Beam propagation in periodic quadratic-index waveguides

Beam propagation in periodic quadratic-index waveguides

Lee W. Casperson

Several techniques are described for studying the propagation of off-axis polynomial Gaussian beams inmedia having straight axes and periodic z variations of the quadratic refraction and loss coefficients. Forsome periodic variations, exact analytical solutions of the paraxial equations are possible, and for sufficientlyslow variations, WKB solutions can always be obtained. All results are expressed in conventional beammatrix form.

1. Introduction

Dielectric waveguiding materials in which the indexof refraction and gain vary at most quadratically in thetransverse direction are generally referred to as lens-like materials, and their ray and beam transmissioncharacteristics are closely related to those of ordinarylenses. In two recent studies, the beam propagationcharacteristics of tapered lenslike media have beenconsidered.1 2 Exact numerical and analytical solu-tions of the paraxial wave equation were obtained for avariety of potentially practical taper configurations.Such tapers have found many applications in couplingbeams from one fiber (or other optical element) toanother. In addition to tapers, other types of z depen-dence may occur accidentally or intentionally in themanufacture of graded-index fibers and other lenslikewaveguiding media. Of particular interest in thisstudy are the beam propagation characteristics of lens-like materials in which the index of refraction and lossprofiles vary periodically in the propagation direction.

The idea that light beams could be guided by se-quences of lenses has long been known.3 4 The earlieststudies of periodic lenslike media related to the devel-opment of gas lenses in which the waveguiding indexprofiles were obtained by introducing periodic tem-perature gradients in the gaseous propagation medi-um.5 -8 Gas lens waveguides are now largely obsoletebecause of less expensive graded-index fiber-opticwaveguides, but interest in periodic lenslike structureshas continued.9-1' Periodically perturbed graded-in-

The author is with Portland State University, Department ofElectrical Engineering, Portland, Oregon 97207.

Received 17 June 1985.0003-6935/85/244395-09$02.00/0.(© 1985 Optical Society of America.

dex fibers are considered to be a possible consequenceof defective fiber manufacturing techniques or distor-tions in multifiber cables. Intentionally introducedperiodic perturbations may also be of value for beamcouplers and mode filters.

For illustration, two specific types of periodic lens-like medium are shown schematically in Fig. 1, and theradius changes in the figure are meant to suggestchanges in the loss and refraction profiles. In the firsttype, the parameters of the waveguide vary smoothlyand continuously with propagation distance, while inthe second type the waveguide is represented as acomposite structure made up of segments that individ-ually are easier to study. Techniques are describedbelow for analyzing numerically the propagation of off-axis Gaussian beams in arbitrary z-dependent lenslikemedia. For a few types of continuously varying peri-odic medium, exact analytical solutions of the paraxialequations can be obtained. Since matrix representa-tions are now available for many kinds of waveguidesegment, very general periodic media can be analyzedexactly by a composite model of the type suggested inFig. 1(b). For sufficiently gradual profile changes,WKB methods can also be applied. At the outset itmay be noted that the assumption of an ideal quadraticlenslike profile can only be valid out to some finiteradius in a realizable medium. However, for simplic-ity, it is assumed here as in previous studies that thepropagating modes are confined entirely within thequadratic-profile region.

The fundamental beam equations and their inter-pretation are reviewed briefly in Sec. II. It is foundthat for periodic lenslike media, the most basic of thebeam equations is of the Hill equation type. In thespecial case that the beam profile varies sinusoidally inthe z direction, the general Hill equation reduces to aMathieu equation, and solutions for beam propagationin sinusoidally periodic lenslike media are discussed inSec. III. While the solutions of the Mathieu equation

15 December 1985 / Vol. 24, No. 24 / APPLIED OPTICS 4395

are most easily obtained numerically, very similar ex-act solutions can be obtained analytically for a newclass of pseudosinusoidal variations discussed in Sec.IV. A method for composite periodic functions ispresented in Sec. V, and an approximate WKB methodfor arbitrary weak periodic variations is described inSec. VI. All the solutions are formulated in terms ofconventional beam matrices, and hence it is straight-forward to analyze or design complex optical systemswhich incorporate one or more segments of periodiclenslike medium.

11. Beam Equations

The reduction of Maxwell's equations to the ordi-nary differential equations of beam optics has beengiven in many places, so it is possible to be brief. Asimilar reduction may be found in Ref. 1 for the funda-mental Gaussian mode and in Ref. 12 for higher-orderbeam modes. For conciseness, several simplifying re-strictions are imposed at the outset. By using theresults obtained here, together with the formulationsjust referenced, it is possible to investigate much moregeneral beam and media configurations.

