ARTICLE IN PRESS

Optics & Laser Technology 42 (2010) 296–300

Contents lists available at ScienceDirect

Optics & Laser Technology

0030-39

doi:10.1

� Corr

ogy, Sic

E-m

journal homepage: www.elsevier.com/locate/optlastec

Phase-locking of a multi-channel radial array CO2 laser

Liang Chen a,b,�, Yude Li a, Kai Jia a, Wei Pan a

a Department of Optoelectronic Science and Technology, Sichuan University, Chengdu, Sichuan 610064, Chinab School of Physic and Chemistry, Henan Polytechnic University, Jiaozuo, Henan 454000, China

a r t i c l e i n f o

Article history:

Received 6 March 2009

Received in revised form

26 June 2009

Accepted 15 July 2009Available online 8 August 2009

Keywords:

CO2 laser

Multi-channel radial laser array

Phase-locking

92/$ - see front matter & 2009 Elsevier Ltd. A

016/j.optlastec.2009.07.011

esponding author at: Department of Optoele

huan University, Chengdu, Sichuan 610064, C

ail address: chenliang822@sohu.com (L. Chen

a b s t r a c t

A theoretical analysis of phase-locking in a multi-channel radial array CO2 laser is presented. The

concepts are based on the theories of injection phase-locking and matrix optics. The mutual optical

coupling occurring within the central region is demonstrated by theoretical deduction. Interrelated

graphs of the coupling coefficient and the distance between the coupling position and the cavity mirror

are calculated by numerical simulation. The results of the analysis are in accord with experimental

results.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

With the rapid development of CO2 laser technology, CO2 lasersystems have been applied in many fields, including industry,military, environment protection, agriculture and medicine. Theoutput powers of CO2 lasers range from only a few mW to over100 kW from laser arrays. However, the poor output beam qualityand large size of the laser array used in many industrialapplications limit its further application.

Using the phase-locking technique it is possible to generatehigher output powers with good beam quality from a laser array.The phase-locking of multiple independent lasers has beenstudied extensively since the phenomenon was first proposed byBasov et al. [1]. In 1992, this method was used in a radial multi-channel CO2 laser array by Yelden et al. [2]. Their laser arrayemployed an unstable resonator configuration. Particularlyattractive features of the concept include major reductions insize, weight, complexity and cost. When the phase is locked, thefocused intensity of the composite beam increases quadraticallywith the number of channels in the array. However, the phase-locking in the radial multi-channel CO2 laser array was analyzedbased on experimental data without theoretical proof.

In comparison with the reports of Yelden [3–5], the CO2 laserarray studied in this paper has been improved. A Gaussian beamcan be obtained by using a stable resonator and the output beamquality is increased further. The principle of the phase-locking inthis radial multi-channel CO2 laser array is proved theoretically.

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).

Coupling coefficients and the light intensity distribution in nearand far fields are determined by numerical simulation.

2. The structure of multi-channel radial array CO2 laser

The structure of Yelden’s multi-channel radial array wasdescribed in their papers [2]. The structure consists of eighttriangular-shaped, nickel-coated aluminum, large-area electrodes,which are arranged in a radial array. Provision is made to watercool each electrode so as to enhance the diffusion coolingproperties of the device. When mounted together in an inter-digital fashion, with RF excitation, eight separate gain regions arecreated. The central region of the structure contains no visibledischarge, but this region is still a discharge area as the results ofthe experiment show.

In this paper, the discharge apparatus employed is similar toYelden’s, But here the optical resonator has been improved. Thedimension of each of the eight separate gain regions is 30�3�1cm3 (l�h�o). The diameter of the central region is 3 cm. Adiagram showing the longitudinal section of the optical resonatorof the laser array is provided in Fig. 1. M1 is a special toric mirrorwith a center hole, which is formed by N concentric loops with thesame curvature radius r1 and the same width. In order to takeadvantage of the gain area, the number of the concentric rings canbe two or multiples there of. This structure can operate stably byregulating the parameters of the cavity.

A three loop structure is used in this paper. M2 and M3 arespherical mirrors. Their curvature radii are r2 and r3 respectively.M4 is the common output mirror. It is essential to make a hole inthe center of M2 for output, in which M4 is embedded. In order tomeet the requirement of parallel output, the two foci of M2 and M3

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L. Chen et al. / Optics & Laser Technology 42 (2010) 296–300 297

are coincident at F. In another word, F is the common focus of allbeams.

The structure of the multi-channel radial array is such that thedistances between mirrors M1 and M2, and M3 and M4 are same.The optical path length between mirrors M1 and M2, and M2 andM3 can be represented by l1 and l2 respectively. The distancebetween the beams, which are parallel to the axis of M1 and themain optical axis, is di. According to the principle of geometricaloptics, beams which are parallel to the axis of M1 should bereflected by M2, to be located in the lower half of M3. The point onM3 is denoted by P. The distance between P and the main opticalaxis of M3 is x.

The cavity parameters are selected as follows:

(1)

TablChar

x (cm

0.68

1.01

1.35

W1,

l1 is 40 cm. The diameters of M3 and M4 are 3 cm.

