Phase coupling of two CO2 lasers

Optics Communications 104 ( 1993) 75-80 North-Holland

Phase coupling of two

OPTICS COMMUNICATIONS

CO 2

lasers

G. L e s c r o a r t , R. M u l l e r a n d G. B o u r d e t Laboratoire d'Optique Appliqude - Ecole Nationale Sup#rieure de Techniques Avancdes - Ecole Polytechnique, Centre de l'Yvette, 91120 Palaiseau, France

Received 19 April 1993; revised manuscript received 8 July 1993

For the first time a theoretical analysis is presented of the phase locking of two lasers sharing one of the mirror cavities using a Rigrod-like approach. Experimental results are in good agreement with the theoretical results.

1. Introduction

Phase-locked laser arrays are of great interest when high brightness, high coherent light sources are required. Intensive researches are focused on the phase coupled arrays of semi-conductor lasers making it now possible to achieve high intensity, narrow bandwidth and quasi-diffraction limited beams. On the other hand, phase locking of high power lasers finds applications ranging from laser machining to CWLIDAR. Various schemes make possible the phase locking of array of lasers including antiguiding techniques [ 1,2 ], diffractive techniques [ 3-5 ] or phase conjugation [6]. Independent sources are more powerful and they can be phase locked by using injection techniques [ 7 ]. In this paper, we present theoretical investigations and experimental results concerning self-injection o f two C02 cw lasers which cavities share a partially transmitting mirror.

2. Theoretical model

The theory of two coupled lasers has been first studied by Spencer and Lamb [ 8 ]. Several authors have then worked out theoretical investigations under some approximations which restrict applications to weak fields or weak coupling or both [ 9,10 ]. We present h6re an exact solution using a Rigrod-like formalism [ 11 ]. The scheme o f the cavity is shown in fig. 1. It pre-

Cavity II A*

A; M2 r2,t

A-

n

A÷

gz' L2

I

Cavity I A-

2

A;

m

A÷

M r,t +

At g~,L M1 r1 ,t

1

Fig. 1. Schematic representation of the two coupled lasers. sents two twin lasers sharing a c o m m o n partially transmitting mirror. All parameters of the lasers are assumed to be identical. The cavity lengths (L1/2 and L 2 / 2 ) are supposed to differ only by a wavelength. The amplitude reflectivity and transmittivity o f the mirrors M, M1 and M 2 are r, t, rl, tl and r2, t2, respectively, the intensity reflectivities are noted R, R1, R E. Let A [ be amplitudes of the fields on both faces of the mirror M ( i = n , m) and on the mirrors M~ and M2 ( i = 1, 2). The intensities are noted I +, and read I + = IA?I 2 We first write the continuity conditions on the coupling mirror M:

A+~=¢ra~ +tA+~ ,

(1)

A E = --ERA + + t A m ,

(2)

E= _+ 1 depending on the direction of propagation of

0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

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the wave with respect to the reflecting side of the mirror. We define g~ and g2 as the real amplitude gains and 01, 02 the phases introduced by propagation along the two cavities I and II. We can then write Am =gl exp(i01 ) A m+

,

(3)

An+ =g2 exp(i02) A~-.

(4)

With eqs. ( 1 ) - ( 4 ) , we are able to define the complex reflectivity of the cavity II and of the cavity I, and the complex transmittivity of the cavity II in the cavity I and of the cavity I in cavity II: A+m Er+g2 exp(i¢2) rm-- Am - 1 +Erg2 exp(i02) '

(5)

(12)

2n vQ, - V~l - J 1 ln(gl ) +arg(rm) = (0, _+ 1 )2n. Av~

Using the Rigrod formula [ 11 ], we can write the output intensity of the laser I as (13)

with g°/+ In ( Rx/R~IRm) ( x / ~ l + x / r ~ ) (l _ ~ ,

(14)

(6) where l is the length of the amplifier medium, and go is the small signal gain which reads (7)

gO go- l+ja,

I~II A2

t Am - 1 +erg2 exp(i02) "

(8)

We assume now the cavity I to be above threshold. This assumption allows us to write that 01 = 2kn and a relation between g~ and rm: g2lrm 12= 1 .

