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o NBS TECHNICAL NOTE 1057

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act of Congress on March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau's technical work is per- formed by the National Measurement Laboratory, the National Engineering Laboratory, and

the Institute for Computer Sciences and Technology.

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THE NATIONAL ENGINEERING LABORATORY provides technology and technical ser- vices to the public and private sectors to address national needs and to solve national problems; conducts research in engineering and applied science in support of these efforts; builds and maintains competence m the necessary disciplines required to carry out this research and technical service; develops engineering data and measurement capabilities;

provides engineering measurement traceability services; develops test methods and proposes

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THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts

research and provides scientific and technical services to aid Federal agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and

economy in Government operations in accordance with Public Law 89-306 (40 U.S.C. 759), relevant Executive Orders, and other directives; carries out this mission by managing the Federal Information Processing Standards Program, developing Federal ADP standards

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The Institute consists of the following centers:

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'Headquarters and Laboratories at Gailhersburg, MD, unless otherwise noted; mailing address Washington, DC 20234. 'Some divisions within the center are located at Boulder, CO 80303.
Beam-Profile Measurement of Laser Pulses

Using a Spatial Filter to Sample the

Hermite Modes for a String of Pulses

Eric G. Johnson, Jr.

Electromagnetic Technology Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued September 1982
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1057

Nat. Bur. Stand. (U.S.), Tech. Note 1057, 44 pages (September 1982)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1982

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

Price $4.50

(Add 25 percent for other than U.S. nnailing)
COMTENTS

Page

1. Introduction 1 1.1 Background Discussion 1 1.2 Organization of This Paper 2 2. The Telescope for Beaniwidth and Phase Front Control

of a Laser Pulse 3 3. The Beam Mismatch Between the Telescope and the Ideal Optical Filter 5 4. Design of the Optical Filter Unit 8

4.1 The Basic Functions for Optical Computing 10

4.2 The Detector Analysis 12

4.3 The Ideal Filter Characteristics 13

4.4 The Nonideal Filter Characteristics 15

4.5 Summary of Results on the Optical Fiber 15

5. Conclusions 15

6. References 17 Appendix A Telescope 18

Appendi x B Mode Mi smatch 22

Appendix C The Unseparated Case Analysis 25

Appendix Program GEN 27

Appendix E Program SELECT 29

Appendix F Program ORTHBAS 33

Appendix G Program CHECK 34

Appendix H Program CONVERGE 36

m

Beam-Profile Measurement of Laser Pulses Using A Spatial

Filter to Sample the Hermite Modes for a String of Pulses

Eric G. Johnson, Jr. National Bureau of Standards

Boulder, Colorado 80303

As a first step in the development of a beam-profile measuring instrument

for laser sources that is capable of determining the distribution of low-order

(less than 25) Hermitian modes in a series of laser pulses, I designed and evaluated the three key parts of such an instrument. First, there is the telescope

system which allows the incident laser beam to be phase, beamwidth, and beam

center matched to the optical spatial filter. Second, there is a brief error analysis of the structure of the mismatch function between the beam out of the

telescope and that expected by the filter. Finally, there is the detailed analysis and design of the computer-generated spatial filter that will cause the incident-laser beam to be cross correlated with the low-order Hermite modes and will

create an array of light spots in the detector (Fourier transform) plane each of

which can be uniquely related to a particular Hermite mode of the original laser

pul se.The principal conclusion is that the Hermite mode analysis can be done with

better than 99 percent separation between modes, provided the phase between modes

is uncorrelated from pulse to pulse when the filter has been fabricated with a two-level, gray-scale structure which samples the profile with either percent,

or lOU percent transmission.

Key words: computer simulation; laser beam profile; mode-matching analysis;

spatial filter, target designators.

1. Introduction

1.1 Background Discussion

Beam-profile measurements of each laser pulse can increase the understanding of how a laser source operates. These measurements can allow eventual real-time control of the source and/or a complete evaluation of the source. There are two basic techniques that

appear to be accurate for beam-profile measuring and that can be designed to allow moderately trained individuals to perform the necessary alignments and sequencing of the measurements: the focal plane [1] and the optical -fi Iter techniques (ref. 2 and this paper). Each

has its advantages and disadvantages. The appropriate one to use depends on the character

of the laser source, the conditions where the measurements will be performed, and the

desired results needed for decisions and/or control.

The focal-plane technique can use a single planoconvex lens around 2 m focal length to

create a fluence pattern (J/cm^-) in the focal plane for the far field of the laser beam.

This pattern is recorded with a TV camera system of appropriate spatial resolution and

responsi vity. The recorded energy density pattern can then be serially transferred to a minicomputer to determine a summary, such as the center of gravity of the beam, the moments

of dispersion of the energy density, and the presence of far-field hotspots due to the laser source and optics. The prime advantages of this technique are: (1) the small number of

optical elements and mechanical parts, (2) the non-predisposition to the characteristics of
the fluence distribution, and (3) the flexibility of using software for summarizing the

results to determine whether or not the laser source is behaving as desired. The main

disadvantages of this system are: (1) the seconds or more it takes to assess the desired

information allow little chance for real-time assessment of individual laser pulses, and (2)

there can be a phase-front curvature that changes the far-field pattern. The latter effect

can be assessed by varying the position of the detector plane. This adjustment requires

stability in the laser source. In brief, the focal -plane technique is best if the beam

profile has temporal detail of no interest or is too variable to allow some optical

processing technique to eliminate the unimportant information. The computer software

permits a statistical reduction of the temporal variations.

The optical -filter technique is complimentary to the focal -plane technique. It works

as follows: The pulse from a laser source goes through an optical filter which causes a complex spatial phase and amplitude modulation to be superimposed on the original laser

pulse. This pattern is then intersected by a lens of one meter focal length whose function

is to form the Fourier transform of the modified beam pattern. At selected transverse

locations in the focal plane of the lens, there are spots of beam intensity which correspond

to strength in key features of the original beam pattern. One example of key features is

the electric field strength of the Hermite modes in the original beam. These spots of

intensities are recorded by an array of detectors whose number can be less than 25.

The prime advantages of the optical filter technique are: (1) the filter can eliminate

the unnecessary information about the beam profile and structure the desired information

into a form that reflects the primary energy content of a laser pulse. This optical

processing happens at the 10 ns rate, therefore the necessary processing is fast enough to allow a pulse-by-pulse assessment of the beam profile from a Q-switched 1.06 urn laser

source. (2) The process is sensitive to the amplitude distribution of the laser beam rather

than to the energy distribution. This means that the analysis is linear in the

electromagnetic field and can thus be made so that we can have a sensitive measurement of

the actual electromagnetic-field details instead of the energy-density details in a laser

pulse. Knowledge of these details can permit diagnostics of the original laser source, such

as the temporal stress changes of the laser rods and the temporal changes in the excitation

processes. The prime disadvantages are: (1) advanced knowledge of the appropriate spatial

filter is necessary so that the optical processing can be optimized. This advanced

knowledge requires a basic structural stability of the laser source system in order for the

reduction of extraneous information to be effective. (2) Because the optical filter

technique is sensitive to the beamwidth, to the curvature of the phase front, and beam

center, it may prove necessary in some measurement systems to include a lens system to match

the laser source to the optical filter. This can imply a significant change in the size and

complexity of the measuring apparatus for that system. This paper addresses such situations

and describes in section 5 how they can be avoided.

1.2 Organization of This Paper

The aim of this paper is to evaluate by computer simulation a beam-profile measuring

apparatus that is made up of three basic components, namely, the telescope unit, the optical
filter unit, and the detectors. Section 2 describes a telescope that can be used to cause

the laser source, a target designator, to have a plane-wave front at the optical filter a beamwidth of 10 nrn and a match of beam axis. Section 3 gives the mathematical formulas for

the mismatch between the telescope and the assumed mode structure of the filter. Section 4 discusses the analysis of the computer-generated optical filter with detectors and the

results of the computer simulation on how well a physical filter will perform the desired

mode analysis. Finally, section 5 evaluates the concepts involved and, as a consequence of

the computer simulations, makes conclusions in the context of these concepts. The

appendices contain the computer programs used in simulation and some other analysis related

to construction of modes and what response the detectors will record. The design is

directed toward 1.06 urn wavelength. If other wavelengths are used then some of the design

parameters will need to be adjusted.

2. The Telescope for Beamwidth and Phase-Front Control of a Laser Pulse

Here we assume four lenses are necessary to get complete control of the beamwidth and

the phase-front curvature of a laser beam. This telescope is necessary only if the phase

front of the laser source is not a plane wave and if the beamwidth, wio, is not 10 mm at the

spatial filter. Figure 2.1 shows the components of the telescope. The f^ are the focal

lengths of the four simple lenses which are chosen to be f2 = 2000 mm, fk = -fi,

fs = 2000 nm, and f6 = -fs. The distances d^ are the spaces between the lenses, the laser

source, and the filter plane. We assume all lenses have an aperture of 120 mm. The q^

represent the complex beam parameters for an ideal Gaussian beam [3]. Thus, we have q^ = i b^+z^, where b^ = w^^(-t/a), ji_is (-Ir' , w^^ is the beamwidth at the n location, and -z^ is

the distance to a point where the Gaussian beam has the width w^. We can use two relationships for the propagation of the laser beam.

Laser

Source

^d.—

q? q- ^4^! q^ q- ^ft q<

•10

Filter

Plane

Figure 2.1. Beam adjustment diagram. Note that d^^, = for the thin lenses and f ^ for the distances between the four lenses.

=
q^+l = qp + dp (Type l--a simple displacement.

fp=0 for this case.

)

qp+j^ = -fp+ fp2 (Type 2--the thin lens law.

fp-qp dp=0 for this case.)

To make efficient computation, we do the following: (Appendix A has the actual com- puter listing.) For processes from left to right of figure 2.1, we use

q = i b + z ^n — n n

This implies that Type 1 has b ^^ = b and z ^^ = z + d when f = 0. Here we set u = 0, ^ "^ ^ n+1 n n+1 n n n n and h = 1. Type 2 has b , = h b and z , = -f "*" l^p ^p » when d = 0. Here we set

u nnnnnnn = f„ - z„ , and h = f ^/(b ^ + u ^). For processes '^ from right to left, we use ^ r = ic + y„ in place of q„, b , and z . Type 1 has c = c ,, and y = y ^i - d when

n — n 'n ^ ^n n' n

-^^ n n+1 'n 'n+l n f^ = 0. Here we set v = and kp = 1. Type 2 has Cp = kp Cp+]^ and yp = fp-kpVp when

dp = 0. Here we set Vp = fp + yp^^ and kp = fnV(Cp+i2 ^ ^^^^^ ^^^^ ^^^ ^^^ ^^^ ^^^ ^^ ^^^ parameters for convenience in computation.

