Laser Resonators with Tilted Reflectors

Robert L. Sanderson and William Streifer

In this paper we solve the mode problem for laser resonators having identical tilted spherical reflectorsof rectangular shape in both stable and unstable configurations. Gaussian quadrature integration is em-ployed to convert the integral equation for the modes into a matrix equation which is solved with thematrix diagonalization program ALLMAT. Plane parallel and aligned concentric resonators have identicallosses; however, the latter are shown to be much less sensitive to alignment. We find that for low lossmodes in the tilted stable resonator the loss can be approximated by the average loss of two aligned reso-nators; the region of validity for this approximation is given. Stable resonator losses increase monotoni-cally with tilt; however, this is not always true for the unstable resonator where the loss may decreasefor small tilts.

1. IntroductionIn this paper we study the effects of mirror misalign-

ment on laser mode patterns and losses. The resultsare limited to resonators having identical rectangularreflectors, which are separated by an isotropic, homo-geneous, nondissipative media. Various reflector curva-tures are considered. To our knowledge, previous in-vestigations have treated only plane reflectors. 1 3

The modes satisfy an integral equation, similar tothat for aligned resonator modes but with an additionalparameter, a, which specifies the tilt. Thus, the modepatterns, E(s) satisfy the equation

yE2ie rikdy El)

2r - (Cbexp { - i [(t/2)(s 2 + t2)

- (2kn/ci)(s + t) - st]j}E(t)dt; (1)

the power loss per transit and resonancedetermined by the eigenvalue -y, i.e.,

condition are

power loss = 1 - I 2.

Each reflector has been rotated through the angle /a,where a is the reflector half-width (see Fig. 1). Otherparameters in Eq. (1) are the wavenumber k, the re-

W. Streifer is with the Electrical Engineering Department,University of Rochester, Rochester, New York 14627; R. L.Sanderson was at the same address when the work was done, andis now with the Eastman Kodak Company, KAD Research Lab-oratory, Rochester, New York 14650.

Received 28 March 1969.

flector spacing d, radius of curvature b, the Fresnelnumber c = ka2/d, and t = 1 - d/b.

In Sec. II, we show that tilted resonators are equiva-lent to aligned resonators with reflectors, asymmetricrelative to the resonator axis. Resonators with planeQ = 1) and concentric (Q = - 1) reflectors are studiedin Sec. III, while in Sees. IV and V stable ( < 1) andunstable (| > 1) resonators, respectively, are con-sidered. We find, as expected, that while aligned con-centric and plane parallel resonators have equal lossesfor equal c, the concentric resonator is much less sensi-tive to misalignment. Furthermore, concentric resona-tor mode losses remain unequal for small tilts which isnot the case for plane reflectors. Stable resonator lossesincrease monotonically with tilt and in some cases canbe determined from a knowledge of aligned resonatorlosses. This is not generally true for unstable resonatorlosses, which may decrease for small tilts. An analysisof the uniform mode in tilted unstable resonators, basedon the Cornu spiral, is included.

All results were obtained using gaussian quadratureintegration and ALLMAT as described in our previouspapers. 4 '5 Unlike the aligned resonator modes, how-ever, the tilted resonator modes are not symmetric.

11. Equivalent Aligned ResonatorThe tilted resonator illustrated in Fig. 1 is obtained

from an aligned resonator with centerline CL and spac-ing d by rotating each reflector through an angle v/a.At the reflector edge, the tangent plane of the tiltedreflector is approximately from the initial tangentplane. Line CeLe which connects the reflector centersof curvature, is the axis of an equivalent, aligned resona-tor and is shifted by A from CL where

(2)= -

(1 - )a

November 1969 / Vol. 8, No. 11 / APPLIED OPTICS 2241

c = c(1 - A/a)2 ,(3)

C = c(1 + A/a)2.

Integral Eq. (1) is rewritten as

yE(v) = { d} exp { - ic(A/a) 2(1 -t) (C)

exp { - i[(f/2 )(u2 + v2) - uv] }E(u)du, (4)

Fig. 1. Resonator geometry.100:

10

U)U)0-j

zIiiU

(-

Fig. 2. Variation of A with 4,

This equation holds for both stable and unstable geom-etries; the positive directions for n and A are shown inFig. 1. For fixed d/ a, the variation of A as a functionof t is plotted in Fig. 2 where we note that A -- for

= 1, i.e., no equivalent resonator exists for the tilted,plane reflector, resonator.