(9)

The intensity reflectivity of the cavity II on the cavity I reads Rm = Irm 12- rZ+g2 + 2Erg2 c0s(02) -1 +r2g22 c0s(02)

(10)

The wavelength 21 which oscillates in this cavity is given by the relation [ 12 ] Ol=-2nL1/21-Jlln(gl)+arg(rm)=2kn,

(11)

where the first term is the geometric propagation, the second is the dispersion induced by the amplifier medium where J~ is the frequency detuning normalized to the homogeneous bandwidth: J l = (Vl-Vo)/Avn, and the last is the change of phase due to the reflection on the cavity II. This is equivalent to 76

--~1 ln(gl)+arg(rm)=2kn,

where V~l and A v~ are the frequency of the oscillating light and the free spectral range of the cavity I, respectively. Let us call vc~ the nearest resonant frequency of the empty cavity I from oscillating resonant frequency V~l. Then eq. (12) reads

I + =~m

A2 - e r + g l exp (i01) r n - A~+ - 1-erg~ exp(i01) '

t--

--2%P~I/AP1

i~ut = ( 1 - R 1 ) I + ,

II~I t+_A + _ t An+ 1 - ~ r g l exp(i01) '

15 December 1993

where gO is the small signal gain at line center (Vo). R m is a function of g2 and 02. Using the well-known relation connecting the intensities of the waves travelling back and forth in a homogeneous amplifier medium I~- I F = R I I ~ a = I m I+m =Rr~Im 2

(15)

we then write Im as a function of R m : gol+ln(~)

I~ =x/~1 ( x / ~ l +X/~m ) ( 1 _

RV/~1Rm ) .

(16)

We have a relation between the amplitudes A ~ and Am: A2 =t-Am =

t

I +erg2 exp (i02)

Am,

(17)

which gives the relations between the intensities t2

I ~ = 1 +r2g 2 +2erg2 c0s(02) Im•

(18)

Using a Rigrod-like method, it is possible to compute the saturated amplitude gain g2 undergone by the impinging intensity I~- after a double past in the amplifying medium and reflection on the cavity mir-

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OPTICS COMMUNICATIONS

ror amplitude reflectivity of which is r2 [ 11 ],

gol+ln(rjg2) I ~ = r 2 ( g E _ r 2 ) ( l + gx/~Er2) .

(19)

When then find with eqs. ( 18 ) and (19) a first relation connecting g2 and 02: t2

l +r2g 2 + 2erg2 c0s(02) ~ gol+ln(~)

X ( x / ~ l + x//~m ) ( 1 __x/R, Rm ) gol+ln(r2/g2) =rE (gE_r2) (1 + gx/~2r2 ) ,

(20)

where Rm depends ofgz and ¢z by relation (10), and all other parameters are fixed. A second relation between g2 and 02 is brought by the dispersion of the light in the cavity II:

02 = - 2nLz/2 z - 6 z ln(gz) + arg(rn) =

-2nVz/AV2 - 6 2 In (g2) + arg(rn).

pute the corresponding value of v~t/Av~. It is then possible to c o m p u t e / r ut and i~ut versus phase 02 by using eqs. ( 13 ), (14) and (24).

2.1. Discrimination of even and odd modes Fader proposes in his model of a two series coupled lasers a discrimination of two modes of the entire cavity [ 12 ]. These modes are named even and odd, and represent the fact that if the two lasers oscillate at the same frequency, the emitted light could be in phase or out of phase. This modes discrimination comes out when we write the phase difference 0 °ut = 0~ut - 0~ ut. 0~at and 0~ut are the phases of the lasers at the output coupling mirrors M~ and M2, respectively. The phases of the lasers are equal at the coupling mirror M, and the output phases of each laser read

01'ut = 0 ~ / 2 , ¢'3~t =¢2/2 +arg( t - ) ,

(21)

Since the laser I is oscillating, r, is real (eq. ( 7 ) ) . We assume now that the cavity II is not oscillating. The frequency of the travelling wave is the frequency v~i of the light injected from the cavity I. Then eq. (21 ) reads 02 = - 2~t(v~l - / J c 2 ) / A v 2 --62 ln(g2) .

(21.a)

The gain g2 could be written as gE=rzgfg~ with SO g~ =x/gE/r2. W e then obtain

a r g ( t - ) is the change phase induced by the transmission through the coupling mirror. The phases introduced by the transmission through the output coupling mirror are not accounted while the output mirrors are assumed to be identical. Then, ¢°ut = 02/2 + arg (t - ) - 0 1 / 2 . We made the assumption that 0~ = 2k~z where k is an integer, so

g f =g~.

