform Subsection

Mirror 32,000 14,400 1,400 9.1

Translation 15,200 4,000 4,000 9.2

Hologram (one wavelength only) 4,500 4,200 200 9.5

Support (static) 5,000 5,000 5,000 9.6

Prefilter (unpolished) 4,000 10,000 500 9.4

Subtotal 60,700 37,600 11,100 —-

Shop Work (20 percent) 12,200 7,500 7,500 —

Labor for Assembly

(about 1 man-year) 100,000 100,000 20,000 —

Grand Total 172,900 145,100 38,600

Rounded to Reflect

Accuracy of Estimate 5200,000 £150,000 $40,000

Figure 10.1. Summary of estimated costs for construction of a standard for beam

profile using the holographic method. We include the optics construction estimates

only.

Various Questions Answered :

1. I have been asked, why build the holographic system. Would not the Hartmann plate

system without optics be cheaper and just as accurate? To answer this question

without direct comparison is not easy. The best we can do here is to list key

differences between the two systems and show what happens. Briefly, the

holographic system has higher accuracy potential for absolute measurements, but is significantly more complex and expensive. See the appendix for comparison.

112
2. It is possible to use this instrument without a beam splitter to look at high

power beams? Yes. The 200 W/cm2 limit is a substantial restriction. If the

designer allows some form of distortion in location of the spots at surface 11 and causes the splitting process at the hologram into nine beams of each beam to take

place anywhere on that hologram, the optics can be changed so that the portion of

the Fourier transform of each beam from the prefilter is no longer the same for

the other beams. In this case, we no longer have the coherent, n2, type of

contribution at the center of the holographic plate. We may have instead just an incoherent, N, type of contribution. This means that the allowed power density on the holographic plate can be increased by N. For this apparatus, we get a 60 kW/cm2 restriction which means a potential 40 MW beam within the 10 cm diameter. In this case the prefilter will have to be water cooled and will have

special reflection characteristics. The errors due to heating are likely to be

significant and will have to be studied to assess their importance. The program,

BEAM, can be used to simulate this configuration so we can have estimates of the

extent of the distortions. Plasma ignition may occur and affect the accuracy of

the measurement. This issue will need careful assessment.

3. What about the polarization? This apparatus can be calibrated for each state of

linear polarization It is most likely that the values for the responsivity of

the apparatus will change with these two polarization states. I do not know

exactly how much change can take place because of this polarization dependence.

We need to perform some preliminary measurements to get an estimate of its

importance. Because the angles of reflection are small (9 = 0.14 rad) and

because we expect the absorption in the mirrors to be less than 2 percent, it is

expected at this time that polarization effects are insignificant compared to the

errors due to rms scattering and due to the detection and amplification processes.

The hologram will also induce some polarization dependence which is also expected

to be insignificant compared to the errors just mentioned. This condition is

expected because this hologram is a weak grating.

4. What about the detection and electronics part of this apparatus? There are significant construction costs for a detection system with 10 ns time response.

Accepting a slower time response could allow use of present TV scanning equipment

at much lower costs.

11. ACKNOWLEDGMENTS

I wish to thank F. Michael Maish of the Scientific Computing Division for helping

me use its 32-bit Interdata computer system. I generated the computer code and

simulation data from this machine. This effort could not have been done without this

facil ity.

I further wish to thank Robert L. Hinebaugh, Major L. Fecteau, and

Dr. Felix Schweizer of CCG for their kind support of this effort.

I have had many useful technical discussions with Dr. Matt Young,

Dr. Stephen Jarvis, and Mr. Aaron Sanders relative to this system.

I wish to thank Phyllis J. O'Rourke and Shelley Etzel for their fine typing of

this complex paper.

I thank Paul Werner for his prototype work, in which we identified the problems

for construction of the bumpy surface hologram. This activity sharpened our understanding of what is needed to get useful, computer-generated holograms.

I thank Matt Weinburg for running the required simulations reported in this paper.

113
12. REFERENCES

[I] Johnson, Eric G., Jr., Laser Beam Profile Measurements Using Spatial Sampling,

Fourier Optics, and Holography, NBS (U.S.), Tech. Note 1009, 96 pages (Jan. 1979),

[2] Panofsky, W. K. H., and Philips, M., Classical Electricity and

Magnetism, pp. 176-182 (Addison-Wesley Publishing Co., 1955).

[3] Coburn, N., Vector and Tensor Analysis, pp. 80-83 (The Macmillan Company, 1955).

[4] Morse, P. M. , and Feshback, H., Methods of Theoretical Physics, Vols. I and II, pp. 805-806 and 810-811 (McGraw-Hill, 1953).

