single longitudinal mode oscillation from a multi-interferometric cavity

Single longitudinal mode oscillation from amulti-interferometric cavity

Yuanhu Wang,* Yanchen Qu, Weijiang Zhao, Chenghong Jiao, Zhiqiang Liang,and Deming Ren

National Key Laboratory of Tunable Laser Technology, Harbin Institute of Technology, Harbin 150001, China

*Corresponding author: [email protected]

Received 10 December 2008; revised 24 January 2009; accepted 11 February 2009;posted 11 February 2009 (Doc. ID 105174); published 3 March 2009

A method to generate stable single longitudinal mode (SLM) radiation from a multi-interferometric cav-ity configuration that can be considered as the combination of one Michelson cavity and two Fox–Smithcavities is presented. A numerical model of the interferometric cavity is investigated to optimize the laserfor mode selection, and experimental verification has been carried out in a tunable TEA CO2 laser. Pulseoutput energy of 300mJ at 10:6 μm has been obtained at repetition rate of 20Hz, corresponding to arepetition of SLM operation of 100%. This result shows that this interferometric cavity gives betterperformance in mode selection than other cavities based on multibeam interference. © 2009 OpticalSociety of America

OCIS codes: 140.3410, 140.3470, 140.3538, 140.3570.

1. Introduction

A tunable single longitudinal mode (SLM) TEA CO2laser is of special interest in areas such as opticallypumped far-infrared lasers, molecular spectroscopy,velocimeters, and remote sensing of the atmosphere[1–3]. Typically, the gain bandwidth of a 1 atm pres-sure CO2 laser is broadened to about 4GHz, so theoutput laser usually contains more than one longitu-dinal mode. There are mainly two ways to obtainSLM operation of TEA CO2 lasers without sacrificingoutput power [4]: either by increasing the gain of oneresonator mode or by decreasing the gains of allbut one mode. The first type includes hybrid andinjection-locking lasers in which a low-pressure cwCO2 laser was used to control the TEA laser output[5,6]. Such laser cavities have been used to generatereliable SLM pulses for matched cavity lengths, butthe tuning range is limited by the cw gain [7,8]. Thesecond technique involves using mode selection com-ponents such as an etalon and other multibeam in-terferometers [9–11]. The reflectivities of these

components or a combination of optical componentsare periodically dependent on frequency. Oscillationon unwanted frequencies can thus be suppressed,making more energy available for the desired mode.The capability of the interferometer to select a singlemode depends on the width of its resonant modes andon their separation (free spectral range). A conven-tional interferometer such as a Fox–Smith interfe-rometer is found to be quite effective for modeselection because of its high selectivity [12]. However,the Fox–Smith interferometer needs a highly reflect-ing surface to couple the light into the interferom-eter, and the low damage threshold of most coatingmaterials limits its application in a high-power laser.

In this paper we present a multi-interferometriccavity (MIC) to obtain SLM pulses of a TEA CO2 la-ser. An additional mirror was added to the open sideof the conventional Fox–Smith cavity, and the MICcould thus be considered as the combination of sev-eral multibeam interferometers. The characteristicsof low threshold and high frequency selectivity in-duced by subresonators make SLM operation of alaser with MIC more feasible. A numerical modelwas developed to describe the behavior of a MICsuch as frequency-dependent reflectivity and mode

0003-6935/09/081430-06$15.00/0© 2009 Optical Society of America

1430 APPLIED OPTICS / Vol. 48, No. 8 / 10 March 2009

discrimination ability. Experimental verification wasalso presented to demonstrate theoretical predictionand the capability of producing single-frequency out-put using the MIC configuration.

2. Theory

The frequency-selective properties of a MIC shown inFig. 1 can be analyzed by deriving the reflected elec-tric field from the interferometer. For this derivationthe wavefront inside the interferometer is approxi-mated by a plane wave since the region where thederivation takes place is close to the beam waist. As-sume the four mirrors, Mj ðj ¼ 1; 2; 3; 4Þ, have an am-plitude reflection coefficient rj and the length of thebeam splitter from each mirror is Lj. Let the incidentfield be given by E0 and the cavity losses be assumednegligible. The electric field starting and finishing atthe beam splitter will experience a change in ampli-tude and phase given by rjδj, with δj ¼ expði2kLjÞ andk ¼ 2πv=c, where v is the frequency of the wave and cis the velocity of light.