¢°~t=O2/2 +arg( t - ) - k n .

A~- =g~- exp(i0E/2)A~-

The interference intensity then reads

=

t exp(i02/2) A gx/~2/r21 +¢rg2 exp(i02) m •

(22)

I~ u` = ( 1 - R E )

t2 g2 rE 1 +rEgZ+2~rg2 cOS(02) Ira.

(25)

It = I , +I2 +2/x/~/~ cos(0°~t) •

(23)

Here appear two solutions for 00% with regard to the parity of k, which define the different modes of the entire cavity: if k is even, the lasers are in phase (even mode), and if k is odd, the lasers are out of phase (odd mode).

(24)

2.2. Results of numerical simulation

From this equation, it is possible to derive the output intensity of the laser II: A~ut =X/1 -r22 A~-,

15 December 1993

Replacing 02 given by eq. (21 a) in eq. (20) we are now able to compute gz as a function of (v~z- v~2)/ A v2. Then, using eqs. (9) and (10), we can compute Rm and g~. Last, eq. (12) makes it possible to corn-

For the computations, we assume vcl--ltc2= It0 where Vo is the frequency at line center, AVl= AVE=AVH= 100 MHz and g°l=0.8. The computations have been performed for two coupling mirror 77

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OPTICS COMMUNICATIONS

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15 D e c e m b e r 1993

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Fig. 3. Intensities of two coupledlasers versusL ~when R = 0.95. reflectivities which correspond to the case of strong coupling (R = 0.7 ) and weak coupling (R = 0.95 ). In each case I~, 12 and It are plotted against the frequency shift of laser I from the line center. The results are represented in figs. 2 and 3. These results are the same as Fader's results [ 13 ]: the intensities of the two lasers tend successively to zero and to a maximum, and they could be identical only when the two lasers are coupled in phase or out of phase, which is in agreement with Chow's theory [ 10 ]. If they are in phase, the total intensity reaches a maximum, while it vanishes when the lasers are out of phase. The width of the resonance depends on the coupling coefficient.

t+: "I Fig. 4. E x p e r i m e n t a l set-up.

amplifiers filled with a mixture of CO2: Ne: He: Xe ( 1 3 : 1 6 : 6 6 : 5 ) with 17 Torr pressure and 0.8 m length. The internal tube diameter is 7 mm, and the output coupling mirrors are ZnSe spherical mirrors 3 m in radius curvature and 0.66 in reflectivity. The lengths of the cavities are 1.42 m. The coupling mirror is a flat ZnSe mirror. In order to select the 10P(20) CO2 lasing line, a diffraction grating is set in the cavity I. It must be noticed that the diffraction grating induces 5% additional loss in one of the cavities and then the two cavities are not exactly identical. Both cavity lengths may be adjusted using PZT translator. The length of the cavity II is adjusted using a PZT translator supporting one of the mirrors. The intensities I1 and/2 are analysed using large area HgCdTe detectors D1 and D2, respectively. The two beams are coherently added using a ZnSe beam splitter. One of the beams is sent to a CO2 wavelength analyser in order to control the lasing line. The other is then splitted into two beams. The first one is directed to a fast HgCdTe detector D4 while the second one could be analysed in two ways by a small area detector D3 ( 50 pm × 50 pm): using a scanner mirror we analyse the far field of the coherently sum of the two beams, or we measure the intensity of the center of the far field (It) versus the length of the cavities.

3. Experimentation

3.2. Results

3.1. Experimentation set-up

The experiments have been performed using the two mirrors reflectivity already studied in the numerical computation. The results obtained for R = 0 . 9 5 appear in fig. 5 where I~ and It are plotted versus the length of the cavity II when the laser I emits at line center. We may

In order to confirm the theoretical results, experiments have been worked out using two CO2 lasers. The experimental set-up used is shown in fig. 4. The experiment uses two equivalent CO2 lasers made of 78

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OPTICS COMMUNICATIONS

15 December 1993

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Fig. 7. Beats between the two coupled lasers when L2 is modulated (R=0.7). I (UA) 8 11

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Fig. 6. Experimental results: I~, I 2 and I t for R=0.7 when L2 is modulated.