[5] Born, M., and Wolf, E., Principles of Optics, pp. 203-232 and 460-490 (Pergamon

Press, Third Edition, 1965).

[6] Cathey, W. T., Optical Information Processing and Holography, pp. 1-73 (John Wiley

and Sons, 1973).

[7] Focke, J., Total illumination in an aberration-free diffraction image. Optica

Acta 1, 161 (1956).

[8] Boshnyak, B. M., and Korolav, A. N., Synthesis of astimgatism-free condensers

consisting of sperical mirrors. Opt. Spektrosk 43, 354, and Opt. Spectrosc,

U.S.S.R., 43, 204 (1977).

[9] See [6] pages 106-109 for Fourier Transform Formulae and pages 116-118 for Imaging

Formulae.

[10] Hudson, Richard D., Jr., Infrared System Engineering (John Wiley and Sons, 1969). j

[II] Daniel son, B. L., Beers, Y., Laser Attenuators for the Production of Low Rower 1 Beams in the Visible and 1.06 pm Regions, NBS (U.S.), Tech. Note 677, 25 pages (Jan. 1976)!

[12] O'Neil, R. W.. Kleiman, H., Marquet, L. C, Kilcline, C. W., and Norham, 0., Beam

diagnostics for high energy pulsed CO2 laser. Applied Optics 13_, 314 (1974).

[13] Brown, B. R., and Lohman, A. W., Complex spatial filtering with binary masks.

Applied Optics 5., 967 (1966).

[14] Gallagher, Neal C, Jr., Angue, John C, Coffield, Frederick E.,

Edwards, Robert V., and Mann, J. Adin, Jr., Binary phase digital reflection

holograms: Fabrication and potential applications. Applied Optics]^, 413

(1977).

[15] Haskell, Richard, Synthetic holograms and kinoforms. Optical Engineering 14-, 195

(1975).

[16] Haskell, R. E., and Culver, B. C, New coding techniques for computer generated

holograms, Applied Optics 11, 1712 (1972).

[17] Harris, Richard E., A pattern generating code, written to produced gyrex machine

code on the CUC 6600. Contact the Cyroelectronics section of the Electromagnetic

Technology Division, Boulder, Colorado.

[18] See Optical Spectra, page 26 (Aug. 1978) and pages 30-34 (Sept. 1978).

114
[19] Denny, Cy, of the Spawr Uptical Research, private communications.

[20] Spawr, Walter J., Pierce, Richard L., Metal mirror selection guide, SOR Report

No. 74-004, Spawr Optical Research, Inc. (1976).

[21] Biddle, Philip E., of the NBS Machine Shop at Boulder, Colorado, and Werner, Paul,

of the Optical Electronic Metrology Section in the Electromagnetic Technology

Division, private communications.

[22] Pickel, Jeff, of Pickel Industries, Inc., private communications.

[23] See various manufacturers' cost lists for such units.

[24] Brown, Hugh B., Markevitch, Bob V. , Radar signal optical processor, paper

presented at Effective Utilization of Optics in Radar Systems,

conference sponsored by BMDATC/SPIE, Huntsville, Ala. (Sept. 27-29, 1977).

[25] Bleha, W. P., Lipton, L. T., Wierner-Avnear, E., Oinberg, J., Reif, P. (i., Casasent, David, Brown, H. B., and Markevitch, B. V. , Application of the liquid

crystal light value to real time optical data processing. Optical Engineering 17,

371 (1978).

[26] See [25] for concepts. Further development would be necessary to apply this unit

to this apparatus.

[27] See [5], page 444.

[28] Gillespie, R. W., Wick, R. V., and Saxman, A. C, Laser Wavefront Profiling

Techniques for Pulse Radiation NTIS Code aAD-AOH 944/6ST Report(1975)

.

[29] Workum, Captain John Van, Plascyk, James A., and Skolnick, Michael L., Laser

wavefront analyzer for diagnosing high-energy lasers, SPIE 141 , Adaptive Optical

Components (1978).

[30] Hardy, John W., and Hudgin, Richard H., A comparison of wavefront sensing systems,

SPIE Ul, 67 (1978).

[31] Wetzstein, Hanns J., A tutorial review of key system aspects of wavefront

measurements, control and conjugation, SPIE 141 , 32 (1978).

[32] Hardy, John W., Active optics: A new technology for the control of light, Proc.

of the IEEE 66, 651 (1978).

13. APPENDIX—THE HARTMANN PLATE METHOD

I define a conceptually perfect system for measuring beam profile using only the

Hartmann plate and the necessary electronics at the required far field of each hole in this plate [28]. To allow as close a comparison as possible to the holographic method,

the key constraints applied to it are also imposed on the Hartmann plate [1].

