modes of resonators with mirror reflectivity modulated by absorbing masks

Modes of resonators with mirror reflectivity modulatedby absorbing masks

Sandro De Silvestri, Vittorio Magni, and Orazio Svelto

The modes of a stable laser resonator containing, near one mirror, an absorbing mask with two apertures havebeen calculated on the basis of scalar diffraction theory and experimentally observed in a pulsed Nd:YAG

laser. The mode structure has been investigated as a function of the mask geometry, and an interpretation in

terms of supermodes is provided.

1. Introduction

Making resonators that can support large volumefundamental modes for generation of high power dif-fraction-limited beams is a central problem in laserdesign. As a possible solution, phase locked arrays oflasers, in which the beams of different emitters arecoherently combined, have been proposed during thelast few years. Research efforts have been devoted toapplying this idea to semiconductor lasers1 and a fewpapers concerning CO2 lasers have also been pub-lished.2 -5 In recent work68 the authors suggested thesegmented-mirror resonator as a way to phase lockindividual emissions from different parts of an activemedium of wide cross section. This novel scheme isshown in Fig. 1, and essentially consists of a stableresonator containing, near one of the mirrors, an ab-sorbing mask with a few apertures. The size of eachaperture is close to that used for selecting the TEMoomode in a standard resonator. It has been provedexperimentally with a Nd:YAG laser that, if the aper-tures are the proper size and are placed at correctdistances, the emission consists of different phaselocked lobes. The modes of this structure have simplybeen interpreted, following the supermode theory9 de-veloped for semiconductor laser arrays, as a combina-tion of different TEMOO-like modes, each oscillating ina separate channel, that exchange energy (and phaselock) through the diffraction. It should be noted thata uniform mirror covered with an absorbing mask isequivalent, as far as the field inside the resonator isconcerned, to a mirror whose reflectivity is spatially

The authors are with CNR Center of Quantum Electronics &Electronic Instrumentation, Polytechnic Institute of Physics, 32Piazza Leonardo da Vinci, 20133 Milan, Italy.

Received 22 February 1989.0003-6935/89/173684-07$02.00/0.© 1989 Optical Society of America.

modulated according to the mask profile. For thisreason resonators containing such mirrors have beencalled segmented-mirror resonators.

In this paper we present theoretical and experimen-tal results of a detailed analysis of the segmented-mirror resonator shown in Fig. 1. The modes of theresonators have been numerically calculated withinthe paraxial approximation on the basis of scalar dif-fraction theory and have been observed in experimentswith a pulsed Nd:YAG laser. Comparison of the theo-retical findings and the experiments have shown excel-lent agreement. These results are expected to provideimproved designs and a better understanding of dif-fraction-coupled laser arrays.

II. Mode Calculation

The structure we have analyzed, sketched in Fig. 1,consists of two spherical mirrors (M1 and M2) of radiiR1 and R2 placed at a distance L and of a suitable maskinserted close to the output mirror M1. We have con-sidered masks with rectangular symmetry so that inCartesian coordinates the 2-D integral equation de-scribing the modes can be separated and reduced to 1-D equations that are the same as those for infinite stripmirror resonators. Therefore, in this section we studyonly strip resonators. The mask is characterized bythe amplitude transmission function r(x), which, ingeneral, is a complex function. According to the scalardiffraction theory and paraxial approximation, thefield amplitudes of the modes at the reference planejust behind the mask (marked by RP in Fig. 1) are thesolutions of the Fresnel integral equation:10 11

au(x) = T2(X1)K(x,x 1)u(X1)dX1, (1)

where u(x) is the field amplitude of the wave incidenton the mask, a is the eigenvalue, and

K(x,x 1 ) = V77XiB exp[-j(7r/XB)(Ax2 - 2xx1 + Dx2

)]

3684 APPLIED OPTICS / Vol. 28, No. 17 / 1 September 1989

MASK-fn

I N

: k Z

L I .wM,

CD)wjCDLn0-J

1.0

0.8

0.6

0.4

0.2

Fig. 1. Resonator containing an absorbing mask with two apertures(top). Sketch of the absorbing mask (bottom).