Fig. 8. Beats between the two coupled lasers (R = 0.95 ) when L~

observe that the experimental results agree well with the theoretical predictions in fig. 3 only in a narrow range surrounding the m a x i m u m value of It. These correspond to the frequency locking ranges. Outside of these ranges the two lasers oscillate with different frequencies (fig. 8) or the laser II may emit other CO2 lines and our theory is no longer valid. In fig. 6 the same results appear when R = 0.7. These curves must be compared with the ones plotted in fig. 2. In fig. 7 are shown the frequency locking ranges for R=0.7. In this case, the two lasers are always oscillating on the 10P(20) line. Indeed, with this mirror, the laser II is near oscillation threshold. Its gain is saturated by its own light and by the light injected from the laser I. Then it must get advantage of the reflectivity of the cavity I which is line selected in order to reach oscillation threshold. In both fig. 5 and fig. 7, we observe the even and odd modes corresponding to in-phase and out-ofphase emission. We have compared the frequency locking ranges width obtained in fig. 7 and fig. 8 with the theoret-

ical values computed using the formula given by Fader [ 13 ]. We find that the locking ranges in terms of AL/2 are 0.24 for R = 0 . 7 (theoretical result is 0.25) and 0.036 for R = 0 . 9 5 (theoretical result is 0.046). Since the two beams are added using a 50% beam splitter, the interferences are constructive in one of the transmitted directions and destructive in the other when the two beams oscillate at the same frequency. Then, in the in-phase mode and assuming the beam intensities to be equal, one of the directions holds the sum of the intensities while the intensity in the other direction is zero. The situation is changed in the out-of-phase mode. When the beams do not share the same frequency, one half of each intensity is transmitted in both directions. Then, both directions have the same intensity. In fig. 9 is shown the far-field of coherent sum of the two lasers when R = 0.95. In this figure we have represented the farfield in three cases: (i) the two lasers emit in phase (even mode), (ii) the two lasers are out of phase (odd mode), and (iii) the two lasers are uncoupled

is modulated.

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I (UA)

been d e m o n s t r a t e d that coherent a d d i t i o n o f the two e m i t t e d b e a m s in the in-phase m o d e can result in twice the intensity supplied by the two incoherent lasers.

10 p20

15 December 1993

- P20 I n p;na s e

Acknowledgements

X (UA)

This work has been s u p p o r t e d by Direction des Recherches, Etudes et Techniques.

Fig. 9. Far field of the two lasers ( R = 0.95 ). (laser I on 1 0 P ( 2 0 ) a n d laser II on 1 0 P ( 2 2 ) ) . In case ( i i ) , the intensity o f the o d d m o d e interference vanishes only at the center because the phase fronts o f the interfering waves are not identical in our exp e r i m e n t a l set up resulting to the different propagation lengths between the output mirrors and the detector.

4. Conclusion Using a Rigrod-like formulation we have developed a theoretical m o d e l for describing the performances o b t a i n e d with phase coupled lasers which cavities share a c o m m o n partially transmitting mirror. This m o d e l agrees well with the e x p e r i m e n t a l results o b t a i n e d using two m e d i u m p o w e r CO2 lasers. It has

80

References [ 1] D. Botez, L.J. Mawst, G.L. Peterson and T.J. Roth, IEEE J. Quantum Electron. 26 (1990) 482. [ 2 ] G.L. Bourdet, G.M. MuUotand J.Y. Vinet, IEEE J. Quantum Electron. 26 (1990) 701. [ 3 ] D. Mehuys, W. Streifer, R.G. Waarts and D.F. Welch, Optics Lett. 16 (1991) 823. [4] G.L. Bourdet, IEEE J. Quantum Electron. 28 (1992) 2033. [5] R.H. Rediker, K.A. Rauschen and R.P. Schloss, IEEE J. Quantum Electron. 27 ( 1991 ) 1582. [ 6 ] S. MacCormack and J. Feinberg, Optics Lett. 18 (1993) 211. [7] G.L. Bourdet, R.A. Muller, G.M. Mullot and J.Y. Vinet, Appl. Phys. B 43 (1987) 203, 273. [8 ] M.B. Spencer and W.E. Lamb, Phys. Rev. A 5 (1972) 893. [9] H. Mirels, Appl. Optics 25 (1986) 2130. [ 10] W.W. Chow, Optics Len. 10 (1985) 442. [ I 1] W.W. Rigrod, J. Appl. Phys. 36 ( 1963 ) 2487. [ 12]A.E. Siegman, in: Lasers (University Science Book, Mill Valley, 1986) p. 468. [ 13 ] W. Fader, IEEE J. Quantum Electron. QE-21 ( 1985 ) 1838.