Before addressing the details of this comparision, I make five points:

1. This discussion examines only how to make the most accurate beam profile measuring

unit that can respond to time changes in the profile that are faster than 1 ms. For those changes slower than 1 ms, the wavefront analyzer unit has superior

characteristics [29]. Assuming this unit can preserve its calibration, it appears

115
to be the best for high accuracy measureinents of phase front because there d^re no significant limitations in the spatial sampling of the wavefront.

2. For the time changes faster than 1 ms, the Hartmann plate is definitely better

than the holographic method if the rms variations along the phase front between

each sampled point exceed ir. The holographic method cannot identify uniquely the

relative phase between sampling holes if this phase exceeds tt. This ambiguity

means that the Hartmann method should be used with laser beams that have rapidly

changing phase structure, i.e., moderate beam quality as well as moderate

directional control.

If the structure of the beam profile is known in advance, then some modified

version of the holographic system could be better than the Hartmann plate because

the number of detectors could be substantially reduced (see discussion of field

instrument in reference [1]).

3. The ultimate accuracy of the holographic method is controlled by the noise of the

electronics. All other potential errors appear to be removable by calibration.

As already discussed in the conclusion and section 7, the noise of the electronics

implies that the phase error can approach 1% even if we have measurements with

0.1% accuracy of the power at each spot. In addition, we have seen [1] that the

dynamic range for the phase must be restricted to j< a <_ tt, if we are to have

uniqueness. These facts represent what the Hartmann plate method should

accomplish to be equal to or better than holographic method. Therefore, I require

the Hartmann plate to have 1 percent accuracy and a dynamic range of tt for the

phase measurement. In this case, the phase will range: -it/2 <_ a _<Tr /2.

4. The sampling holes at the plate are the same as those in the prefilter, namely

1 mm diameter. This diameter represents the current capability for direct and

consistent drilling in the substrate. If the diameter must be made smaller, then

both methods can be adjusted to reflect the change. The critical conclusions in

this exercise are unchanged by the reduction of ttns sampling diameter.

5. To compare the two methods we design the Hartmann apparatus to the above accuracy

and sampling conditions. This exercise shows that the Hartmann technique has

intrinsically a lower spatial sampling than the holographic method for comparable

accuracy. For simplicity, I ignore the noise uf the electronics for the Hartmann

system. If such information becomes available on that noise before the

appropriate beam profile devices have been selected, then that data should be

used.

I now discuss the Hartmann plate. I follow these four steps:

1. I use the far-field criteria of [1], namely:

V 2 /Z < 7r/10

with k - 2ir/A to fix some distances. This implies Xz >_ 5(mm)2 with

d^ = 1 mm. Thus, z >^ 0.5 m for X - 10.6 ym, and z >_ 5 m for X = 1.06 ym. In this analysis, we use Xz = 5(mm)2 as the smallest allowed distance for the far

field of each hole.

116
2. The phase equation relates to the displacement of the Airy disc at the far field

as a = kd]^ x/z. We only look at the x coordinate for this discussion. The

maximum displacement has a = ir/Z, and implies x = X]^ = 1.25 mm. We estimate the

accuracy for measuring this position with a square array of detectors. Each

detector has a net dimension of 10 pm for each coordinate. This spatial quanta

implies a position accuracy of 0.5 percent for the +1.25 mm range. We conclude

that the accuracy of the Hartmann system is comparable to that of the holographic

apparatus (using the above far-field criteria). We continue this evaluation to see what this comparable accuracy implies.

3. The diameter, d2, of the Airy disk at this distance is

d2 = 2.44 Xz/di = 12.2 mm.

4. The dynamic range, s, of each sampling cell in the far field is given as

s = 2x1 + d2 = 14.7 mm.

The cell size deduced in step 4 is only possible if there are shields separating each

cell to prevent overlap of the Airy patterns. If the shields cannot properly work,

then the size of each cell must be increased. For example, allowing at least three

rings of the Airy pattern in the cell dimension instead of the single ring causes a nev s, namely:

2x^ + (3.24/1.22)d2 2.5 + 32.4 = 34.9 mm.

These cell diameter results imply the allowed sampling interval for the Hartmann

plate equal 15 to 35 mm. Remember, s = 5 mm for the holographic method. Thus the

Hartmann method has 1/3 to 1/7 as much detail as that for the holographic method.

If we want comparable sampling frequencies for both the Hartmann and the

holographic methods, then we must use lenses in the Hartmann system. We need one lens

for each sampling hole. That action allows us to get the far field at the focal plane

of each lens and hence reduces the sampling interval. Remember,

s = 5 mm = (az)(0.5 + n) where n is the factor designating the number of rings within

the sampling area. The table below summarizes the results.