00 1 2 3

is the kernel for the round trip in the cavity withoutany mask. In this expression X is the laser wavelengthand the coefficients A, B, and D are the elements of theray transfer matrix pertaining to the reflection frommirror M1 followed by the propagation for a distance L,reflection from mirror M2, and another propagation fora distance L to the reference plane:

A = 1- 4L/R 1 - 2L/R 2 + 4L2/R1R2,

B = 2L(1 -LIR 2 )

D = 1- 2L/R2.

Note that the geometrical phase shift exp(-j 47rL/X)due to a round-trip propagation does not appear ex-plicitly in Eq. (1).11

For a resonator containing a two-aperture absorbingmask the transmission r(x) is zero everywhere exceptin correspondence to the two transparent regionswhere r = 1. For our numerical simulation we haveassumed R1 = R2 = 8 m, L = 0.5 m, a 100% mirrorreflectance, and a wavelength X = 1.064 ,um (Nd:YAGlaser). The mask geometry is sketched in Fig. 1 andconsists of two rectangular apertures of size a by c in anabsorbing screen separated by a strip of width b. Thedimension c in the transverse direction y is of no con-cern in the mode calculation and is considered in theexperimental section (III) of the paper. The modeprofile and the losses were calculated for many differ-ent masks, in particular we have analyzed the behaviorof the losses as a function of the aperture width and oftheir separation.

Equation (1), which is not amenable to a closed formsolution, was solved numerically by the Prony methodwith search, which simultaneously provides a few ofthe first eigenvalues and eigenfunctions.121 3

Figure 2 shows the round-trip losses r = 1 - Ia12 ofthe strip resonator for the first lowest order even-parity and odd-parity modes as a function of the aper-ture separation b for a given aperture width a = 1.1mm. A single aperture of this size placed on the reso-nator axis allows the selection of the TEMo mode.

APERTURE DISTANCE, b (mm)

Fig. 2. Round-trip losses of the first even-parity and odd-paritymodes as a function of the aperture distance b for a mask withaperture width a = 1.1 mm. The numbers near the curves refer to

subsequent figures that show the mode profiles.

(a)

0.8

0.6

0.4

-2 -1 0 1 2

(c)

(b)

I

-2 -1 0

(rad)1 2

(d)

Fig. 3. Mode intensity profiles generated by the mask with aper-tures of width a = 1.1 mm separated by b = 0.4 mm (point 3 in Fig. 2).(a) Computed near field, (b) computed far field; (c) and (d) corre-sponding experimental profiles. The shaded area in (a) representsthe part of the beam intercepted by the absorbing mask. The

horizontal scale in (c) is 0.44 mm/div, and 0.54 mrad/div in (d).

1 September 1989 / Vol. 28, No. 17 / APPLIED OPTICS 3685

IIZ

ZI

The two curves of Fig. 2 oscillate and intersect periodi-cally in a way that closely resembles the behavior of thelosses of unstable resonators. Thus the odd- andeven-parity modes alternatively assume the role of thefundamental (minimum losses) mode of the cavity.Figures 3(a), 4(a), 5(a), and 6(a) present the modeprofiles corresponding to the points marked by 3-6 inFig. 2, which correspond to maximum discriminationbetween the fundamental and the first higher-ordermodes. Figure 3(a) depicts the mode profile obtainedfor small separation (b = 0.4 mm) between the aper-tures, which, as expected, is similar to a Hermite-Gaussian mode of order 1 (TEM1). By increasing theaperture distance, the losses of the even mode decreaseand reach a minimum; the mode profile correspondingto this minimum (point 4 in Fig. 2) is shown in Fig. 4(a).It can be seen that the mode consists of three clearlydistinguishable lobes and that the mask absorbs main-ly the central lobe. Comparing the mode profiles ofFigs. 3(a)-6(a) to the plot in Fig. 2, we can extrapolatethat, as a general feature, the number of lobes consti-tuting the fundamental mode, for an aperture of 1.1mm, increases by one at each intersection of the losscurve, whereas the wave transmitted through the maskis always made by two lobes whose shape only slightlychanges as the aperture distance changes.

