Theory of 1-N-way phase-locked resonators withperturbations in the array elements

Eric K. Gorton and R. Michael Jenkins

The recently proposed 1-N-way resonator based on beam splitting and beam combining effects in rect-angular cross-sectional multimode waveguides offers a valuable way in which N low-power laser ele-ments can be combined in a coherent fashion. We develop a theory of such resonators in the presenceof perturbations in the N-element array. We demonstrate that despite the presence of perturbationsthere is only one possible mode of the resonator. The theory is used to provide an understanding of theeffects of a number of possible perturbations that could arise as a result of manufacturing processes andoperational effects. The results give scaling laws for the design tolerances on array element mirror tilt,array element optical path length control, and the effects of array element malfunction and their need forgain balance.

OCIS codes: 230.0230, 140.4780, 230.7370.

1. Introduction

Methods to phase lock low-power lasers in a coherentfashion have received much interest of late. Oneparticular technique that has received attention isthe use of novel resonator structures to enforce laser-element to laser-element mode mixing. One suchstructure is the Talbot cavity,1,2 which allows thecoherent mixing of modes from differing elements ina laser array. This leads to low-loss resonatormodes with both phase and frequency locking acrossthe array. Such designs, however, suffer from anumber of difficulties. The mode mixing is not uni-form across the array. This lack of uniformity re-sults in perturbations that are due to manufacturingtolerances that cause array elements to lose theircoherence with their neighbors. Also the near- andfar-field intensity distributions of the modes from thearray resonator structure are difficult to use in apractical way.

The use of a 1-N-way phase-locked array resonatorhas been proposed by several researchers.3–7 Com-parison of this resonator with that of the Talbot cav-ity8 indicates that mode mixing from any arrayelement to any other array element is uniform so thatone can expect better modal stability as well as the

structure supporting a mode whose near- and far-field intensity patterns are easier to interface withother optical elements in any integrated system.

In this paper we follow the earlier theory on 1-N-wayresonators9 and extend this theory to take into accountthe possible phase and gain or loss perturbations in theindividual array elements. We begin by outlining theway in which the proposed resonator operates. Wedevelop the resonator equations and indicate themethod by which they can be solved. An analyticsolution for the general case of these resonators withN-array elements, each having its own individualphase and gain terms included, is provided. Havingestablished the solution for the resonators we turn ourattention to the effects of the misalignment of the mir-ror at the array end of the resonator and provide ananalytic solution for the resonator and hence a designtolerance as a function of array number. We also ex-amine the effects of random phase errors that can arisein the array elements and show the need to maintainsufficient control on the phases of the elements in thearray. Examination of the possible gain or loss per-turbations leads to the idea of graceful degradation forthe resonator as individual elements fail. An exami-nation of the effects of random gain or loss variationsis also conducted. Finally, conclusions are drawn onthe derived properties of the perturbed 1-N-way reso-nator.

2. 1-N-Way Resonator Operation

When an optical field described by the mode of asquare cross-sectional �2a � 2a� guide propagatesinto a rectangular cross-sectional �2a � 2b� multi-

E. K. Gorton �ekgorton@dera.gov.uk� and R. M. Jenkins�rmjenkins@dera.gov.uk� are with the Defence Evaluation Re-search Agency, St. Andrews Road, Malvern, Worcestershire WR143PS, UK.

Received 5 February 2001; revised manuscript received 10 July2001.

20 December 2001 � Vol. 40, No. 36 � APPLIED OPTICS 6663

mode guide at the center of the input aperture of themultimode guide, the subsequent field propagates inthe rectangular guide in such a fashion as to reimagethe original optical field at an axial distance given byL � �2b�2��, where � is the wavelength in the core ofthe multimode guide. During propagation to thislength there are many intermediate axial distancesat which the optical field in the rectangular guide canbe represented by an array of N images of the originaloptical field. For particular values of wavelengthand rectangular waveguide width the value of N isdetermined only by its distance from the initial rect-angular waveguide aperture. The optical field asso-ciated with each element of this array is an exact copyof the original optical field apart from differences inphase.9,10 Each N copy of the original field has anintensity that is equal to that of all other copies in thearray. This equal division of the original field into Ncopies of itself is referred to as an N-way split field.This N-way split field is a property of the modes of therectangular cross-sectional waveguide.5–7,9,10 Infact, any particular N-way split field occurs twice �ataxial distances L�N and �N � 1�L�N� between theinjection aperture of the rectangular guide and reim-aging distance L.