Xz =

n =

X -

resolution error

using 10 um =

one ring

1.7 (mm)

2

2.44

0.43 mm

1.2 percent

three rings

0.72 (mm)2

6.48

0.18 mm

2.5 percent for full scale

These results show that the Hartmann technique is about 1/2 to 1/3 as accurate as the

holographic method. (See references [30], [31], and [32] for additional perspective.)

The ability to check the calibration status in real time of an apparatus is one important difference between the Hartmann and holographic unit. The Hartmann unit

cannot be so checked but the holographic unit can be.

117
To calibrate the Hartinann unit requires (1) a known wavefront that can illuminate

at least two holes in each coordinate, (2) the assumption that the unknown and known

laser beains do not distort the unit during a measurement, and (3) a means to displace

the calibration beam for illuminating other holes in the plate without shifting in the

direction of the phase front. This calibration process is not precise because the

results have no direct estimates of errors.

In contrast, the holographic unit permits a real-time confirmation of the errors.

The measurements of signals where four beams contribute provide significant redundancy

of information about the beam profile. By subtracting a calculated response using the

measured results from the appropriate single and two beam signals to predict the

expected signal at a chosen location where four beams contribute from the measured

response at the same position, we have one data point for a consistancy test in beam j profile measurements. Extending this procedure to all measured signals in real time | gives us direct error estimates in the beam profile during a laser pulse.

In summary, the Hartmann unit does not generate the required error signals;

therefore, it is not an appropriate device for a standard. Under measurement A

situations where the conditions for errors in measurement are known (namely, the

Hartmann plate which has been calibrated by the holographic method), then the Hartmann

unit could be used for beam profile measurements.

118
NBS-114A (REV. 9-76)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-1015

Ifo, 3. flSfiCipiMtt** Ac<S6s5i<3ri Wo,

4. TITLE AND SUBTITLE

Design of a Reflection Apparatus for Laser Beam Profile

Measurements

5. Publication Date July 1979

$. F^feap^^ OfpRization CkHle

7. AUTHOR(S)

Eric G. Johnson, Jr.

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

m, n^6<m^^g,/mtii mn no.

11. Contract/Grant No. Contract 78-109

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS rsfr«(, city, State, ZIP) Sponsored in part by: DoD

Calibration Coordination Group

13. Type of Report & Period Covered

14. %(»)serlflg Agmcy Code

'-^^ ^-^'^

--Iiii'in 15. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or less factual summary ot most significant information. If document includes a significant bibliography or

literature survey, mention it here.) Measurement of both the irradiance and phase front (the beam profile) in real time from the output of a laser has interest for control of that beam and for efficient energy and economic design of the source and the

resulting optical systers. The National Bureau of Standards (NBS) has begun a program to build a unit that

:an measure, at numerous wavelengths from 1.06^11 to 10.6 /^m, a selected spatial sanp le of the beam profile.

Phis device would have the following features: (1) The different carrier wavelengths use the same apparatus

3y changing two mirrors. (2) The beam profile is sampled simultaneously with no time-shift distortions.

i(3) The output data streams documenting the sampled beam prof I le are continuous and are distorted only by the

finite number and the time constants of the detectors. (4) The phase-front information is generated before

the detectors create the data streams. (5) The apparatus uses mirrors and a reflection hologram tfiaiT IT

Itomputer generated. (6) The unit is calibrated plecewise over the range of relative phase and irradiances for

jach pair of neighboring sampling holes which are 5 mm apart. (7) The resulting calibrated unit can measure

l)rofiles near 10 cm In diameter with phase-front variations of less than 5 wavelengths. (8) The expected

i|-esponse time for measurements as controlled by the electronics Is of the order of several tens of lanoseconds. The design analysis reported here Includes: (1) the theory which uses Fourier optics concepts with

>ff-axis reflections and rough surfaces to provide the basis for accurate corrputer simulation of laser beams;

[2) the program, BEAM, which generates the expected behavior of the apparatus under variation of laser ravelength, physical dimensions for curvatures, hologram structure, and changes in positions of the various

:o(iponents; (3) the simulation results which demonstrate the expected characteristics for the apparatus; and

,i:4) the key element in the apparatus, namely the reflection hologram, which requires discussion of the design,

|:onstruction, and testing of this element.

The Hartmann plate method is described briefly so that a comparison between It and the holographic method

:an be made. The comparison shows why the holographic method Is best for a standard for irradiance and

!>hase-front measurements.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated by semicolons)

Key words: Beam profile; calibrated system; holography; irradiance;

laser diagnostics; phase front.

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