As in most previous studies, it is assumed that thedominant transverse electric field components are gov-erned by the wave equation

Fig. 1. Schematic representation of (a) a continuous periodic lens-like medium and (b a composite periodic medium consisting ofsimpler uniform and tapered segments. The radius changes are

meant to suggest changes in ithe loss and refraction profiles.

tion of off-axis mismatched Gaussian beams, but witha slightly more complex substitution higher-orderpolynomial Gaussian beams can also be described.1 2

The results in the present case are

Q2 + ko dQ +kok2 =

dSQS + ko - = 0,

V2 ' + k 2 t' = 0, (1)

where the prime is a reminder that Eq. (1) applies tothe complex amplitude of the electric field. For thesimplified y-independent model of interest here, thequadratically varying propagation constant can bewritten

k2 (x,z) = ko(z)[ko(z) - k 2(z)x 2], (2)

where all the coefficients are in general complex. Formisaligned media one would need an additional termlinear in x, and similar terms in y could also be includ-ed as appropriate.' Equation (2) corresponds to a z-dependent medium in which the quadratic gain andindex profiles may be independently specified. For awave propagating primarily in the z direction, a usefulsubstitution is

E = A(x,z) exp[-i J ko(z/)dz'] (3)

Then Eq. (1) reduces to

2 2ik OA i AO A-kok 2 x 2A 0, (4)ax2 caz dz

where A is assumed to vary so slowly with z that itssecond derivative can be neglected. This last assump-tion is the familiar paraxial wave approximation, andfrom this point onward all solutions (except the WKBresults of Sec. VI) will be exact.

Equation (4) is a partial differential equation, but itmay be reduced to a set of ordinary differential equa-tions by means of the substitution

A(x,z) = exp[-i(Qx 2 /2 + Sx+ P)]. (5)

This substitution leads to a description of the propaga-

dP = _Q s2

dz 2ko 2k,

(6)

(7)

(8)

In the more general situations additional terms andadditional equations would be obtained. It is impor-tant to note, however, that Eq. (6), which must besolved first, would not change its form.12

The complex beam parameter Q is related to thephase front curvature R and the le amplitude spotsize w by means of the familiar formula'3

Q 1 2iho R kOW2 (9)

The complex displacement parameter S is related tothe transverse displacement da of the center of theamplitude distribution and the displacement dp of thecenter of the phase distribution according to the for-mulas

da =-SilQi, (10)

d = -SrQr, (11)

where the subscripts i and r refer respectively, to theimaginary and real parts. The imaginary part of thecomplex phase parameter P can be interpreted as acorrection to the on-axis field amplitude resultingfrom the changing beam structure and position. Simi-larly, the real part of P can be interpreted as a phasecorrection.

Equations (6)-(8) may be solved in sequence and themain emphasis here is on exact solutions of Eq. (6).This is a Ricatti equation, and the first step in solvingit is to introduce the well-known variable change14

ko drQ =__.r dz

(12)

4396 APPLIED OPTICS / Vol. 24, No. 24 / 15 December 1985

With this substitution the Ricatti equation is trans-formed to a linear differential equation with noncon-stant coefficients

+ [koz dj + k2 (z)r = 0, (13)

where the z dependences of the coefficients are notedexplicitly for emphasis. This equation has been ob-tained here from an exact reduction of the paraxialwave equation. The same equation may also be ob-tained from the paraxial ray equation, and in that caser would be interpreted as the z-dependent transversedisplacement of a propagating light ray. While ko andk2 may be allowed to be complex in wave optics studies,special care would be required to apply Eq. (13) in a rayoptics analysis of media having a gain or loss profile.

Solutions of Eq. (13) are much easier to obtain if ko isa constant, and in that case one finds

d2 r k2 (z)-+ r . (14)dz2 k0 14

It is important to note, however, that with an appropri-ate change of variables Eq. (13) can always be trans-formed into Eq. (14), and this transformation is dis-cussed in Appendix A. Hence, there is no loss ofgenerality in using Eq. (14) in place of Eq. (13).

When k2 is a periodic function of z, Eq. (14) may berecognized as a Hill equation. A more common nota-tion for such an equation is

d 2 + f(x)y = , (15)

where f(x) is periodic. This type of equation was firstinvestigated by Hill in connection with the theory ofthe moon's motion,15 and such equations arise com-monly in many branches of physics and engineering.Analytic solutions of the Hill equation are known foronly a few special forms of f(x), and the followingsections apply those solutions to the problem of peri-odic graded-index waveguides.

Ill. Sinusoidal Variations

The simplest appearing Hill equation is one in whichthe periodic function f(x) varies sinusoidally as shownschematically in Fig. 1(a). In this case the Hill equa-tion reduces to the Mathieu equation

d'y + (p - 2q cos2x)y = 0. (16)

This equation was introduced by Mathieu in his studyof the vibrational modes of a stretched membranehaving an elliptical boundary,'6 and it has been consid-ered as a possible model for periodic graded-indexmedia.7 ,9"10 In spite of the simple form of the Mathieuequation, the solutions are not easy to evaluate. Formost purposes they must still be obtained numericallyor from the various published graphs and tables.'7Nevertheless, this equation may be used to indicateschematically the solution procedure for rays andbeams propagating in arbitrary continuous periodicmedia.