The MIC could be regarded as a combination of oneMichelson cavity (formed by M1, M2, M4), and twoFox–Smith cavities (one composed of M1, M3, M4and the other of M2, M3, and M4). The returning elec-tric field from the Michelson interferometer is

EM ¼ E0ðRBSr2r4δ2δ4 þ TBSr1r4δ1δ4Þ: ð1Þ

Here RBS and TBS represent the reflectivity andtransmissivity of the beam splitter, respectively.The normalized intensity reflection coefficient ofthe grating consisting of N similar equidistant par-allel lines is given by [13]

r1 ¼ sinðkapb=2Þ sinðNkdp=2Þðkapb=2Þ sinðkdp=2Þ

ð2Þ

with Pb ¼ sinðθi − θbÞ − sinðθr − θbÞ and p ¼ sin θi−sin θr, where θi is the angle of incidence of the

grating, θr is the angle of reflection, and θbis the blaz-ing angle. a is the half-width of a groove on the grat-ing, and d is the distance between two adjacentgrooves. In order to improve the ability of frequencyselectivity of this cavity, a low-finesse etalon withfree spectral range of about 4GHz was inserted inarm 2 (the arm that contains Mi is called arm i).To calculate the returning electric field from thetwo Fox–Smith interferometers, we use the self-consistent wave method, in which the contributionsfrom all the possible optical paths are considered bymeans of a stochastic approach [14,15]. It can be con-sidered that a wave making a total of n trips in thetwo Fox–Smith interferometers will make m trips inone interferometer and (n −m) trips in the other.Then the number of possible ways that the wavemakes a total number of n trips in the two Fox–Smithinterferometers withm trips in one Fox–Smith inter-ferometer is n!=m!ðn −mÞ!. The total electric field inthe two Fox–Smith interferometers is given by

En ¼ E0

X∞n¼0

Xnm¼0

n!m!ðn −mÞ! ðRBSr1r3δ1δ3ÞmðTBSr2r3δ2δ3Þn�m: ð3Þ

This is a geometric series that can be simplified to

En ¼ E0

1 − r3δ3ðRBSr1δ1 þ TBSr2δ2Þ: ð4Þ

The total returning electric field from the two Fox–Smith interferometers EF, after considering factorscoupling the laser into and out of the two Fox–Smithinterferometers, is therefore given by

EF ¼EnðRBSr1r3δ1δ3 þ TBSr2r3δ2δ3Þ× ðTBSr1r4δ1δ4 þ RBSr2r4δ2δ4Þ: ð5Þ

The resultant reflected field finishing at the beamsplitter, ER, is the summation of the returning elec-tric field from the Michelson interferometer and thetwo Fox–Smith interferometers:

ER ¼ EM þ EF ¼ E0r4δ4½TBSr1δ1 þ RBSr2δ2 − r1r2r3δ1δ2δ3ðRBS þ TBSÞ2�1 − r3δ3ðRBSr1δ1 þ TBSr2δ2Þ

: ð6Þ

10 March 2009 / Vol. 48, No. 8 / APPLIED OPTICS 1431

Now the multi-interferometric system can be consid-ered as a mirror with reflecting coefficient r ¼ ER=E0. With the loss at the beam splitter assumed neg-ligible (RBS þ TBS ¼ 1), the intensity reflectivity R ¼rr� can be determined by

R ¼�R1R2R3 − 2RBSR2

ffiffiffiffiffiffiffiffiffiffiffiffiR1R3

pcosðδ1 þ δ3Þ

− 2TBSR1


pcosðδ2 þ δ3Þ

þ�RBS

ffiffiffiffiffiffiR2

pþ TBS

ffiffiffiffiffiffiR1

p �2

− 4RBSTBS


psin2 δ1 − δ2

2

�

×�1� 2RBS


pcosðδ1 þ δ3Þ

− 2TBS


pcosðδ2 þ δ3Þ

þ�RBS


pþ TBS


p �2

− 4RBSTBSR3


psin2 δ1 − δ2

2

�−1; ð7Þ

where Rj ¼ rj2 represents the reflectivity of Mj. Thisallows us to calculate the overall intracavity spectraldistribution SðνÞ in the absence of gain at the inter-section of the cavity optical axis and the beamsplitter. The electric field amplitude Ez of the zth cir-culating beam after z − 1 round trips is given by