The reflector separation d' of the equivalent resonatoris given, within the parabolic approximation, by

d'= d + A2/b,

so that

d' A2

= 1 b = -b2

When the tilt is small, say A < a, the tilted resonator isequivalent to an aligned resonator with asymmetricreflectors relative to the resonator axis. Since thereflectors extend a - A and a + A from the axis, twoFresnel numbers are defined as

10-

\ 1w, f=I

C= 1/36

i' I I I ! l : i I I0.1 1.0 i0 ....

C

Fig. 3. Comparison of mode-0 losses for tilted plane and con-centric resonators.

100

10

U)

_0

zhiU

a.

k(a - A)2

Ca = 'd'

and

k(a + A) 2

cp d' 0.

We note that since a << b and A < a, d' d and ' so that

0.1

n="/36'

LI=A;

1.0 , , , , , ''16C

100

Fig. 4. Comparison of mode-1 losses for tilted plane and con-centric resonators.

2242 APPLIED OPTICS / Vol. 8, No. 11 / November 1969

>.11 1 1 1 1 1 1 1 11 1 1 1 1 1 1 0 I I , , * -'-!'. ' ' ' ' ' ' "

..1.0

..1

the curves for id 0, but in order to detect them, manymore points would have to be computed.

Our results for - = /36 agree with those publishedby Fox and Li' and not with those of Ogura et al.3

This discrepancy is not the result of inaccuracy as sug-gested by the latter authors; rather it appears due to theadditional condition of zero field at the reflector edgewhich they impose on their expansion functions.

0.1 +-0.1

Fig. 5.

10001

1.0 C 0 100

Comparison of mode-2 losses for tilted plane and con-centric resonators.

where the substitutions

u = t + ci(A/a)

and

v = s + cd(A/a)

have been made. The effect of tilt is thus equivalentto changing the integration limits of Eq. (1) and addinga constant phase factor to the eigenvalue.

Ill. Tilted Plane and Concentric ResonatorsIn Figs. 3-5, we plot percentage power loss vs c for

modes 0, 1, and 2, respectively, for fixed tilts. Themodes are numbered in order of increasing loss. Curvesare presented for both plane ( = 1) and concentric

= -1) reflectors with tilts: = 0, X/36, X/ 10, andX/3.6. For = 0, the losses depend only on I 1 andonly a single curve is required. As expected, the lossesgenerally (but not always) increase with increasing tiltand decrease with increasing c. Furthermore, losscurves for modes 0 and 1, with plane reflectors (Q = + 1)and = /10 or = /3.6 approach each other withincreasing c as reported by Fox and Li.' We note thatfor the parameters chosen, the loss curves do not crossand the eigenvalues yo and -y, approach a common valuein magnitude only. The same competition phenomenonoccurs between modes 1 and 2 with X = X/10 and t = 1,but does not occur between any modes of the concentricresonator (Q = -1) for the tilts considered herein.

The curve for v = 0 exhibits slight ripples which havenot been reported previously. The existence of theseripples was verified by performing the calculation,which is accurate to four places,4 for many closelyspaced values of c. Such ripples probably also occur in

U100)U) -

0

UJ

AX v

a- 0

hi

Uz- 10

0.0 0.4 0.8 .12R/)i

/c = 4}1 -/

/ //

/

. I I tvy f p . . + a c I n,

Fig. 6. Alignment sensitivity for plane ( = + 1) and concentric( = -1) resonators for c = 4 and 8.

MODE 2

70 +

U)7U)0

ci:

S

a.

60+

MODE I50i

40+

0.02 0.04 0.06 0.08 0.1

I/ x

Fig. 7. Loss variation with tilt for plane resonator modes 1 and2 for small tilts.

.16 .20 .24 .26

November 1969 / Vol. 8, No. 11 / APPLIED OPTICS 2243

100-

a)

10--j -

/3.6 A-

'i=O,= I

a= /36't 1

/

INTENSITY

TILTED O 6 2 I INFINITEREFLECTOR , Rj ' REFLECTOR

MODE ~ ~ NE , ; M OD E

RELECTOIR'\MODE

-1.0-.9-.8 -.7 -.6 .5 -.4 .3 ~ 1 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0-.157

X/a

Fig. S. Comparison of mode-0 patterns for a confocal resonatorQ = 0) with infinite reflectors; aligned finite reflectors, c = 4;

and tilted finite reflectors, c = 4 and A/a = 0.1.57.

tive to tilt, i.e., the patterns are quite similar to those ofan aligned resonator with the same , but have theirorigin shifted by A. This is illustrated in Fig. 8 for c =4.0, = 0.0, = /10, and A/a = 0.157. The powerloss, however, changes from 0.41% for the aligned caseto 1.64% for the tilted one. Note that the aligned mode

100-

10.Woof

U) -e) -

0 -_

z -

U -

a-

I0-

Unen0-J

zWU

wa.

