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AF:T/GEP/?H/8ID-2 (D

MODE ANALYSIG IM A MISALIGNED UNSTABLE RESONATOR

THESIS

AFIT/GEP/PH/81D-2 Richard w. Berdine ILt USAF

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• d •.—•«• »• i • ii • 11 »• m —

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"-*• • - - ..—.-^~——*.-~..»-»^W- — -" • ----

AFIT/GEP/PK/81D-2

MODE ANALYSIS IN A

MISALIGNED UNSTABLE RESONATOR

THESIS

Presented to the Faculty of the School of Engineering

of the Air Force Institute of Technology

Air University

in Partial Fulfillment of the

Requirements for the Degree of

Master of Science

by

Richard W. Berdine, B.S.

lit USAF

Graduate Engineering Physics

December 1981

plööesslöi) For

' I I. '1

i .

at - Codes_

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>ist : Special

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=3B>^H

PREFACE

The purpose of this study was to analyze the effects of mirror mis-

alignment on the transverse modes and beam steering of an unstable laser

resonator. The analysis was developed for any general unstable resonator

design with rectangular apertures and did not allow for inclusion of a

gain medium. The final result, a computer program, can be used to cal-

culate mode eigenvalues and subsequently the intensity and phase in the

plane of the feedback mirror for a desired mode. The slope of the phase,

from which a beam steering angle can be determined, is also calculated

for the lowest loss mode. The resulting tilt in the phase front is due

to diffraction, and is consequently a beam steering angle additional to

the geometric misalignment. The code is basically an extension, or mod-

ification, to a previous computer model developed by J.E. Rowley, and

follows similar work done by P. Horwitz.

Although a major portion of this work may be found elsewhere, speci-

fic details and applications throughout the text are generally not avail-

able. They are provided here in order to present a complete and clear

progression of the analysis. A large number of equations and derivations

are required since the topic is analytical in nature rather than experi-

mental. This sometimes leads to trivial substitutions and algebraic

steps being included for the sake of continuity, but it is hoped that

these are minimal. Physical interpretations and definitions are included

where possible for better understanding.

I would like to express my gratitude to my advisor, Lt. Col. John

Erkkila, for his time, patience, guidance, and especially his enthusiasm

which inspired me continually throughout.

ii

— - - • -

Table of Contents

Preface ii

List of Figures v

Abstract vii

I. Introduction 1

Background 1 Objectives . 6 Assumptions 7 Procedure and Organization 8

II. Development of the Integral Equation from the Fresnel - Kirchhoff Diffraction Formula .... 9

1-D Analysis 9 Introduction of Phase Lag 12

III. The Integral Equation Appropriate to the Misaligned Resonator 19

Effects of Mirror Misalignment 19

IV. Solution of the Generalized Integral Equation .... 2k

Stationary Phase Approximation 2k The Polynomial Equation 28 Second Order Approximation . 30

V. Beam Steering 33

Geometrical Beam Steering 33 Beam Steering Angles Due to Diffraction 35

VI. Results, Conclusions, and Recommendations 38

Results 38 Conclusions 39 Recommendations ..... kO

Bibliography k6

Appendix A (Solution of definite integral in equation (2.23) ) • ^

Appendix B (Final form of the integral equation) 5*

Appendix C (The polynomial equation) ......... 53

iii

J.—~-.—^-—....-•...,. • ... .. ^ .^ .... ...... .. ^_

Appendix D (The plots of higher order modes)

Appendix E (Program listing)

Vita

55

61

75

iv

_ rtfiM—i ililHiilli

List of Figures

Figure

1-1 Stable and unstable resonator geometries

1-2 Equivalent lens system of an unstable resonator

2-1 Geometry of rectangular plane mirrors .

2-2 Typical unstable resonator geometry (for phase lag)

3-1 Geometry of an unstable laser resonator (misaligned) .

5-1 Equivalent asymmetric cavity of the misaligned resonator

6-1 Intensity plot for the lowest loss mode with 6 = 0.0 .

6-2 Phase plot for the lowest loss mode with 6 = 0.0

6-3 Intensity plot for the lowest loss mode with 6 = 0.2

6-4 Phase plot for the lowest loss mode with 6 = 0.2

6-5 Intensity plot for the lowest loss mode with 6 = 0.5

6-6 Phase plot for the lowest loss mode with 6 = 0.5 .

6-7 Pnase slope vs. mirror misalignment (magnification =2.0 and Equivalent Fresnel number =9.6)

6-8 Phase slope vs. mirror misalignment (magnification =2.9 and Equivalent Fresnel number = 16.4)

D-l Intensity plot for mode #2 and 6 = 0.0

D-2 Phase plot for mode #2 and 6 = 0.0

D-3 Intensity plot for mode #2 and ° = 0.5

D-4 Phase plot for mode #2 and 6 s 0.5

D-5 Intensity plot for mode #3 and 6 = 0.0

D-6 Phase plot for mode #3 and 6 = 0.0

D-7 Intensity plot for mode #3 and 6 = 0.5

D-8 Phase plot for mode #3 and 6 = 0.5

Fage

2

3

10

11

20

35

41

41

42

42

43

43

44

45

56

56

57

57

58

58

59

59

Figure

D-9 Intensity plot across the feedback mirror only using the first order approximation

D-10 Phase plot across the feedback mirror only using the first order approximation

Pa,~e

60

60

vi

, • - •--—--• • - • -• • JA •"- — -•*---• ' '

ABSTRACT

The integral equation that describes mode structure of an unstable

resonator with rectangular apertures is developed from scalar diffraction

theory. This equation, modified to account for misalignments, is solved

by applying the asymptotic methods developed by Horwits. A second order

approximation of the method of stationary phase is then employed to cal-

culate phase and intensity values for all points in the output plane.

The phase front is also curve fitted to a straight line over the geomet-

rical region for the lowest loss mode. From the slope cf the straight

line, a direction of propagation can be attributed to the wave. This is

a diffracted beam steering angle and is additional to the geometric steer-

ing angle (i.e., the beam steering angle due to the geometric misalign-

ment of either or both mirrors).

Plots of intensity and phase for various degrees of misalignments

are presented as results of a computer program that utilizes the derived

expressions. Also included are graphs of the phase slope versus mirror

misalignment.

vii

•*-*—-*-->- —

MODE ANALYSIS IN A

MISALIGNED UNSTABLE RESONATOR

I. Introduction

Background

In any laser cavity there are two modes of oscillation: longitudi-

nal (or axial) and transverse (or radial). The longitudinal modes cou-

pled with a specific transverse mode determine the frequency of radiation

emitted, while the transverse modes alone determine the intensity distri-

bution across the laser beam. In either case, the existing modes can be

found by applying the appropriate boundary conditions to the wave equa-

tion. In a stable resonator with spherical mirrors and edge effects

(diffraction effects due to the finite size of the mirrors) neglected,

the transverse solutions to the wave equation become Hermite-Gaussian cr

Laguerre-Gaussian functions for mirrors with either rectangular cr cir-

cular symmetry respectively (Ref. 1:1324).

For the unstable resonator, edge effects must be included since the

output is a diffraction-coupled beam passing around, rather than through

the output mirror. An iterative approach to finding the resonant modes

of the cavity accounts for these edge effects and was introduced by Fcx

and Li (Ref. 2). This method is to first assume an initial field distri-

bution in the resonator and then apply the Fresnel-Kirchhoff diffraction

formula to the problem of infinite strip mirrors.

Figure 1-1 shows differences between a stable resonator and an un-

stable resonator. Any ray originating in the stable resonator and strik-

ing one of the mirrors will remain inside the cavity, even for an infi-

jm • i>i .

STABLE

Mirror 1 Kirror 2

2a,

Mirror

Fig. 1-1. Typical stable and unstable resonator geometries.

nite number of passes. In the unstable resonator however, the ray will

eventually pass out of the cavity. For the case under consideration, a.

assumed large in comparison with beam spot size on mirror 1, the ray will

pass around the much smaller feedback mirror. It can therefore be seer.

that only edge effects from the smaller mirror need to be considered.

The more familiar criterion for stable and unstable resonators is

given by the product of g and g , where

1 -1 1,2 (1.1)

th Here L = cavity length and E- = radius of curvature of the i mirror

Also, R. is defined as positive for concave mirrors and negative for

conve-:. For stable resonators the product of the g parameters lies

inside the range of

ataMMMlfe* L .,...

Co. -t

1

UNSTABLE RESONATOF

7

ill

F-i r'

T 2a, r

EQUIVALENT LENTS SYSTEM

T 2a- t

Fie. 1-2. Eauivalent lens system of an unstable resonator,

0 < 'ias

(1.2)

''nstable resonators however, are characterized by either g g > 1 or

6,ga < ° •

The equivalent lens train of a typical unstable resonator is shown

in Figure 1-2 . As stated earlier, an initial field distribution is as-

sumed at P and propagated through one round trip to P' using the

Fresnel-Kirchhoff diffraction formula. With the restriction that the

field reproduce itself after the round trip and introducing phase lag due

to mirror curvature, the resulting integral equation is

v g(x)

-l

N -it(v - x/M)2 g(y) e K- ' ' dy (1.3)

Here x is the spatial coordinate, y is a dummy variable of integra-

----- — • -- I • I »—I-''

tion, i = -J^l , M = cavity magnification, g(x) is the field distribution

on the output mirror, and t is defined in equation (E.10). The trans-

verse field distribution of the resonator modes are then given by the

eigenfunctions of the integral equation. The multiplicative constant, v,

is the eigenvalue associated with the eigenfunctions and gives the dif-

fraction loss and phase shift of the mode (Ref. 1:1325).

Rowley (Ref. 3), following the analysis of Horwitz (Ref. 4), developed

a computer code to numerically solve the integral equation. He assumed

that the field on the mirror before the round trip, g(y), consisted of a

unit amplitude cylindrical wave plus a series of edge diffracted waves,

and is given by

N g(y) = 1 + nSa cnHn(y) (1.4)

(Ref. 3:17). Therefore, the field on the mirror after the round trip,

g(x), would differ only by the multiplicative constant v .