(2) r1 ¼ 460 cm, r2 ¼ 45.57 cm, r3 ¼ 15.38 cm. (3) M1 has three concentric loops with the same radius of

curvature, r1 and width, 1 cm. And the radii of the threeconcentric loops are 2, 3 and 4 cm respectively.

Three Gaussian beams can be obtained in one channel. From theparameters above, Table 1 may be obtained. This resonator hastwo substantially different transverse dimensions, because of thespecial toric mirror is used. It is stable in the radial direction andunstable in the azimuthal direction. The analogous azimuthallyunstable resonators can produce output beams with high quality[6–8].

3. Analysis of phase-locking characteristic

3.1. The principle of phase-locking in the laser array

From [2] it was concluded that the central region was initiatingoptical coupling between the multiple channels, and therebyacting as a core-oscillator injector in a phase-locked oscillator/regenerative amplifier system. The region between M3 and M4 isthe common discharging area of the laser array. At the initial timeof discharge, the paraxial rays caused by spontaneous emission inthe common discharging area are approximately parallel to themain optical axis. These rays converge at F after being reflected byM4 and being amplified in the common discharging area. The

Fig. 1. The longitudinal section diagram of the laser array.

e 1acteristic parameters of the laser array.

) d (cm) |(A+D)|/2 W1 (cm)

2 0.3870 0.3874

3 0.2738 0.3785

4 0.1170 0.3715

W2, W3 and W04 are the radii of the light spots which poured into M1, M2, M3 and

spherical wave which passed through F is injected into eachchannel to excite the oscillations. The pointolite on F has the samefunction. To simplify the problem, the effect of the commondischarging area on phase-locking is represented by the pointoliteon F.

In this laser array, the main restriction on the output mode isimposed by the edges of the electrode. Its eigen mode is aHermite–Gaussian beam. It can be expressed by the weightedsums of fundamental mode Gaussian beams and high-order modeHermite–Gaussian beams as follows

expðikffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ z2p

Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ z2p ¼

X1m¼0

X1n¼0

Amnw0

wðzÞHm

ffiffiffi2p

x

wðzÞ

!Hn

ffiffiffi2p

y

wðzÞ

!

�exp �x2 þ y2

w2ðzÞ

� �exp �i½kzþ

k

2RðzÞðx2 þ y2Þ� � ðmþ nþ 1ÞjðzÞ

� �ð1Þ

where

RðzÞ ¼ z 1þpw2

0

lz

� �2" #

w2ðzÞ ¼ w20 1þ

lz

pw20

!224

35

Hence a part of the energy exiting the pointolite on F mustcouple with the energy of the eigen mode in the form of afundamental mode Gaussian beam in each channel. The ampli-tude distribution and the curvature radius of equiphase surfaceare transformed by refraction and diffraction. The signal is alsoinfluenced by the magnification caused by the excited particles inthe common discharging area. It tends to approach fundamentalmode Gaussian beam distribution after many cavity roundtrips.Therefore the coupling between the spherical wave (as the signal)and the fundamental mode Gaussian beam (as the eigen mode)can be considered approximately as a coupling between twofundamental mode Gaussian beams in the phase-locking process.The coupling rate of two fundamental mode Gaussian beams isused in the theoretical analysis. We choose the geometry, size andcurvature radius of the equiphase surface of the spherical wavetransmitting in each channel as its beam parameters. For thesimplification of analysis, only the radial direction is considered inthis paper.

3.2. The definition of coupling coefficient

The coupling coefficient of two fundamental mode Gaussianbeams can be expressed as [9]

c00 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi2

ww0q

rð2Þ

where

q ¼1

w2þ

1

w02þ i

k

2

1

R�

1

R0

� �ð3Þ

w, w0, R, R0 are the radii of the light spots and the curvature radii ofequiphase surface respectively.

W2 (cm) W3 (cm) W04 (cm)

0.3631 0.1086 0.1083

0.3549 0.1019 0.1017

0.3483 0.0939 0.0937

M4 respectively. x and d respectively represent the value of them when i ¼ 1, 2, 3.

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The energy coupling coefficient can be expressed as

Z ¼ c00 c00� ð4Þ

The principle of phase-locking in the laser array shows that thetwo beams involved in the process of coupling are the sphericalwave emitted from F and the fundamental mode Gaussian beamof the laser array. The spot radius w0 of the spherical wave isrepresented by its geometry size transmitting in each channel.The radius of curvature of the equiphase surface R0 can becalculated using the ABCD laws of spherical wave.

