theory of 1- n-way phase-locked resonators
TRANSCRIPT
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Theory of 1-N-way phase-locked resonators
Eric K. Gorton and R. Michael Jenkins
A 1-N-way resonator based on beam splitting and beam combining effects in rectangular cross-sectionalmultimode waveguides was recently proposed. Such a resonator structure offers a valuable way inwhich N low-power laser elements may be combined in a coherent fashion. We examine the case ofpassive 1-N-way resonators. We develop a theory of these 1-N-way structures to show that there is onlyone possible mode of these resonators. The theory is used to give a scaling law for the design tolerancesof the beam splitting and beam combining region of the resonator.
OCIS codes: 140.0140, 140.4780, 230.7370.
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1. Introduction
Recently there has been much interest in mecha-nisms that will permit the phase locking of low-powerlasers in a coherent manner. One particular mech-anism that has received attention is the use of novelresonator structures to achieve laser-element tolaser-element mode mixing such that phase locking ofarrays of low-power laser elements may be obtained.One such structure is the Talbot cavity,1,2 which per-mits the coherent mixing of the modes from variouselements in the laser array. This mixing leads tolow-loss resonator modes with both phase and fre-quency locking across the array. Such designs, how-ever, suffer from a number of difficulties. The modemixing is not uniform across the array. This lack ofuniformity results in perturbations owing to manu-facturing tolerances that cause array elements to losetheir coherence with their neighbors. Also, thenear- and far-field optical distributions from the ar-ray are difficult to use in a practical way.
More recently, the use of a 1-N-way phase-lockedarray resonator was proposed.3–7 Comparison ofthis resonator with that of the Talbot cavity8 indi-ates that mode mixing from any array element tony other array element is uniform, so one may ex-ect better modal stability, near- and far-field pat-erns, and thus easier interfacing with other opticallements in any integrated system.In this paper we provide a theory of passive 1-N-
E. K. Gorton [email protected]! and R. M. [email protected]! are with the Defence Evaluation Re-earch Agency, St. Andrews Road, Malvern, Worcestershire WR14PS, UK.Received 2 February 2000; revised manuscript received 19 June
000.
16 APPLIED OPTICS y Vol. 40, No. 6 y 20 February 2001
ay resonators. We begin by outlining the way inhich the proposed resonator operates. We develop
he resonator equations and indicate the method byhich they may be solved. An analytic solution for
he general case of these resonators with N arraylements is provided. Having established the solu-ion for the resonators, we turn our attention to theffects of the length of the rectangular guide on theode loss of the resonator as the number N of ele-ents in the array are varied. A scaling law is thus
erived. Finally, conclusions are drawn on the de-ived properties of the passive 1-N-way resonator.
2. 1-N-Way Resonator Operation
When an optical field described by the mode of asquare cross-sectional ~2a 3 2a! single-mode opticalwaveguide is propagated into a rectangular cross-sectional ~2a 3 2b! multimode optical waveguide atthe center of the input aperture ~or on the axis! of therectangular guide, the subsequent optical field prop-agates in the rectangular guide in such a fashion as toreimage the original optical field at an axial distancegiven by L 5 ~2b!2yl, where l is the wavelength inthe rectangular guide.5–7 In propagating 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 guide aperture. The optical field associatedwith each element of this N array is an exact copy ofthe original optical field, apart from differences inphase.9,10 Each of the N copies of the original fieldhas an intensity that is equal to that of all othercopies in the array. This equal division of the orig-inal field into N copies of itself is referred to as an
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N-way split field. This N-way split field is a prop-erty of the modes of the rectangular cross-sectionalwaveguide.5–7,9,10 In fact, any particular N-wayplit field occurs twice @at axial distances LyN and
~N 2 1!LyN# between the injection aperture of therectangular guide and the reimaging distance L.
The position in the aperture of the rectangularguide of the nth copy ~n # N! of the split field iscentered at y 5 yn, where yn 5 ~2n 2 N 2 1!ry2 andr 5 2byN 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 origin at the center of the rectangularguide. The walls of the rectangular guide lie at y 56b. These two N-way splits are not entirely equiv-lent because the phase of the optical field of eachopy of the original optical field in the N elementrray differs. The phases of these N copies in theplit field are related by pfn, where fn 5 1y~2N! 1
~N 1 1!y4 1 n~n 2 N 2 1!yN at the first split and by2pfn at the second split.