Letting

cnHn(y) = ^„(y) + \Gn(y) - (1.5)

the integral equation to be solved is >

N v Cl + J, Ca^x) + bnGn(x)]3 =

ys fe-it(y - X/M)2ü + ji[anFn(y) + bnGn(y)]} dy (1.6)

-1

The method of stationary phase is used to approximate this integral.

'•'—-—*—-—'•*•• *• • ..••... .-.-. 1. . -. .-•-- n, ,1, -—• — — - - • -- •---*•'

't states that an integral of the fori

.b

•itp(y) I =

/••"' q(y) dy (1.7)

car. be expressed as a series when t is large and q(y) is a slowly vary-

ing function. The first two terms of the series are given by

I a e -ir/U Q(y ) e-itp(y°> / *j .

q(b) -itp(b) _ q(a) -itp(a) pTb] p'U; (1.8)

(Ref. 3:19; 14:1073). This is a first order approximation to the inte-

gral equation and is used in Chapter ^ for determination cf eigenvalues.

A second order approximation is also given in Chapter 4 and is required

for intensity and phase calculations outside the mirror edges. In equa-

tion (1.8) the prime indicates the derivative of the function with re-

spect to y, and y0 is the stationary phase point, i.e.,

p'(y0) • o (1.9)

Before equation (1.6) can be solved, explicit forms for Fn(y) and

Gn(y) are needed. They are

F (y) "'n-i exp[-it(l - yA'n)/Mn-1J

^int 1 - y/Mn (1.10)

and

H^BMHBA

G (y) =-J*nzl e>-d:-it(l - y/M r/Mp-V {un) n V ^iTTt 1 + y/M"

(fief. 3:20; 4:1531). where the variable M is defined in equation

(3.12). Horwitz determined the functional form of the P"n's and Gn*s

by an asymptotic expansion of the integral equation (fief. 4). The spe-

cific set of functions were then chosen for their reproducibility, i.e.,

after making the stationary phase approximation to equation (1.6) the

same set of functions are arrived at. Equating coefficients then leadc

to a 28 + 1 degree polynomial in v, which is solved by a generalized

root finding routine. Once the eigenvalues have been determined, the

field is calculated by first specifying a particular eigenvalue or mode

and then calculating the constants a^ and tu (a detailed solution is

, left to Chapter 4).

A limitation to this analysis is the fact that it is only valid for

cavities with perfectly aligned mirrors. Since mirror alignment is crit-

ical to the effectiveness of any laser system, it is desirable to be able

to determine the sensitivity of the system for a specified misalignment.

This sensitivity is measured in terms of the power out, beam steering,

and mode distortion (i.e., the beam quality). Several investigations in-

to mode distortion due to mirror misalignment have been reported (Ref. 5i

6,7,8), however, only a few have dealt with the problem of beam steering

(Ref. 8).

Objectives

The purpose of this study is to analyse the effects of mirror mis-

alignment on the transverse modes and beam steering of an unstable laser

resonator. The analysis is developed for a bare strip resonator with

^^^^

r rectangular aper-^ures following similar work done by Korwitz (Ref. 5).

A computer program was written (a modified version of a code devel-

oped by Rowley) to facilitate the study. Tne computer model calculates

node eigenvalues and subsequently evaluates the intensity and phase in

the plane of the feedback mirror for a desired mode. The slope of the

phase, from which the direction of propagation can be determined, is al-

so calculated for the lowest loss mode. The resulting tilt in the phase

front is due to diffraction, and is a beam steering angle additional to

zhe geometric misalignment.

Assumptions

The basic assumptions of Reference 3 s^e still invoked since the

analysis is an extension of this previous work. In addition to these is

the added restriction of small misalignments (Assumption 5). They are

(Ref. 3=2) :

1. That scalar diffraction theory can be used if a) the diffract-

ing aperture is large compared to wavelength and b) the diffracted

fields are not observed toe close to the aperture. The first constraint

is easily attained at optical wavelengths and the second is achieved at

moderate cavity lengths.

c. That the diffraction integrals and mode eiger.functions are sep-

arable allowing a 1-D strip resonator to be used in the foregoing anal-

ysis. When the mirror separa-„ion is very mucn larger than the mirror

dimensions, the problem of the rectangular mirrors reduces to a one-di-

mensional problem of infinite strip mirrors (Ref. 2:^57).

3. Diffraction effects from enly the feedback mirror are considered

if the larger mirror is considered infinite in comparison with the beam

spot size on that mirror.

4. That the modes in the strip resonator consist of a fundamental

cylindrical wave modified by a finite number cf edge diffraction effects.

This assumption is supported by early analysis of unstable resonators

(Ref. 9:279).

5. That the optical axis (the line joining the centers of curvature

of the two mirrors) does not pass too close (within a Fresnel zone) to

the edge of the small mirror (Ref. 5"-l67).

Procedure and Organization

In Chapter 2 the Fresnel-Kirchhoff diffraction formula is modified

to include phase lag due to mirror curvature for a strip resonator; and

in Chapter 3 the resulting integral equation is generalized to include

effects cf small misalignments. Chapter 4 then deals with its solution

using the stationary phase approximation. In Chapter 5 the diffracted

beam steering angle is presented and the magnitude compared to the geo-

metric beam steering angle, while the results and conclusions are left

to Chapter 6.

• -* ' ' • 1) ' I»"«..

II. Development of the Integral Equation

from the

Fresnel-Kirchhcff Diffraction Formula

Equation (1.3) is derived by first reducing the problem of rectan-

gular mirrors tc a one-dimensional problem of infinite strip mirrors.

This analysis follows the work of Fox and Li (Ref. 2). The next step is

to account for phase lag due to mirror curvature, which can be found in

References 3 and 4 . Although this derivation can be found in the ref-

erences cited, it is included here for easy access and completeness of

the study.

i

1-D Analysis (Ref. c-.k^o-h^, ^S^-^-36)

In a Cartesian coordinate system, the field E du3 to an illumi-

nated aperture A is given by the surface integral

Ep(x.y) ik f , y) e"lkr (1 + cosG) dS (2.1)

where Ea is the aperture field, k is the propagation constant, r is

the distance from a point on the aperture to the point of observation,

and 6 is the angle which r makes with the unit normal to the aper-

ture. From Figure 2-1, where propagation is from right to left, this car.

be written as

c a

(1 + £' ) dx_ d\ (2.2)

. -— • — •

Fig. 2-1. Geometry of rectangular plane mirrors.

Also X is the wavelength of the radiation and r is the distance de-

fined by

• \A£ + (xg - x:):

(y, *!>' (2.3)

If L is large compared to the dimensions of the diffracting aperture,

mirror 2, then the binomial expansion of the above square root can be

approximated by retaining only the first two terms of the expansion, i.e.

r*iO 4(x?p)g4

p I P'

Mirror 1 Mirror 2

Fig. 2-2. Typical unstable resonator geometry.

will be more sensitive to this round-off, therefore equation (2.4) is

substituted into equation (2.2) to become

B(x1(ya) = . -ikL le

XL E(x2,y2) e *'• Or. - ar,)"] ZL'

-c -a

dxa dy,

[2.5)

The resonant modes of any laser cavity are characterized by repro-

ducible fields after only one round trip through the resonator. From

Figure 2-2 this is given by

YE(x'.y') = E(x .yj (2.6)

Here B(x'ry*) is the field after the round trip at F' , E(x ,y ) is the

original field at P , and y is a complex constant. When the mirrors

have rectangular geometry the fields are best represented in a Cartesian

11

coordinate system where the two ortrv gonal components become separable.

E(x,y) = U(x) U(y) (2.7)

Substitution of equation (2.7) into equation (2.5) yields

a c

U(xl)U(yi) = ^fe—J U(xB) e ^ dxj U(ys) e ^ dy£

"a "c (2.8)

Restricting the analysis to a one-dimensional strip resonator with the

integral defined over the region of the diffracting aperture, mirror 2,

the equation for the one-way diffraction of a wave from P to F be-

comes

ikL r

yse 2 j ik

U(xJ = J^e " /u(xj e a dx= (2.9)

IK/ \ I

- 2L(X2 " X^ 'I' V

0(x) = kC^ -A(,:) + A£ -A(.xJ] (2.11)

or

0(x) = kCRj - J?.\ - x\ + Rs - ^R; - y.\ ] (2.12)

The maximuir, free space region between plane P^ and the i mirror is

A^, while A(x^) is the free space region between plane F^ and the i

mirror at some distance x^ . It should be emphasized that this is the

pnase lag for propagation in one direction only.

We can assume the paraxial approximation to be valid over the entire

mirror if the radius of curvature is large and the physical dimensions

are small compared to the resonator length. Therefore, if the above

square roots are expanded in a binomial series as before, only the first

two terms are retained.

>/l - xf/'Rf * 1 - *l/2Zl (2.13)

Equation (2.12) is now written

0(x) = k(x^/2R1 + x^/2R£) ',2.14)

Writing i/Rj in terms of the g parameter from equation (l.l), the

phase lag takes the form

0(x) "^pcj(l - gx) + xj(l - g8)] (2.15)

Thi? phase term, seen to be positive, iß actually a phase advance-

13

... -

merit; but when added to the negative ;)hase of equation (2.9) it gains the

awkward notation of phase lag. The total phase for the one-way diffrac-

tion of a wave as it applies to a resonator with spherical mirrors is

then

0. otal jf_v\ + y.% - 2Xlx£ - x»(l - gaj - x|(l - g )]

(2.16)

After combining some terms, the diffraction formula for propagation from

F to P is given by

a ikL f

» = #•* 2 J -a„

ik, 2L(XI6I + x£g£ - 2Xlxs)

U(xJ e dx„

(2.17)

Similarly equation (2.10) becomes

»a ikL F ik/ ,2 ,

*3 - *i*i>

"ai (2.16)

If the reproducibility argument of equation (2.6) is invoked, i.e.,

Y*U(xy = U(xj) (2.19)

then equations (2.17 - 2,19) can be combined to yield the round trip dif-

fraction formula as it applies to a laser resonator with spherical mir-

rors and rectangular apertures. In the equation below, the constant

phase term has been absorbed into the complex constant y" to become \ .