0.1

X 0.2

/ = 0.05

,/a = 0.0

1.0

O.I0.1

Fig. 10. Loss vs

g ',/.=0.5

I\ \

\\\

\'\

I0.

l1 /a= 0.05/1 a .

1

/a= 0.05

- I - i i i.O10

c for the 1-mode of aligned and tilted confocalresonators ( = 0).

100-

0.1-~..1- .0

C

-410

Fig. 9. Loss vs c for the 0-mode of aligned and tilted confocalresonators ( = 0).

In Fig. 6, the change in power loss, relative to thealigned case, vs tilt for fixed c(c = 4.0 and 8.0) is plottedfor the 0-mode. It is evident that the concentricresonator (Q = -1) is much less sensitive to alignment(tilt) than the plane resonator (Q = + 1). Furthermore,with increasing c the sensitivity to misalignment de-creases for t =-1, while the reverse is true for t = + 1.Power loss vs tilt for modes 1 and 2, with c = 4.0 and

= + 1 is plotted in Fig. 7. In this case, the powerloss initially exhibits a slight decrease before increasingwith increasing tilt. Similar, and therefore more pro-nounced, behavior for tilted unstable resonators isreported in Sec. V.

IV. Tilted Stable ResonatorsIn studying tilted stable resonators, we find that the

patterns of well-confined modes are relatively insensi-

10-UnU)0-J

I-zwUWa.

</a=0.5

\ \/a =0. 2

1.0

0.10.1

0.0

i.0C

Fig. 11. Loss vs c for the 0-mode of alignedtors for t = 0.8.

10

and tilted resona-

2244 APPLIED OPTICS / Vol. 8, No. 11 November 1969

l l l l l , l l l l l ) l l l l l

I I I i i s 111 1 1 1 1 1 1 11

I---L A/azO.5

III

and(C ,j)

-y(3)E(s) = J (q3) K(s,t)E(t)dt,

an expressionsults, viz.,

for y(a,3) in terms of y(a) and a(3) re-

toU)U,

0

z

liia:

10

1.0:

A.l

1.0

2-(aO) y() + y(a)-

0oo:

.=0.0

,o

0. - 1.0 10C

Fig. 12. Loss vs c for the 1-mode of aligned and tilted resona-tors for -= 0.8.

and the infinite mirror Gaussian-Hermite mode are notidentical.

If we assume that the mode pattern is unchanged forsmall tilts, aside from the shift A, it is possible to derivea simple, approximate expression for the power loss.We write Eq. (4) as

y(a4,)E(s) = | K(s,t)E(t)dt

o r(cb= {f i + f K(st)E(t)dt, (5)

J(Cab o

where K(s,t) represents the kernel in Eq. (4), E(s) isapproximated by the mode pattern of the aligned resona-tor and y(a, 3) explicitly shows the dependence on theintegration limits. Now by replacing s and t by - sand -t, respectively, and noting that K(-s,-t) =K(s,t) we obtain

(. 1~c2) 0o ,(aO)E(-s) = {f i + f K(s,t)E(-t)dt. (6)

fo f _(C,)}

Equations (5) and (6) are added if the aligned mode issymmetric, E(-t) = E(t), or subtracted if E(-t) =

-E(t) to obtain

I(cal) C(Cai)2y(cc,1')E(s) = J K(s,t)E(t)dt + | K(s,t)E(t)dt

a(Cn d if (Coi v)

and if we invoke

0

, o.

al

as

f- 1.0

0.I1

0.1 i.0 to 100

Fig. 13. Mode-0 loss vs c for aligned resonators for t = 0, 0.4,0.8, and 1.0.

100-

U)U)0-j

2-ZU

Xi

10

1.0

= 1.0

0.1 10 100

f (c1A)

'y(a)E(s) = -)f (Conb

K(s,t)E(t)dt Fig. 14. Mode-1 loss vs c for aligned0.8, and 1.0.

resonators for t = 0, 0.4,

November 1969 / Vol. 8, No. 11 / APPLIED OPTICS 2245

100-

T A/a=0.2

I

u.l L .. .| ! 4 4 i 4 C

0,011 1 1 1 1 1 1 1 1 1 s I

01 . . . 111 . .1 11 . . ., .

are given in Figs. 13-16. Equation (7) is accurate towithin 10% of the true loss when the mode numbersatisfies

(n + 1) < 2 (I - = 2c= l7r a 7r

(8a)

where a = (1 - 02) '. A similar restriction on Gaussian-Hermite expansions of aligned confocal modes is dis-cussed in Ref. 5. By substituting for A/a in Eq. (8a),an equivalent inequality in terms of the tilt 7/ isobtained

(8b)Xq': c( - r1 _ (n -1)r -1]._< 1-c >1X- 27r ( 2c8

0 1.0l I I I I I I I I I i l l l l l l l l | I I I I - I I I

C

Fig. 15. Mode-2 loss vs c for aligned resonators for = 0, 0.4,0.8, and 1.0.