In the authors' opinion, also mentioned in a previouspaper, 6 the modes of this resonator can be interpretedfrom a similar viewpoint to that used to develop thetheory of supermodes of phase-locked laser array.

(a)

0

. (mm)

(c)

0

0U,

U,

0

c

(b)

a I6

1,_

-2 -1 0 1 2

e (mrad)

(d)

Fig. 4. Same as Fig. 3 with a = 1.1 mm and b = 0.88 mm.

(a)

0x (mm)

(C)

_=____

|__

:__

_Li£__


(a)

-2 -1 0x (mm)

(c)



(b)

0

e (mrad)

(d)

(b)

0

e (mrad)

(d)

z

I

z.11

Iz

z

II

z

IIz

zII

2

081

I-

z

z

06

04

0.2

-2 -1 0x (mm)

(a)

'6ILiJ

1 2

Fig. 7. Mode intensity profile (wave incident on the mask) andphase in the resonator containing one off-axis aperture. The maskis the same pertaining to Fig. 4 with one of the apertures blocked.The reference surface for the phase is the surface of the mirror nearthe mask. The shaded area represents that part of the beam inter-cepted by the absorbing mask; the corresponding losses are 56.9%.

zU

II

0x (mm)

0

>-0z

zU,

0

(b(0 -

8

6

-2 -1 0

x(mm)

1 2

Fig. 8. Supermodes in the resonator containing a mask with a = 1.1mm and b = 0.88 mm (same as that of Fig. 4). The intensity profilesare calculated by superimposition of the off-axis mode of Fig. 8 onthe mirror image respect to x = 0. The even supermode (a) and theodd supermode (b) are obtained, respectively, by subtraction and

addition.

35

30

The resonator with the two-aperture mask is similar toa two-channel diffraction-coupled laser array in whichthe overall mode can be approximated as a linear com-bination of the fundamental modes pertaining to eachchannel that are assumed to interact via the radiationdiffracted from the channel aperture into the otherchannel. The mode profile of a resonator with an off-axis diaphragm, which is substantially a misalignedstable resonator, is smooth and similar to a Gaussiancurve, not centered on the aperture but displaced to-ward the axis, so that a large amount of the energy ofthe mode impinges on the mask near the resonator axisand is absorbed. An example of this off-axis mode,obtained with the mask corresponding to the case ofFig. 4 after blocking one of the apertures, is shown inFig. 7. When two of these off-axis modes are allowedto oscillate simultaneously by the two-aperture mask,they interfere to produce the fringe patterns displayedin Figs. 8(a) and (b) according to whether the two off-axis modes are summed or subtracted. A comparisonof Figs. 8(a) and (b) and the calculation of the energyabsorbed by the mask reveals that, for the given aper-tures' distance, the even-parity supermode is the low-est loss mode (losses of 40.5% against 50.8%). Also, theintensity profile of this supermode closely resemblesthat shown in Fig. 4(a) obtained by solving Eq. (1).More generally, the number of interference fringes,their position, the losses of the resulting supermode,and the discrimination between modes depend on thedistance between the two off-axis modes and thereforeon the distance between the two apertures.

This picture is similar to that proposed for the dif-fraction-coupled phase-locked semiconductor laser ar-ray.9 14"15 It should be noted, however, that in our casethe off-axis modes cannot be simply described asGaussian beams with spherical wavefronts, and it isnot easy to derive a direct formula to predict the dis-

25

j-E

20

15

10

5

00 2 4 6 8 10 12

n

Fig. 9. Square of the center-to-center spacing between the twoapertures (d) corresponding to the minima of the loss curve in Fig. 2as a function of the integer n that numbers the sequence of theminima (dots). The solid line is an interpolating straight line. Thevalue of d' is divided by the wavelength X and the resonator length

L.

tance between the interfering off-axis modes like theone reported for the semiconductor array.14"15 We cannevertheless try to extend empirically the formula re-ported in Ref. 14 to our case by writing the followingequation which must be satisfied for maximum modediscrimination:

2

=2n, (2)

where dm = b + a is the center-to-center spacing of thetwo apertures corresponding to the minima of losses(see Fig. 2), n is an integer number (even or odd for adominant even or odd mode, respectively), and is asuitable constant dependent on the resonator geome-try, whose value lies for diode laser arrays between 1and 2. This simple picture has been compared withthe exact-mode calculation by plotting, as shown inFig. 9, the value of d2 /XL, as a function of the integer


l.