The integral equation is

14

Y'JvxlJ = TT- / / U(XJ e ' = i/j J{: -a, -a

i^(xf~ + y2e _ 2y x )

dx» ^x (2.20)

This equation, although derived from the geometry of Figure 2-2, is

equally valid for the resonator of Figure 1-2 and, in general, any laser

resonator. This can be seen from the sign convention chosen for tne ra-

dius cf curvature, R* . In the above example R, and R are defined 3 2

as positive (refer to page 2) leading to a direct substitution, from

equation (2.14) to equation (2.15), of 1/R^ in terms of gj_ . From

Figure 1-2 however, equation (2.14) takes the form

0(x) = k(x;/2Rj - x|/2R2) (2.21)

But with 1/R = (g - l)/L , the phase again takes the form of equation

(2.15). Thus equation (2.20) is a general result where tne sign cf R.

has been absorbed into the g parameter.

The next step is to consider diffraction effects from the feedback

mirror only. This permits tne limits of integration over P to go to

infinity since the beam spot size on mirror i is assumed small compared

to mirror dimensions. If in addition to setting the limits of integra-

tion over mirror 1 to ± °° , we let

X'a = x (2.22)

x-, = y

35

the integral equation becomes

*//

a,, oo .

(x*gj + x£g2 - 2xjx) ?; YU(x) = fr I | U(y) e ^

-a — oo

e 2L dxt dy (2.23)

The justification for equations (2.22) is seen by following a single

ray from mirror 2 to mirror 1 and then back again. As the ray strikes

mirror 1 the point of incidence must be the same as the point of reflec-

tance, i.e., x = x' . However, the ray leaving mirror 2 will in gener-

al be displaced from the original position giving x£ f- x'p . Therefore,

if x' is the x position at which the field is to be calculated after

the round trip and x2 = y (where y is a dummy variable of integration

over the diffracting aperture), equation (2.20) becomes equation (2.23)

above.

The interior integral is extracted with its evaluation relegated to

Appendix A; the result being the complete kernel of the integral equation.

ixr r 2^[(2g1g£- l)(x2+r) -2xy1

lzfrgl e 01 w>

Now if

and

g = 2g1g2 - 1 (2„25a)

F2 a? F^2g7-2Tti2

fU(x) =

im My* + - 2>:v ! U(y) e dy (2.26)

-a.

All quantities are the same a? defined previously with the addition of

F^ . Here F^ is the ordinary Fresnel number of the smaller mirror. 2 2

The ordinary Fresnel number is physically interpreted as the additional

path length per pass in half wavelengths for a ray traveling from one

miz'ror's center to the other mirror's edge, compared to one traveling

from mirror center to mirror center (Ref. 11:159-161).

Normalizing the coordinate system such that a = 1 , the resulting

integral equation is

YU(x) = JiF / U(y)

•;

•iTrFLg(xs + yE) - 2xy] dy (2.2?)

Further simplification is obtained by defining

% = %» ~ V"' (2.2b)

and

U(x) = e e(l gix) (2.29)

'eq

Several different interpretations for the equivalent Fresnel number,

, are available (Ref. 11:159-161; 12:360). Here the equivalent

Fresnel number is defined as the distance, in half wavelengths, between

the outer edge of the output mirror and the nearest point on the outgoing

geometrical wave when that wave just touches the mirror center (Ref. 12:

360). That geometrical wave is assumed to be a cylindrical wave of the

1?

TTiWIIll li ^—

:orm

-iTTN„_x2 e e(l (2.30)

Substitution of equations (2.28) and (2.29) into equation (.2.27) yields

YgU) e -*bi - *)»• _ ( •if'«-^=

/ ViF / g(y) (

-1

- iTTF[g(xs + ys) - 2xy] e dy (2.31)

After some manipulation, detailed in Appendix 5, this equation sim-

plifies tc the final form

vg v /it I f \ -it(y - x

• •*>* d»

-1

where from Appendix 3, t = nMF and v = yvM .

18

(2.32)

1 ii^M^t r .

Ill. Ine Integral B-mation Appropriate

to the

Misaligned Resonator

In Chapter 2 the integral equation was developed for the case of a

perfectly aligned resonator. It has been shown (nef. 5:li~-16&\ 10:2241-

2242) that the equation appropriate to the general nisaligned resonator

differs fron the usual one only in the limits of integration. This is

true regardless of which mirror is tilted.

The limits of integration are determined by the angle through which

the mirror is tilted, which mirror is misaligned, and the cavity geome-

try. The results of this section (i.e., the limits of integration) per-

tain only to the geometry of Figure 3-1 and cannot be generalized to in-

clude other configurations.

Effects of Mirror Misalignment •

This analysis is basically a geometry problem with the development

restricted to misalignment of the feedback mirror.

In Figure 3-1 the optic axis of the perfectly aligned resonator is

the line bcde. When mirror 2 is tilted around point c by an angle 6, the

optic axis of the resonator becomes hgfe. The center of curvature of

mirror 1 is at e, while the centers of curvature of mirror 2 before and

after the tilt are d and f, respectively. The angle between the new op-

tic axis and the old is ffl. For small angles 0, CD is given by

6P„ • äf/ea = g _ g _ L (3.1)

19

ill -| ;•-—^ - •- • '~ —->-•- , • • . ... i • ill »JliJll

Mirror 1

2a,

Mirror 2

Fig. 3-i. Geometry of an unstable laser resonator.

This, however, neglects the fact that R by convention is negative.

Therefore, substitution of a negative R„ and rewriting in terms of the

g parameter, equation (3»1) becomes

CO = z*± - l

i -gl4 (3.2)

The degree of tilt, as expressed in distance across the feedback

mirror, is given by the line eg . This distance is

eg - ©(Rj - 1') * co(Rj - L) (3.3)

If we again write P3 in terms of g^ and use 4k;e results of equation

(3«2), it becomes

20

»a M •I • ta

— eLgg gxg~ - 1

eg = r-rrh (3-

—

ror 2, the extent of the asymmetric mirror is from -1 + 5 to 1 • 6 ,

The limits of integration are now

a = -1 + ö (3.9a)

and

ß = l+o (3.9b)

Although 6 is in general a linear function of G for any cavity con-

figuration, its exact form must be determined from the specific geometry,

The generalized integral equation is now written as

P

vs(x) • M I «(y) e"it(y ' */*)* dy (3.lc)

The series functions Fn and Gn also have minor changes due to

the misalignment, i.e.,

,^sh, I /M expL-it(P - x/K") /H

arid

/Kn-i expL-it(a - x/fry/VL ]

If mirror 1 had. been misaligned by tba same angle 6, then it would have

been found (Ref. 8:579) that

6R R ,, — 12 hh , 0 1 v 06 ' L - Ra - B, " slSs - 1

(-l3)

It is evident from equations (3.4) and (3.13)> that the geometry of Fig-

ure 3-1 is less sensitive to misalignment (by a factor of gx) for a tilt

of mirror 2 vs. the same tilt of mirror 1 .

23

— I«A—, I I ,| J

IV. Solution of '.he Generalised

Integral Equation

In Chapter 3 the integral equation was modified to include effects

of small misalignments. This chapter is concerned witn its solution by

applying the method of stationary phase.

Since the problem is basically algebraic and requires numerous man-

ipulations, the scope of this section is limited to a detailed outline

of the solution.

Stationary Phase Approximation

The integral equation to be solved is

yiIJg(j)e-it(y-xAO:

J ' ' a

where g(x) is the field distribution on the output mirror after the round

trip, g(y) is the original field distribution, and v is the eigenvalue

associated with the eigenfunctions.

Following Chapter 1,

N g(y) =1+1 cnHn(y) (Jf.2)

n-i

where it was assumed that g(y) consisted of a unit amplitude cylindrical

wave plus a series of edge diffracted waves. The basis for this assump-

tion is that the original field is composed of the primary cylindrical

wave plus diffraction effects from the previous N reflections. Tne

series terminates with H,.(y), the last function to have ar. effect or. the

ZU

field. The addition of one more function (or reflection) to the series

would add only to the amplitude and not the spatial distribution or shape

of the field, i.e., H\.+1(y) is a constant. A good approximation is to

let

N > In(250 Kei)

In H (4.3)

(Ref. 4:1533).

Combining equations (l.j))i (4.1), and (4,2) yieldr

v [1 + I Ca^FnCx) + cnGn(x)l] n=i

J¥f *-u iti.y - >: 0!)8{1 R S [a,,? (y) + b G (y)]]dy (4.4) This is equivalent to equation (1.6) with the limits of integration

changed to account for the misalignment. Substitution of the Fn's and

Gn's from equation (3.H) allows the stationary phase approximation to be

applied, refer to equation (1.8). Defining the quantities

o ?; e-it(y - x/M)2

yfifjl+rnti) +bnGn(y)Je-ii^'-^)\ly

(4.5a)

(4.5b)

equation (4.4) becomes

v 11 + I UrFn(x) * b G (x)Jl n- j n = 0

(4.6)

25

•.

The problem is to now apply the stat' Dnary phase approximation to tne

individual In's and then sum the results. Starting with I_ and com-

paring with equation (1.7), it is seen that

ana

q(y) = \/ —

p(y) = (y - X/H)1

(4.7*)

(4.7b)

Also, from the stationary phase point definition, equation (l«9)i

yn = xfa (4.8)

Employing equation (1.6) and with some manipulation, it can be shown that

lo- » - M- V 4iirt -it(P - x/v,)z -it(o - X/M)!