In the computation, we assume fp, bi, zi, cio, and yio are specified. For illustration

purposes, we set di = d9 = 100 mm and define M = (bi/cio)^'^ as the magnification change

necessary to match the input beam at 1 to the output point at 10. The computation is an iteration process, where d3 and d? are set until |b5/c5-l| is less than 10"^. The condition

of self consistency requires that the quantities, bs = ifzf^/Wi)^ and C5 = (fe fs/D^)^,

with the definitions D2 = f2 + fi+ - d3 and D4 = f6 + fs - d2, are equal, since they are the

same when the telescope is properly adjusted. We start the iteration process with the

control parameter T = >^d7d3 set to 1200 mm to get a convenient dynamic range for d? and da

of about 800 mn. Thus, we start d; = \A ' and d3 = T/M ' . The new d-, are constrained to be d,^ d,^ = t2 where the new d,^^ is related to the ol d d ^ by d

^^ ^3 = d ° [b /c l^^^.

3 3 3 3 5 6

We reset dy = 67 and da = ds for each complete new calculation until bs = ce, as desired. When the above is done, then we find ds = yg - Z5.

In this analysis, we provisionally restrict-^ < M < 1.67, and IzJ < 100,000 rrni as the

allowed range of deviation for the laser source beam parameter relative to those of the

optical filter. If the source parameters are outside this range, it will be necessary to choose different optics to make the intended measurement sequence. For this magnification

range, the range of d3 is 930 to 1700 mm and d? is 850 to 1550 m. If we design the beam

path so that two flat-turning mirrors at 45° angles are used for each distance adjustment,

then the travel distance can be 400 im or less. Under these conditions, the range of d5 is

2900 to 3100 mm. The computer program in appendix A generates the values of bp, Zp, the convenience

parameter p = b •^nnn^ + z '^, the inverse range^rparameter e - z /p , and the beamwidth para- nn'^n

meter Wp = (X pp/n bp)^''^- for all n values, as well as the round-trip phase, Zp - yp, discrepancy and the round-trip beamwidth discrepancy bp - Cp. The latter parameters permit a
visual check on the consistency of the computation. In addition, these results are computed

with individual changes by 1 mn in d^, in d^, and in d^. Further changes are made of 2 percent in b^ and 100 nri for z^.

This computer program establishes that a telescope can match the phase front and the

beamwidth to the optical filter described in section 4. The position adjustments are very

sensitive. Thus, they will need accurate control to 1 urn over a 1-cm travel range. The

gross adjustment of 400 mm is set at discrete locations of 4 mm spacings. A machined strip

of 100 slots provides this accurate spacing. To illustrate the expected results, appendix A

has listed a base case with M = 1 and z^ = -10,000 mm. Mo further data will be described

since this paper establishes feasibility only.

Reference 4 addresses the effects of spherical aberration, which should be small for

the lens with focal length of 2 m. The accuracy of the computer-generated optical filter

will be a more significant source of error. If spherical aberration proves to be a problem,

we would use two thin lenses with anti reflective coatings and glass with an index of 1.5 to reduce both the reflections and the spherical aberrations for 1.06-um wavelength.

Figure 2.2 shows a top view of how the lens and mirrors could be located to realize a telescope in a space of about 1200 by 1200 by 200 mm^. Figure 2.3 shows a three-dimensional

view of the telescope. Notice that the d^ distance must be greater than the 100 rmi designated in the previous discussion because the last mirror which turns the beam out of the

plane of the telescope toward the filter requires a space of 200 mn or more. This turning

mirror is necessary to make a compact structure of both the filter and the telescope. The

optical filter will have a lens of 1 m focal length that is anti reflection coated and may be

made up of two lenses to make the spherical aberration zero.

3. The Beam Mismatch Between the Telescope and the Ideal Optical Filter

Unless the laser source is a device that has been manufactured with the same physical

laser cavity dimensions, and with the same type of other equipment such as the optics,

electronics, flash lamp, and the optical resonator for each unit; the beamwidth and phase

front from a given unit will not be known without some measurement strategy to establish the

proper adjustment of the telescope described in section 2. Therefore, the purpose of this

section and appendix B is to indicate the type of formulas needed to define the measurement

strategy. It should be obvious that these suggested formulas are complex and will need more information and work as well as a computer to perform the necessary processing of the final

results so that the beam mismatch can be minimized properly.

The design issues to allow computer processing are not addressed within this paper.

The derived formulas assume that the primary modes of the laser source are represented by a few Hermite modes so that the results are not abstract. If this assumption proves to be

false, then it will be necessary to do a new mismatch analysis in the terms of a more appropriate description of the natural modes for the laser source. For now, we do our analysis

on the basis that a few Hermite modes characterize the modes of a laser source with sufficient precision.

Using the scalar theory [3], we find that the electric field at the filter can be

represented in two ways, namely (1) a complete Hermite expansion showing the coefficients
1200 mm

Beam

Center

1000mm M

t -

M-^y *-500mm— ^ -^500 mm-» i Ij^^^'i"*

<7 da

^t I —750 mm — M,

M- -^ 750 mm —"

*

M

I —500mm— ^ —500mm-*

8

6

M9

M11 1-0

t fa da ^ -^800 mm— ^ iST'l

10

-tiM

Figure 2.2. Top view of telescope showing nominal dimensions

for optics at extreme positions.

Exit Port to Telescope

Loser

Entrance Port

to Telescope

Deflecting

Mirror

Figure 2.3. Telescope three-dimensional diagram.
that would be measured by an ideal optical filter and (2) a complete Hermite expansion showing six parameters that reflect the change from the ideal match of the optical filter.

Equation (3.1) shows the former and eq (3.2) shows the latter. Thus, we have

E (x,y,2) = I A^^ H^ (/2~x7w^) H^^(/27/w^) go, (3.1)

n,m

where

g

= exp [-(x^ + y2)/w^2]_

Here, x and y are the coordinates for the surface of the ideal optical filter. (I use the

word ideal in this section because the actual filter will have inaccuracies which will be

described in section 4.) We assume the z coordinate is zero in this analysis. The w^ is

the beamwidth that the ideal filter uses in its decomposition of the electric field. The

coefficients A = R exp (j_ S ) have the unknown phase factors, S^^, and the measured

amplitudes, R^^^. The electric field is now given in terms of the coordinates of the laser beam in the

neighborhood of the ideal optical filter as

E (x, y z) = ). B H (/2x7w ) H (/27/w )g, (3.2) nmn n m n n , m ^

"

where

g

- exp n ki - (x- x y^) (l/w ^ _ i k/2R )],

and where x = -xq + x c^c? - y CiS2, y = -yo + xs2 •• yc2, and Y = -x siC2 + y siS2.

Here ci = cosOi, si = sinOi, C2 = cos02. and S2 = sin02. The 0^ and 02 are the usual

spherical coordinates o and ^j), respectively. For this expansion in eq (3.2) to be valid,

assumed wo is independent of z because 0^ and Og are small angles of rotation, Xq, y^ are

small distances for displacement, and the phase mismatch, r2 = (kw-:^/2Ro) is also small.

It is convenient to define the six parameters of adjustment by using

w = w^(l +r),c =c=l, s =G,s=o,r, T 1 12 112 2 2

we

pQ = •^ Xo/Wf 3""^ finally qo= ^Z yg/w . (3.3)

Appendix B shows briefly how the coefficients ^^^^ can be related to the B^^ coefficients. Here we note from eqs (B.8) and (B.9) that relationship is in the form

A = I C , , B , , , (3.4)

nm , , nmn m n m ' n m

where C^^^i^^i can be known by the analysis in appendix 3. The actual measurement has

7
Here

R^ = I C , , C „ .. D , , „ „ . (3.5) nm ,,,,,, nmn m nmn m n m n m

n m n m

D , , „ „ = < B , , B n"ni" >. (3.6)

The brackets represent some form of averaging of the phase from each mode of the laser

source. If the laser has truly Hermite modes, then a reasonable hypothesis is

D = k2 6"1 6'"1 . (3.7)

n m n m nm n m

'

This means that each mode has a uniform random phase relationship to the other modes. Other

hypotheses will be necessary if the basis modes of the laser source are not these Hermite

modes but are some linear combination of them which then can be treated as containing random

phase relationships between the new modes (see section 5 on this point). Additional information is needed to complete this mismatch analysis. At this time we do not have that information, therefore we stop the discussion here.

In summary, we have just shown the type of analysis necessary to unravel the measurements of R^ from the actual mode strength, namely, the B set. Various testing will have

nm ^ ' -^ ' nm ^ to be done to determine what is the appropriate assumptions on the character of Dpnin'm' ^^"^ each type of laser source system.

4. Design of the Optical Filter Unit

In section 4.1 we develop an almost orthonormal set of over-complete functions that can

be used to understand the detector system, the ideal filter, and finally the nonideal filter. In addition, some discussion is made in appendix C to identify how this set can be

changed to better reflect what happens in a laser source.

In section 4.2 we develop the formulation to simulate how the detector unit can be represented and how it can operate to measure the individual modes resulting in a prescribed

pattern in the focal plane of the lens. We also define the conditions that the detectors

must satisfy.

In section 4.3 we set up the ideal optical filter analysis using the Hermite functions

as the modes.

In section 4.4 we do the nonideal filter analysis to detemine the problems in realizing

an optical filter. Here we learn what assumptions and procedures are necessary to get the

appropriate mode analysis.

Section 4.5 summarizes the results of these sections. The appendices D, E, F, and G have computer programs used to check the ideal and nonideal filter analysis. They are referenced in the appropriate subsections.

Figure 4.1 shows a side view of the optical filter unit. The tube contains a lens of

1000 mn focal length which is coated to minimize the effects of reflection at 1.06 urn wavelength. There is a filter plate located in front of the lens (illustrated in fig. 4.2).
/

Adjustable -1 cm

Detector Plane

Focal Position

of Lens

E

E

O

O

o

Filter Plate

Optically

Processed

Laser Beam

h^- 120mm

-Piano Convex Lens

(cooted to minimize

reflections)

Laser Beam

from Telescope

Figure 4.1. Side view of optical filter unit showing functions and nominal dimensions.
70 mm

Sample hole

(0.02 mm )^

Figure 4.2. Filter plate (front view). Illustration of size

and type of sample holes. Actual pattern is complex.

The device performs the optical computing on the incident laser beam from the telescope.

There is an array of detectors at the end of the tube located in the focal plane of the

lens. This array maybe a CC device or a TV scanning unit that has uniform response at the

1.06 Mm wavelength. The filter is a complex pattern of grid holes which samples the

incoming beam and causes both phase and amplitude modulation of that beam.