The position in the aperture of the rectangularguide of the nth copy �n � N� of the split field iscentered at y � yn, where yn � �2n � N � 1��2 and � 2b�N is the distance between centers of adjacentelements of the array of the N split fields. Here y isthe coordinate in the aperture plane of the rectangu-lar guide with its origin at the center of the rectan-gular guide. The walls of the rectangular guide lieat y � b. The two N-way splits are not entirelyequivalent since the phases of the optical field of eachcopy of the original optical field in the N-elementarray differ. The phases of these N copies in thesplit field are related by ��n, where

�n � 1��2N� � �N � 1��4 � n�n � N � 1��N

at the first split and by ���n at the second split.Thus if one truncates the rectangular guide at the

first split, reflects the field and imposes phase shiftsof �2��n on appropriate elements of the array, thefield that propagates in the reverse direction in therectangular guide will reconstruct the original opticalfield at the initial rectangular guide aperture. Thusa resonator can be formed that would split the outputof the single axial guide equally between the N ele-ments of an array of guides similar to that at theinput aperture of the rectangular guide with no ap-erture loss and then, after reflection and phase cor-rection, recombine them into the original single axialguide again with no loss. Such lossless operationhas been shown to be obtainable5–8 within paraxialapproximation in which the rectangular waveguide’smode propagation constant is taken as that given inRef. 11.

Accordingly the proposed 1-N-way resonator isshown schematically in Fig. 1, which provides thearrangement for the array square-section optical

waveguides with corrective phase plates, the rectan-gular guide propagation region, and the on-axissquare-section guide that receives the combined fieldfrom the rectangular guide. The resonator is com-plete with mirrors, one or both of which can be madepartially transmitting. It is within the square-section waveguides that form the array that low-power laser elements can be placed to allow thestructure to operate as an active laser. If the outputof such a laser were to be used with fiber-optic sys-tems, the output from the resonator could be takenfrom the on-axis guide. The introduction of low-power lasers into the array elements of the proposed1-N-way resonator structure could result in the in-troduction of both phase and amplitude perturba-tions within the N-array elements. Our aims in thispaper are to examine the mode properties of suchperturbed resonator structures and to examine nec-essary design issues.

3. Theory of Perturbed 1-N-Way Resonators

To develop the theory for the 1-N-way resonators wefollow the method given in a previous paper.9 Sincethe method is outlined in some detail in Ref. 9, wepresent only a brief summary of the method here bygiving the important relations required for progressbefore further development of the theory. Underparaxial approximation we have fundamental modefields in the on-axis guide and the guides of theN-way array given by10

E1� y, a� �1

�acos��y

2a� , � y� � a, (1)

and under the same approximation the modes of themultimode guide are given by10

Eq� y, b� � �1

�bcos�q�y

2b � if q is odd

1

�bsin�q�y

2b � if q is even� � y� � b.

(2)

Fig. 1. Schematic of a 1-N-way resonator.

6664 APPLIED OPTICS � Vol. 40, No. 36 � 20 December 2001

A fundamental mode of amplitude am emitted fromthe mth member of the array couples to the modes ofthe multimode guide with coupling amplitude

Cmq � y�ym�a

y�ym�a

E1� ym, a� Eq� y, b�dy. (3)

The field then propagates in the multimode guide oflength d by use of the mode propagation constant

�q �2�

� �1 �12 ��q

2b�2 . (4)

The propagated field then couples to the on-axisguide at z � d by use of the coupling amplitude

cq � y��a

y�a

Eq� y, b� E1� y, a�dy. (5)

The fundamental field now propagates in the on-axisguide of length l�, with propagation constant10

�a �2�

� �1 �12 � �

2a�2 . (6)

The mode reflects from the mirror with amplitudereflectivity r2 and propagates back along the guidelength to recouple to the multimode guide with thecoupling amplitude again given by cq. The field con-tinues to propagate in the multimode guide withpropagation constant �q and then couples to the ntharray element with coupling amplitude Cnq. The re-sulting fundamental field now propagates in an ac-tive array element of length l and unperturbedpropagation constant �a to reflect at the mirror ofamplitude reflectivity r1 and to return to the exit ofthe array. During a round-trip propagation throughthe array element the field encounters the phaseplate that has a single-pass phase shift of ���n andphase and gain perturbations particular to this ele-ment of the array. We write all these gain andphase shift contributions �including the phase plates�in one term, �n�, defined as

�n� � �2�i�n � �n. (7)

In Eq. �7�, �n is a complex number, the real part ofwhich represents the fundamental mode gain duringa round trip in the array element and the imaginarypart represents the phase perturbation experiencedby the mode as a result of element-to-element differ-ences in optical path length. Such phase perturba-tions could arise because of manufacturing tolerancesor active processes in the gain medium.