The Mathieu equation possesses a variety of period-ic, aperiodic, and unstable solutions depending on thevalues of the parameters p and q. From a comparisonof Eqs. (14) and (16), it may be seen that use of theMathieu equation is equivalent to assuming that thequadratic term in the complex propagation parameterhas the periodic z dependence

k2 (z)= p - 2q cos2z,

ko (17)

where p and q are in general complex constants. Withan obvious renormalization, the Mathieu equationrepresentation can, of course, apply for arbitrary spa-tial modulation frequencies.

The Mathieu equation has been introduced herebecause of its popularity and simplicity (apparentrather than real) and also to illustrate the solutionprocedures that for other periodic functions may bequite straightforward. Since Eq. (14) is an ordinarylinear second-order differential equation, it followsthat for any given values of the parameters p and q thegeneral solution may be written in the form

r(z) = au(z) + bv(z), (18)

where u(z) and v(z) are imagined to be linearly inde-pendent Mathieu functions and the coefficients a andb must be determined from initial conditions. Simi-larly, the rate of change of the parameter r(z) is

r'(z) = au'(z) + bv'(z). (19)

If the values of r(z) and r'(z) at the plane z = z are r1and r', respectively, it follows from Eqs. (18) and (19)that the general equations for r(z) and r'(z) can bewritten in the usual matrix form:

Fr(z)1 -A(z) B(z) iFrl1Or'(z)= [C(z) D(z)JLrl']

where the matrix elements areu'(Z1 )V(Z) - V'(Z)U(Z)

V(Z)U'(Z1 ) - U(Z1 )V'(Zl)

(20)

(21)

v (z = )u(Z() - U(ZO)v(Z)B U) (z1 )u'(z1 ) - U(Z1 )U'(Z1 ) (22)

u'(z1 )V'(z) - V'(ZIU'(Z)

- v- (Z1 )u'(z1 ) - U(Z1 )V'(Z1 )

D~)=v(z1)U'(z) - U(Z1)V'(Z)D~ =V(Z 1)U'(Z1) -u(z1)V'(z1 )

(23)

(24)

To reduce these matrix elements further, one wouldhave to make use of any special properties of the func-tions u(z) and v(z). However, for the Mathieu equa-tion, the results remain somewhat complicated, and itseems as easy to integrate the equation numerically.

In the case that the second-order paraxial equationis understood to represent the propagation of lightrays, Eq. (20) describes completely the ray propaga-tion process in periodic quadratic-index media. Simi-larly, however, it follows from Eqs. (12) and (20) that


C (z-) =

P(z) - P(z1 ) = - In A + BQ(z1 ) l S2 (z,) B2 L ho J 2k0 A + BQ(z,)/ko

(29)

V VP V V V H V The conditions in which the determinant is unity areconsidered in Appendix B.

Equations (25), (28), and (29) provide a complete0 ) 2 64q 6 0 description of the amplitude distribution, phase front

DISTANCE 1-1 curvature, and displacement of Gaussian beams pro-pagating in periodically varying lenslike materials. Itwas suggested at the beginning of this section that forthe media of interest Eq. (14) might take the form of aMathieu equation. However, this suggestion was onlymade for purposes of illustration, and the precedingresults apply for arbitrary z-dependent graded-indexmedia. When the paraxial equations actually take the

G 2 q 6 8 0 form of the Mathieu equation, it is most straightfor-DISTRNCE .. ) ward to obtain the solutions by direct numerical inte-

, , , , , gration.(0 A If\/\r Al /\ 18 As an example, several numerical solutions will be

shown for the Mathieu equation

0 2 q 6 8 10DISTANCE (Cr)

Fig. 2. Normalized beamwidth w(z)/w, as a function of distancealong a periodically modulated quadratic-index waveguide with themodulation frequency (a) y = 0.5Syo, (b) y = 1.0 ,yo, and (c) ay = 1.5yo

(after Ref. 18).

the propagation of the complex beam parameter Q isgoverned by the Kogelnik transformation's

Q(z) - C + DQ(z1 )/k0 (25)

hk A + BQ(z,)/ko

A transformation of the same sort applies to the com-plex displacement parameter S(z). To see this, wefirst write Eq. (7) in the form

dlnS Qdz ko

(26)

C + DQ(z,)/ko

A + BQ(zl)/k,

where Eq. (25) has also been used. But it follows fromEqs. (21) to (24) that C is the derivative of A, and D isthe derivative of B. Hence, Eq. (26) can be written

d lnS _d F BQ(zl)1___= -In I A+ . (27)dz dz L ko

Using the facts A(z1) = 1 and B(zj) = 0, Eq. (27) may beintegrated to obtain the displacement transformation

S(z) = S(z,)[A + BQ(z 1 )/ko]-. (28)

Similarly, if the determinant of the beam matrix isunity, Eq. (8) may be integrated to obtain the phasetransformation

d2 +F(1 +2cosz r = 0dz2 k2 (30)

in which the quadratic term in the index profile isstrongly modulated. The coefficient F in real (nonab-sorbing or amplifying) graded-index media corre-sponds to the unmodulated index ratio n2 /no, and forconsistency with previous experimental and theoreti-cal studies the value adopted for this ratio is 25 mm-2.If the waveguide were not modulated, it would followfrom Eqs. (6) and (9) that the steady-state spot size canbe written

(31)

With the values no = 1.5 and X = 1 gim, this steady-statespot size is ws = 0.0065 mm, and this value will be thebasis for our normalization of the spot size in periodi-cally modulated waveguides.