(a)

05

on

cIn00-J

0.4

- even modes

---- odd modes 1110.2 | I I , 1.

0.8 1.1 1.4 1.7APERTURE WIDTH, a (mm)

Fig. 10. Round-trip losses of different modes as a function of theaperture width a for aperture separation b = 0.7 mm. The number

near a curve refers to Fig. 11 which shows the mode profile.

that numbers the sequence of minima in Fig. 2. It canbe seen that the points are well aligned on a straightline described by

m = 3.0 n,

0x (mm)

(c)

UU,

Z

-2 -1 I 0 1 2

e (mrad)

(d)

(3)

where an even or odd value of n corresponds to the evenor odd dominant mode, respectively. Comparing Eq.(3) with Eq. (2) a value = 0.75 is obtained. Thisresult confirms the validity of the intuitive picture of adiffraction coupled array. It is, however, clear that theexact mode shape and power loss can only be obtainedby solving Eq. (1).

As an alternative picture we can consider a perturba-tive approach by hypothesizing that the modes are thesum of a few Hermite-Gaussian modes of the stableresonator without diaphragm. As an example, theprofile shown in Fig. 3, obtained when the aperturesare very close, can be regarded as a slightly distortedTEM1 mode selected by the mask that suppresses theTEMo. Since the Hermite-Gaussian set is complete,any field profile can obviously be expanded on thatbasis; this picture, however, is convenient and the su-perimposition of a few Hermite-Gaussian modes givessensible results only if the perturbation introduced bythe mask is small. To describe the results for largeaperture separation, as for Figs. 4-6, a large number ofHermite-Gaussian modes must be involved, andtherefore it becomes difficult to interpret the action ofthe mask simply as a selector of the Hermite-Gaussianmodes.

When the size of each aperture is different from thevalue (a = 1.1 mm) previously considered, the modeprofiles become more complex and the results lessintuitive. Figure 10 illustrates the behavior of thelosses as a function of the aperture dimension a withthe central absorbing region of the mask kept at aconstant size b = 0.7 mm, which approximately corre-sponds to the first minimum of the even mode of Fig. 2.Figure 10 shows that in this case the dominant mode isalways of even parity and that mode crossing betweeneven parity modes is now present. For small values of

Fig. 11. Mode intensity profiles generated by the mask with aper-tures of width a = 1.65 mm separated by b = 0.7 mm (point 11 in Fig.10). (a) Computed near field, (b) computed far field; (c) and (d)corresponding experimental profiles. The shaded area in (a) repre-sents that part of the beam intercepted by the absorbing mask. The


a (<1.34 mm) the dominant mode is essentially similarto that depicted in Fig. 4. As a becomes larger thelosses of this mode reach an almost constant value, butanother mode, made of five distinguishable lobes in-stead of three, becomes the fundamental. At eachintersection of the loss curves the number of lowest lossmode lobes increases by two: the central lobe is alwaysabsorbed by the mask and the others are transmittedthrough the apertures. As an example, Fig. 11 showsthe mode profile corresponding to the point marked by11 in Fig. 10. It can be seen that the relatively lowlosses are due to the fact that, after each round trip, thefringe pattern is such as to concentrate most of theenergy on the apertures and only a small amount on theabsorbing mask.

To complete the analysis we also investigated thebehavior of the mode profiles and of the losses as afunction of the width b of the absorbing area betweenthe two apertures while keeping constant the overalldimension 2a + b = 2.8 mm. The loss curves for thefirst even and odd modes are presented in Fig. 12, andan example of the mode profile, corresponding to thepoint marked by 13, is shown in Fig. 13. As in the casespreviously analyzed, the loss curves intersect and ateach crossing point the number of principal lobes of


(b)06

I-I

1.0(b)

0.8

Ln

vf)0-J

0.6 -

0.4

0.2

010

AI %I

I I

II1I1

_

I

- even modeodd mode

I I

0.4 0.8 1.2 1.6

APERTURE DISTANCE, b (mm)

Fig. 12. Round-trip losses of the first even-parity and odd-paritymodes as a function of the aperture distance b for a mask with theoverall dimension 2a + b = 2.8 mm. The number near one of the

curves refers to Fig. 13 which shows the mode profile.

the fundamental mode increases by one; the output is,however, always constituted by two lobes.