4.1 The Basic Functions for Optical Computing

We define the following as one possible set of basis functions that may be used in

optical computing. This set is not unique. Others are possible and should be considered

when the exact conditions in the laser source dre known. For this study, we assume (1) that

the low order (n < 5) Hermite modes contain 95 percent of the laser energy, and (2) that the

detectors are placed at equal spacing in the focal plane of the lens. Thus, we define

H(n,k,x) ^ U(n) H (/2 x /w^) exp [-(x/w )2]

n f a

* [cos (2Trkfx) (1 + (-l)")/2 + sin (2ukfx) (l-{-l)")/2] , (4.1)

where U(n) = [ /2" 2 cN/(*^ w^n'2")] '

, and where k > only. The c is the spacing of the

grid holes in the nonideal filter. The c = 0.02 mm for our test. M is the number of grid

holes in the range < x < L. Here, L = 35 iin and, hence, N = 1750. The w^ is the filter

beamwidth which is 10 mm for our design. The f = d/Az, where d is the distance between

detector elements, z is the focal length of the lens, and X = 1.06 um the wavelength. The

d = 10 w^, where w^ is the Gaussian sampling width of the detector element. Finally, the w^

is defined by

10
1 _ 1 1

d f C where w = A z/(w ), and w^ is set equal to 0.02 \tm. It is a concidence that w^ and c are

the same. It represents the type of dimensions that can be realized for fabrication elements in both detectors and in grid holes of the film. If we define a weight function,

W(x) = exp [2 (x/w^)2], (4.2)

and allow an integration over the range -«> < x < +«', the following results are approximately

true to a part in 19"^ or better for all n and k > 0. Thus,

oo k m

/ d H(n,k,x) W(x) H(ni,ki) =6 5 F, (4.3)

ki ^i

where

F = cN[l +

5"^ (-1)"],

1< and 5 =0 unless k = ki

.

If the range of integration is ±L instead of ±«', then the result of eq (4.3) is not exactly true. For the low-order Hennite modes, namely n < 4, the computer results show that

it is valid. Also, for low-order modulation of k < 8, the computer results show ttiat it is valid. If the final system tends to use n > 5, then it becomes necessary to develop a different basis that reflects the truncation at x = L. Appendix C suggests a way this can be

done.

There are two features of this basis set, namely, the functions are symmetric in x, H(n,k,x) = H(n,k,-x), and we assume that the y coordinate has the same basis functions, thus

H(n,k,y). If we assume the changes in H(n,k,x) are small compared to the grid size, then we can make the discrete structure with X£ = (^+l/2)c where < i > N-1, and we can presume

that the basis filter design can be covered with the range of x being < x < L and y being

< y < L. The other quadrants can use the same grid structure in the fabrication of the

filter. Thus, we have the filter function described with the aid of the function T(x) as

5 3 T(x) = I \ H(n,k,x) S(n,k) W(x), (4.4)

n=0 k=0

where T{x) > and T(-x) = T(x). We cut off the n and k in our computer simulation work, a decision reflected in eq (4.4). The y-coordinate function has the same form. The total

filter can be described as

T{x,y) = T(x) T(y)

for this analysis. This assumption of form allows separation of the two dimensions and

extensive simplification in the structure of the computer-generated models. If this proves

to be invalid, then the problem requires significantly more speed in computation. For this

11
paper, we ha^e assumed these conditions so that we can define what will be the problcns in

the eventual fabrication of the nonideal optical filter. The nonseparability of coordinates

can then be addressed as necessary in a later study.

The discrete function analysis gives us the result

I H(n,k,x^) W(x^) H(ni,ki,x^) =

6|^ fiJJ |

{l+6^(.l)")

£=0 1 ^

for the range of < k < 9 and < n< 5. Appendix D has the computer programs that generate

the H(nikiX£) set. Appendix F has the computer program that selects the strength of the set

used in eq (4.4) by determining the value of S(n,k) for the nonideal case. We discuss this

situation in more detail in section 4.4.

We have a set of basis functions which can be used provided the modes n > 5 are not present in the primary beam. If they are present, then the procedures indicated in appendix

C become necessary. For the rest of this section we assume the energy content of the laser source is such that only the n < 5 is true for both the x and y coordinates. Using refer- ence 5 we have the electrical field at the optical filter given as

5 U,(x,y) = I A U(n) U{m) H (/2" x/w.) M (/2"y/w.) G(x) G{y),

1 ^n,m n f„ n f n,m=0

where

G(x) = exp [-x2/w^2]^ (4^5)

The electrical field at the detectors is given as

V^(x^,y ) = ~- /dxdy T(x,y) U (x,y) exp l—-^- (xx^ + yy^)]. (4.6)

By using the separability assumption for the filter, we construct the following forni for U2,

U2(xiyi) = -.— ^ A V (xi) V (yi) w 2

"^ ^ ^ iXz n,m n,m n ^ m ^ f ^2

wi th

V (x ) = J^-- / dx U(n) H (/Z x/w.) T(x) exp [-(x/w.)^] exp[-2Tr i xx Az]. (4.7)

" 1 w^/F n f

^^ f 1

These V^lx^ are convenient to simulate the detector.

4.2 The Detector Analysis

We look at V^(xi) and form a convenient strategy for evaluation of the finite spot size

for the detector. We assume that each detector has a spatial response that is Gaussian;

thus, we convolve the response function to define an effective detector response at position

"j" as

12
K(n,j) = —1— /dx V (x ) exp [-(x -jd)2 /w 2]. (4.8)

s

This form is convenient for the computer simulations and adequately represents what

happens at a given detecto>^. In TV-type scan systems it may prove necessary to develop a weighting algorithm for the response from each of the detector elements in the chip to simulate a Gaussian response profile.

If we use eq (4.7) and integrate over xi, make the definition

00 F(n,j) = S(nJ) [1+5 J (-1)"] = / dx T(x) H(nJ,x)/Nc,

and note that K(n,-j) = K(n,j) for n even, and K(n,-j) = -K(nJ) for n odd; we get for

J > 0.

K{nJ) - F(nJ) [Mcv^//ttw^] * [(l+(-l)") - i(l-(-l)")]/2. (4.9)

We can identify the set of Fourier-type transform relationships as

CX) S(nJ) [1+5^ (-1)"]= ^ / dx T(x) H(nJ,x) (4.10)

and eq (4.4)

We can further modify this structure to reflect the fact that our analysis is over a finite range with discrete elements. Thus, we get

5 9 T(xJ = I I H(m,k.xJ S(m,k) W(xJ

m=0 k=0

and

1. n 9^-1

S(n,k)[l+6 ^-1)"] =£ 1 T(x.) H(n,j,xJ. (4.11)

"^ ji=0 ^

^

4.3 The Ideal Filter Characteristics

The ideal filter has S(n,k) = except for those cases shown in table 4.1. The relative size of each coefficient R(0) to R(5) was chosen to maximize the strength of the individual modes for each Hermite structure as well as allow separation between the modes. The

R(5) to R(9) were chosen to allow the T(x) be positive definite for all x. A single scale

factor remains by which each coefficient can be multiplied. These coefficients allow T(x)

to be less than 1.04 for all x. This scale value was chosen to minimize the error of the

nonideal filter. Its value should be 1.00. For the purpose of this report, the explicit

scale change is unnecessary. Thus, to get the correct R(0), etc. for the ideal filter

divide the values given by A = 1.04. The programs D, E, and G are used to generate the

13
Table 4.1. The Strengths of the Ideal Filter

(A = 1.04)

S(0,0) = 0.116 = R(6) * A S(2,0) = 0.0408 = R(7) * A S(4,0) = 0.04 = R(8) * A S(6,0) = 0.04 = R(9) * A S(0,2) = 0.01687944 = R(0) * A S(l,3) = 0.022859717 = R(l) * A S{2,4) = 0.023948275 = R(2) * A S(3,5) = 0.027209995 - R(3) * A S(4,6) = 0.027589743 = R(3) * A S(5,7) = 0.029858334 = R(5) * A

Table 4.2. The Ideal Filter Mode Mixing

S(l<,n) Elements

Table 4.3. The Mode Mixing for the Monideal

Filter with a Two-Level Quantization

1 +1.2E-3 -1.7E-2 -6.0E-3 -6.8E-5 5.7E-4

+1.8E-2 1 -l.lE-2 -1.3E-2 +5.1E-3 2.6E-3

-7.1E-3 -2.8E-3 +1 -2.6E-2 1.8E-2 2.0E-2

5.1E-3 6.5E-3 1.2E-2 1 4.8E-3 -8.1E-4

1.4E-2 l.lE-2 -4.3E-2 2.4E-2 1 -3.5E-2

7.7E-2 1.9E-2 -5.0E-2 -2.0E-2 -2.2E-2 1

n=

k

Pure

1 2 3 4 5

2 1 -8.9E-7 -4.1E-7 +1.4E-5 +5.3E-5 -1.5E-4

3 -3.7E-7 1 1.4E-5 4.9E-5 -1.3E-4 3.5E-4

4 -2.4E-6 1.3E-5 1 -1.5E-4 -4.6E-4 9.5E-4

5 6.2E-6 -3.7E-5 -1.2E-4 1 8.8E-4 -2.6E-3

6 2.2E-5 -8.4E-5 -3.9E-4 8.6E-4 1 -4.5E-3

7 6.3E-6 3.2E-5 -1.2E-4 -3.5E-4 9. IE -4 1

14
expected self consistency for S(n,k), given the values used to generate the T(x) from table

4.1. The basic conclusion is that 'node rnixing is less than 0,45 percent and this is primarily due to the fact that the n = 5 mode has truncation error. It is possible to eliminate this mixing by suitable orthogonality procedures. This would be equivalent to a rigorous account of the finite range of each mode, namely L = 35 vm. I did not do this in this

report since the prime emphasis here is to identify the key areas of error and to try to devise a technique for elimination or reduction of those errors. The fabrication of the

nonideal filter completely masks these errors so they are ignored at this time. Once a technique dealing with the errors generated by the nonideal filter has been determined, then

it may prove worthwhile to do the orthogonality. Finally, the choice of order and strength

for each mode has been arbitrary, it may prove necessary to define other orders and

strengths to minimize cross correlation effects in nonideal filters.

4.4 The Nonideal Filter Characteristics

The nonideal filter has one modification from the ideal filter where T(x) ranges from

the values to 1 and has T(x) either or 1 at each cell location. This implies either

percent or 100 percent transmission. It is technically possible to construct quantized filters with three levels, namely percent, 50 percent, or 100 percent and so forth. However,

it is fairly easy to fabricate a filter of 100 percent or percent transmission; therefore,

it is desirable to consider such a nonideal filtet". Using the program "Check" as shown in

appendix 3 with the nonideal filter generated by the program in appendix E, we see in table

4.3 a mixing between the modes of up to the 3 percent level. If we assume that the phase

correlation between modes is random, then the mode mi;<ing has ar\ error of about 1 percent

for the determination of the energy in each mode. If there is no phase averaging, then it

becomes necessary to do one of two steps; consider a new set of modes which can have a ran- dom phase relationship between the modes or improve the nonideal filter by using a three or more level quantization process instead of the two level quantization. In the latter case,

we accept that the problems of fabrication, such as variable thickness transmission films, can be solved economically or that the cell size c = 0.02 can be changed to c = 0.02*S and

the number of elements becomes H' - N/S. Here, S = 3 allows a four-level gray scale. The

basic two-level quantization is done in "Select" by the requirement

N M Y(x. ) = \ T(xJ -

\ E(xJ, (4.12) ^ £=£l Jl=Jll+l

which is a running sum. Here E(Xj, ) = 1 whenever Y(Xj,,) > 0.5, otherwise E(x ) = . It

is not known at this time if this is the best possible two-level sche^ne for quantization.