100-

)U)0-J

zLu 10

i:Lda-

1.04--0.1

Fig. 16.

1.0

1.0 c Io

Mode-3 loss vs c for aligned resonators for t0.8, and 1.0.

100

= 0, 0.4,

The power loss is then given by

1 I7(as,)I2 1 - Y'y(a) + 0(3)12= 4[(1 - hy(a)l2) + (1 - I1(v3)2)] + fl-(Ce) - (/3)12.

Since for low loss modes the last term is negligible

- Iy(as3)l2 j ![( - 17(a)12) + (1 - .7y()j2)]; (7)

i.e., the power loss of the tilted resonator is approxi-mately the average of the losses of two aligned resona-tors, one with half-width (a - A), the other (a + A).It should be emphasized that the above analysis de-pends on the tilted mode pattern being quite similar tothe aligned pattern except for the shift A.

By computing loss curves for various tilted resonators,we can empirically determine the limitations of Eq. (7).These results are illustrated in Figs. 9-12, where therelative shift A/a rather than the tilt /X has been heldconstant for each curve. Losses for aligned resonators

Clearly, the Approximation (7) is best for low ordermodes, large c, small , and small shift A/a. Note thatFigs. 9-12 include cases where Eq. (7) does not apply.Furthermore, the approximation (7) appears to dependonly on ,but the shift A and thus the losses dependupon the sign of t.

For a given tilt 27/X and Fresnel number c, the shiftA/a, given by

A 27r

a (1 -t)c \/'

and the losses are greater for t > 0 than for t < 0.Thus, for a given I I resonators with t > 0 are more sen-sitive to misalignment than those with t < 0.

XA XR

REFLECTOR 2

VIRTUAL \\CENTER "\

X,, X .'

Ed -|' d

Fig. 17. Spherical wave approximation to the uniform intensitymode of the tilted unstable resonator.

S (A)SWL)

0.8 12I

04

-0. -04,

-0.2 .-Oa. 0.4 0.8 A

I I-04

-0.8

(a)

- .o 0'8C(A I0.4 0.8

Y 12

-04

-0.8

(b)

Fig. 18. (a) Cornu spiral analysis corresponding to a loss maxi-mum for the aligned resonator (line lo). Increasing tilt resultsin lines 1l and 12. (b) Cornu spiral analysis corresponding to aloss minimum for the aligned resonator (line lo). Increasing tilt

results in lines lo and 12.

2246 APPLIED OPTICS / Vol. 8, No. 11 / November 1969

100:

10-u)U)0

I-z

ci:Lua-

1.0

ll l , , l , l , ll e | * E i E | l!

10o

.f=0.8

OOT

U)U)0-J

2rz(L

90+

.Xv A/a=0.2

a/a= °/ °

/a=O

80+

100

90

80

70

60-

50-U)U)0-J 40-

z

i:Lu 30-a-

2 4 6 8 10Cp

Fig. 19. Loss vs cp for aligned and tilted unstable resonators for= 6.0.

100 -

9()-

8C)-

70-

- \ \I

6 0-

\- MODE2

M 0'-7 MODE I

MODE 0

-4 6 10 2 CP,0

Fig. 21. Loss vs cp for the tilted unstable resonator for 4 = 1.2and A/a = 0.1.

2 )-I100-

90-

80-

70.

60-

50+4 6

8 CI 012 14 16 18 20

Fig. 20. Loss vs cp for the aligned unstable resonator for 4 = 1.2.

U)U)0.42-

z

cL

a.

V. Tilted Unstable ResonatorsThe Cornu spiral analysis presented in Ref. 5 for the

uniform mode of an aligned unstable resonator can beextended to the tilted geometry by shifting the sphericalwave center to the equivalent resonator axis as shownin Fig. 17. For a uniform spherical wave, the distribu-tion on reflector 2 is

(XR) i[iexp [- ik(r + 1)d]] exp [-(ikp2d)xR2]2M~~~~s

X f exp [i(7r12)s1]ds, (9)

40+

\. '-tMODE 2

\\ ,tL MODE I30+

)E 0

20±

2 4 6 8 10 12 CP

Fig. 22. Loss vs cp for the tilted unstable resonator for 4 = 1.2and A/a = 0.2.