A propagating off-axis light ray or polynomialGaussian beam in an ordinary z-independent graded-index medium oscillates about the z axis with thespatial frequency

7y = (n2/n0 )'/2 , (32)

and for these examples the oscillation frequency is yo =5 mm-'. The actual oscillation in a modulated wave-guide are typically much more complicated than thesimple sinusoidal variations of frequency o thatwould be found in a z-independent waveguide. Nev-ertheless, yo is a convenient reference frequency, andresonance effects are found to occur when the modula-tion frequency is close to oyo.

Figure 2 shows the spot size oscillations that areobtained for modulation frequencies of -y = 0.5 7o, oy =1.0 -yo, and y = 1.5 yo. It is clear from this figure thatthere is a resonance effect when y = 1.0 yo, and thisresonance causes the spot size fluctuations to increasewithout limit. This type of instability has been notedpreviously in a study of laser beam propagation in a


(a) 2

0

(b) 5

irE 4

, 3

2 2

Ez1

(c )2

10

0

W, = x (no)1/ 12

7rno n2

(a to

(b)

qo

15

0

-80

0 2 4 6 8 10

DISTAN4CE 11)

DISTANCEE r,-)

0 2 4 6 8 10DIS1ANCE t-)

Fig. 3. Displacement of the amplitude center of the beam awayfrom the waveguide axis in micrometers as a function of distancealong a periodically modulated quadratic-index waveguide with themodulation frequency (a) -y = 0.5yo, (b) y = 1.0j'o, and (c) y = 15yo

(after Ref. 18).

r(z) = a (1 + 0 cos-z) cos( F 2 G sinz1+G ( Yy(1-G ) 1+Gcos-yz

- 2 tan [(1 -G)' 1 ' tan ( ]

11 + G cosyz) s ( F/2 J G sinyz1+G in (y(1-G2) 11+Gcosyz

I- 2 tan [(1 - G) tan (Z)] (34)

where a and b are arbitrary constants. Various exten-sions and special cases of Eq. (34) are considered inAppendix C. The lenslike media governed by Eq. (33)are only of interest if the periodic z dependences canresemble the actual z dependences that one mightexpect to encounter in practical waveguides. Thisseems to be the case though, and the periodic term inEq. (33) may be nearly sinusoidal or it may have nar-row maxima or minima, depending on the values cho-sen for the parameters F and G. A more detaileddiscussion of the possible forms of the periodic term isincluded in Ref. 19.

The matrix elements corresponding to Eq. (34) canbe most easily obtained from a comparison with thegeneral results of the previous section. Thus, from acomparison of Eqs. (18) and (34) one finds that theperiodic functions u(z) and v(z) are

u(z) = + G cosyzz cos F"' G sinz 21 + G / \y(l - G2) F1 + G cosyz (1-G) 12

X tan-1 [(1- G 2)' tan (T)}[ 1 + 0 \2IYZ1 1)

periodically perturbed plasma waveguide.10 Similarresults are obtained for the beam displacement, andplots of the displacement of the center of the ampli-tude distribution away from the z axis are shown in Fig.3. Again there is a resonance effect when -y = 1.0 yo.

IV. Pseudosinusoidal Variations

The previous section emphasized ray and beampropagation in periodic media governed by the sim-plest appearing paraxial equations. In particular thecomplex Hill equation was considered to be reducibleto a Mathieu equation. However, except for someinitial general considerations the solutions for beampropagation had to be obtained numerically. For thisdiscussion it is assumed that the paraxial equation, Eq.(14), takes the specific form'9

d2+r F + ' G cos-yz dz2 [(1 + G cosYz)4 1 + G cos-yz I (33)

where in general F, G, and y may be complex. Whilethis new equation appears slightly more complicatedthan the similar Mathieu equation, it has the advan-tage that the general solution can be written explicitlyin terms of elementary functions. In particular, thesolution of Eq. (33) is

v(z) = (1 + G cosyz" sin ( F1/2 G 0 sin-yz 2Z 1) + = I +y(G - G2) 1 + G cos-yz (1 - G2)1/2

X tan-, [(1 - G2)1/' tan[ 1+0 ~ ~ 2 )JJ (36)

Using these formulas, the matrix elements are given byEqs. (21)-(24). To actually evaluate the matrix ele-ments, however, it is necessary to compute the deriva-tives of u(z) and v(z). For this purpose it is helpful toreplace Eqs. (35) and (36) by the equivalent forms

u(z) = (1 + G cosy7z) cos [F1/2 U dz' 11 + G L o (1 + G cos-yz')'J

u(z) =(1+ G cos7z sin F1/2 Z dz'1 + G ) L ° (1 + G cosyz')2J

Now the derivatives are found to be

U Z)= - yG sin-yz Cos[____________

1 + G (1 + G)(1 + G COSyz)2 sin[ ],

v'(z = - yG ~+ sin[ ] + F 1 cos[1 + 0 ~(1 + G) (1 + cos-yz)'2

(37)