Ez ¼ E0rz−1rz−14 exp½4πcL4iðz − 1Þ=ν�: ð8Þ

Hence the overall intracavity spectral field distribu-tion can be derived as

SðνÞ ¼E0

�1 − ½rr4 expð4πcL4i=νÞ�z−1

�

1� rr4 expð4πcL4i=νÞ: ð9Þ

The overall intracavity spectral distribution IðνÞ atthe point P can be calculated as the module of SðνÞ:

IðνÞ ¼ SðνÞSðνÞ � : ð10Þ

The spectral dependence of the power circulating in-side the laser cavity, as indicated by IðνÞ, is shown in

Fig. 2 for the experimental MIC parameters given byTable 1. The effects of the grating have been takeninto account. The SLM operation is ensured by thelarge spacing in intensity between the preferredmode and its side modes.

To obtain good overlap between the interferencewavefronts, the transverse modes of each laser cavityshould be matched. Assume the curvatures of M2,M3, and M4 are ρ2, ρ3, and ρ4, respectively. Considerthe grating as a plane mirror. One gets the followingexpressions for beam waists ω01, ω02, and ω03 of themain cavity (formed by M1 and M4), subcavity 1(formed by M2 and M4), and subcavity 2 (formedby M1 and M3), respectively:

πω201

λ ¼ ½ðL1 þ L4Þðρ4 − L1 − L4Þ�1=2; ð11Þ

Fig. 2. Calculated result of the resonator: (a) dashed and solidcurves correspond to the intensity reflectivity of the MIC and cir-culating intensity of the laser in a range of �5GHz, respectively.The dotted curve represents the bandwidth of the gain. (b) Thedashed curve corresponds to the circulating intensity of the laserwithout mode selector in a range of �500MHz; the solid curve, thecirculating intensity with the MIC; and the dotted curve, the in-tensity reflectivity of the MIC.

Fig. 1. Schematic diagram of the MIC. BS, beam splitter.


πω202

λ ¼�ðL2 þ L4Þðρ2 − L2 − L4Þðρ4 − L2 − L4Þðρ2 þ ρ4 − L2 − L4Þ

½2ðL1 þ L4Þ − ρ2 − ρ4�2�

1=2; ð12Þ

πω203

λ ¼ ½ðL1 þ L3Þðρ3 − L1 − L3Þ�1=2: ð13Þ

For our configuration, the beam waists are on thegrating for the main cavity and subcavity 2. There-fore, in subcavity 1, the distance from the beamwaistto M2 is

Lω ¼ ðL2 þ L4ÞðL2 þ L4 − ρ4Þ2ðL2 þ L4Þ − ρ2 − ρ4

: ð14Þ

In order to perfectly match the transverse mode,all beam waists should be equal and the beam waistin subcavity 1 should be in such a position that thegrating corresponds to the beam splitter, whichmeans

Lω ¼ L2 − L1: ð15Þ

From Eqs. (11)–(15), we get the simple equations formode matching as follows:

ρ2 ¼ L2 − L1 þðL1 þ L4Þðρ4 − L1 − L4Þ

L2 − L1; ð16Þ

ρ3 ¼ L1 þ L3 þðL1 þ L4Þðρ4 − L1 − L4Þ

L1 þ L3: ð17Þ

For our configurations ρ4 ¼ 5m, then M2 can be sub-stituted as a plane mirror and M3 can be with curva-ture of 20m.