All attempts to improve the scheme have failed because the rate of computation has been too slow. New time-efficient computation strategies are necessary.

4.5 Summary of Results on the Optical Filter

Briefly, this section has demonstrated that it is possible to devise an optical filter

that can do dii^ect mode analysis of laser beams. There are some technical difficulties that

15
still need further work to make the construction of these filters as efficient as possible.

Appendix C suggests what needs to be done is devise the most appropriate modes for a particular laser source. The comparison of the results for two-level quantization with the

nonquantized version suggests that it is very desirable to devise a means to fabricate at

least a three or more level quantization process and/or devise a time-efficient strategy for

minimizing the consequence of the quantization process.

5. Conclusions

We have learned the following:

1. It is possible to construct a telescope which will allow both a variation in the beamwidth and the adjustment of the phase-front curvature. This unit is necessary only if

the laser source has a beamwidth and phase front that vary significantly compared to the final desired accuracy from the quality assurance measurements on a particular

laser source. It is necessary to build a matching telescope for a laboratory beamprofile measuring apparatus so that each source can be assessed to determine if the source beam profile can be evaluated adequately and to establish what the need is for

such a telescope in a field instrument.

2. The nonquantized version of an optical filter can perform mode analysis with separation

of better than Q.l percent of the amplitudes. At least one such device would be of

great value to allow a determination of the Hermite mode structure for various laser

sources. Once these structures are known, it is possible to devise optical filters

which do not require telescopes, which require only a few detectors, which require no extensive computation, and hence can be devised to allow construction of a GO-NO-GO

type measurement system. The latter unit will be valuable for field sites using personnel who have very little training. Technical realization of the nonquantized optical filter would be done with multiple exposures of high resolution (3000 lines/mm or better) film.

3. The two level quantization process produces extensive mode mixing (about 8 percent).

To make a quantized optical filter useful, a new condition becomes important. A mode

analysis other than Hermite may become appropriate. These new modes are defined by the

condition that the phase relationship between modes is random. This condition will

allow mode separation to better than 1 percent which is more than adequate accuracy for

a GO-NO-GO type instrument.

4. If the particular laser source has no stable mode structure, then it becomes necessary

to use an artificial mode set such as the truncated Hermite modes in conjunction with a quantized process that is more than two level. In this way we can construct a GO-NO-GO

measuring device.

In summary, this paper has demonstrated that the optical processing can be used to do

mode analysis and that the optical filter must be designed uniquely for each generic

laser source (instrument). In addition, it can be concluded that fast computer techniques need to be developed to devise the optimum-quantized optical filter for two dimensional beam patterns without any significant symmetries.

16
The following persons have been helpful toward completing the design of this system:

Don Boyle and William Hall have helped by allowing me to use their computer sytem to perform

the simulations and by providing consultation in the use of the machine. William Hall has

provided the necesary skills to get the final prografn converted into FORTRAN from Basic so that it would work 30 times faster and thereby will allow me to look at level quantitizations. Aaron Sanders and Matt Young have been helpful as active sounding boards to react to the numerous ideas that were eliminated and to the ideas that resulted in the formulation

presented in this paper. Edie DeWeese and Susie A. Rivera did the excellent typing and com- position work on the paper. Dean Eicher supplied the fine drawings.

This effort was funded partially by the Combined Calibration Group of the Department of

Defense.

6. References

[1] Day, G. W. ; Stubenrauch, C. F. Laser far-field beam-profile measurements by the focal

plane technique. Nat. Bur. Stand. (U.S.) Tech. Note 1001; 1978 March.

[2] Johnson, E. G, , Jr. Laser beam profile measurements using spatial sampling, Fourier

optics, and holography. Nat. Bur. Stand. (U.S.) NBS Tech Note 1009; 1979 January.

[3] Kogelnik, H. ; Li, Ti . Laser beams and resonators. Proc. IEEE VI -54: 1312-1329; 1966

October.

[4] Smith, Warren J. Optical elements--lenses and mirrors. The Infrared Handbook, Chapter

9, Sections 9.8.1-9.8.4 and 9.9.1, pp. 9-14 to 9-18.

[5] Cathey, W. Thomas. Optical information processing and holography. John Wiley & Sons;

1974. p. 9. [6] Handbook of Mathematical Functions, Nat. Bur. Stand. (U.S.) Appl. Math. Ser. 55; 1964

June. pp. 773-788.

[7] Schultz, Martin H. Spline analysis, Prentice Hall, Inc.; 1973.

[8] Prenter, P. M. Splines and variational methods. John Wiley & Sons; 1975.

17
Appendix A. Telescope

10 REM THIS PROGRfln SETS UP THE TELESCOPE FOR BEftM MEnSUREMENTS

20 CLOSE 1 39 DIM XCie.Z)

49 OPEN "CEN:",!.!

53 DRINT "NUrBER OF U(1),Z(1) PAIRS"

60 INPUT N

70 FOR I-l TO N 80 PRINT 'READ IN PAIRS"

90 INPUT X(I,n.X(I,2)

108 NEXT I 110 FOR I=N+1 TO 10

120 X(I.1)=0

130 NEXT I 140 K8-1

150 DIM BC10),Z(10),C(10),Y(10).U(10),H(10).F(1ID).KC1Q),V(10),D(10)

160 DIM U(10)

170 P9=3. 14153

180 DATrt 0,2030,0,-2000,0

190 DATA -2030,0.2000.0,20013

200 U(1)=X(K8,1):Z(1)=XCK3.2)

210 K3=K3+1

220 IF KB"! I GOTO 1530

230 A=.10S4E-2

240 B(l)=P^>^U(n'^2/A

250 IF IJC1)=0 GOTO 1530

260 U(liD) = 10 273 C(ll3)=P^•-^<Ul:lO)'^2/A

280 M=SQR(B(1)/C(10))

230 Ml-'BQR(M)

300 T=1203

305 RESTORE

310 FOR 1=1 TO 10

320 REftl F(I)

330 IF Fn)=Q GOTO 3G0

340 D(n=0

3SQ GOTO 400

3GQ IJ':i)=0

370 v(n=0

330 HCI)=1

330 K(I)=1

400 HEXT 1 410 D':i) = ie0

420 D<:9) = 1G0

430 DC7)=ni:KT

440 D':3)=T/i11

4?.f) FOr-^ 1 = 1 TO 4

<m Gosi;3 750

.17B HEXT I 400 FOR J=l TO 4

430 I = 10-

J

500 G03IJB 360

513 NEXT J L20 Ii3=0

530 Bi=!3C5)/C(S)

e-^0 E2=131-l

5'j0 IF AS3(D2)<.lE-3 GOTO G'^O 5:0 N8=II3-H: IF N3>6 GOTO 1550

570 B3=SQR(E1)

530 B4=3QR(B3)

5y9 D(3J-B4:.D(:3)

6:JQ D(7J=D(7)/B4

610 PRIHT "CONVERGENCE VALUE B2=".B2

620 I^o: G03U3 750

630 1=4: casus 750

649 1=7: GC3U3 330

650 I=S: G03UB 860

18

1
£iQ GOTO SZa

GV9 DC5)=Yi:6)-ZC5)

bOa FOR I<7=1 TO 7 698 REM TEST GENERATION DflTrt 7Qe H3=l<7 7 IB G03IJB 1390

720 NEXT K7

738 PRINT "CURRENT RUN CftSE. I<3=",K8

74Q GOTO 200

750 REM SUBROUTINE FOR LEFT TO RIGHT GENERftTION

7ba 11=1+1

770 IF F(n=0 GOTO 030

730 U(I)=F(I)-Z(n

730 HCI)=F(I)-^2/(B(D'^2+U(I)'^2)

£J60 B(Il)=H(I)>l:B(I)

818 Z(I1)=-F(I)+H(I)>:-U(I)

823 GOTO 850

830 B(I1)=B(I)

840 Z(ir;=Z(I)+D(I)

8-Ja RETURN

StiQ REM SUBROUTINE FOR RIGHT TO LEFT GENERATION

876 11=1+1

E30 IF F(l)=0 GOTO 940

838 V(I)=F(I)+Y(in

SQ0 K(I)=F(I)^2/(C(Il)i>2+V(I)^2)