Thus, if one truncates the rectangular guide at thefirst split, reflects the field, and imposes phase shiftsof 22pfn on the appropriate elements of the array,the field propagating in the reverse direction in therectangular guide will reconstruct the original opticalfield at the initial rectangular guide aperture. Thusa resonator may be formed that will split the mode ofthe initial single axial guide equally between the Nelements of an array of similar single-mode guideswith no aperture loss and then after reflection andphase correction recombine them into the originalsingle axial guide, again with no loss. Such losslessoperation has been shown5–8 to be obtainable withinthe paraxial approximation in which the rectangularoptical waveguide’s mode propagation constant istaken as that given in Ref. 11.
Accordingly, the proposed 1-N-way resonator isshown schematically in Fig. 1, including the arrange-ment for an array of square-section opticalwaveguides with corrective phase plates, the rectan-gular guide propagation region, and the on-axissingle-mode guide that receives the combined fieldfrom the rectangular guide. The resonator is com-pleted with mirrors, one or both of which may bemade partially transmitting. It is within thesquare-section waveguides that form the array thatlow-power laser elements may be placed to allow the
Fig. 1. Schematic of a 1-N-way resonator.
structure to operate as an active laser. If the outputfrom such a laser were to be used with fiber opticalsystems, then the output from the resonator could betaken from the on-axis guide. Our aims in this pa-per, however, are to derive the mode properties ofsuch passive resonator structures and to examinenecessary design issues.
3. Resonator Theory
The mode supported by the on-axis square single-mode guide and by the square single-mode guides inthe N-way array is of the form11
E1~y, a! 51
ÎacosSpy
2aD , uyu # a, (1)
and the modes of the rectangular guide are given by11
Eq~y, b! 5 51
ÎbcosSqpy
2b D q odd
1
ÎbsinSqpy
2b D q even, uyu # b. (2)
In both Eqs. ~1! and ~2! the y coordinate is measuredacross the appropriate waveguide aperture with theorigin at the center of the appropriate waveguideaperture. We note that it is necessary to examineonly the y dependence of the mode fields because thedimensions of all the square guides and the rectan-gular guide are the same in the orthogonal directionto y in the guide apertures. Thus there will be nodiscontinuity, and mode coupling from square guideto rectangular guide and back in this orthogonal di-rection is guaranteed to have no effect on the field.
To analyze the propagation of the fields within theresonator we imagine the optical field of the mode ofthe square guide as being emitted from the mth mem-er of the array into the rectangular guide with aomplex amplitude am. This mode will couple to the
modes of the rectangular guide by means of the am-plitude
Cmq 5 *y5ym2a
y5ym1a
E1~ym, a!Eq~y, b!dy. (3)
In Eq. ~3! ym is the position of the center of thewaveguide of the mth member of the array based onhe y coordinate in the rectangular guide. The fieldhat is due to this one array element in the rectan-ular guide can be written as
F~y, z 5 0! 5 am (q51
q5`
CmqEq~y, b!, (4)
where we take the z coordinate to lie in the moderopagation direction, which is the resonator axis.e define the z 5 0 position to be at the right-hand
side of the rectangular waveguide in Fig. 1. Thisfield may be propagated through the rectangularguide of length d by use of the mode propagationconstant11
bq 52p
l F1 212 Slq
2bD2G . (5)
20 February 2001 y Vol. 40, No. 6 y APPLIED OPTICS 917
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The coupling of the resultant field at z 5 d into then-axis guide is governed by the amplitude
cq 5 *y52a
y5a
Eq~y, b!E1~y, a!dy. (6)
This amplitude is zero unless q is an odd integer.Using Eqs. ~4!–~6!, we therefore obtain for the field atthe entrance of the on-axis guide owing to the mthelement of the array in terms of the mode of theon-axis guide
F~y, z 5 d! 5 am (q51
`9 cq Cmq exp~2ibq d!E1~y, a!, (7)
where the prime on the summation sign indicatessummation over odd values of q only.