Combining these facts, we determined that, after around trip, the portion of the emitted field amplitudeof the nth array element that is due to the contribu-tion of the mth element at the array–multimodeguide interface is

anm � am r1 r2 exp��2i�l � l���a�Bnm exp��n��, (8)

where we have identified the element-to-element am-plitude coupling term to be

Bnm � �q�1

�� cq Cmq exp��i�q d� �

r�1

�� cr Cnr exp��i�r d�.

(9)

For a self-consistent mode of the resonator to exist wemust have

�an � �m�1

N

anm, (10a)

or

g�an � �m�1

n

Bnmam exp�i�n�� (10b)

where � is the constant round-trip mode gain and g �exp�2i�l � l���a��r1r2. The set of simultaneousequations for the N-array element amplitudes givenby Eqs. �10a� and �10b� can be represented in matrixform

g�a � B � a, (11)

which is the eigenvalue and eigenvector problem.Here a is an eigenvector that contains the individual

field amplitudes of the array elements and B

is anN � N matrix of field amplitude cross-coupling termswhose n,mth element is Bnm exp��n��.

The nontrivial solution of Eq. �11� requires us todefine an N � N determinant D whose n,mth elementis Bnm exp��n�� but whose diagonal �n, n� elementsare given by Bnm exp��2�i�n� � g�. We note thatBnm

2 � BnnBmm results in

D � DN �j�1

N

Bjj exp��j� �, (12)

where DN is an N � N determinant whose diagonalelements �n, n� are given by 1 � g��Bnn exp��n�� andall the other elements are 1. This form of determi-nant has been evaluated previously.9 For this casewe found that

D � ��1�N� � g��N � � g��N�1 �j�1

N

Bjj exp��j� � . (13)

Since the nontrivial solution of Eq. �11� requires thatD � 0, from the solution of Eq. �13� we found that theround-trip mode gain is given by

� �1g �

j�1

N

Bjj exp��j� �, � � 0. (14)

The eigenvector associated with the first value of � isfound to have its components related in the followingmanner:

am � an

Bmm

Bnm. (15)

20 December 2001 � Vol. 40, No. 36 � APPLIED OPTICS 6665

The zero value of � in Eqs. �14� implies 100% moderound-trip loss and is associated with an infinite setof eigenvectors with components related to each otherin such a way that the internal field in the multimodeguide at the on-axis guide is zero. However the in-ternal field can take on values across the rest of themultimode guide aperture at this point. It is clearfrom this that even with perturbations in the indi-vidual array elements there is only one physical modesolution in this resonator. The resonator mode am-plitude round-trip gain is thus given by the first partof Eqs. �14� and the array element field amplitudesfor the resonator mode are related by Eq. �15�. Sub-stitution of the unperturbed expression for �n� ��n� ��2�i�n� into Eqs. �14� results in the expression

� �1g �

j�1

N

Bjj exp��2�i�j�, (16)

which is the result previously obtained for the unper-turbed array case.9

4. Phase Perturbations

One possible phase perturbation that could occur inthe manufacture of 1-N-way resonators in semicon-ductor technology arises from the fact that the chip onwhich the structure is grown is cleaved to provide themirror at the output end of the N-way array. Angu-lar mismatches between the pattern laid down dur-ing the device growth period and the semiconductorchip cleave plane could amount to a few milliradians.To examine the effect of such angles on the resonatormode it is necessary to add a phase error that reflectsthe effective plane mirror tilt away from the resona-tor axis. Such a phase error can be added easily byincorporation of the additional round-trip phase intothe perturbation expression of Eq. �7� for each arrayelement. The phase error for a particular elementdepends on the position of that element so that theadditional round-trip phase change for a tilt angle of� is given by

�ntilt � 2

2�

�� yn � y1�� �

4��n � 1�

��. (17)

Thus, Eq. �7� becomes

�n� � �2�i�n � 4�i�n � 1�

��. (18)