ß - x/M x/M (4.9)

When written in terms of Fn and Gn from equation (3.11)t this becomes

10 = 1 + Fx{%) +G,(x) (4.10)

Applying the same technique to Ia, where

P

I, = yJZji^r^) 4 biGalv)'i e-it(y - x/H)" dy (*.li)

a

the approximation yields

26

Il = aaF8(x) •• b/Jjx) + l-1(x)LaJlF1(ß) + b^O)]

+ Ga(x)CaxFj(a) + bjG^a)] (4.12)

One more iteration is necessary in order to show a general trend. Again,

using the stationary phase approximation to evaluate I , the result is

I - a_Fa(x) + b _G.(x) + F,(x)[a_F-_(ß) + bG(i:)] S"3 1- "~ £ 2 2s

•» Ga(x)La/8(a) + bsGs(a)] (4.13)

From equations (4.10), (4.12), and (4.13) the resulting summation car. be

generalized to include all In's, i.e.,

N N I In = 1 + F (x) + G (x) + I [anFn+1(x) + VWx)] n=u * * n=i

N +Fi(x> n?i [ar-Fn(e) + W#J

Gi(x) n=i ta^F^(a) + bnGr.(°)n (4.14)

The equivalence of equations (4.6) and (4.14) gives rise to

v [1 + E C«nFn(x) + brGn(x)]] = 1 + Fx(x) + Ga(x) n-l

+ nJ, C%,rn+1(x) + bt,Gn+l(x)]

+ rx(x) ji LVnf^ + bnGn^)3 • O^x) £ C^P^a) - bnGn(a)]

(4.1:0

27

The Polynomial Equation

From equation (4.15) a- polynomial equation with determinable coef-

ficients is developed. This eigenvalue polynomial (where v is the

eigenvalue) is of order 2N + 1 and is solved numerically. Each root

is then associated witi. a different resonant mode of the cavity.

Referring to equation (4.15)i the first step is to equate coeffi-

cients. For n f 1 we have

va-„ , = a_ - a-, n+1 n £*

and

/•* (4.16a)

vb ^ = t = b.TvN_n (4.16b)

n+l n Tl

Here the right equality is obtained from

r v a = — = —s i4. If i n+i v ..n

and letting n = N - 1 , i.e.,

,n-l a-, a„v i n ,,n-N M-7=r-iirT-^ ^-lc)

The next step is to equate constant terms, which gives

V * * +5!^N+1 + Wl ^-1U)

Here the x dependence has been dropped since, by definition, the last

functions to contribute to the spalial distribution of the field are

FN(x) and G„(x), or that Fj.+t(

x) ^^ G..+>(x) axe constant for all x

28

-—»^-*—• •- — •'- —-—'-—— --.,..•••. -• — — - -

Equating coefficients of F-(x) and G.(x) allow- us to write

N va i S * + nil [ar.Fn(fr) 4 Wp)" &.20a)

and

vb = = 1 + I LonFn(c) + b_G (a)j (4.20b) n=i ' " "

Substitution for a , b , a , and b from equation (4.16) and rewriting

equation (4.19) we obtain

VN . 1 + Ev^-n [ai.Fr(p) + Vn(ß)] (4,21a)

bj.vN = 1 + Ev1*"" [aNFn(a) + ty^a)] (4.21b)

V =1 +aNFN+l + Vu+1 ^-21c>

In the above equations the summations are understood to be from n = 1

to n = N . Although not readily apparent, equations (4.21) can be com-

bined to yield a polynomial equation in v of order 2N + 1 . The ex-

pression is left to Appendix C, where it is seen that the coefficients

are calculated from a knowledge of Fn(a), Fn(ß). G (a), G(ß), P„+1, and

G,,+1. Therefore, by specifying the quantities M, 6, and K (.refer to

equations (3-9), (3.11), (3.12), and (4.3) ), the roots of the polynomial

are eventually determined.

Once the eigenvalues have been calculated, the constants a and n

b are determined for a particular mode (again refer to Appendix C).

The resulting field is then evaluated at incremental positions across the

feedback mirror, i.e.,

29

"—*-"- - iii*-- '-

•' —•—•'-•' -

N g(x) = 1 •+ I [a F (x) •» b 3 (x)] (4.22)

Given the field, g(x), ihe phase is easily calculated fron

0(x) = arctan(^) (4.23)

where g(x) is the complex number X + iy

x

Second Order Approximation

A higher order approximation to the method of stationary phase is

required, due to the singularities involved, whenever y approaches tne

endpoints of the integral. For I (refer to equation (^.8) ) and the

case of a perfectly aligned resonator, this occurs when x approaches

the shadow boundaries.

Since the v's were determined from a valid first order approxima-

tion over the region a < x < ß , then the eigenvalues are acceptable for

all x . This is justified by noting that V is a constant. The series

constants a and b are alsc valid from the first approximation, re- n n

fer to Appendix C, where it is seen that they are a function of the con-

N—n stants V , F (a), G (a), F.,..,. and G..,. . This implies that in order

+o calculate the phase and intensity for all x values, a new approxima-

tion to the integral

I -itp(y) - fq >) e"i1 dy (4.24)

is all that is required.

30

•1 ......—-^- j^-^m—-^. . -. —~»-M«~>—fc»^-.•-.. ... - .. ' ma^tMtn ii • n

The higher order approximation I J equation (4.24) can oe simplified

• • ' i:.

u(x) = e ytpM(x) c-p "{>: (4.25a)

i.nc

v(x) - ^TZ P'(x) (4.25b)

Depending or. the location of the stationary phase point, y , the apprcxi

nation take? one of three forms. They are: 1) for y S a

I = U(P) [E*[VO)] - ^-jHj - "(a) [E*Lv(a)] - ^-y^J v'*.2d)

2) for a < v < p o

./l± *

sufficient to say that the method of solution is similar to that of the

first order approximation.

.

32

-^. —-.^- l^t...,-- . -• .^—-^«««»A^.. .-.•,- _„ .... -J*Mjfc—••- - i-K ,• • - —- - -•--•!'- • -••-.-

"1 V. HEAP! :TEERING

The major effects of mirror misalignment, in unstable resonators are

beam steering- and mode distortion. Mode distortion is easily verified

by observation of the results presented in the next chapter. The geomet-

rical beam steering angle, y , is however not determined due to the math-

ematical construct of the misaligned resonator, i.e., the misaligned re-

sonator is modeled as an aligned asymmetric resonator (see Figure 5-l).

Additional information is required in order to calculate y (refer to

Chapter 3) since this analysis depends only on a knowledge of 6, M, and

N . eq

A diffracted beam steering angle, y,, is observed by comparison of

Figures 6-2, 6-4, and 6-6. The phase fronts are seen to shift slightly

with variations in the parameter 6. Since the normal to the phase front

determines the direction of propagation, a straight line curve fit of the

phase is desired.

This chapter calculates order of magnitude quantities for y and

9 (the mirror tilt angle). Also, a least squares curve fit is discussed

for determination of y,.

Geometrical Beam Steering

The beam steering angle due to the geometric misalignment is the

angle which the new optic axis makes with the old. From Figure ~}-\

\'--"f^; (5.1)

Combining equations (3>5), (3»7)» and (2.25) the tilt angle of the mirror

33

-- * -» —

can be written as

0 = si£7 —r~- $ (5-2)

Substitution of equation (5«2) into (5-l) gives Y in terms of 6. g

Y„ • Si." 1 1- g,g 1°E

a '(M - 1):

—H (5-3)

In order to make a comparative analysis later, we specify M = 2.0,

N =9.6, and choose L to be 2.0 meters. From equations (B.6) and

(2.2S)

&i&s " KM + 1/M +2) = 1.125 (5. 1 there is no longer an axis within the

34

i i • ^ •i Ml fil

2a,

Mirror 2

Mirror 1

Fig. $-1. The equivalent asymmetric cavity of the misaligned re- sonator as discussed in Chapter 3.

resonator with the symmetry of the original resonator, and there is no

possibility of eysiting modes characteristic of the aligned unstable re-

sonator (Ref. 8:579-580). With a£ = 0.015 meters, Ygmax is equal to

0.5 mrad. This is equivalent to a mirror tilt of approximately 1 mrad

as calculated from equation (5.2).

Beam Steering Angles Due to Diffraction

The diffracted beam steering angle is first determined by finding

the slope of the phase over the geometrical region. Refering to Figure

5-1, the region is seen to be from -M+6M to M+öM. This is easily ob-

tained by noting that the cavity magnification is a constant depending

only on R , R , and L. The transverse magnification of the perfectly

aligned resonator is defined as

35

M =£L= X1 (5.7) a2

where the dimensions are normalized such that a = 1.0 . When mirror 2 s

is misaligned the equivalent resonator of Figure 5-1 has the same cavity

magnification, and is given by

M = if^= r^V (5.8)

or xs = M + 6M. The extent of the geometrical region in the negative

direction can similarly be shown to be -M + ÖM.

A straight line curve fit of the phase is achieved by the method of

least squares. Given n sets of points (x,y), where y is the phase

in radians and x is the normalized distance, the best straight line

y = mx + b is determined by solving the two normal equations

mn +m £ XJ = £ yj

(5-9)

(Ref. 15:683), where the summation is from j = 1 to j = n . The dif-

fracted beam steering angle is then

tanYd*Yd=f ." (5-10)

The quantities d and r are the change in y and x respectively in

MKS units, i.e., d is obtained from e = e y and r = a2x. Therefore,

the diffracted beam steering angle is

36

^-^-^....-^^J_,-J.—_^__^—.. . ....... , ..... ... • n m. ill

.JSL 2Tra2x (5.11)

Prom the results of the next chapter, the slope (y/x) is determined

from the phase calculations for various values of 6 and plotted in

Figures 6-7 and 6-8. Analyzing the case of M = 2 and N =9.6 (Figure eq

6-7)i (y/x) is seen to be approximately O.lA . Also, from equations

(2.28) and (2.25)

- = aP(M ~ 1/M)

a %glKpnL 'eq" (5.12)

which gives

-5E a0(M - 1/K (5.13)

Using the values from the previous cavity, a = O.CI5 m and L = 2.0 n,

YJ ~ 7 urad. The maximum value of 0.14 is seen to occur at 6 = .225, amax therefore, Y is equal to 0.225 y or w 0.1 mrad. It is clearly

€> Smax seen that the diffracted beam steering angle is approximately 7% of

the geometrical beam steering angle, and is a significant contribution

when considering propagation over long distances.