3. Experiment

The tunable TEA CO2 laser used in experiment hadan active volume of 1:8 cm × 1:8 cm × 50 cm and wasoperated at a repetition rate of 20Hz with aCO2:N2:He gas mixture of 1∶1∶3 at atmosphericpressure. The parameters of the optical elementsin the experiment were in accordance with the dataused in calculation previously. A diffraction gratingof 150 grooves=mm, which was used to line tunethe laser, was placed at a Littrow angle, and an aper-ture of 10mm diameter was employed to yield thefundamental transverse mode of the laser. The polar-ization of the laser was determined by the laser-headBrewster windows, which paralleled the direction of

the grating grooves. The output energy of the laserwas measured with a Newport 818E-20-50L pyro-electric energy detector, and the wavelength wasmonitored with a spectrum analyzer. A HgCdTe de-tector with bandwidth of 100MHz was used tomonitor the laser pulse, and a Tektronix TDS7254Bdigital oscilloscope was used to record all the signals.

The distances of the three mirrors and the gratingfrom the beam splitter were fine tuned in order to getoverlapping of the resonant frequencies of each of thefour subresonators formed by mirrors M1 −M3,M1 −M4, M2 −M3, and M2 −M4 only at a desiredwavenumber within the active medium gain band-width. For comparison, the spectra for the corre-sponding Michelson interferometer, Fox–Smithinterferometer, and MIC were examined. The spec-tral mode structures of pulses from each cavity wereanalyzed with a Fourier transformation of the pulses.All the cavities operated at a wavelength of 10:6 μmand in approximately TEM00, as observed on heat-sensitive paper. When the laser cavity consisted ofonly the grating and an output mirror, M4, the outputpulses typically contained multilongitudinal modes,and the energy was approximately 380mJ when theexcitation voltage was set to be 25kV, as shown inFig. 3. The Fourier transformation clearly indicatedat least five longitudinal modes separated by about130MHz in frequency, corresponding to the longitu-dinal separation decided by the resonator length.

The number of modes was obviously reduced in thecase of the Michelson cavity, Fox–Smith cavity, andMIC resulting from the interferometric effect. How-ever, SLM oscillation from the Michelson cavity wasnot perfect especially in the case of high excitationvoltage as shown in Fig. 4 because of only two-beam

Table 1. Parameters for MIC Calculation

Parameters Symbol

Wavelength of the incident field, λ ðμmÞ 10.6Length of arm 1, L1 ðcmÞ 15Length of arm 2, L2 ðcmÞ 9Length of arm 3, L3 ðcmÞ 5Length of arm 4, L4 ðcmÞ 100Reflectance of the beam splitter 70%“First-order” reflectance of the grating (M1), R1 95%Reflectance of M2, R2 99%Reflectance of M3, R3 99%Reflectance of the output coupler (M4), R4 80%


interference. Furthermore, the reflectivity of thebeam splitter had to be chosen very carefully to pre-vent parasitic oscillations. By using the Fox–Smithcavity or the MIC, the mode beating effect in the de-tected pulses was clearly quenched, and SLM opera-tion could be obtained stably. At this time the pulseshape changed into a smooth trace, and its Fouriertransform consisted only of fundamental frequency,as shown in Fig. 5. But when the laser operates withthe Fox–Smith cavity, the excitation voltage shouldbe set to a low value to keep the beam splitter frombeing damaged. Hence the output energy was onlyabout 40mJ as the SLM oscillated steadily. The pulseenergy from the MIC was found to be greatly in-creased to approximately 300mJ with 100% SLM op-eration asM3 andM4 were fine tuned by piezoelectrictransducers, and there was not any damage to opticalcomponents.

4. Conclusion

We have reported a multi-interferometric configura-tion to achieve single longitudinal mode operation.Single-mode operation has been theoretically pre-dicted using a stochastic method and assuminginfinite summation of the circulating wave. Experi-mental verification was carried out using a TEACO2 laser. In the experiment, pulse output of300mJ has been obtained at repetition rate of 20Hz,corresponding to the repetition of SLM operation of100%, and no damage to any of the optical compo-nents was observed. Both theory and experiment de-monstrate that the laser achieves enough modediscrimination to obtain SLM oscillation. The MIChas superior overall performance to the traditionalMichelson cavity and Fox–Smith cavity in terms ofmode selectivity and increased output energy, andit has wide applications.

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Fig. 3. Typical pulse shape when the laser operated without themode selector.

Fig. 4. Pulse shape from the laser oscillation with a Michelsoncavity.

Fig. 5. Shape of a SLM laser pulse using the MIC.


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single longitudinal mode oscillation from a multi-interferometric cavity

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