It has been shown8 that, when d � L�N � dN, then

Bnm � N�1 exp�i��nm�, (19)

where �nm � ��N � 1 � 2n�2 � �N � 1 � 2m�2��4N �1�2. Substitution of Eqs. �18� and �19� into Eqs. �14�gives

� �

exp�i�4��

��

�

2N�gN �

n�1

N

exp��4�in�

� ��

exp��i�2N �

gN

�1 � exp��4�iN�

� ��1 � exp��4�i

�

� � . (20)

Thus the resonator mode intensity has a round-tripgain of

���2 �R1 R2

N 2

sin2�2�N����

sin2�2�����, (21)

where Ri � �ri�2 are the mirror reflectivities for the

intensity. Equation �21� is an effective expressionfor a diffraction grating with N apertures and is plot-ted in Fig. 2 for a number of values of N when R1 �R2 � 1.

Equation �21� has a maxima at � � q��2, where qis an integer and we find ���2 � R1R2. The minimaof Eq. �21� occur at � � p��2N, where q � p�N andp is an integer and ���2 � 0. The first minimumoccurs at � � ��2N, which gives a measure of thewidth of the mode gain–loss curve. We note, how-ever, that the angle at which the two outer arrayelements first become out of phase also provides ameasure of the width of the mode gain–loss curve.This angle is given by � � ��4�N � 1�. To resolvethe dilemma of which expression for the curve widthbest expresses our device design needs we solve Eq.�21� numerically11 for the angles at which the gain isreduced to 95% and 90% of the maximum value.Figure 3 shows that the 1��N � 1� scaling fits betterat low N compared with the 1�N rule for our chosengain reductions.

A second type of phase perturbation could arisebecause of refractive-index changes in the elements ofthe array as a result of the active gain medium.These could arise because of differences in heating or

Fig. 2. Resonator mode loss as a function of tilt for some arraynumber values.

6666 APPLIED OPTICS � Vol. 40, No. 36 � 20 December 2001

cooling the elements or variations in pumping theactive medium. In semiconductor technologies elec-trical contact resistance can vary between elementsleading to variations in pump electron density andhence refractive index and gain. We note that, sinceBjj � BN�1�jN�1�j, by using Eqs. �14� and �19� for Neven we can write

� �1g �

j�1

N2

Bjj�exp��j� � � exp��N�1�j���

�

exp��i�2N �

gN �j�1

N2

�exp�i�j� � � exp�i�N�1�j���, (22)

where we have used �j� � Im �j. Equation �22� in-dicates that symmetric pairs of array elements inter-fere with each other to reduce the overall mode gain.Evaluation of the mode intensity gain proceeds fromEq. �22� to give

���2 �R1 R2

N 2 �� �j�1

N

cos �j��2

� � �j�1

N

sin �j��2 . (23)

To demonstrate the effect that phase perturbationshave on the resonator mode gain we chose a set ofrandom numbers from a normal distribution withzero mean and a standard deviation of sp. We usethese random numbers to represent the phase per-turbations �j� in Eq. �23� to provide a probability den-sity plot for the resonator mode gain. A typical plotis shown in Fig. 4. It can be shown that Eq. �23� canalso be written as

���2 �R1 R2

N 2 �N � 2 �j�1

N�1

�k�j�1

N

cos��j� � �k�� . (24)

If we take cos��j� � �k�� � K, where K is the mostprobable value for cos��j� � �k��, we find that

���2 � R1 R2�K � �1 � KN � . (25)

Equation �25� confirms the results from many com-puter simulations that the resultant distribution ofmode gain is only weakly dependent on N and be-comes less dependent as N increases. This is par-

ticularly true for values of K � 1 that occur when theprobability distribution for the phase perturbations isnot spread over a large range.

5. Gain Perturbations

Perturbations in the gain in each element of the arrayfrom the norm can occur in real lasers as a result ofa number of mechanisms. Variations that are due totemperature as a result of differing pumpingstrengths or on differences in cooling will result ingain variations among the array elements. Differ-ences in the make up of the gain medium �e.g., dopinglevels� or the strength of the pump source could occurbecause of manufacturing issues. As mentioned inSection 4, electrical contact resistance can vary be-tween elements leading to variations in pump elec-tron density and hence refractive index and gain.