(38)

(39)

(40)

where the empty brackets correspond to the bracketedquantities in Eqs. (37) and (38) or the equivalent quan-


(35)

tities in Eqs. (35) and (36). Using Eqs. (37)-(40) thepropagation matrix elements can now be obtained di-rectly from Eqs. (21) to (24). With these elements thepropagation characteristics of arbitrary off-axis raysand polynomial Gaussian beams are completely char-acterized by Eqs. (20), (25), (28), and (29). Whilethese procedures may seem a bit cumbersome, theresults are exact and explicit, and they are much fasterto evaluate than the numerical solutions which wouldotherwise be required.

In concluding this section it may be well to empha-size that the solvable Hill equation with pseudosinu-soidal coefficients is in many respects very similar tothe relatively unsolvable Mathieu equation. Thus, forsmall values of G Eq. (33) reduces to

d'rdz2 + F + (-y' - 4F)G cosyzjr = 0. (41)

If y is set equal to 2, this isd'rd-2 + [F + 4(1-F)G cos2z]r = 0, (42)

which is of the same form as the Mathieu equationgiven above as Eq. (16).

V. Composite Periodic Profile Variations

In the previous sections it has been assumed that thegain and index profiles of the periodic waveguidingmedium vary continuously along the waveguide axis.While this kind of behavior would be expected for mostpractical periodic media, it is important for complete-ness to also consider the possibility of periodic mediain which the profiles alternate abruptly between ana-lytically. characterizable segments as suggested in Fig.1(b). In this case it is also possible to obtain analyticsolutions for the parameters of a propagating light rayor polynomial Gaussian beam at any position along thewaveguide.

As the simplest possible illustration of these com-ments, one may consider a waveguide which includessegments of length da and quadratic propagation coef-ficient k2a alternating with segments of length db andcoefficient k2b. The transformation matrix for onecomplete period of this waveguide is

where a5 = cos[(A + D)/2]. From the known ray orbeam parameters at the boundaries between stages it isstraightforward to obtain these parameter values atany intermediate points along the waveguide.

In the simplest possible illustration given above itwas assumed that the waveguide consisted of alternat-ing segments of uniform guiding media. It is apparentthough that more complex waveguides can be modeledusing the same techniques. For the example sketchedin Fig. 1(b), one stage of the waveguide may be repre-sented by a product of four matrices, and the matricesfor several types of tapered medium have been report-ed. 2

VI. Approximate Solutions

As noted above, ray and beam propagation in contin-uous media is governed by an equation of the form ofEq. (15). It is just for equations of this form that theWKB method provides approximate solutions, provid-ed f(x) does not change too rapidly and does not passthrough zero. More specifically, it is essential that thefunction f(x) must change negligibly in one wavelengthof the oscillatory solution. If this condition is satis-fied, the approximate solution of Eq. (15) can be writ-ten 2

y(x) (f(x)V-"4 {a cos I [(x')]1" 2dx'

+ b sin I V(x')]112dx'1 (45)

where as usual the coefficients a and b are determinedby initial conditions.

Using Eq. (45), it can be shown that the matrix formof the solutions to Eq. (14) is2

F k2(Z) 1/4 fz[ k2(Z)') 11/2A = ] Cos [ dz',

J 2J zZL)k 0JFk2(Z)k2(Zl) 1 /4~sn a k2(Z') 1/2

B =[ k 2 Isin J k dz',

k 2(Z)k2(Z) 1/4 k(Z') 1d2I k0 2 sin ko d ,

k J Zf:

(46)

(47)

(48)

{A B) =cos(k 2 b/kO)1/2db

(C DJ -(k2b/ko)1/2 sin(k2b/kO)'

cos(k2a/k0 )'2da

L-(k2a/kO) 1/2 sin(k2a/kO) '/2da

(ko/k2b)"12 sin(k2b/ko)2dbl

12db cos(k2b/ko)'12dbJ

(ko/k 2a)'/2 sin(k2a/ko)'/2dal

cos(k2a/kO)"/'daJ

If the fundamental propagation constant h0 alsochanged between segments, it would be necessary toincorporate in this result the transformation matricesfor the dielectric boundaries.