This field may be propagated with propagation con-stant ba through length l9 of the on-axis guide, re-flected from the mirror of amplitude reflectivity r2,and returned to the exit of the on-axis guide. Thisreturn field is written as
F~y, z 5 d! 5 am r2 exp~22il9ba! (q51
`9 cq Cmq
3 exp~2ibq d!E1~y, a!. (8)
The field recouples to the rectangular guide accordingto the amplitude relation of Eq. ~6!, so after the fieldhas been propagated through the rectangular guideto the array it can be written as
F~y, z 5 0! 5 am r2 exp~22il9ba! (q51
`9 cq Cmq
3 exp~2ibq d! (r51
`9 cr Er~y, b!exp~2ibr d!. (9)
The field can then be coupled to the nth element ofthe array by use of the amplitude rule of Eq. ~3!.Propagation in the array element involves a doublepass through the phase plate, which imposes a single-pass phase shift of 2pfn, a phase shift that is due tolength l of the guide and reflection r1 from the mirror.The resultant field amplitude that is being returnedfrom the nth element of the array to the rectangularguide will be
F~y, z 5 0! 5 am r1 r2 exp@22i~l 1 l9!ba#
3 (q51
`9 cq Cmq exp~2ibq d! (
r51
`9 cr Cnr
3 exp~2ibr d!exp~22pifn!E1~y 2 yn, a!. (10)
hus, after a round trip, the portion of the emittedeld amplitude of the nth array element that is due to
the contribution of the mth element is
anm 5 am r1 r2 exp@22i~l 1 l9!ba#Bnm exp~22pifn!,
(11)
18 APPLIED OPTICS y Vol. 40, No. 6 y 20 February 2001
where we have identified the element-to-element am-plitude coupling term to be
Bnm 5 (q51
`9 cq Cmq exp~2ibq d! (
r51
`9 cr Cnr exp~2ibr d!.
(12)
For a self-consistent mode of the resonator to existwe must have
san 5 (m51
N
anm
or
gan 5 (m51
n
Bnmam exp~22pifn!, (13)
where s is the constant round-trip mode transmis-sion and
g 5 s exp@2i~l 1 l9!ba#yr1 r2.
The set of simultaneous equations for the N-arraylement amplitudes given in Eq. ~13! may be repre-
sented in matrix form:
ga 5 B<
z a, (14)
where a is an eigenvector that contains the individualfield amplitudes of the array elements and B
'is an
N 3 N matrix of field amplitude cross-coupling termswhose n, mth element is Bnm exp~22pifn!. Equa-tion ~14! may also be written as
B<
z a 2 ga 5 0
r
~ B<
2 g! z a 5 0, (15)
here the n, mth element of the new B'
2 g matrix isagain Bnm exp~22pifn! but the diagonal ~n, n! ele-ments are given by Bnn exp~22pifn! 2 g.
We proceed to solve the eigenvalue problem of Eq.~15! by recognizing from Eq. ~12! that
Bnm2 5 BnnBmm. (16)
This relation allows determinant D of the B'
2 gatrix to be written as
D 5 DN )j51
N
Bjj exp~22pifj!. (17)
DN is an N 3 N determinant whose diagonal ele-ments ~n, n! are given by 1 2 gyBnn exp~22pifn!, andall other elements are 1. By subtracting the penul-timate row of DN from the last row and evaluating,one obtains
DN 5 aN DN21 1 aN21DN219, (18)
where an 5 2gy@Bnn exp~22pifn!# and DN9 is an N 3N determinant whose diagonal elements are given by1 1 an, with n running from 1 to N 2 1. The final
m
n
roarzaisr~E
Td
diagonal element is equal to 1, and all other elementsof DN9 are equal to 1. By subtracting the penulti-
ate row of DN9 from the last row we find that
DN9 5 aN21DN219. (19)
By repeated application of Eq. ~19! and examinationof D29 it can be shown that
DN9 5 )j51
N21
aj. (20)
Equation ~18! now becomes
DN 5 aN DN21 1 )j51
N21
aj. (21)
By repeating the algorithm of Eq. ~21! and by exam-ination of D2 it can be shown that
DN 5 F1 1 (k51
N 1akG )
j51
N
aj. (22)
Using Eqs. ~17! and ~22! and substituting for thevalues of an, we find that
D 5 ~21!NF~g9s!N 2 ~g9s!N21 (j51
N
Bjj exp~22pifj!G ,
(23)
where g9 5 exp@2i~l 1 l9!ba#yr1r2. Inasmuch as theontrivial solution of Eq. ~15! requires that D 5 0, we
find from the solution of Eq. ~23! that the round-tripmode transmission is given by
s 51g9 (j51
N
Bjj exp~22pifj!,
s 5 0. (24)
Fig. 2. Mode loss as a function of rectangul
The eigenvector associated with the first value of s isfound to have its components related in the followingmanner:
am 5 an
Bmm
Bnm. (25)
The zero value of s in Eq. ~24! implies 100% modeound-trip loss and is associated with an infinite setf eigenvectors that have components related to onenother in such a way that the internal field in theectangular guide at the on-axis guide aperture isero. However, the internal field can take on valuescross the rest of the rectangular guide aperture. Its clear from this that there is only one physical modeolution in this resonator with a mode amplitudeound-trip transmission given by the first part of Eq.24! and array element field amplitudes related byq. ~25!.