910 C(I)=K(I)*CCI1)

920 V(I)=F(n-K(I)*V(I)

338 GOTO 360

940 C(I)=C(I1)

950 V(I)-Y(IJ)-D(D

3S9 RETURU

970 REM SUBROUTINE TO GENERi^TE LEFT flND RIGHT BEAM TRACE

i80 FOR 1=1 TO 9

930 G0SU3 753

1000 HEisn" I 1810 FOR J=l TO 9

1028 I=10-J

1030 GOSUB 860

1040 he:<t j

1050 RETURN

1060 REM PRINT BLOCK NUM3ER 1 1070 PRINT ON CD "TELESCOPE PROGRAM , VERSION 2.^27/81

1030 PRINT ON (l)GETDi^TEC;(0)."DriTE OF RUN HERE"

1090 PRINT ON Cl)G£TTIMES(0),'Tir£ OF RUN HERE"

1180 PRINT ON C1)'FD<ED VARIABLES ARE IN MM UNITS. THEY ARE"

1110 PRINT ON (1)»D(1)=''.D(1)

1120 PRini ON Cl)"D(9)=",DO)

1130 PRINT ON (1)"IJ(1O)='.U(10)

1140 PRINT ON (n"T IS THE DISTANCE CONSTRAINT FOR D(3),D(7)'

1150 PRINT ON C1)"T=".T

.1160 PRINT ON (n°THE MAGNIFICATION SET FOR TELESCOPE IS ".M

1170 PRINT ON (1)"TH£ ACTUAL MAGNIFICATION IS ' 1130 M3=SQR(B(1)/C(10))

1190 PRINT ON (1)"M3=",M8

1200 RETURN

1210 REM PRINT BLOCK NUMBER 2

1220 FOR 1=1 TO 10 1230 PRINT ON (1)"B .Z .D ,F FOR I-",I

1240 P=B(I)'^2+2(I)'^2:E=Z(I).^

1250 U1=SQR(A*P/(P9*B(I)))

1260 PRINT ON ( 1) ,B( I) .Z( I) .D( I),F(I)

1270 PRINT ON (1) "INVERSE RANGE. WIDTH . PHASE DIFFER. DELTA B

1280 PRINT ON (1) E.U1.Z( I)-Y( I) .B( I)-C( I) 1290 ne:<t I 1300 RETURN

1310 PRINT ON (1)"BASE DATA CASE": RETURN

1320 PRINT ON (1) "CHANGE D(3)': RETURN

1330 PRINT ON (1) "CHANGE D(!5)": RETURN

19
1340 PRINT ON (1) "CHANGE DC')": RETURN

1358 PRINT ON (1) "CHANGE BCD": RETURN

1368 PRINT ON ( 1) "CHANGE Z(l)": RETURN

1378 PRINT ON (1)"BASE DATA CASE REPEAT ': RETURN

1388 REI1 SUBROUTINE FOR GENERATING THE TEST CHANGES

1398 ON N8 GOSUB 1458,1478.1488,1490,1500,1510.1520

1400 GOSUB 970

1410 GOSUB 1060

1420 ON N3 GOSUB 1310,1320.1330,1340,1350,1360.1370

1430 GOSUB 1210

1448 RETURN

1450 D3=D(3): REM BASE DATA FOR SAVING

1460 D5=D(5):D7=D(7):B3=B(1):Z8=Z(1): RETURN

1470 DC3)=D3+1: RETURN

1430 D(3)=D3:D(5)=D5+l: RETURN

1430 D(5)»D5:D(7)='D7+l: RETURN

1300 DC"')=D7:B(n = ,93:KB8: RIZTURN 1510 BC1)=B3:Z(1)=100+Z8: RETURN

1520 Z(l)-Z8: RETURN

1530 CLOSE 1 1548 STOP

1550 PRINT "NO CONVERGENCE"

1560 END

TrL5SC0?E P^DSl&M t VERSIDN 2/27/31

QiZ-OiVBl 3AT-r-af—ItHM ^r^£

D9:^4:07 TI«^E 0= ^U"^ HE\E —FIXED ^A^1A-51:-S-A^f^N-MM-tJNlTS-t^f^ir-Att —

D(l)= 100

DC9)= 130

T IS TH»E 3IST4N:E CDNST^AINT FD? 3(3), 3(7)

T= 1?00 —THf-MAG^I^J'-AT«N 5^T-F5^-^ft-e5-C-0Oi -1-5 1 THE A:TJAL HA3NiIFi:MID'^ IS

M9= 1 —^^se-^ATA -c-ftST- -

B fZ •3 .F FD^ 1= 1 2^326?. 22 -10 33D 100 —I^VE-^5f-^.A*«^-»-Wi3Ti f-i»HA5€ -^i^ = f^» 3EiTA -3 -.13290?07E-5 13.5579&1 -. 1 55 55^SSF -3 .19515ifllE-3

3 tl »3 .F F3^ 1= 2 — 2^5?^-2i22 ^993-3 D 2000

I>JV5R5E RAM3E* 4I3H « ^H&SE 3IFFE^, 3ELTA B -.1323233ZE-5 13. 55^876 -. 1 59 55 <^S SE -3 .19515<^llE-3

—^ —-»l r^ f-F f^\ i- 3

12.105452 -1995.5222 llb^.S314

1>1V = RSE -lANSE, WI3TH « PHASE 31F = E^, 3 E L T 4 "i — -.53132325-3 10i556«76 .-21^?A2?5f -^ - -. 10332*>P5-7

3 .Z .3 tF FD^I= t, 12.105452 -^31.14383 3 -2330

-i^VE^Sr ^A>JGr» ^*^-3T^ 1 -»r1AS£ ^i F^-E^ » 3ELTA ^

-. 12029134E-2 4.395S94 . 214?42 85E -9 -.103325^E-7

I »J .F F3^ 1= 5 35-.43^-a6B 4421*7732- 2556.3929

IMy/E^SE RftNSEf <^I3TH t ^HftSE 3IF = :-^, 3E.Tft 5 -.732n)35E-3 4.396594 3 -. 332 5 4871 E -7

_^ ,£ t3 y-f F3^ -!-^ 6

35.433C63 1234.5397 3 -2000

20
I^J\/tR5E R4N3Ef **I')H « !>H4SE DIFFER* )ELTa B -^^^-??0>7^—? ^91«^-95-8 3 •33 2 54^71-r^^-

6 .Z •) fF F3^ 1= 7 13.=;<»S719 763.52317 123S.3897

.130?3D58£-2 3.8182958 -. 25329<»27£-9 -. 1

1

56259 lE-7

5 fZ f3 fF FQ^ 1= 5

1 i^^Ah^9 1 99 9. 91 ^^ 3 2^3^

IMVE^SE ^ftNSEt WIDTH « ^H^SE DIFFER. DELTA ?. .*9999335E-3 1D.3030D1 -. 25329<»27E -9 -. 1

1

562591E-7

-3 rZ T^ »-F F^^—F« 9 295262.22 -99.99977 100

INVERSE RAMSEf WIDTH t PHASE DIF=E^t DELTA B —Ti+*7052^5-6 l-^^^>t>9^ r^^^6r3^^^^£—

4

-.^Sl-^*^^

B tZ tD tF Fj^ 1= 10

295262.22 . 229 5<» 685E-3 D 20DD

-i-^VE^5E -RAM5t^-.H-DfH > P HASE D IF= E^^ i&k^A-B

.25330237E-1^ ID •8763D897E-5 .25192453E-3

21
Appendix B. Mode Mismatch

Reference 6 shows the form for generating the Hermite modes, namely

oo G = exp[2pu - u2] = I H (p) u"/n!

n=0 " :b.i)

Thus, H^(p) = (9y"G|^j_Q. This means we differentiate the generator, G, n times and then set

u = to identify the H^(p) mode.

Reference 6 also shows the normalization of Hermite function under the integration.

Thus, we have

n „n,

/ dx /2/(w,/^) H (/2x/w,) H (/2x/w,)exp( -ZxVw/) = 5' fn 2 ! (B.2) fn f f mn

The eqs (3.1) and (3.2) can be rewritten in terms of the generator eq (B.l) as (suppress

unnecessary variables)

"1 "2 /2" E(xiy) = i (3 ) O ) gB exp[-U32-u42+2_(xu3 + yuj

"1^2

n in1"2

n 1 n ?_ /J

I (9„) (9 J goA„,„„exp[-M32-q^2h2-- (xM3+yu4)

nin2 U3 u^' nin2

:b.3;

We define

with

_ r 2dxdy ,_. w = J 2"- exp(S),

ITW^

;b.4)

S

-- i\d - (x^-+y2)/w^2 _ (ij^?-+u2^+U3^-+u-,^-)

-(x2+y2)(i/(i + r^)2 - 2^2r2)/w^2

+ (2/'2/w JCu^x+u^y + (xu3^-yu^)/(l+rl)], (B.5)

We can apply eq (B.2) by multiplying eq (B.4) and by integrating the resulting equation.

The results are

r- n m

"1 ""1 A = /^ ), (9 ) (3 ) (3 ) (9 ) B W/D , n^n^

3.6)

with D^p^ = (2)"''"'^n!m! . After a straightforward but tedious integration over x and y in the

W, we find W takes the form, W = WoWi, where we have the many definitions given below:

22
Wo = explg^), g^ = 2(u2U4 + uius)

Wi = exp(g^)/(l+f7)^/2

f7 = fi + f2 + fif?. - f72/4

g^ = -f6 - (Ul+U3)2 fl/(l+fl) - (U2+U4)^ f8/(l+f8)

+ [2 f4(ui+U3) + f42]/(i+fi)

+ 2(U2+U4)[f5 + 2(ul+U3)f3/(l+fl)]/(l+f8)

+ [f5 -H 2(ui+U3)f3/(H-fl)]2/{l + f8)

f8 = f2 - f3VC4(l+fl)],

here

^2 ^ (mT^-i^2)(c2^^ c,2s22) -1.

f. = (-2(l+ri)2

C1C2

2ir2)(S2C2)Si2,

kw.

2^4 = 2[-n-- r- - 1]U3 + -T- Ui,S2 - i S1C2

/2

"^ ^^ 2(l+r ) ^ " y2npoCiC2 + qoSz),

2fc = 2uJ.r- 1) - 2

l+r,

U3C1S2 2."^ + W^ S, So

"^ ^^ZU+fY' i''2^^<=l0C2 - P0C1S2)

2x^ 2' ^6 = 71-^77 ("3P0 + ^4^0^ + (^(l+r ) ^ " ii^2Hpo^+qo ;b.7)

We can eliminate Wo in eq {B.6) to get

DA = I B , ,0 )" (9 )"" O +2U3)"0 +2u,)"'Wi

nrn nm ,

, n m Mo Mu u, ^ m. ;b.8)

23
To go further we would assume the fact that 81, 82, n, rz, Po, and qo are small and

consistently retain only the necessary terms. For our purposes this exercise is

unnecessary; therefore, we stop our analysis and just show the definition of C^^n'm' ^^

C .

. =^0 )"'(9 r'o + 2U3)"(9^ + 2u,)\. (B.9)

nmn'm' D M3 u^ m^ ^2

24
Appendix C. The Unseparated Case Analysis

The new formulas of similar form to text that would be used in the more complicated

analysis are given as:

Ui(x,y) = I A^ H^(x,y)

n and

Here we def i ne

where

U2(xi.yi) =TT7 1 A V (xi,yi)(Nc)2. (C.l)

I Az n n n

^(JiJz.") " 7^17 -^ dx^dy^ V^(xi .y^ )G[j ^ J2 .^i .yJ.

s

G(Ji,J2.Xi,yi) = exp [- -K {(x^ - Jid)2 + (y^ - J2d)2}], {C.2) ^

and

-2_2tt V^{xi,yi) = / dx dy H^(x,y) T(x,y) expl-^^ (xix+y ly) j/(Nc)2. {C.3)

The filter function has the form

+5 5 T(x,y) = \ y

^r. , H (x,y) B{x,y) (C.4)

• • c n n.^1^2 "

with < T(x,y) < 1 for all x,y and B(x,y) = exp [TT-rix^+y^) ]. Note, T(x,y) is a real

Wc

number while H^ and C^ inzi f^^y be complex numbers. The Gaussian responses give

K(JiJ2n) =^7^'^^^'^^ H^(x,y) W( j ^ J2)/B(x,y )

,

and

W{Ji,J2) = exp [-2TTf ^(xji + yJ2)]. (C.5)

We make the discrete analysis assumptions

^SL " c(U^), and y_ = (c(I+^). (C.6)

Thus, the complete orthogonal function transformation sequence is given as

25
N/2-1

K{JiJ2,n) = I W(JiJ2)T(x^,y_)H (x^.yJ/LN^Blx^.yJ], (C.7)

and

5 5 5

I I I

'^ Ji=-5 J2=-5 n=0 -'^i'^^ - i I

T(x ,y ) = I I

) C • H (x y )B(x y )/W(JiJ2). (C.8)

Here we assume H^(x£,yj^^) are real functions and have

C* = C n,-Ji,-J2 nJi.J2*

The H (x„,y ) d.vo. chosen to be such that

n X -^

"^ " ^^ ^^^'^J^J^'^J W(-Ji+Ii,-J2+J2)/M^. (C.9)

^ n 'l 1,1

Thus, F = is ji ^ ji and J2 ^ J2 for a11 n and n. If ji = Ji and J2 = J2, the F is if

n / n. Also, F is zero if (ji) = jgln) and (j2) / jj^ln). Finally, if (ji) = Ja(n),

(J2) = j[)(n), then we require F = 1. These values imply 11 x 11 x 6 x 6 constraints on the

form of Hn(xj^,y^).