Once the transformation matrix for one stage of thewaveguide is known, it becomes straightforward toobtain the transformation for an arbitrary number ofstages by means of the theorem 2 0

(A B = 1 JA sin(st) - sin[(s - 1)] B sin(sto)]

(C D sino tC sin(s0) D sin(s) - sin[(s - 1)691


(43)

, (44)

_

.

k2(z) 11/4 rz h2 (z') 1/2D = k 2(ZI) Cos Jzl ko dz'. (49)

Equations (46)-(49) are applicable to any slowly vary-ing quadratic-index fiber guide. For example, with z-dependent media governed by the Mathieu equationas represented in Eqs. (14) and (17), the matrix ele-ment A becomes

A = (p- 2q cos2zi) cos J (p - 2q cos2z')112 dz', (50)

and similar results apply to the other elements. Theintegrals in these matrix elements can be evaluated bymeans of widely tabulated elliptic integrals.21

VII. Summary

Longitudinal variations of the refraction or loss pro-files are often introduced intentionally in the manu-facture of lenslike media, and sometimes they occuraccidentally during the fabrication process. In thispaper we have investigated several techniques for ana-lyzing the propagation of polynomial Gaussian beamsin periodic lenslike media. Among these techniquesare (1) direct numerical solution of the beam equa-tions, (2) identification with the widely studied Math-ieu equation for media with sinusoidal perturbations,(3) identification with a new analytically solvable Hillequation, (4) decomposition of the periodic profile intodiscrete solvable segments, and (5) approximate solu-tion using a WKB method. All these methods lead toray or beam matrices of standard form, so that sectionsof periodic media, once analyzed, may be easily includ-ed as elements in more complex optical systems. Ex-cept for the WKB method all the techniques listedabove yield exact solutions of the paraxial ray or beamequations. We have also included as appendices briefdiscussions on analytically eliminating any z depen-dence of one of the propagation constants, on the uni-modularity of the resulting matrices, and on solutioncharacteristics of the solvable Hill equation.

Appendix A: Equivalent Media

As noted in the text, it is most straightforward toanalyze quadratic-index waveguides in which the koterm in the complex propagation constant is indepen-dent of distance z. However, this is not a fundamentalrestriction, because any lenslike medium having a zdependent ko can be transformed into another lenslikemedium in which ko is a constant. 2 2 The most funda-mental equation governing beam propagation in lens-like media is Eq. (14), which we write in more detailedform as

(Al)d +ko() dr(z)] + k2 (Z)r(z) = 0.

It is postulated here that the z dependence of thecomplex variable r can be identical to the z' depen-dence of another variable r' which is a solution of theequation

provided that the new coefficients ko0 (z') and k2'(z')together with the new independent variable z' are suit-ably defined. In particular, it is always possible totransform the lenslike medium described by Eq. (Al)to another medium in which ko' is independent of z'.

The transformation just described consists of therelationships

ko'(z') = ko[z(z')] dz(z')

k 2'(z') = k2 [Z(z')] dzz'

(A3)

(A4)

(AS)r'(z') = r[z(z')],

where the function z(z') is a completely arbitrary con-tinuous monotonic function. The validity of thistransformation can be confirmed by direct substitu-tion into Eqs. (Al) and (A2) and use of the chain rule

d dz' ddz dz dz'

(A6)

One thus concludes that there is an infinite set oflenslike media that are equivalent in the sense that thedependent variables transform in the same way be-tween two reference planes.

Of special interest is the transformation which takesa z dependent ko(z) and yields the arbitrarily specifiedconstant ko'(z') = ko'. From Eq. (A3) this constraintyields

dz' ho'dz ko(z)

(A7)

The integral of this equation is the length transforma-tion

°' = kodz)o k(z) (A8)

With Eq. (A4) the new quadratic term in the propaga-tion constant is

k2'(Z') = k 2(z)k 0(z)/k 0'. (A9)

Using these substitutions, Eq. (A2) can be written inthe simpler form

d 2 r'(z) k2'(z')~~+ r'(Z') = 0.dz' 2 kol

(A10)

This is the equation form included in the text as Eq.(14) with the restriction that ho be independent of z,and the same equation was also a basis of our recentstudies of tapered waveguides. In this Appendix it hasbeen shown that this form is actually general, since anyz dependence of ko can always be transformed away.

Appendix B: Unimodularity of the Beam Matrices

It is well known that many of the ray matrices andbeam matrices encountered in practical optical prob-lems are unimodular. That is, they obey the relation-ship

dz [o(z') dr'(z')] + k 2'(z')r'(z') = 0, (A2) detM= A D| = AD-BC= 1.CD


(Bi)

This result is never essential for analyzing an optsystem, but it may lead to simplifications of somthe resulting formulas. In the present study, forample, use of Eq. (Bi) led to the simple phase transmation formula given in Eq. (29). Because of sreductions it is always helpful to know whethermatrices under study are unimodular. Thus it is]sonable to examine the conditions in which the meces governing ray and beam propagation in z-derdent graded-index media are unimodular.

As noted in the text, it follows from Eqs. (21) to that the matrix elements for an arbitrary z-dependmedium always obey the relationships

dAdz

dB = D.

y(z) = a I1 + G cos-yz) o F 12 G sin-yz 2(1 + G Y(1 - G2 ) 1 + G cos-yz (G2 - 1) 1/2

Xah[(G 2- 1)1/2 ('yz ]\ (1 + G cos-yzX tanh 1+ G tan 2 )] +b ( 1+G

X sin F 12I G sin-yz _ 2

( -y(l- G2 ) 1 + G cosyz (G2 - 1)1/2

X tanh1 [(G 1 tan (i-)J}) (Cl)

If G is equal to unity, Eq. (34) reduces to the form

(B2) y a(1 + = osaoz ) c{[tan (Yz ) + ( i )J}

(B3) + b (1 + CO5YZ ) sin {F2- [tan (i 2) + 3 ( )]}(C2)

(14) is applied to Eq. (20) one also

dC _k 2(z)dz k,

dD k2(z)=- B.dz ko

Using Eqs. (B2)-(B5), one finds-(AD - BC) = D -+ A -- B --C -dz dz dz dz dz

k2(z) k 2 (z)AB-C= DC AB+ AB-CD

= 0.