4. Rectangular Guide Length Design Tolerance
When the rectangular guide has a length given bydN 5 LyN, the correct design length, then it can beshown that8
Bnm 5 N21 exp~ipfnm!, (26)
where
fnm 5 @~N 1 1 2 2n!2 1 ~N 1 1 2 2m!2#y4N 2 1y2.
In this case we have, from Eq. ~24!,
s 51g9 (j21
N
Bjj exp~22pifj! 51g9
expS2ipN D . (27)
hus, when the rectangular guide has the correctesign length, the mode suffers a round-trip loss of
1 2 usu2 5 1 2 R1 R2, (28)
ide length for some values of array number.
ar gu20 February 2001 y Vol. 40, No. 6 y APPLIED OPTICS 919
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which is just the loss expected from the mode if theinternal resonator losses are zero. In Eq. ~28!, Ri 5uriu
2. It is expected, however, that as the rectangularguide length departs from this ideal length the inter-nal mode losses will increase. Figure 2 shows howthe mode losses vary as the rectangular guide lengthd is varied about the zero loss ~R1 5 R2 5 1! lengthdN.
We note that the analytic theory developed so far inthis paper delivers an identical result to that ob-tained previously for N 5 8 by numerical methods.8The computer processing time with the analytic the-ory is at least 100 times faster for this value of N,12
and it becomes increasingly more efficient as N isincreased. This is so because it becomes necessaryto involve increasingly more higher-order modes ofthe rectangular guide as N increases. Analysis ofthe overall widths of the plots of Fig. 2 provides thescaling relation
ddplot
dN5
2N
. (29)
We have defined ddplot as the range over which eachof the plots of Fig. 2 has been generated. In eachpart of Fig. 2 this overall plot width is represented bythe whole length of the abscissa.
At any particular loss point on the main character-istic the width of the loss characteristic is a constantfraction f of the overall plot width, so
dddN
5 f2N
or
dd 5 f2r2
l. (30)
It may be considered that a design aim for theresonator be a mode loss less than 10%. At thisloss point, f > 3.5 3 1022. We conclude that theabsolute tolerable error, dd, in the rectangularuide length is independent of N but that the frac-ional error ddydN does depend on N through the
dependence of dN on N. For less than 10% modeloss the rectangular guide length must be controlledto dd , 0.07r2yl.
5. Conclusions
In this paper we have shown that an analytic resultfor the round-trip transmission of the modes of a1-N-way resonator can be obtained in terms of theself-coupling coefficients Bjj and the imposed phaseshift in the array element. The result shows thatthere is only one physical resonator mode whoseround-trip losses are zero when the rectangularguide has the correct design length ~LyN!. The
nalytic solution obtained agrees exactly with pre-ious numerical results and provides for muchaster computer run times. This becomes increas-ngly significant as larger values of N are employed
20 APPLIED OPTICS y Vol. 40, No. 6 y 20 February 2001
ecause one requires that increasingly larger num-ers of modes in the rectangular guide be taken intoccount. The provision of an exact value of zero forll but one of the N possible mode eigenvalues per-its the unambiguous interpretation of the results.mbiguities may arise when numerical routines as-ign small but nonzero values to these eigenval-es.12
The theory has also permitted identification of thearray element field amplitudes in terms of any oneelement amplitude. These field amplitudes are re-lated only by their self-coupling and cross-couplingterms. We have used the theory to examine how themode losses depend on the rectangular guide lengthfor a range of values of N. For reasonable resonatorosses we found that the absolute errors in rectangu-ar guide length that can be tolerated scale as r2yl.his value is independent of array number. We notehat the theory appears to be compatible with per-urbations in array elements.
The authors acknowledge several useful discus-ions with their colleagues A. J. Davies of Royal Hol-oway, University of London, and J. Banerji of thehysical Research Laboratory, India. J. Banerjilso kindly provided results from some numerical cal-ulations.
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