We can construct a set of H^ functions by differentiation by x and y of a single function, D, and impose the constraint that all functions are nonzero in a finite region of x and y and by the constraint that

\ 0(x^,y_)2 w{j,Ji) = (CIO)

for the allowed set j,ji, namely, < Ijl < 5, and < Ijil < 5.

It is easy to show that the desired H^ functions are a linear combination of those

functions generated by differentiation of D. The basic problem is devising the approp-^iate

D function. To succeed will require computer simulation and appropriate application of

spline theory for two dimensional variable [7,8] and a decision of the reference electric

field that should be used to characterize the particular laser source. One example of such

a reference would be the Hermite TEMoo mode that has been heavily truncated by the

transverse dimensions in the laser source.

2.6
Appendix D. Program GEN

15 CLOSE^s'^'''^" ^^" "^ ^''""^ •'^'^ ^^^^ ^^ HCN,K.L)

16 CLOSE 4

20 OPEN "MB: -,5.1

25 OPEN "CEN:'.4,1

27 REU 5

48 DIM Hid, 81)

76 GOSUB lUB

74 N8-N-1

79 FOR L-6 TO N8

7? GOSUB 1428

78 GOSUB 3886

79 PRINT -L-.U-TIME IS '.GETTirESCB)

82 UF 5

83 REU 5

84 CLOSE 4

85 PRINT -FILE GENERATED*

86 CLOSE 5

87 STOP

1118 REM INITIAL BLOCK

1111 DIM H(7),C(9),S(9),U(7)

1112 C-.2E-1

1113 C3-C+2

1114 IJ1-.2E-1

1115 C2«2*C

1116 iJ2-18

1117 2-1888

1118 Al-l.e64

1119 N-1750

1128 D1«18)4(U1

1121 G5«C2*S0R(2)/IJ2

1122 F-Dl/fil

1124 tB-(l/U2)*2

1125 PI -3. 141592654

1128 U4•(Pl*Ul/fil)'^2

1138 U5<U3H)4

1132 (J6*2«U4

1134 F1«P1*2*F

1135 GOSUB 1488

1218 REM SET GENERATOR FUNCTIONS

1212 C5-C0S(Fl>tC)

1214 S5-SIN(F1KC)

1216 H(8)-l

1218 Vl-EXP(-C3*tJ5)

1228 V2«EXP(-C3«U6)

1222 V3-V1'^2

1224 V4»V2+2

1226 H(l)—G5/^

1228 C(l)-C0S(Fl*C/2)

1238 sen—SIN(Fl*C/2)

1232 C(8)-l

1234 5(8) -8

1236 V5-EXP(-C3*U5*.25)

1238 V6>1

1248 V7-EXP(-C3*US*.25)

1242 V8«l

1258 RETURN

1318 REM HERMITE SET N-0 TO 7

1312 H(1)-HC1)+G5

1314 FOR Nl«2 TO 7

1316 H(Nl)-H(l)*H(Nl-l)-2*(Nl-l)*H(Nl-2)

1318 NE.<T NI 1328 REM COS AND SIN SET

1322 CB.C(1)*C5-S(1)*S5

1324 S(1)-C(1)*S5+S(1)«C5

27
132G C(l)-C8

1328 FOR K-2 TO 9

133B C(K)-C(K-l)*C(l)-S(K-n*S(l)

1332 S(K)-S(K-l)*C(l)+C(K-lJ»t<S<l)

1334 NEi-CT K 1338 V5-V5*V6

1348 V7-V7tfV8

1342 V6 -76*73

1344 V8-Ve*V4

1358 RETURN

1488 REM U(N) SET

1482 U(8)-SQR(2>«:*N*SQR(2/1»l)AE)

1483 FOR Nl-1 TO 7

1484 u(n1)-u(n1-1)*sqr(1/(ni*2))

1485 ne:<t N1

1486 RETURN

1428 REtl GENERf^TE HI (J) FOR EACH L CPSE

1422 GOSUB 1318

1424 Hl(l,81)-V7

1426 FOR Nl-0 TO 7

1427 H2-(-l)'H<l

1428 H4-H(Nl)>t(U(Hl)«VS

1438 FOR K-B TO 9

J432 J-N1+K*8+1

1414 IF N2<S GOTO t44t

1436 H1(1»J)-H4«C(K)

1438 CQTO i4i8

1449 HHl,i)9H4*%tKi^

U51 HEXT K

U5S NEXT H\

I46S RETUtN

3c)bd ktil UU1FILE

3818 MfiT OUTFILE ON (5) HI 3812 MfiT Hl-(B)

3828 RETURN

28881 END

28
Appendix E. Program SELECT

10 REh PROGRfln SELECT RCH)

28 PRINT "THRESHOLD VftLUE, Tl-7"

38 INPUT Tl

40 CLOSE 5

58 CLOSE 4

60 CLOSE 3

70 CLOSE 2

88 N3-1

98 OPEN "FILTER ",4.0

180 OPEN STfl4".2,0

118 RED 2

128 OPEN •CEN:".3.1

130 PRINT ON C3)"PR0GRflM SELECT". GETDflTE$(0).•TI^E IS',GETTirE$(0)

140 OPEN "ri0:".5,0

150 DIM A(1750).E(1?'50),R(9),H1(1.81).U(7).K(5)

160 DIM ASCII, 250). A4(ll, 10)

170 PRINT "REftD IN DATA FROM FILTER YES-l,NO-0"

180 INPUT ft5 190 IF A5«l GOTO 1060

200 PRINT "BASE VALUES. R(6) .R(7).R<8),R(9)-?'

210 INPUT R(6).R(7).R(8),RO)

220 PRINT ON (3) "BASE VALUES R(6) TO R(9) -".RCei ,R(7).R(8),R(9)

230 N-P50

240 N2-N-1

250 C-.2E-1

260 LJ2-10 270 P 1-3. 141592654

280 GOSUB 2100

290 FOR Nl-1 TO 7

308 U(N1)-U(N1-1)/SQR(2«N1)

318 NEXT Nl

328 FOR Nl-8 TO 5

330 PRINT K(';Nl;')-7"

340 INPUT K(N1)

350 R(N1)-K(N1)*U(N1)/D7 ^ 360 PRINT ON (3) "K(N1) .R(N1).N1-".K{N1).R(N1).H1

378 NEXT Nl

398 SrISt "SB.START. INCREMENT. NUMBER OF CASES. THRESHHOLD. Tl400 INPUT S8.S9.N9.Tl

410 PRINT "READ IN HI (I. J) WLUES YES-1 NO-B"

420 INPUT NB

430 IF N8-8 GOTO 1668

440 REU 5

450 FOR L2-0 TO 174

460 FOR Ll-1 TO 10 470 MAT INFILE ON (5) HI

480 GOSUB 1610

490 FOR Nl-0 TO 5

500 N4-N1+1

510 K-Nl+8*(Nl+2)+l

520 A4(N4.L1)-HI(1.K)

530 NE>a Nl

540 A4(11.L1)-H1(1.B1)

550 NE>a LI 560 MAT OUTFILE ON (2) fi4 570 NEXT L2

588 FF 5

598 REU 2

688 L3—

1

618 L-B

628 FOR L2-B TO 174

630 MAT INFILE ON (2) M

640 FOR Ll-1 TO 10

650 L-L+1

660 L3-L3+1

29
678 FOR K-1 ID 11 68B ft5(K,L)-ft4(K.Ll)

698 NEXT K 788 IF L<258 GOTO 730

718 MAT OUTFILE ON (5) fl5 728 L-B

738 REM

748 NEXT LI 758 NEXT L2

768 CLOSE 2

778 PRINT ON (3) "FILE TRANSFER TirE IS VGETTIhESCB)

780 UF 5

798 BF 5 888 IF N3>8 GOTO 1008

618 N3-1

828 GOSUB 1660

838 FOR L2-0 TO 6

848 MfiT INFILE ON (5) A5

858 FOR Ll-1 TO 250

868 L-L1-1+258*L2

878 fl(L)-8

888 FOR K-8 TO 9

898 Kl-K+1

988 ft(L)"fi(L)+fl5(Kl,Ll)«R(K)

918 NEXT K 928 fi(L)-A(L)/fi5(ll.Ll)

938 GOSUB 1530

948 NEXT LI 958 NEXT L2

968 PRINT ON (3) 'LARGEST AC;L3i •)-",A8

970 PRINT "LARGEST A( 'jLS; ")-*,A8

980 PRINT 'SMALLEST A(';L5; )-'.A9

990 PRINT ON (3) "SMALLEST A(";L5; )"",A9

1000 S8-S8-S9

1010 FOR JB-l TO N9

1020 S8«S8-i-S9

1030 GOSUB 1710

1040 NE.XT J0

1050 GOTO 1210

1060 REM READ IN N, R(N),fl(L) ETC.

1070 REU 4

1080 INFILE ON (4) N,U(6),U(7).R(6).R(7),R(B),R(9),D7

1090 PRINT ON (3)"N,U(6),UC),R(6)-R(9).D7-'

1180 PRINT ON (3) N,U(6).UC»).R(6),R(7),RCB),RO),D7

1110 PRINT ON (3)"R(N1),U(N1),K(N1).N1"

1120 FOR Nl-0 TO 5

1130 INFILE ON (4) R(N1),U(N1)

1140 PRINT ON (3) R(N1).U(N1),R(NI)*U(N1)«D7.N1

1150 ne:<t Nl

1160 N2-N-1

1170 FOR L-0 TO N2

1180 INFILE ON (4) ft(L).E(L»

1190 NEl-a L 1208 PRINT ON (3)' FILTER TRANSFER Tirt IS SGETTIhCfCB)

1218 PRINT "NEU K(N)-1, NEU SCALE-B, NO— 1* 1228 INPUT A5

1238 IF A5-1 THEN 320

1240 IF A5-0 THEN 390

1260 REU 5

1270 GOSUB 1328

1280 CLOSE 5

1290 CLOSE 4

1300 PRINT "DONE'.'TIft IS ".GETTIMESCB)

1310 STOP

1320 REM PRINT TO FILTER DATA

1330 REU 4

1350 PRINT "SELECT SB SCALE AND Tl THRESHHOLD VALUES-'

1360 INPUT Sa.Tl

30
1361 FOR 1-8 TO 9

1362 R(I)-R(I)*S8

1353 ne;<t I 1370 OUTFILE ON (4) N,U(S) ,U(7).R(6) .R(7) .R(B),R(9) .D7

1375 CLOSE 3

1380 FOR Nl-0 TO 5

1398 OUTFILE ON <4) R(N1).U(N1)

1400 ne:<t N1

1410 N2-N-1

1420 fl7-0 1430 FOR L-B TO N2

1440 L4«N2-L

1450 A(L4)-S8*fi(L4)

1460 A6-A(L4)

1470 GOSUB 1970

1480 ne:<t L 1490 FOR L-0 TO N2

1500 OUTFILE ON C4) fi(L),E(L)

1510 ne;<t L 1520 RETURN

1530 REM SUBROUTINE FOR SIZE

1540 IF fl8>fl(L) GOTO 1570

1550 flB-fl(L)