November 1969 / Vol. 8, No. 11 / APPLIED OPTICS 2247

50-I

401

V)

0-j

zLu

(L30-1

""ZMODE 2

MODE 0

r . . . . . .

"I II I I . . I

20-

\I

= iexp [- ik(rc + 1)dl]l Xf[2cfp/7r]-i

L ;1J2M [2cap/7r] 2

X exp [- i(r/2)s2 ]ds. (11)

The integral in Eqs. (10) and (11)

(As+ gr

Y = J e-128 ds

is given in terms of Fresnel integrals

(n40- / ' , \.. MODE 2

J4 x,/ '2-

z

U300--

20'

4 6 b 102 CP

Fig. 23. Loss vs cp for the tilted unstable resonator for 4 = 1.2

and A/a = 0.5.

where

C= - (-/) [1 + a A(l- A/a)]'

52 = (~ 7 A~ -[1-a(1 + A/a)]'

XR is a coordinate on reflector 2 measured from theequivalent resonator axis, ca and cp are defined by (3),M= p+ ,p= (Q2-1)and

2(4 1)

The integral in Eq. (9) is approximately constant, i.e.,independent of x,, when either c is sufficiently large or

M1f>> 1. Equation (9) then becomes

[iexp [- ik(r, + 1)d] [

L _]2M exp [-(ikp/2d)xR21

XJ [2cp/r] exp [- i(7/2)s2 jds, (10)

and a uniform distribution with quadratic phase shift is

a mode with eigenvalue

Y = C(s+) - C(s-) - i[S(s+) - S(s-)3 (12)

From the well-known properties of the Cornu spiralanalysis,| Y12 and therefore 1,y12 are proportional to thelength of the straight line, 1, from s- = -(2cap/7r) 1 tos+ = (2c3p/7r)i on the curve of C(s) vs S(s). Thus, inthis case, to determine the variation in 71j2 with changesin c or tilt, we simply convert these to variations in ca,and c,6 and determine the change in length of 1.

Consider Fig. 18 (a). The solid line lo is for an alignedgeometry with c chosen such that the losses are a mini-mum. As the reflectors are tilted, the line shifts to l,and then 12. These lines are approximately paralleland 12 < 11 < lo. Thus, the tilt has the effect of increas-ing the loss. Conversely, if c were chosen at a lossmaxima, see lo in Fig. 1S(b), the tilt serves to increase 1,i.e., 12 > 11 > lo so that the loss decreases with tilt.These results are confirmed in Fig. 19 where we plotcalculated mode losses vs c for t = 6.0 and A/a = 0,0.10, and 0.20. Note that the excursions betweenmaxima and minima decreased with increasing tilt.Furthermore, the loss minima do not occur at the sameceq values as for the aligned resonator.

The situation for smaller values of ill or t and highermodes is quite complex even for small shifts, A/a < 0.2.We thus present without further comment calculatedresults for three modes and t = 1.2 with A/a = 0.0,0.10, 0.20, and 0.50 in Figs. 20, 21, 22, and 23. Themodes were distinguished at points of equal loss byplotting the locus of real and imaginary parts of y asdiscussed in Ref. 5.

Part of this material was presented as paper FE16 atthe March 1969 Meeting of the Optical Society ofAmerica [J. Opt. Soc. Amer. 59, 510A (1969)]. Thiswork was supported by the National Science Founda-tion.

References1. A. G. Fox and T. Li, Proc. IEEE 51, 80 (1963).2. W. H. Wells, IEEE J. Quantum Electron. QE-2, 94 (1966).3. H. Ogura, Y. Yoshida, and J. Ikenoue, J. Phys. Soc. Japan

20, 598 (1965).4. R. L. Sanderson and W. Streifer, Appl. Opt. 8, 131 (1969).5. R. L. Sanderson and W. Streifer, Appl. Opt. 8, 2129 (1969).

C. J. Bartleson, Kollmorgen Corp., Newburgh, N.Y., has been selectedfor the Journal Award by the Society of Motion Picture and TelevisionEngineers for his paper, "Color Perception and Color Television,"published in the January 1968 issue of the Journal of the SMPTE.

2248 APPLIED OPTICS / Vol. 8, No. 11 / November 1969

MODE 0