Ill. Experimental Results

The results of the numerical analysis were experi-mentally checked with a flashlamp pumped Nd:YAGlaser. The laser rod was 76-mm long by 6.35-mm indiam; the resonator was 0.5-m long and made up of two8-m radius concave mirrors in accordance with thevalues used for the simulation; the mask was placedalmost in contact with the 60% reflectance output mir-ror. Note that a reflectance value less than one doesnot alter the mode profile but only affects the eigenval-ue by a constant factor. The different masks weremade by chemical etching on a 100-Am thick temperedsteel plate with a precision of h15 um. The transversedimension c [see Fig. 1(b)] of the rectangular apertureswas set equal to 1.3 mm to effectively select the TEMomode in the y transverse direction. The optimumvalue of c was established experimentally by findingthe size of the square diaphragm that, inserted on axisnear the output mirror, provides an output beam ofminimum divergence. The mode pattern was record-ed using a linear array (reticon) of 512 photodetectorsspaced at 25-,gm intervals, which was placed along thecentral line of the mask. The near field was obtainedby 1:1 imaging the output of the laser on the detectorby a 242-mm lens, and the far field by placing thedetector in the focal plane of an 816-mm lens.

To adequately compare the experiments with the-ory, the far-field pattern has to be examined as well asthe near-field profile, because it provides details of thephase of the emitted lobes. A central peak in the farfield, as known, denotes an even-parity mode (twolobes locked in phase), a zero at the center reveals anodd mode (two lobes of opposite phase). A few experi-

Fig. 13. Mode intensity profiles generated by the mask with aper-tures of width a = 0.77 mm separated by b = 1.26 mm (point 13 in Fig.12). (a) Computed near field, (b) computed far field; (c) and (d)corresponding experimental profiles. The shaded area in (a) repre-sents the part of the beam intercepted by the absorbing mask. The


mental near- and far-field patterns are shown as (c)and (d) sections, respectively, of Figs. 3-6, 11, and 13and compared with the calculated theoretical shapes,plotted in sections (a) and (b). For any mask tested,we have found excellent agreement between theoryand experiment. A small discrepancy is however ap-parent, especially in the far-field pattern: the pointsof zero intensity predicted by theory are always abovethe base line in the experimental picture. The mainreason lies in a limited spatial resolution of the detec-tion system due to: (i) small lens aberration, (ii) oper-ation of the diode array near the edge of silicon absorp-tion spectrum.

The performance in terms of output energy is limit-ed by the high power losses. As an example, the out-put energy at 19 J input for the mask with a = 1.1 mmand b = 0.88 mm, which produces the even mode of Fig.4, was 24 mJ. When one of the two apertures wasclosed, the threshold slightly increased from 4 to 5 J,and the output energy was reduced to 10 mJ, as aconsequence of the halved mode volume. These datacan be compared to those of the TEMoo mode (energy22 mJ, threshold 4J) generated in the same resonatorby substituting for the two-aperture mask a singlerectangular diaphragm of identical width (a = 1.1 mm)


0

x (mm)

(c)

0

e (mrad)

(d)

- - l

(a)

IIV)Z

ZI

I-

I

0

13

I

placed on the resonator axis.The limited amount of energy obtained with the

absorbing masks considered in this paper invites studyof more appropriate spatial modulation (both in am-plitude and phase) of the reflectivity, leading to areduction of the resonator internal losses.

IV. Conclusions

Two-aperture segmented-mirror resonators havebeen extensively investigated. The numerical simula-tions have shown some peculiar properties of the novelstructure, such as mode degeneracy and crossing of theloss curves, and the experiments have confirmed thetheoretical results. Some physical insight can begained describing the modes as the coherent combina-tion (supermode) of fundamental modes of a mis-aligned resonator with a single aperture. These re-sults should be useful, we believe, for betterunderstanding and improved designs of diffraction-coupled semiconductor or CO2 laser arrays.