1560 L3-L

1570 IF ft9<fl(L) GOTO 1600

1580 fl9«fl(L)

1590 L5-L

1600 RETURN

1610 fl4(7.Ll)-HlCl,l)

1620 A4(8.L1)-HH1.3)

1630 fl4(9,Ll)-Hl(1.5)

1640 fl4(10,Ll)-Hl(1.7)

1650 RETURN

1660 REU 5

1670 FF 5

1680 FF 5

1690 BF 5

1700 GOTO 800

1710 REM GENERATE E(L) GIVEN fl(L). SB

1720 A2-0

1730 T2-1

1740 fi7-B 1750 FOR L-0 TO N2

1760 L4-N2-L

1770 fi6«S8*A(L4)

1780 GOSUB 1970

1790 fi2-fl2+ft7'^2 1800 NE'.-a L

IBIB El"fl7/H

1820 E2-fi2/N

1830 E3-SQR(E2-Elit€l)

184a PRINT ON (3) 'ERROR EUE3, VEl.ES. 'SCfiLE-'.SB

1850 PRINT ON (3)'TirE '.GETTirtSCB)

1860 PRINT ON C3) 'THRESHOLD VfiLUE-*,Tl

1870 RETURN

1880 BF 5

1890 FF 5

1900 AB'l

1910 L3 —

1

1920 L5-1750

1930 fl9«l 1940 88 «0 1950 B9«0

1960 RETURN

1970 REM SUBROUTINE TO SET THE FILTER

1980 REM START AT L4-N2 UITH E(N2)-a

1990 IF L>0 GOTO 2030

2000 E(L4)-0

2010 A7-A7+A6

31
2020 RETURN

2030 ft?'ft?+ftG 2040 IF fi7>Tl GOTO 2078

2058 E(L4)-B

2068 RETURN

2070 E(L4)-1

2080 fl7-fl7-T2 2898 f^ETlMH

2100 REM SUB

2UB D7-C)*t^«SQR(t/fn<iA

2128 U(9)-S0l|(gl«n

111 38 RETURN

£14a CLQH I 1190 OPEH 'CEHlMa

Bl6kJ FOk L-0 TO N2

2170 print on (3) a(l).e(l).l;

2180 ne:<t L 2190 CLOSE 3

2200 END

32
Appendix F. Program ORTHBAS

1 REM PROGRnn ORTHBRS

18 CLOSE 5

26 CLOSE 3

30 OPEN r«:',5,2

48 OPEN "CEN-.'.S.l

58 REU 5

55 PRINT X1,X2,X3-?"

56 INPUT X1.X2.X3

57 PRINT ON (3)"Xl.X2.X3-",Xl.X2i.X3

68 PRINT ON (3) 'TIME START '.GETTIMESCB)

78 DIM H(l.Bl)

71 DIM S(3325)

79 K—

1

88 FOR J-XI TO X2

81 FOR J1«J TO X3

82 K«K4-1 83 S(K)-e

84 NEXT Jl

85 NEXT J

86 Kl-K

87 PRINT "KI-".Kl

88 STOP

98 FOR L-B TO 1749

95 MftT INFILE ON (5) H

188 K —

1

110 FOR J-Xl TO X2

112 X-H(1.J)/H(U81)

115 FOR Jl-J TO X3

128 K-K+1

125 S(K)-S(K)+H(1.J1)*X

130 NEXT Jl

135 NEXT J tis MINT *L**,L.cErTtrct(ei

140 NEXT L

15t REU 9

169 CLOSI S

J65 OPEH •MTN0M*9

17f FOR K'l TO Kl

169 OMTPILI Oh (SI 91(KI/879

Ibl Kklhl UN i3)'i>i';Ki')»'.S(K),

185 NEXT K

186 PRINT ON (3) -END OF DflTrt OUTPUT .GETTItt$(0)

280 STOP

210 END

33
Appendix G. Program CHECK

10 REM PROGRfln CHECK AN© REDUCE OF S SET WITH ERROR CHECK

20 CLOSE 4

25 CLOSE 5

30 CLOSE 3

40 OPEN "MB:', 5.2

50 OPEN -CENt'.S.l

60 OPEN -FILTER". 4,9

65 REU 5

66 PRINT ON (3)"TirE START". GETTIME$(0)

67 PRINT "TirE STftRT".GETTIhE$(0)

68 PRINT ON (3)"PR0GRflM CHECK DOTE OF RUN IS ".GETDfiTE$(0)

70 REU 4

71 DIM HlCl.ei).U(7).RC9),E(1750).fi(1750).El(1750)

72 INFILE ON (4) N.U(6) .U(7).R(6).R(7).R(8),R(9).D7

73 PRINT ON (3)"N.U(6).U(7).R(6)-R(9),D7"

74 PRINT ON (3) N,U(6) .U(7) .R(6) ,R(7) .R(B),R(9) .D7

75 FOR Nl-0 TO 5

76 INFILE ON (4) R(N1),U(NH

77 PRINT ON (3)"R.U,K.U1TH N1-".N1

78 PRINT ON (3) R(N1)*U(N1) .R(N1)*U(N1)*D7

80 NEXT Nl

81 N2-N-1

85 FOR L-0 TO N2

90 INFILE ON (4) fi(L).E(L)

95 NEXT L

100 REU 4

105 CLOSE 4

110 DIM Sl(1.81).S(l.ei).fl2(1750)

120 FOR L-0 TO N2

125 MPT INFILE ON (5) HI

130 FOR J-1 TO 80

134 G8-ECL)-fi(L)

135 S(l.J)-S(l.J)-tWl(l,J)*G9

136 NEXT J

140 NEXT L

145 REU 5

146 PRINT ON (3) "TIME FOR S".GETTIMES(0)

147 PRINT -TirE FOR S'.GETTIMESCB)

148 N8-H/2

150 FOR Nl-0 TO 7

151 N3•(-1)'^N1

152 FOR K-0 TO 9

153 J-K:«<8+N1+1

154 IF K>0 GOTO 156

155 GOSUB 1000

156 S(1.J)-S(1.J)/N8

158 NEXT K

160 NEXT Nl

165 PRINT "SET FOR REDUCE IS I FOR NO fiND FOR YES"

166 INPUT R

168 IF R-1 GOTO 200

169 GOSUB 2000

200 FOR L-0 TO N2

201 El(L)-0

205 MAT INFILE ON (5) HI 208 FOR J-1 TO 80

210 E1(L)-E1(L)+H1(1,J)*S(1.J)

212 NEXT J 213 E1(L)-E1(L)/Hl(l.ei)

215 NEXT L 216 REU 5

270 PRINT ON (3) "TIME FOR S1".GETTIME$(0)

271 PRINT -Tirt FOR SI ",GETrirt$(0)

300 MAT Sl-(0)

310 FOR L-0 TO N2

315 MAT INFILE ON (5) HI 320 FOR J-1 TO 80

34
325 S1(1.J)-H1(1.J)*E1(L)+SI(1,J)

326 NE>a J 330 NEXT L 348 REU 5 350 FOR m-0 TO 7

355 NS-C-D+Nl

360 FOR K-0 TO 9

365 J-l+K*8+Nl

370 IF K>0 GOTO 375

371 G05UB 1100

375 Sl(l,J)-Sl(l,J)/Ne-S(UJ)

380 HEXT K 385 NEXT Nl

389 PRINT ON (3) 'S VALUES '.GETTirE$(0)

390 MAT PRINT ON (3) S

391 PRINT S AND SI COrFLETED'.GETTIMESCB)

394 PRINT ON (3) "DIFFERENCE IN S VALUES DUE TO TRANSFORM"

395 MAT PRINT ON (3) SI

396 OPEN "TRUE". 4,0

397 A3-0

398 REU 4

399 PRINT ON (3) "L^A.E.El.E-EI LIST " 400 REM

402 FOR L-0 TO N2

403 OUTFILE ON (4) A(L) .E(LJ .El(L)

404 PRINT ON (3) L,A(L),E(L) ,E1 (L) ,E(L)-E1 (L)

405 NEXT L 500 PRINT "DONE".GETTIME$(0)

510 REU 4

520 REU 5

530 CLOSE 3

540 CLOSE 4

545 CLOSE 5

550 STOP

1000 REM

1001 IF N3<0 GOTO 1010

1005 S(l.J)-S(l.J)/2

1006 GOTO 1015

1010 S(l.J)-0

1015 RETURN

1100 REM

1101 IF N3<0 GOTO 1110

1105 Sl(l,J)-Sl(l.J)/2

1106 GOTO 1115

1110 Sl(l.J)-0

1115 RETURN

2000 REM REDUCE OF AREA IN COfUTATION

20B1 FOR Nl-8 TO F 2002 FOR K-0 TO 9

2003 J-9*K-mi-|

2004 IF K<2 OR %>? GOTO »«

20B9 GOTO 2Blf

20B6 SU'J)»f

2087 HE'XT K

2009 J-8*(Nl+2)+Nl-l

2010 S(l.J)-0

2011 ne:<t Nl

2015 RETURN

6000 END

35
Appendix H. Program CONVERGE

le REM PROGRPin CONVERGE

15 DPiTfl 2.U2.1«3.e,3.1.3>1.3*l

16 CLOSE 2

17 OPEN "CEN:".2.8

20 60SUB 11B0

25 PRINT 'START TIhE '^GETTirESCB)

30 PRINT "READ INITIAL FILTER YES-1,N0-B. ITERATION LEVEL L9'

35 INPUT 18,19

48 IF I8«0 GOTO 90

45 GOSUB 1000

46 GOSUB 1250

50 GOSUB 1500

54 PRINT ON (3)'S UITH R',GETTiriE$(0)

55 GOSUB 1600

60 GOSUB 1700

64 PRINT ON (3)*S UITHOUT R'.GETTIME$(0)

65 GOSUB 1800

70 GOSUB 1600

75 GOSUB 2000

80 GOSUB 1400

85 GOTO 95

90 GOSUB 1250

91 GOSUB 1300

92 I9-I9+1

95 GOSUB 1900

485 I9-I9-1

486 PRINT ON (3) • ITERATION '/ALUE ISM9.GETTIME$(0)

487 PRINT 'ITERATION VALUE IS ", 19,GETTIME$(0)

490 IF I9>0 GOTO 50

495 GOSUB 1280

500 GOSUB 1200

510 PRINT "TIME STOP',GETTII-E$(0)

515 STOP

1000 REM SUBROUTINE TO READ FILTER AND PRINT STATUS

1001 PRINT 1000'

1010 CLOSE 4

1020 OPEN 'FILTER •.4.0

1025 REU 4

1030 INFILE ON (4) N.U(6) .U(7) .R(6).R(7) .R(B) .R(9).D7

1035 PRINT ON (3) 'N.UCe) .UC).R(6)-R{9) .D7"

1040 PRINT ON (3) N.U(6).UC).R(6).R(7).R(8) ,R(9).D7

1045 FOR Nl-0 TO 5

1050 INFILE ON (4) R(N1).U(N1)

1055 PRINT ON (3) "R. U.K. UITH Nl-'-Nl

16GQ PRINT G;-1 (3) R CMi ; ,'J '.N 1) ,Ri;MU^'<ri?/UCMl)