From Eq. �14� it is straightforward to show that,when d � L�N � dN,

���2 �R1 R2

N 2 � �j�1

N

exp �j��2

, (26)

where �j� � Re �j is the amplitude gain perturbation.We can see from Eq. �26� that, if a fraction f of thearray elements is damaged such that �j�3 �� whilethe remainder of the elements have an unperturbedamplitude gain �, then

���perturbed2

���2� �1 � f �2 exp�2��, (27)

where ���2 is the unperturbed resonator mode gain��j � 0� as given by Eq. �16�. This result demon-strates that as array elements fail the overall gain ofthe resonator mode declines in a graceful fashionaccording to a �1 � nfailed�N�2 rule. This might be apessimistic rule since we have assumed that elementfailure is equivalent to an infinite element absorp-tion. This decline in resonator mode gain can becompensated for if the remaining array element gaincoefficients � can be increased.

To demonstrate the effect that random array ele-ment amplitude gain variations �j� could have on theoverall resonator mode gain, we follow the method

Fig. 3. Comparison of scaling laws for small-gain reductionscaused by tilt.

Fig. 4. Resonator mode gain probability density plot for normallydistributed random phase errors in the array elements: sp �0.1�, N � 16.

20 December 2001 � Vol. 40, No. 36 � APPLIED OPTICS 6667

adopted for phase perturbations and chose a set ofrandom numbers from a normal distribution withmean amplitude gain � and standard deviation s�.We used these random numbers in Eq. �26� to providea probability density plot for the resonator mode gain.Typical plots are shown in Fig. 5.

It can be seen from Fig. 5 that the resonator modegain exceeds the mean value of the array elementgain more often than it falls below this value forsufficiently large values of s�. This is generally truewhen s� � 0.1, the distribution being more or lesssymmetrical about the mean value when s� � 0.1.

6. Conclusions

We have obtained an analytic expression for the 1-N-way resonator mode gain in the presence of any arrayelement-to-element perturbation in either phase orgain. The theory shows that there is only one modesolution in any case and that this solution depends onthe self-element coupling coefficients and the phaseand gain properties of the array elements. The ar-ray element electric field amplitudes that enter the

array are found to be related by the self-element andinterelement coupling coefficients only and do notdepend on any perturbation effects in the array.Under the conditions that multimode guide length dis within one third of its design length dN, theseself-element and interelement coefficients areequal.8,9 This, together with the present result onarray perturbations, indicates that the 1-N-way res-onator design is highly effective in mixing the modesfrom each array element, equally splitting the modesamong the array elements and hence locking themtogether even under extreme perturbation condi-tions.

The perturbation theory has been used to providean analytic result for the resonator mode gain whenthe array mirror is misaligned with respect to theresonator axis. The result obtained is equivalent tothat obtained for a diffraction grating with N aper-tures and can be used to set design tolerances on thearray mirror tilts acceptable for low-loss resonatoroperation. The scaling law obtained for these low-loss conditions requires that the tilt be less than��20�N � 1�. Work on random phase errors thatcould arise because of active processes in the gainmedium or because of manufacturing techniques in-dicates that the resonator would continue to operatewith decreasing gain as the phase errors grow. Forlow-loss conditions to prevail the phase errors shouldbe maintained at or below the 0.1� level. This tol-erance appears to be independent of the array num-ber as the theory shows, in the first approximation,that the tolerable error is only weakly dependent onN.

Turning to possible gain perturbations in the arrayelements we have been able to show that there is agraceful degradation mode of operation as elementsin the array fail catastrophically. The failure mech-anism is assumed to be total absorption so that lesssevere failure mechanisms would have less effect onthe resonator mode gain as a whole. Compensationfor array element failure might be possible if theremaining array element gain coefficients can beraised sufficiently. By examining the possible fluc-tuations in the array element gain coefficients abouta mean value we have also determined that symmet-rical resonator mode gain distributions prevail belowvalues of the array element amplitude gain standarddeviation of s� � 0.1 while the resultant resonatormode gain distribution becomes skewed towardhigher values of resonator mode gain as s� increasesabove this value.

The authors acknowledge several useful discus-sions with their colleagues A. J. Davies of Royal Hol-loway, University of London, and with J. Banerji ofthe Physical Research Laboratory, India.

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Fig. 5. Probability density plots for the resonator mode gain inthe presence of array element gain perturbations: N � 16.

6668 APPLIED OPTICS � Vol. 40, No. 36 � 20 December 2001

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20 December 2001 � Vol. 40, No. 36 � APPLIED OPTICS 6669