VI. Results, Conclusions

and

Recommendations

Results

The main result of tnis study is the development of the computer

program, BARC 2, as listed in Appendix E along with a description of in-

put variables. This code calculates intensity and phase in the plane of

the feedback mirror by specifying Ö, H, and N . Also, the slope (y/x ) eq

is determined for the lowest loss mode only, where the diffracted beam

steering angle is related to the slope by equation (5«^3)«

Plots of intensity and phase for the first three modes of a cavity

with H = 2.0 and H = 9.6 are at the end of this section and in Ap- eq r

pendix D. For various 6, the intensity and phase distortion is clearly

evident. The phase of the lowest loss mode (Figures 6-2, 6-4, and 6-6)

is seen tc remain relatively uniform, but has a definite slope or direc-

tionality associated with it as 6 changes. This slope is plotted in

Figure 6-7 for a range of c between 0.0 and 0.25 •

For the cavity configuration chosen in Chapter 5f the diffracted

beam steering angle is a significant portion of the total beam steering

angle for specific values of 6. Again referring to Figure 6-7, for c =

0.01125 the slope is 0.1304 . This gives y « 6.5 urad and y =

0.0112S y •-' 5.6 urad, which shows that y, is of the order of y Cmax d g

when 6 * 0.01 . This value is, however, totally dependent on cavity

geometry and must be evaluated for each resonator as discussed in Chap-

ters 3 and 5« More importantly, > is seen to vary with no apparent

regularity as 6 varies.

38

..••i, J

Conclusions

By comparison of the phase and intensity plots with those in Refer-

ence 5, the basic conclusion reached is that program BARC2 produces valid

results. Also, for the case of a perfectly aligned resonator, i.e. 6 =

0, the results are consistent with those of the previous study (Ref. 3)>

As shown in Chapter 3» the geometrical beam steering angle is a lin-

ear function of the mirror tilt angle for small misalignments. This has

been confirmed from a previous analysis by Krupke and Sooy (Ref. 8). Al-

so, the diffracted beam steering angle is of the order of the geometrical

beam steering angle for 6 « 1.0; however the exact limit depends on

which mirror is misaligned and the cavity geometry.

The irregularities associated with the diffracted beam steering

angle versus mirror misalignment (Figures 6-7 and 6-8) is a result of the

structure in the phase. Had the feedback mirror been illuminated by a

plane wave of infinite extent, the diffracted beam steering angle would

have been zero for all 6 since the mathematical model is simply a trans-

lation of the diffracting obstacle (see Figure 5-1). Referring to Figure

6-2, when the mirror is translated to the right (the dashed lines indi-

cate the-position of the mirror) it intercepts a further advanced wave

on the right than on the left, and a definite slope is observed in the

phase front. Specifically, at 6 RJ 0.2 (again refer to Figure 6-2) the

extent of the mirror becomes -0.8 to 1.2 where a peak in the phase

is intercepted on the right and a minimum on the left. This produces the

maximum beam steering angle of Figure 6-7.

Further inspection of Figure 6-2 would require y, to occur at amax

6 ä 0.9 • This is exactly the case and the slope at this point is 0.519 .

The reason fcr terminating the plots in Figures 6-7 and 6-8 at 6 = 0.25

39

•^—»:— . ... . ... . • ^.-—.••^t-r , ^^

is due to the excessive amount of computer time required to generate them.

Therefore, relative values for the diffraction beam steering -angle can be

estimated from the phase of the perfectly aligned resonator.

Recommendations

The computer model, developed from the previous analysis, calculates

intensity and phase in the plane of the feedback mirror for the case of a

bare strip resonator. A beam steering angle due to diffraction is also

determined for the lowest loss mode.

An obvious extension would be to determine beam steering angles for

all possible modes of the cavity. In addition, the model might be modi-

fied to account for the presence of a saturated or non-uniform gain medi-

um. Since the analysis was restricted to mirrors with rectangular aper-

tures, the case of circular mirrors would be another topic for investiga-

tion.

Still another area of consideration would be the analysis of mirror

tilt and its effect on higher order aberrations; or the change in the

phase slope vs. mirror misalignment curves for higher Fresnel numbers.

40

mm •*--•-—• ••'•• - . — .. • -

«»• l.»0 mo* r.oc WIT«! .0000 ItOOC • i

no« ciotm».!*' 1.091 -.U2IE-01

•1.00 .00 1.00

flJRROR PLRNE

Fig. 6-1. Intensity plot for the lowest loss mode with 6 0.0 .

i -1.00 -1.00

«CO: 0.00

«90= t.oo OELTB' .0000

ROOC • 1

«one ciocimiLug 1.091 -.G02K- 01

MRROR PLANE

Fig. 6-2. Phase plot for the lowest loss mode with 5 =0.0 .

*n

J

tw 9.80 •wo* :.oo

OCLTB» .7000

H8K » t

HOOC CIK»«'.uc .0*6 -.S'.STt •01

.00 l.OO i.oo

HIRROR PLANE

Fig. 6-3. Intensity plot for the lowest loss mode with .6 = 0.2 .

tef a. 60 tl«0= 2.00

OCLTR= .tooo

root » 1

HOOE f ItXHVR^lKi 1.026 -.SM7C-01

.00 1.00 too

MIRROR PLfiNE

Fig. 6-4. Fhase plot for the lowest loss mode with 6 = 0.2

42

«— . -

Fig. 6-5. Intensity plot for the lowest loss mode with .6 = 0.5 •

' 500.00-] oca- i.sc

l»0= t.oo

•

OCLTR* .6000

nore » 1

naoe cioc«»«L.ue. I.0G3 -.U1BC-01

• 1 \ Ü 100.00- 3

0)

!

•y .oo-

1

-100.00-

-fOO.OC -1.00 -1.00

"••!••' .00

• '- T 1.00

• 111•1(ii111 2.00 1 ,....,... -,——,

3.00 «.00 s.oc

1 MIRROR PLANE: Fig. 6-6. Phase plot for the lowest loss mode with 6 =0.5

^3

-- - •uattH - - — •

o o mt o 10 u> o u> Csl a en Csj

%

a

ii II hi II n a o O X _j IU a D u UJ z E c z o

Q z

8

J," ' '' " ' "f •'" • •' ' T MM —

BdOlS

8

UJ

CD *—i

_l (X en

a; o tu a:

3SbHd Fig. 6-7. Phase slope vs. mirror misalignment, 6. INCDEL is the in-

crement value of 6 for which the slope was calculated.

44

o n w* o o V o> 10 in

o IS CM a

• II & bl ii n a a X _i tu a o I ) UJ z c c z o o

z *—•

£~ 8 1 I I » t » T 'T f T -T"1 * I t I" | t » T1 f •« t -^ >' T 'I 'T 'T

V•

3d01S 3SüHd

IC

o

Fig. 6-8. Phase slope vs. mirror misalignment, 6. INCDEL is the in- crement value of 6 for which the slope was calculated.

45

BIBLIOG'-APHY

1. Kogelnik, H. and T. Li. "Laser Beams and Resonators," Proceedings IEEE, Vol. 54, No. 10: 1312-1329 (Oct. I966).

2. Fox, A.G. and T. Li. "Resonant Modes in a Maser Interferometer," Bell System Technical Journal. Vol. 40, No. 2: 453-^88 (March I961).

3. Rowley, James E. Computer Analysis of Modes in an Unstable Strip Laser Resonator. MS Thesis. Wright-Patterson AFB, Ohio: School of Engineering, Air Force Institute of Technology, Dec. I98O.

4. Horwits, P. "Asymptotic Theory of Unstable Resonator Modes," Jour- nal of the Optical Society of America, Vol. 63, No. 12: 1528-15^-3 (Dec. 1973).

5. . "Modes in Misaligned Unstable Resonators," Applied Optics. Vol. 15, No.l: 167-178 (Jan. 1976).

6. Perkins, J.F. and C. Cason. "Effects of Small Misalignments in Empty Unstable Resonators," Applied Physics Letters. Vol. 31» No. "y. 196- 200 (Aug. 1977).

7. Hauck, R., H.P. Kortz, and H. Weber. "Misalignment Sensitivity of Optical Resonators," Applied Optics. Vol. 19, No. 4: 59&-601 (Feb. I960).

8. Krupke. W.F. and W.R. Sooy. "Properties of an Unstable Gonfocal Re- sonator C0„ Laser System," IEEE Journal of Quantum Electronics. Vol. QE-5, No. 12: 575-586 (Dec. I969).

9. Siegman, A.E. "Unstable Optical Resonators for Laser Applications," Proceedings of the IEEE. Vol. 53, No. 3= 277-287 (March 1965).

10. Sanderson, R.L. and W. Streifer. "Laser Resonators with Tilted Re- flectors," Applied Optics, Vol. 8, No. 11: 2241-224? (Nov. I969).

11. Siegman., A.E. and R. Arrathoon. "Modes in Unstable Optical Resona- tors and Lens Waveguides," IEEE Journal of Quantum Electronics, Vol. QE-3, No. 4: I56-I63 (April 1967),

12. Siegman, A.E. "Unstable Optical Resonators," Applied Optics, Vol. 13, No. 2: 353-367 (Feb. 19?4).

13. Beyer, H.H. CRC Standard Mathematical Tables (25th Edition). Boca Raton, Florida: CRC Press, Inc., I98O.

14. Butts, F.P. and P.V. Avizonis. "Asymptotic Analysis of Unstable La- ser Resonators with Circular Mirrors," Journal of the Optical Soci- ety of America, Vol. 68, No. 8: 1072-1078 (Aug. 1978).

46

•• 1

15« Krevszig, E. Advanced Engineering Mathematics (3rd Edition). New York: John Wiley and Sons, Inc., 1972.

47

^ ===== •MB

APPENTIX A

Appendix A is devoted to the evaluation of the definite integral in

equation (2.23).