(B4)

(B5)

Similar modifications are helpful for special valuesof the parameter F. For example, if F is a negative realnumber, Eq. (34) can be replaced by

/1 + G cosyz'\y(z)=ak 1+G )

X cosh (-F)' 1 /2 f G sin-yz 2( - G2 ) i1 + G cosyz (1 -G2)1/2

X tan- 1+G tan ( 2 )}) + b (1 + G cosyz

I (-F)1/ 2 f G sin-yz 2

-y(l - G2 ) 1 + G cos-yz (1 -G

X tan' [(1 . G 2)1/2 tan (z(B6)

Therefore, the determinant of the beam matrix is con-stant, even in a z-dependent medium. But from Eqs.(21) to (24) the initial value of the matrix elements at z= z is A = 1, B = 0, C = 0, and D = 1. Since thedeterminant is constant and has an initial value ofunity, it follows that the matrices for z-dependentgraded-index media governed by Eq. (14) are alwaysunimodular.

This work was supported in part by the NationalScience Foundation.

(C3)

If F is set equal to zero the general solution becomes

yWz = a 1+ G cosyz\( 1 + G )

+ b (1 + G cosyz[ 1 G sin-yz 21 + G Ly(1 G2) ] 1 + G cosyz (1 - G2)112

X tan' [(1 - G2)1/2 tan t'Z (C4)

where the constant b has been replaced by bF-112.In the limit that y goes to zero Eq. (34) reduces to

Appendix C: Pseudosinusoidal Simplifications

The analysis in Sec. IV has been based on a certainsolvable Hill equation for which the solutions can beobtained analytically. Although the general solutionof that equation has been given analytically in Eq. (34),it may be worthwhile to consider explicitly some of thenonobvious special cases and extensions of that resultfor particular values of the parameters F, G, and y.First of all, it may be noted that, if G is a real numbergreater than unity, Eq. (34) can be more convenientlywritten:

- F/2z 1y(z) = a cos I I +b[sin L F"2z

1(1 + G)21

where b has been replaced by minus b. In this limitthere is no reason not to also set G to zero since it addsno generality to Eq. (33). Thus Eq. (C5) simplifiesfurther to

y(z) = a cos(F112 z) + b sin(F"12z), (C6)

and this is the standard result for a ray or beam propa-gating in a z-independent lenslike medium.


But when Eq.obtains

(C5)

References1. L. W. Casperson and J. L. Kirkwood, "Beam Propagation in

Tapered Quadratic Index Media: Numerical Solutions," IEEE/OSA J. Lightwave Technol. LT-3, 256 (1985).

2. L. W. Casperson, "Beam Propagation in Tapered QuadraticIndex Media: Analytical Solutions," IEEE/OSA J. LightwaveTechnol. LT-3, 264 (1985).

3. G. Goubau and F. Schwering, "On the Guided Propagation ofElectromagnetic Wave Beams," IRE Trans. Antennas Propag.AP-9, 248 (1961).

4. J. R. Pierce, "Modes in Sequences of Lenses," Proc. Natl. Acad.Sci. U.S.A. 47, 1808 (1961).

5. D. W. Berreman, "A Lens or Light Guide Using ConvectivelyDistorted Thermal Gradients in Gases," Bell Syst. Tech J. 43,1469 (1964).

6. E. A. J. Marcatili, "Modes in a Sequence of Thick AstigmaticLens-Like Focusers," Bell Syst. Tech. J. 43, 2887 (1964).

7. P. K. Tien, J. P. Gordon, and J. R. Whinnery, "Focusing of aLight Beam of Gaussian Field Distribution in Continuous andPeriodic Lens-Like Media," Proc. IEEE 53, 129 (1965).

8. Y. Suematsu, "Light-Beam Waveguide Using Lens-Like Mediawith Periodic Hyperbolic Temperature Distribution," Electron.Commun. Jpn. 49, 107 (1966).

9. M. S. Sodha, A. K. Ghatak, and D. P. S. Malik, "ElectromagneticWave Propagation in Radially and Axially Nonuniform Media:Geometrical-Optics Approximation," J. Opt. Soc. Am. 61, 1492(1971).

10. M. D. Feit and D. E. Maiden, "Unstable Propagation of a Gauss-ian Laser Beam in a Plasma Waveguide," Appl. Phys. Lett. 28,331 (1976).

11. L. Ronchi and C. Garbarino, "Gaussian Beams in Periodic Lens-like Media," Opt. Acta 29, 1171 (1982).

12. L. W. Casperson, "Beam Modes in Complex Lenslike Media andResonators," J. Opt. Soc. Am. 66, 1373 (1976).

13. H. Kogelnik, "On the Propagation of Gaussian Beams of LightThrough Lenslike Media Including Those With a Loss or GainVariation," Appl. Opt. 4, 1562 (1965).