3.3. Calculation of the energy coupling coefficient

The positive propagation direction is ordered from M1 to M2,M3 and M4. The center of the spherical wave caused byspontaneous emission inside the common discharging area issituated around F on the main axis of the laser array, and thedistance between the spherical center and the reflection point on

Fig. 3. The changes of the coupling coefficient with the dist

Fig. 2. The changes of the coupling coefficient with the dist

M2 is L0. Taking M2 as reference plane, the optical matrix for aroundtrip can be expressed as follows

m ¼1 0

�2=r2 1

!1 l2

0 1

� � 1 0

�2=r3 1

!1 l1

0 1

� �a b

c d

� �

�1 0

�2=r1 1

!1 l1

0 1

� �ð5Þ

After N�1 roundtrips, the paraxial rays of the spherical wavearrive at point E between M1 and M2 in the negative direction. Thedistance from point E to M2 is DL. The transfer matrix can beexpressed by

A B

C D

� ��N

¼1 Dl

0 1

� �mN�1 1 0

�2=r2 1

!1 L0

0 1

� �ð6Þ

After N�1 roundtrips, the paraxial rays of the spherical wavearrive at point Q between M1 and M2 in the positive direction. The

ance in space between M1 and M2 after two roundtrip.

ance in space between M1 and M2 after one roundtrip.

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L. Chen et al. / Optics & Laser Technology 42 (2010) 296–300 299

distance from point Q to M1 is DL0. The transfer matrix can beexpressed by

A B

C D

� �þN

¼ 1 Dl0 01 1 0

�2=r1 1

!1 l1

0 1

� �mN�1

�1 0

�2=r2 1

!1 L0

0 1

� �ð7Þ

where, N represents the times of roundtrips. N ¼ 1, 2, 3y.According to Eqs. (6) and (7) and ABCD laws of spherical wave,

the radius of curvature of the spherical wave between M1 and M2

can be given as

R2 ¼AR1 þ B

CR1 þ D¼

B

Dð8Þ

Fig. 5. Computer modeling of composite profiles and beam quality. (a) and (b) ¼ near

(d) ¼ near field and far field output intensity profiles for 24 phase-locked beams.

Fig. 4. The changes of the coupling coefficient with the dista

The spherical wave is restricted by the size of reflectors in thislaser array. The beam radius between M1 and M2 is equal to thewidth between the axis of the discharge channel and the fringerays. j0 is the slope of the fringe rays, j the initial slope, r theradius of beams, and r1 the width of the concentric loops of M1.

The fringe rays can be calculated byr

j0

� �¼

A B

C D

� �7N

0

j

!.

So

r ¼ Bjj0 ¼ Dj ð9Þ

furthermore, jrr1/B can be gotten from rrr1.The radius of the light spot and the curvature radius of

equiphase surface of fundamental mode Gaussian beam on M1 areused to figure out the light spot and the curvature radius of

field and far field output intensity profiles for 24 nonphase-locked beams. (c) and

nce in space between M1 and M2 after three roundtrip.

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equiphase surface between M1 and M2 according to ABCD laws ofspherical wave. These can be calculated as follows

w ¼ w11

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

2Dz

r1

þDz

r1

� �2

þlDz

pw21j

!224

35

vuuut ð10Þ

R ¼r2

1 � 2r1Dzþ ½ðDzÞ2 þ ðlr1Dz=pw11Þ2�

1þDz½1þ ðlr1=pw11Þ2�

ð11Þ

where Dz ¼ Dl when the direction is positive, otherwise Dz ¼ Dl0

By substituting Eqs. (8–11) into Eq. (4), the energy couplingcoefficient between the spherical wave and the eigen mode can beobtained. The relationship between the energy coupling coeffi-cient and coupling position is shown in Figs. 2–4. The three beams(i ¼ 1, 2, 3) in one channel are represented by three differentsymbols. The figure shows that the coupling coefficient changeswith the distance between the coupling position and M1 after N

roundtrips. For N ¼ 1, 2, 3, the coupling coefficients in the positivedirection are not less than 0.59, 0.5 and 0.7 respectively. Theoscillation of phase-locking can be constructed by the couplingbetween a spherical wave and a fundamental mode Gaussianbeam.

4. The output light intensity of the laser array

The eigen mode of this resonator is a fundamental-modeGaussian beam. The general expression describing the distribu-tion of light field [10] is

eðr; zÞ ¼ A0o0

oðzÞ exp �r2

o2ðzÞ

� �exp½�ijðr; zÞ� ð12Þ

where

oðzÞ ¼ o0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

z

z0

� �2s

; RðzÞ ¼ z 1þz0

z

� �2� �

; jðx; y; zÞ

¼ k zþr2

2RðzÞ

� �� arctan

z

z0; z0 ¼

po20

l

The limiting condition of the margin should be considered. Thecalculated near and far field intensity distributions of funda-

mental-mode Gaussian beam with and without phase-locking areshown in Fig. 5.

5. Conclusion

Based on the theories of injection phase-locking and matrixoptics, a theoretical analysis method of phase-locking is obtained,which can be used in multi-channel radial array CO2 laser. Energycoupling coefficients are calculated and simulated. The lightintensity distributions in the near and far fields are also simulated.

The results show phase-locking can be obtained by thecoupling between the spherical wave emitted from the commonfocus F and the eigen mode of the laser array. The results oftheoretical analysis are in accord with the experiment. The theorypresented in this paper may be used as a universal theory forother injection phase-locking systems.

Acknowledgment

The work was fully funded by the National Natural ScienceFoundation of China under Grant 60278020.

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