~ ^r(x?6i + * e. n J e

•> ~\ ik/ p 2 2x,x; - Tfr(xfgj + y*g. 2L 2xxy) dxa (A.l)

The exponentials can be combined to yield

, _ Ü1 g (xa + Y2)

i 2L fes^ y ;

XL e 2L-2xigi " 2xi^x + y^ dx. (A.2)

Completing the square in the exponent, the integral becomes

_L 2L 2 XL

ik / z . e\ ik(x + y)

^ I e L 61

dx.

(A.3)

Instead of (A.3), we write the integral as

XL Pl 2L-D2V s y (* 1 ad8-.-»

2S1 JJ r IK,- /—

expi.- Yl-xWSi x + il*"! ^ •J J dxx Vs?

(A.4)

Now, if we let

and

-„—,.» ,.^.

k _ 2rr p " L XL

iX Vg - g g*J

48

(A.5)

• * in- • 1 ••,•_, .a. _-_^^-

then the resulting integral is

oo

fe-igV; J Ti dV (A.6)

where the constant term in front of (A.4) has been dropped. With the ad-

ditional definitions

CD = N/P"V

dt» = S dV (A.7)

the definite integral of (A.6) is written

JH[ VPg

oo

-lor , o r . 2 e dm 2 / -Iü> . / rr e düj = V/TTT [A.8)

(Ref. 13:38o); evaluation of.the integral can be found in any handbook of

mathematics. Substituting the value of ß, from (A.5), into (A.8) yields

the constant

LX 2igj

(A.9)

With the integral evaluated, (A.k) becomes

ZXLf expi. I- &ga 2LL-fc2 2Si (A.10)

If the exponential is expanded with a common denominator of 4Lg f then

49

... „^ ...... . ' ... .^.— -.^ ttt< ,, 1,^1^ „v.-

——

1 (A.lü) takes the final form of the kernel when the substitution of 2rr/X

is made for k, i.e.,

2ES7 eXP(" 2XI^[(2glE= • 1)(X2 + y"} " 2xy]1 (A.11)

50

••*••• * , — - . !

APPENDIX 5

Before equation (2.3l) takes the form of equation (l.3)< substitu-

tion of some variables must be accomplished.

Yg(x) e - Vi." J g(y) e

-1

,F(M - ^)y2 -i"F[g(xs + y2) - 2xy]

Given

ay

(B.i)

N/g +1 - vg -1 (B.2)

(Ref. 11:157) and solving for g, we get

M = UR + 1 + Vg - 1 g + 1 - (g - 1

M = g + Vg2- 1

(B.3)

(B.4)

(M - g)2 = g2 - 1

MS + 1 2M

(B.5)

(B.6)

Substitution of this in equation (B.l), the integral becomer.

Yg(x) = %/iF g(y) e - iftOl - £)(y2 - x2) 4 (x2 4 yS)(M + i) . 4xy]

-a

dy

(B.7)

51

•:; • - '---•-•^•"'-

-1TTF(£ + Mys - 2xy) Yg(x) = yftF g(y) e dy (B.8)

•/•

-l

-inMF(y - x/M)2

I g(y) e Vg(x) = Sf j g(y) e dy (B.9)

If we define

t = rrMF (B.10)

then equation (B.9) can be written

-it(y - X/M)2

Ys(x) •-- v/~ /g(y) e dy (B.ll)

-l

Letting

v = YVM" (3.12)

equation (1.3) is arrived at, i.e.,

-it(y - x/M) ?

vg(x) = x/^ | g(y) e dy (B.13)

-1

52

•—•* •-- ••'• — -- • J—•

APPENDIX C

The coefficients of the polynomial are not readily determined from

the form listed below; however, with the following definitions it is the

most concise form available.

a

a

vN"n F (a) n

E vN"n F (g) n^

I vN~n G (a) n

VN"n G (g) n

Here it is understood that the summation is from n = 1 to n = K

The polynomial is now written as

^+1 " ** - v-V, • CQ) • /(Fß - FN+1 * Ca - W

(Cl)

+ v^Ga - FaV + KGZ ~ FeV

W3a " G^ - WFa " F^ = ° (C2)

Also, the F..+, and G..+1 are the series functions with the x depen-

dence neglected.

The constants a and b are then calculated, once the eigenval- n n

ues have been determined, by specifying a particular mode. The first two

equations of (4.21) are combined to yield

53

1

s 1 + A 2 vK~n F (a)

FN+1

2 vN_n G (a) + nv ' F, N+1 E vN"n F (a)

TV N+1

(C3)

Here again the summation is from n = 1 to n = N .

The a^'s and b.,'s are coupled through the third equation of

(4.21), and from equations (4.16) the resulting series constants are de-

termined, i.e. ,

bn = °S

>,N-n,

,N-n

l(v-l) bn S+l N+1

(C4)

54

APPENDIX D

The plots presented here are a few of the higher loss modes, and

are included for additional verification of the computer program by com-

parison with the graphs found in Reference 5« All but the last two plots

are a result, of the second order approximation. Figures D-9 and D-10

are seen to be the intensity and phase across the feedback mirror only,

and are identical in structure to Figures 6-1 and 6-2.

55

• -- • •*-- • •• • -~—•-— - .-•-

Fig. D-l. Intensity plot for mode #2 .

Q 100.00- vyvu

NEQ= • .BO

•IS» t.00

OCUTSt .0000

MDC - I

nooe EiocavoLUCi • BOM 5S16£~0!

! Of -1.00

nlRRDR PLRNE

Fig. D-2. Phase plot for mode #2

56

..- •Hli^li^iii^i^i^B^l^

Kl» • .•0

two« I.M or:.!«» .soon MOM » f

loor rior»«n.uet .7W? . 7».0»-01

MIRROR PLflNE

Fig. D-3. Intensity plot for mode #2 and c = 0.3 .

Mf8= i.to «KM f .00

OELTIIc .5000

nooc » t «00t C10tK»BLUC

.79»! .708OE at

.00 ..00 t.oo

MRROR TLRNf:

Fig. D-4. Phase plot for mode #2 and 6 = 0.5

57

HL-d^B .A.

r=^T=;-:•-••.-".

Fig.D-5. Intensity plot for mode #3 and 6 = 0.0

ICSx I.M wo* f.00 00.«= .moo IWDC • 9

.7451 .tit!

-1.00 .00 1.00

11RR0R PLANE

Fig. D-6. Fhase plot for mode #3 and 6 s o.O

58

i iir •• i' IIIMÜ

Fig. D-7. Intensity plot for mode #3 and 6 - 0.5

M0.00-

-100.00-

-1.00 -1.00 1.00 t.oo

rlRROR PLANE

wo-- s.»0 n«oc t.oo OCLTR« .5000

KODE • J

tlODf E10t«»«LUCi .6«»: .9831

Fig. D-5. Fhase plot for mode #3 and 5 - 0.5

59

mm Mi mt- Z.00 OELT«= .0000 BOOT » 1

nooe ti«««v.ut i i.osi -.boree-o»

MIRROR 1.00

PLANE

Fig. D-9. Intensity plot across the feedback mirror only, us- ing the first order approximation.

10.00-1 aeo- a. to IW* t.oo

Ofl10: .0000

«OOE » ]

o.oo- nooe etoENvmuci

I.OSI .60?« -01

5 -10.00- A A A | UJ

VMA IM eta 1\ "W it\ AMV IV £ -"•«>• V\ /W /'

PHfi

V u 1/ -SO.Off- n

-1.00 -.so .00 .(0 i.oo i.'so 2.00 z.so

fllRROR PLANE

Fig. D-10. Phase plot across the feedback mirror only using the first order approximation.

60

r^r^^rm*j4H

APPENDIX E

A listing of program BARC2 employing the analysis and derivations of

the preceding chapters is given here. Also included is a list of input

variables required for program operation. Unless otherwise stated, the

input variables follow normal Fortran convention for being either real or

integer values.

3ARC2 Inputs (variables are listed in order required) -

MAG: Cavity magnification (real)

NEQ: Equivalent Fresnel number (real)

DELTA: Mirror misalignment (offset in fraction of mirror radius)

I.'BIG: Desired number of terms in field series

MTEST1: Input 0 to list eigenvalues.

MTE5T2: Input 0 to continue with other options, 1 to dc new cav- ity (MAG,NEQ,DELTA), or 2 to exit,

MODE: Desired mode number for phase and intensity calculations (1 to 2*NBIG + l). The eigenvalues, X, are listed accord- ing to loss, i.e., mode #1 is the lowest loss mode.

MTE3T3". Input 0 to calculate intensity and phase, 1 to continue with other options, or 2 to exit.

MTEST4: Input 0 to calculate intensity over expanded range. If MTEST^ equals 0, control is to subroutine ALLINT with variables INCX and MTEST5 skipped.

INCX: Increment value of x for phase and intensity calculation

MTEGT5: Input 0 to list field, phase, and intensity across the feedback mirror.

MTFT>T6: Input 0 to plot, constants SL. and b versus NBIG, 1 to return, or 2 to exit (return is to MTE3T2), If MTEST6 equals 0, the plots are generated and control is automatically returned to MTE3T2.

Kote - INCX is the integer number of points between consecutive whole x values. For example, INCX = 100 specifies 100 points between x = 0 and >: = 1 for the phase and intensity calculations.

61

^••M

Subroutine ALLINT Inputs -

XMIN: Minimum x value over which intensity and phase are cal- culated.

XMAX: Maximum x value over which intensity and phase are cal- culated.

INCX: Increment value of x

NTEST1: Input 0 to list field, phase, and intensity.

KTEST2: Input 0 to plot intensity and phase.

Program operation is returned to MTEST6 in BARC2.

Note - For the best results let: XMIN = -MAG + DELTA*MAG -0.5

and XMAX = +MAG + DELTA*MAC +0.5

- Since ZCPOLY (the root finding routine) limits the degree of the polynomial to 49, the maximum value for NBIG is 2^ .