1065 ne:<t N1

1070 N2-N-1

1075 FOR L-0 TO N2

1080 infile on (4) acd.ecd

1085 ne:<t L 1090 REU 4

1095 CLOSE 4

1099 RETURN

1100 REh SUBROUTINE TO SET INTITAL CONDITIONS

1101 PRINT '1100"

1102 N4-1750

1105 DIM H(1.81).S(1.8l).E(N4).A(N4).El(N4)

1106 DIM U(7).R(9)

1110 CLOSE 5

1115 CLOSE 3

1120 OPEN M0:".5.2

1125 OPEN 'CEN:'.3.1

1130 REU 5

1135 PRINT ON (3)'TI^E START". GETTirE$(0)

1140 PRINT ON (3) 'PROGRAM CONVERGE. DATE IS •.GETDATE$(0)

1145 RETURN

36
w

1288 REM SUBROUTINE TO CLOSE PROGRAM

1281 PRINT 128B*

1285 RED 5

1218 CLOSE 5 1211 PRINT ON (3)"fi(L).E(L).El(L).L"

1212 FOR L-e TO N2

1213 PRINT ON (3) flCL) .E(L) .E1(L),L

1214 ne:<t L 1215 PRINT ON (3)'TirE STOP',GETTirE$(0)

1228 CLOSE 3

1222 CLOSE 2

1225 RETURN

1258 REM SUBROUTINE TO OPEN TRUE

1251 PRINT '1250'

1255 CLOSE 4

1265 OPEN 'TRUE ',4,0

1266 REU 4

1278 RETURN

1288 REM SUBROUTINE TO CLOSE TRUE

1281 PRINT '1280"

1285 REU 4

1298 CLOSE 4

1295 RETURN

1308 REM SUBROUTINE TO READ TRUE

1381 PRINT '1300'

1305 REU 4

1310 FOR Nl-0 TO 9

1315 infile on (4) r(n1)

1328 ne:<t N1

1325 FOR J-1 TO Bl

1338 INFILE ON (4) S(l.J)

1335 ne;<t J 1348 INFILE ON (4) S2,S3.N

1341 N2-N-1

1345 FOR L-0 TO N2

1358 INFILE ON (4) fi(L) .E(L) .El (L)

1355 ne:<t L 1368 RETURN

1408 REM SUBROUTINE TO PRINT TO TRUE

1401 PRINT '1400'

1405 REU 4

1418 FOR Nl-B TO 9

1415 OUTFILE ON (4) R(N1)

1420 NE:<T Nl

1425 FOR J-1 TO 81

1430 OUTFILE ON (4) 8(1. J)

1435 ne:<t J 1440 OUTFILE ON (4) S2.S3.N

1442 PRINT ON (3) N4.N.N2-'.N4.N.N2

1445 FOR L-0 TO N2

1450 OUTFILE ON (4) fl(L) ,E(L) .El(L)

1455 ne:<t L 1460 RETURN

1500 REM SUBROUTINE TO GENERATE S.S2.S3 GIVEN E(L) fl(L)

1501 PRINT '1500'

1505 S3-0

1510 S2-0

1511 REU 5

1515 MAT S-<0)

1520 FOR L-0 TO N2

1525 MAT INFILE ON (5) H

1530 FOR J-1 TO 80

1534 A7-E(L)

1535 S(l.J)-S(l.J)4H(l.J)*A''

1540 ne:<t J 1545 S3-S3+(E(L)'^2)«H(l.Bl)

1550 S2«S2+(A(L)'^2)*H(l.Bl)

37
1555 ne:<t l

1566 REU 5

1561 N8-N/2

1562 FOR Nl"8 TO 7

1363 N3•(-1)'^N1

1564 FOR K-B TO 9

1565 J-8*K+N1+1

1566 IF K>B GOTO 1572

1567 IF N3<0 GOTO 1570

1568 S<l,J)-S(l,J)/2

1569 GOTO 1572

1570 S(l.J)-0

1572 S(1.J)-S(1.J)/NB

1573 ne:<t K 1574 HEXT Nl

1575 S2-S2/N8

1576 S3-S3/N8

1577 RETURN

1600 REM SUBROUTINE TO GENERATE El GIVEN S

1601 PRINT "leao'

1605 FOR L-0 TO N2

1610 El(L)-0

1615 MfiT INFILE ON (5) H

1620 FOR J- 1 TO 60

1625 E1(L)-E1(L)+H(1,J)«S(1.J)

1630 NEi-CT J 1635 E1(L)-E1(L)/H(I.B1)

1640 ne:<t L 1645 REU 5

1650 RETURN

1700 REM SUBROUTINE TO REMO'iE R(N1) FROM S

1701 PRINT M700"

1705 S(1.1)-S(1.I)-R(6)

1706 S(1.3)-S(l,3)-R(7)

1707 S(1.5)-S<1.5)-R(8)

1709 S(1.7)-S(1.7)-R(9)

1710 S(l,17)-S<l,17)-R(0)

1712 S(1.26)-S(1.26)-R(l)

1714 S(l,35)-S(l,35)-R(2)

1716 S(1.44)-S(1.44)-R(3)

1718 S(l,53)-S(1.53)-R(4)

1720 S(1,62)-S(U62)-R(5)

1725 FOR K-0 TO 9

1726 FOR Nl-a TO 7 1727 J-8*K+N1+1

1728 IF K>1 fiND K<B GOTO 1735

1729 S(l.J)-B

1735 IF NK6 GOTO 1740

1736 S(l.J)-0

1740 NEi-a Nl

1745 ne:<t K 1750 RETURN

1800 REM SUBROUTINE TO PRINT S

1801 PRINT '1800"

1810 MAT PRINT ON (3) S

1815 S5-0

1820 FOR J"l TO B0

1825 S5 -354^(1, J )'^2 1630 ne:<t J 1831 FOR J-1 TO B

1832 S5-S5+S(1.J)'^2

1833 he:<t J 1640 PRINT ON (3) "S+Z.SS-'.SS

1850 RETURN

1900 REM PRINT R,S2.S3,S4

1901 PRINT '1900"

1905 PRINT ON C3)"R SET",GETTIME$(8)

1908 S4-0

1910 FOR Nl-0 TO 9

38
1915 PRINT ON (3)"R(';N1;*)-'.R(N1)

1926 S4-S4+R(N1)*2

1921 IF NK6 GOTO 1925

1922 S4«S4+R(N1)'^2

1925 ne:<t N1

1930 PRINT ON (3)"E+2 SUM S2-';S2.'fl+2 SUM S3-':S3

1935 PRINT ON (3) 'R+2 SUM S4«*;S4

1940 RETURN

2000 REM GENERATE E(L) GIVEN fl(L).El(L).R8

2001 PRINT •2000"

2004 REi^D L6.R7

2005 R8«L6-1

2006 PRINT ON (3) QUANTT7Fn L6-'.L6, "RELflXftTION R7«',R7

2010 L5-l/1?8

2915 J5M/L6

2913 L8«l

2B20 fl7»0 2321 B7»9

2325 FOR L-0 TO N2

2033 L4=N2-L

2335 flS=fi(L4)-El(L4)«R7

2040 fi7-fl7+ft6 2045 B1«L3

2053 B2=L7

2055 FOR J-1 TO L6

20SQ B1=B1-J5

23E5 E2=B2-L5

2070 IF fl7>Bl GOTO 2089

2075 ne:<t J 2080 E(L4)-B2

2LJ1 B7«B7+B2

2035 A7-A7-B2

2;j35 HE:<\ L ^, ^. 2C33 PRINT ON f.3) "SUM OF E(L) IS .B7,*N0RMfiL B-'/N -•.B7/N

2E33 PRINT HN (3)"REmiNDER A7"".ft7

2G?3 PE^.""^

2'::3e i:i-;o

39
NBS-n4A (REV. 2-8C) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS TN-1057

2. Performing Organ. Report No. 3. Publication Date

September 1982

4. TITLE AND SUBTITLE

Beam-Profile Measurement of Laser Pulses Using A Spatial

Filter to Sample the Hermite Modes for a String of Pulses

5. AUTHOR(S)

Eric G. Johnson, Jr.

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP)

10. SUPPLEMENTARY NOTES

[^J Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

As a first step in the development of a beam-profile measuring instrument

for laser sources that is capable of determining the distribution of low-order

(less than 25) Hermitian modes in a series of laser pulses, I designed and

evaluated the three key parts of such an instrument. First, there is the

telescope system which allows the incident laser beam to be phase, beamwidth,

and beam center matched to the optical spatial filter. Second, there is a brief error analysis of the structure of the mismatch function between the

beam out of the telescope and that expected by the filter. Finally, there is

the detailed analysis and design of the computer-generated spatial filter that

will cause the incident-laser beam to be cross correlated with the low-order

Hermite modes and will create an array of light spots in the detector (Fourier

transform) plane each of which can be uniquely related to a particular Hermite

mode of the original laser pulse.

The principal conclusion is that the Hermite mode analysis can be done

with better than 99 percent separation between modes, provided the phase

between modes is uncorrelated from pulse to pulse when the filter has been

fabricated with a two-level, gray-scale structure which samples the profile

with either percent, or 100 percent transmission.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

computer simulation; laser beam profile; mode-matching analysis; spatial

filter, target designators.

13. AVAILABILITY

llf] Unlimited

[^ For Official Distribution. Do Not Release to NTIS

[x"! Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C

20402.

[2^ Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

43

15. Price

$4.50

'' U.S. Government Printing Office 1982 - 676-774/39 Reg. 8 USCOMM-DC 6043-PPO
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