This paper has demonstrated the possibility of in-creasing the mode volume in a resonator by insertingan appropriate mask. The high internal losses ob-tained by using an absorbing mask suggests an investi-gation of novel schemes for reflectivity modulation.

References

1. G. Harnagel, D. Welch, P. Cross, and D. Scifres, "High PowerLaser Arrays: A Progress Report," Laser & Appl., 5, 135-138(June 1986); sections on semiconductor laser arrays in Confer-ence on Laser and Electro-Optics Technical Digest Series 1987,Vol. 14 (Optical Society of America, Washington, DC, 1987).

2. D. G. Youmans, "Phase Locking of Adjacent Channel LeakyWaveguide CO2 Lasers," Appl. Phys. Lett. 44, 365-367 (1984).

3. L. A. Newman, R. A. Hart, J. T. Kennedy, A. J. Cantor, and A. J.De Maria, "High Power Coupled CO2 Waveguide Laser Array,"Appl. Phys. Lett. 48, 1701-1703 (1986).

4. R. A. Hart, L. A. Newman, and J. T. Kennedy, "StaggeredHollow-Bore CO2 Waveguide Laser Array," in Conference onLaser and Electro-Optics Technical Digest Series 1987, Vol. 14(Optical Society of America, Washington, DC, 1987), pp. 354-356.

5. D. Cantin, M. Piche, and G. A. Heckman, "Phase-Locking ofCO2 Laser Array Using Diffraction Coupling," in Conference onLaser and Electro-Optics Technical Digest Series 1988, Vol. 7(Optical Society of America, Washington, DC, 1988), pp. 82, 83.

6. S. De Silvestri, P. Laporta, V. Magni, and 0. Svelto, "Segment-ed-Mirror Phased-Array Lasers," Appl. Phys. Lett., 51, 1771-1773 (1987).

7. S. De Silvestri, P. Laporta, V. Magni, and 0. Svelto, "Modes ofSegmented-Mirror Laser Resonators," in Conference on Laserand Electro-Optics Technical Digest Series 1988, Vol. 7 (Opti-cal Society of America, Washington, DC, 1988), pp. 16-17.

8. S. De Silvestri, P. Laporta, V. Magni, and 0. Svelto, "Modes ofResonators with Modulated Reflectivity Mirrors," in SixteenthInternational Conference on Quantum Electronics TechnicalDigest (Japanese Society of Applied Physics, 1988), pp. 98-99.

9. E. Kapon, J. Katz, and A. Yariv, "Supermode Analysis of Phase-Locked Arrays of Semiconductor Lasers," Opt. Lett. 9, 125-127(1984).

10. P. Baues, "Hyugens' Principle in Inhomogeneous, Isotropic Me-dia and a General Integral Equation Applicable to Optical Re-sonators," Opto-electronics 1, 37-44 (1969).

11. A. E. Siegman, Lasers (Oxford U.P., London, 1986), pp. 7 9 2 - 7 9 3 .12. A. E. Siegman and H. Y. Miller, "Unstable Optical Resonator

Loss Calculations Using the Prony Method," Appl. Opt. 9,2729-2736 (1970).

13. W. D. Murphy and M. L. Bernabe, "Numerical Procedures forSolving Nonsymmetric Eigenvalue Problems Associated withOptical Resonators," Appl. Opt. 17, 2358-2365 (1978).

14. J. Katz, S. Margalit, and A. Yariv, "Diffraction Coupled Phase-Locked Semiconductor Laser Arrays," Appl. Phys. Lett. 42,554-556 (1983).

15. J. Katz, J. Cser, and W. K. Marshall, "Diffraction-Coupled RealIndex-Guided Semiconductor Laser Arrays," in Conference onLaser and Electro-Optics Technical Digest Series 1986, Vol. 7(Optical Society of America, Washington, DC, 1986), pp. 78-79.

The authors wish to express their gratitude to PaoloSironi for his valuable contribution both to the theo-retical and experimental work.


modes of resonators with mirror reflectivity modulated by absorbing masks

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