62

— • --- -•--- - - -• ; • • .». -^«»--^- —- ~~r.-.*A..J-

100 no 120 130 140 ISO 160 170 180 190 200 :MO 22 0 230 240 250 260 270 280 290 300 310 320 330 = 340= 350= 360= 370 = 380 = 390 = 400 = 410 = 420 = 430 = 440- 450 = 460 = 4 70 480 = 490 = 500 = 510= 520= 530= 540 = 550 = 560= 570 = 580= 590 = 600 = 610 = 620 630 = 6A0- 650 = 660 = 670 = 680= 690- 700 = 710 = 720« 730=

PROGRAM HARC2(DATA, INI III »UUTPU1 »TAPE8-0UTPUT . TAFF4-DAIA> COMMON STOREX»X1NTENI700)»PHASE2C700) REAL NEQ» MAG. MSUHN (51) » MSUPN (51 > COMPLEX EYE.COEEC.n.LAMBDA(51)«CONSTA(M i-fll LD(700) COMPLEX CL

740= WRITENPIG 800= WRirE(B,993) 830-993 F()RMAT< IX,»INPUT ZERO TO LIST EIGENVALUES :*,/> 820= REAM ».MTEST1 P3o= WRtTE(8,979)MIEBTl

850=f 860=C COMPUTE COEFFICIENTS OF THE POLYNOMIAL 870-C P(Z)=COEF(1>*Z**NMEG 4 COEF(2)*2**(MDEG-15 + ... + 080 = C COEF(NHEG>*Z + COEF(NUEG+1) R9()=C 900=C************************************************************* 910= ALPHA*-1.*DELTA 920 = BETA=1.+BELTA 930= COEF (1>=CMPLX > 1010= AN4»ALPHA*(1.-1»/MSUPN(I+1>) 1020= AN5=L 1030- AN6=ALPHA-BETA/MSUPN(I-F1 ) 1040= FPETA( t >-=--(CEXP(AN2#AN3**2)/AN3>/ANl 1050= FALPHA«T)=- /BETA/ ( -AN1 ) 1100= FRETA(M)=FAL.PHA(M) 1110= liPETA ( h ) =CEXF' ( AN2*ALPHA**2 ) /ALPHA/AN1 II 20= GALPHA ( M > =GHETA ( M ) 1130= C0EF(2)=-(F BETAd>+OALPHA+l.> 1140= L1=NPIG>2 1150= NA=1 1J'60= NB=1 1170= HO 21 1=3,LI 1180= X2=CMPLX(0.»0.) 1190--- Y2=X2 1200- HO 18 JA=1,NA 1210= KA=NA-JA+1 1 220= XI =FPET A ( JA ) »GALPHA < KA ) -FALPHA ( JA ) *GBETA (KA ) 1230= X2=X1+X2 1240=18 CONTINUE 1250= IFU.EG.3) GO TO 20 1260= HO 19 JP=1,NP 1270= KH=NB-.JBil 1280= Y1= F AL FHA(JB)*GBE I A < KB)-FBETA(JB)*GAL PHA < KP > 1290= Y2=Y1+Y2 1300=19 CONTINUE 1310= NP--NK + 1 1320=20 Z1=FPETA(1-2)-FBETA(1-1 ) 1 330-- Z2 = GAL PFIA (12) -GAl PFIA < I - 1 ) 1340- COFF(T >-X2 + Y2 + Zl + Z2 1350= NA=NA+1 1360=21 CONTINUE 1370= L2=NHIGF3

64

1300= 1390« 1400 = 1410= 1420= 1430 = 1440 = 1450 = 1460 = 1470 = 1480 = 1490 = 1500 = 1510= 1520 = 1530 = 1 540 = 1 550= 1 560= 1570= 1580 = 1590 = 1600 = 1610 = 1620= 1630 = 1640= 1650 = 1 660 = 1670 = .1 680= 1690- 1700 = 1710 = 1 720» 1 730= 1740 = 1/50= 1760= 1770 = 1780= 1 790 = 1800 i 1R10 = 18;.'0 = 1830- 1840« 1850 1860 = 1870 = 1880 = 1890 = 1900 = 1910 = 1920 = ]930 1940 = i?5= 1960 = 19/0 = 1 980 = 1°90 = •'000 =

NA = NBIG--1 NH=NBIG~1 DO 20 I-L2«NC0EF X2=CMF'LX LAMBDA« I.) = IAMBDA(K)-CDUM * CLI»l AMBDA(I)rSMAG«EOPM

333 F ORMAT'1X,I9,1X.4CG14.7.1X>»/) 1=11

70 CONTINUr EVPH*ATAN2(AIMAG(LAMBDA(NDEG) ) rREAL(LAMBDA(NDt B) > >*UIO. /PI SMA==Rt At. ( LAMBDA < NDEG ) ) **2+AIMAG < LAMBDA < NDLCi) > #*2 SMAG-SHRF(SMA) IF(MILS!1 A 0.0) MRCTE(8f333)NDEG»l AMBDA(NDEQ)iBMAGtEVPH

_-~. •

65

--• "*• A

2010=C***********************************#»********#**************»*» 2020-C 2030--C NOW CALCULATE IHE CONSTANTS FOR THE FUNCTION SUM FOR A PAR- 2040-C 1ICULAR KODE. THEN LOOP TO CALCULATE IHr. FIELD ft! A SELECTED 2050--C NUMBER OF POINTS. :.>060=c 2070"C**************************************************'*»*********** 2080=45 WRITE(8»99/) 2090=997 FORMAIdX» »INPUT 0 TO CONTINUE» 1 TO MO NEW CAV1IY, OR 2 TO I x J T : * 2100- I ,/> 2110= READ *»MTEST2 2120= WRIT! (8.979>MTEST2 2130- IF(MTEST2-1) 100*777*888 2140--100 WRI IE(8»V9'.j) 2150*995 I ORMAT ( 1X»*INPU1 DESIRED MODE NUMBER:*»/) 2160- RE AH »»MODE 2170= URITE(8f979)MOBE 2100= LABEL(15)=10H MOPF * 2190= ENCODE ( 10» 98/»I. ABEL( 16 > )MOUE 2200=987 FORMAT MOPE »ROOT 2230= X2=CMPLX(0. .0. ) 2240 = Y2=X2 2250 = Z2 = X2 2260= DO 40 1=1»NUIB 2270= RINDEX(I)=I 2280= AN1=R00TX*(NBIG-1) 2290= X1=AN1*FALPHA(I > * ( FilOOT-1 . >/FBETA(M> 2300 = X2=X1+X2 2310= Y1=AN1*6ALPHA*6BETA(M)/FBETA(M) 2340= Z2=Z1+Z2 2350=40 CONTINUF 2360= DO 41 I=1»NBIG 23/0^- AN1=R00T**(NBIG- T ) 2380= CONSTH(I)=AN1*(X2*1.>/(R00T*«NB1G-Y2+Z2) 2390= CONST A < I ) = < AN1 * (ROOT -1 . )-CONSTB(l >*GBETA(M> ) /FBETA(M) 2400=41 CONTINUE 2410= WRITE(8»982) 2420=982 FORMAT(IX.«INPUT 0 TO CALC INTENSITY, 1 TO CONTINUE« OR 2 10 EXIT! 2430= 1 *./) 2440= REAIi * »MI ES I 3 2450 = WRI IE < B»979 ) MTE51 3 2460= ENCODE < 10»

2640* 2650 =

2670« 2680 = 2690 = 2700» 2710 = 2720 2730 = 2740 = 2750" 2760= 2770= 2780= 2790» 2800 = 2810 2820 2Ü30 2840 2850 2860 2870 2880 2890 2900 2910 2920 29 30 2940- 2950 = 2960 = 2970= 2980 = 2990= 3000= 3010 = 3020» 3010 = 3040» 3050= 3060 = 3070 = 3080 = 3090 = 3100 = 3110 = 3120 = 3130 = 3140 3150« 3160 = 3170 = 3180 = 3190 = 3200 = 3210 = 3220 = 3230 3240 = 3250= 3260 3270 =

104 992

= 950

554 556

= 31

= 32

557

= 37

38 102 991

103

INCREMENT üF X FüR PLOTS»! )

0 TO LIST FIELDi PHASE» ANB INTENSITY

TO

*./)

)

'AN4 'AN 3

CALL ALL. I Nl < MAG , MSHBN . M8UFN i CONST A , CONS 1 B . I . NBIG . ROO I f 1 ALF'HA.BF TA.l ABEL»DELTA»MODE) GO TO 102 WRIT!(8.992) F ÜRMAI(IX»»INPUT READ *.INCX WRITE(8.979)INCX URITE950) FORMA 1 < IX»»INPUT READ *»MTEST5 WRITE(8»979>MTEST5 IF*i NHATA=0 X=ALFHA BRIGHT=0. NHAIA = NIiATA-Fl STOREXCNDATA)=X X1=CMPLX(0..0.) X2=X1 DO 32 1=1«NBIG AN1=CS0RIEL*PI*T/MSUBN(I)) AN2»-T*l YE/MSUBN ) IF(THETA.GT.300.) N2=N2+1 XINTEN 1 It) Rl rilRN» OR 2 10 EXITJ

1 *./) READ »»MTEST6 »RITEHTE3T6 IF(MT ES I 6 1 ) 103.45.868

I ABEL(9>«10HC0NSTAN1 • I ABE: I. ( I0)»1 OH

67

. •••'-• • •*• - - j_^.

3280= 3290 = 3300 = 3310 = 3320- 3330 = 3340 = 3350= 3360 = 33/0 = 3380 = 3390= 3400 = 34)0 = 34?0 = 3430 = 3440= 3450= 34*0 = 34 70= 3400 = 3490 = 3500= 3510= 3520 = 3530 = 3540 = 3550 = 3560 = 35/0 = 3580 = 3590 = 3600 = 3610= 36 20= 3630 = 3640 = 3650 = 3660 = 3670 = 3680 = 3690 = 3700 = 3710= 3720= 3730= 3740 = 3750 = 3760= 3770 = 3780 = 3/90 = 3800= 3810 = 3 U 20» 3830 = 3840 = 3850= 3860 = 38 70= 3880 = 3890 = 3900 = 3910-

L ABEL (11 > =• 1 OH MOM (CONS I LABEL(12)=10HANT A>»*2 lit) 42 I - If NU IG PCONA( 1 >~REAL (CONSTAU ) > **2 i AiMAG( CUNM A( J > >**.'