14. F. B. Hildebrand, Advanced Calculus for Applications (Pren-tice-Hall, Princeton, N.J., 1965), p. 50.

15. G. W. Hill, "Mean Motion of the Lunar Perigee," Acta Math. 8,1(1886).

16. E. Mathieu, "Memoire sur le mouvement vibratoire d'une mem-brane de forme elliptique," J. Math. Pures Appl. 13, 137 (1868).

17. N. W. McLachlan, Theory and Applications of Mathieu Func-tions (Oxford U. P., London, 1951), Part 2.

18. J. L. Kirkwood, "Propagation of a Gaussian Beam in TaperedOptical Fibers and Concave Metallic Strip Waveguides," M.S.Thesis, U. California, Los Angeles (1983).

19. L. W. Casperson, "Solvable Hill Equation," Phys. Rev. A 30,2749 (1984); 31, 2743 (1985).

20. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford,1970), p. 67.

21. I. S. Gradshteyn and I. W. Ryzhik, Table of Integrals, Series,and Products (Academic, New York, 1972), p. 156.

22. S. Yamamoto and T. Makimoto, "Equivalence Relations in aClass of Distributed Optical Systems-Lenslike Media," Appl.Opt. 10, 1160 (1971).

0

Patter continued from page 4394

Long-gain-length solar-pumped box laserA new laser cavity configuration efficiently couples solar radiation to the

laser mode volume. C3F7I gas is optically pumped by two xenon-arc solarsimulators, creating a population inversion and subsequent lasing from atomiciodine at 1.3 ,um. The laser cavity is a stainless-steel box with internal high-reflectivity mirrors that guide the laser mode volume through the opticallyexcited C3F7I gas. Lasing output powers of -300 mW have been achieved fordurations of 150 msec.

Previous solar-pumped gas laser systems have been limited to laser gainlengths of less than 10 cm, requiring very high solar concentrations to achievelasing. This new system allows lasing at substantially lower solar simulatorintensities (150 suns) and much longer laser gain lengths (60 cm).

The laser was constructed using 0-ring grooves to allow a vacuum seal withthe quartz glass plates on each side of the stainless-steel laser frame. Brewsterwindows in the upper left-hand corner allow external laser cavity mirrors to bemounted and easily aligned. The back cavity mirror has maximum reflectiv-ity at 1.3 ,m, and the output mirror has a reflectivity of either 97 or 85%. Ineach of the three internal corners of the laser cavity high-reflectivity dielectricmirrors are placed. These mirrors allow the laser optical path (mode volume)to be aligned with the incoming solar radiation pattern. The laser beam isdetected by a germanium linear-array detector, which is used to resolve thelaser beam profile. Two xenon solar simulators produce a maximum of 5 kWeach of light. A mechanical shutter is used to allow rapid excitation of thelaser cavity. The input simulator light is concentrated in the form of adoughnut.

The system allows long gain lengths at much lower solar concentrations,substantially increasing the practicality of solar pumping. The system wasoriginally developed for studies of power transmission over long distancesthrough space.

This work was done by Russell J. De Young of Langley Research Center.No further documentation is available. This invention is owned by NASA,and a patent application has been filed. Inquiries concerning license for itscommercial development should be addressed to the Patent Counsel, LangleyResearch Center, Mail Code 279, Hampton, Va. 23665. Refer to LAR-13256.

Optical scanner for linear arrayAn optical scanner instantaneously reads contiguous lines forming a scene

or target in the object plane. The reading may be active or passive and thescans continuous or discrete. The scans are essentially linear with scan angleand are symmetric about the axial ray. A nominal focal error, resulting from acurvature of the scan, is well within the Rayleigh limit. The scanner wasspecifically designed to be fully compatible with the general requirements oflinear arays.

The essential elements of the optical system are shown in Fig. 7. The cornermirrors M,, M2, M3, and M4 are perpendicularly oriented so that any rayincident on the front surface of the scan mirror will be directed by the cornermirrors along a parallelogrammic path arriving at the back surface of the scanmirror where it will be reflected in a direction parallel to the incident ray.Except for the imaging lens, all the optical elements are plane mirrors. Therotation of the scan mirror is the only motion required.

LINEARARRAY

Fig. 7. Optical system and path followed by a fan of axial rays, Lo,and some arbitrary fan of rays, Li, are shown.

Since the angle of incidence on the front and back surfaces of the scan mirroris equal, it follows that the incident and emergent rays must be parallel.Moreover, since each ray must pass through the center of the scan, there is alateral shifting of frames so that each ray leaves the system collinear with theaxial ray. Effectively then, the front surface of the scan mirror scans theobject plane frame by frame while the back surface of the scan mirror descanseach frame onto the image plane. All this indicates that each scan line couldbe identified with a particular scan angle.

continued on page 4407


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Beam propagation in periodic quadratic-index waveguides