42 PCONB(T)=REAI. (CONSTBU)> **2 F-AIMAG < CONG T T< (I))**2 CALL HGRAPH(RTNBEX.F'CONA»NB:iG»L.ABEL»1»-1,11) LABEL(12>«10HAN1 B>**2 CALL HGRAFFURINUEX.F'CONB-NBJR.LABF I»1»-1i11 ) WRITS(8.984)MODE

984 FORMAT«IX»»COMPLETED PLOT (Jf CONSTANTS» MODE =*»I2»/> GO 10 45

970 FORMAT COMMON STOREX< ''00 ) »PHASE < 700 )» XINTEN< /OO) rPHASI.2 > 700) REAL MSUBNC51 ) »MSUPNCH1 > »INARG.1 »INARG2» INARG3» INARG4. TNARGS RE AL J NARG6»INARG7»INARG8.MAG » M1 NU COMPLEX APART 1»APART 2»BPART1»BPART2»ALLFUN»CONST A(51 ).ROUT »EYf COMPLEX AF UN. BE UN. SPNTUO. SPNTU1 »UOX »OUT CON»F REST, • CONST I< (51) COMPLEX EYEFAC»EYEF1»EYEF2»SPNTU2»VAR4»FIEIDC700I •

********************************************** :t*l**»t**»*Tin»»»*»*M

THIS SUBROUTINE FOLLOWS PROGRAM BARC ANB COMPUTES BEAM INTENSITIES IN THE: OUTPUT PLANE:, t INTERMEDIATE POINTS FOR EVALUATION ARE TNPUT WHILE ALL OTHER REQUIRED QUANTITIES ARE CARRIED THROUGH IN THE ARGUMENT LIST AS FOLLOWS.'

MAG= CAVITY MAGNIFICATION MSUBN= ARRAY FOR FAR UAL SUMS OF INVERSE POWERS OF MAG MSUPN* ARRAY FOR MAG TO SOME POWER CONST« ARRAY OF CONSTANTS IN THE ASYMPTOTIC SERIES r= QUANTITY DEFINED IN BARE. PER HORWII/ NBIG- 4 TERMS IN THE SERIES ROOT« MODE EIGENVALUE LABEL• PLOT LABELING ARRAY

***************************************** NBATA-1

BRIGHT *>0. SX1=0.0 SX2=0.0 SY1=0.0 sxr=o.o HO 5 T=1.10

5 MCOUNT

3920=901 F IIRMAI ( 1 X.MNPUI HJN AND MAX X VALUES ANIi * POINTS DLTWtlN: **/) 3930" KLAU ».XM1N.XMAX.1NCX 3«40= WRITE (8»90?)XM1N.XMAX.1NCX 3950*902 F -UK-MAT ( lX»*INPur VALUES ARE : *»F5.2i2X»F5.2»2X»I5»/) 3960*903 FORMAT < IX. »INPUT VALUE IS : *,T5./> 3970* UKITF(8.V04) 3980*904 FORMAT«IX»»INPUT 0 TO LIST FIELD» PHASEi AND INTENSITY : *>/> 3990» PFAD *. MTEST1 4000= WRITE(8.903) NTEST1 4010 = IF(NTESTl.NE.O) BO TO 20 4020» WRITE(8.905) 4030*90S FORMAT*. 4040= .t tlX»*PHASE2 (LiEG)*./) 4050*20 X-XMIN 4060=50 ALLFUN*(0..0.) 4070= DO 310 1=1.NBIG 4080= M1NV=1 ./MSUPN< I ) 4090* VAR1=2.*(1.+1./MSUFN(2*I>/MSUBN/(BETA-STAPHA*MINV)*

4560 =

45 70= 4500 = 4590» 4600= 4610 = 4620 = 4630 = 4640 = 4650 = 4660 = 4670 = 4680 = 4690 = 4700 = 4/10--- 4720= 4 730= 4740 = 4750 = 4760 = 4770= 4 780= 4790 = 1800= 4810= 4820« 4830 = 4840 = 4850 = 4860= 4870= 4880 = 4890 =

4900* 4910 = 4920 = 4930 = 4940 = 4950= 4960 = 4970 = 4

5200«

5210 = 5220» 5230« 5240* 5250= 5260 = 5270= 5280= 5290« 5300» 5310= 5320 = 5330 = 5340 = 5350=: 5360* 5370 = 5380= 5390= 5400 = 5410 = 54 20= 5430 = 5440= 5450= 5460 = 54 70 = 5480= 540 = 5500 = 5510 = 5520 5530 = 5540 = 5550= 5560= 5570= 5580= 5590= 5600 = 5610= 5620= 5630= 5640 = 5650= 5660 = 5670= 5680= 5690 = 5700 = 5710= 5720 = 5 730 = 5740= 5750 = 5760 = 5770- 5 780 = 5790- 5800 5810 5820 5830

PHASE (HI) »PHASE (IH ) +360 . PHASE( 1 + 1 ) «PHASE < 1 t 1 ) -360.

IU1 545 1=J1.J2

IF(PHASE(Jl>.GT.0.0) IF(PHASE(Jl).11.0.0) CONTINUE K=M2 IF(K.tL.NC0UN1(1)) 60 TO 540 CALL HGF*PI 5X1=--SX1 + SIORt.X( I ) SX2*SX2+ST0R£X

5640* 5850a

5080= 5890 = S900* SS» 10 = 5920' 5930" 5940* 5V5o = 5960 = 59/0" 5980* 5990 = 6000 = 6010" (',020= 6030 = 6040= 60'>0

6480 = 6i vo- 6500= 6510* 6520= 6530= 6540= 6550- 6560= 6570= 6580 6590 = 6600= 6610 = 66?o= 6630- «S640- 6650= 6660 = 6670 = 6680= 6690 = 6 700= 6710 = 6 720 = 6730= 6740= 6750= 6 760 = 6770 = £.780 = 6790 = 6800 = 6810 = 6820= 68.50 = 6840" 6850* 6060 = "»70«

0 = 6890 = 6900 = 6910= 69'20 = 6930* 6940= 6950 = 6960 = 69 70= 6900 = 6990 = 7000 = 7010» 7020* 7030= 7040 = '050« 7060 = 7070* 7080- 7090 = 7100» 71 10-

20 II ' SF(I) .GT.- SFMAN1 ) 80 10 3« PRINT*»• SCALE! SCAI f FACT UK ERROR ... '

30 SFNICE a BF BO ru 4 0

= c NEED TO USF. THE NEXT LARGER SCALE FACTOR

SFNICE = SF

7120= IF (REAL (7.Z) .L'l .0. ) CERF =--CERF 7130= 70 RFTURN 7140= E NU 7150=r:*****j******* ************************ ************ ************ 7160=- COMPLEX FUNCTION FRESMX) 71/0^ COHPLEX EYEtZfCERF 7180= EYE= 7190= 2«SQR1 /2. 7200= F RESl=(1 .+EYE)/2-*CERF(Z> 7210= FRESL*CONJQ

VII A

Richard K. Berdine, son of James E. and Betty R. Berdine, was born

th the 8* cf April 1952 in Creston, Iowa. He graduated from East Union

High School in Afton, Iowa on 25 Kay 1970. He then attended Iowa State

university from September 1970 until May 1975 when he graduated with a

B.S. degree in Engineering Science. After 2\ years in private industry

he attended Officers Training School and was commissioned on 29 June

1978. His first assignment was to Aeronautical Systems Division, and

after two years was reassigned to the Air Force Institute of Technology

at Wright-Patterson AFB, Ohio.

Permanent Address: F.O. Box 5 Afton, Iowa 5O83O

75

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SECURITY CLASSIFICATION OF THIS PAGE (Whrn Date Enter id)

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AFIT/GEP/PH/81D-2 AO l. r.OVT ACCESSION NO 3 RECIPIENT'S CAT ALOG NUMBER

4 TITLE (and Subtitle)

MODE ANALYSIS IN A MISALIGNED

UNSTABLE RESONATOR

5 TvPE OF REPORT 6 PERIOD COVERED

MS Thesis

6 PERFORMING O^G REPORT NUMBER

7. AUTHORf»; S CONTRACT OR GRANT NUMBER'S;

Richard W. Berdine lLt USAF

9 PERFORMING ORGANIZATION NAME AND ADDRESS

Air Force Institute of Technology- ^FIT/EN) Wright-Patterson AFB, Ohio 4f#33

ID PROGRAM ELEMENT. PROJECT. TASK AREA 6 WORK UNIT NUMBERS

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Dean for Research and Direc^^#T^!^tT'^r" Professional Devclcpmenr WRIGHT-PATTEI

OF TECHNOLOGY (ATC) •Oil AfS, Q.1 4.^133

IS*. KEY WORDS (Continue on reverse side if necessary- and identify by block number)

Misaligned Resonator Unstable Resonator Tilted Reflectors Beam Steering Mode Analysis

20 ABSTRACT (Continue on reverse side It necessary and identity b\ hlork number)

„The integral equation that describes mode structiire of an unstable resonator with rectangular apertures is developed from scalar diffraction theory. This equation, modified to account for misalignments, is solved by applying the asymptotic methods developed by Horwitz. A second order approximation of the method of stationary phase is the employed to cal- culate phase and intensity values for all points in the output plane, The phase front is also curve fitted to a straight line over the geomet-

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rical region for the lowest loss mode« From the slope of the straight line, a direction of propagation can be attributed to the wave« This is a diffracted beam steering angle and is additional to the geometric steering angle (i.e., the beam steering angle due to the geometric mis- alignment of either or both mirrors).

Plots of intensity and phase for various degrees of misalignments are presented as results of a computer program that utilizes the derived expressions. Also included are graphs of the phase slope versus mirror misalignment.

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