## Abstract

Experimental results on 1D and 2D phase-locking of laser diode arrays are
presented. Attention is paid to the employment of the arrays consisting of wide
aperture lasers diodes. Selection of the “in-phase”
supermode, preferable for most of the cases, is attained in the external quarter
Talbot (L_{c}=Z_{T}/4=d^{2}/2λ) cavity due to
the output mirror tilt at the angle φ_{m}=λ/2d.
Analysis of the parameters that influence on the phase-locking is given. Our
experiments confirm theoretical predictions of the system stability and adequate
selectivity for the laser diode array fill factor (FF) FF=0.6.

©1999 Optical Society of America

## 1. Introduction

Considerable recent attention [1–12] has been focused on phase-locking of semiconductor lasers. Particular emphasis is placed on phase-locked semiconductor laser diode arrays (LDA) because of high efficiency of the system (≥30%), small size and scaling up possibility.

The array of N emitters allows for oscillation of N cavity supermodes. Single supermode oscillation is required to achieve diffractional limited beam. Theoretical simulations [1–3], earlier experiments with single mode emitters [4–6] displayed multimode regime preference, resulting in 3 and more times of diffractional limit excess in the output beam divergence. Employment of a spatial filter [7–9] prevented from undesired supermodes generation. Though this method proved to be reasonable and experimentally tested, it still resembles a complicated pinhole technique when the profile of the filter matches precisely distribution of the supermode allowed to oscillate.

Significant improvement of the selection is realized in the so called Talbot cavity
in which the effect of self imaging of a periodical array of emitters occurs at
Talbot Z_{T}=2m(d/λ)^{2} m=1,2,3⋯ and
sub-Talbot (2m+1)Z_{T}/2 planes [3,10].

Earlier successful experiments were carried out employing Talbot cavity technique [2] to phase-lock 20 [9], 30 [11] small aperture (S<10 μm) emitters spaced at d<50 μm (FF=S/d<0.1). Despite the obvious advantages, there exist few problems with Talbot cavity technique employment.

- High intensity is obtained if the cavity sustains single supermode, that, in turn takes place if FF<<1 (for example FF=0.08 [11]). Low FF, in turn, results in low output power emitted by phase-locked LDA of a fixed length.
- Small value of FF results in multilobe far field pattern decreasing the power in the central lobe.
- “In-phase” and “out-of phase” supermodes are not discriminated simultaneously even if the cavity round trip length is L
_{c}=mZ_{T}. - Placing the mirror at L
_{c}=Z_{T}/4 it is possible both to select single “out-of-phase” supermode and to slightly decrease generation threshold (if compared with L_{c}=Z_{T}/2) [12]. The problem in this case consists of two-lobe far field pattern that again is not optimal for high intensity radiation.

The problems mentioned above are widely known and suggest, probably, the solution of compromise depending on the aims to be achieved. The aim of this paper consists of the illustration of the actual feasibility of 1D and 2D phase-locking of LDA with wide aperture emitters placed with relatively high FF=0.6.

## 2. Experimental results review

In the experiments [13–15] on phase-locking, we employed LDA’s with wide aperture emitters (S=120 μm), spaced at d=200 μm (FF=0.6) (fig.1).

Attention was paid to the peculiarities of phase-locking if individual diodes emit
multimode radiation. Our approach was aimed at the employment of powerful arrays.
Phase-locking was demonstrated for the LDA of N=8 emitters in a cavity of length
L_{c}=Z_{T}/4=2.5 cm. Fig.2 displays far field pattern for “out-of
phase” supermode.

The parameters of the pattern are as follows:

- 2 central lobes exhibit twice diffraction limit excess;
- 2 lateral lobes are measured to have Full Width at Half Maximum (FWHM) δΨ=0.5 mrad in agreement with diffractional limit for the synthesized aperture Nd=1600 μm, λ=0.8 μm;
- contrast parameter V = (I
_{max}- I_{min})/(I_{max}+I_{min}) = 0.97 revealing high coherence of the output beam; - intensity of the “in-phase” lobes is approximately 10 times less than that of “out-of-phase” supermode confirming high selectivity of the cavity.
- the observed far-field patterns were obtained at injection current 0.5 a (I
_{th})<I<1.5 a (3I_{th}) per single diode, where I_{th}=0.5 a is the threshold injection current per single diode for phase-locked mode oscillation in the external cavity (out-of-phase supermode).

In most of the applications the “in-phase” cavity supermode is
preferable as it has one central lobe in which most of energy should be
concentrated. For the quarter Talbot cavity the feedback is sufficient to sustain
the oscillation, but the “in-phase” supermode has the highest
generation threshold in such a cavity. Tilting the output mirror it is possible to
tune the desired “in-phase” supermode. The possibility of
“in-phase” supermode selection in the Z_{T}/4 cavity
with the tilted mirror (φ_{m}=λ/2d) was discussed in
paper [16]. From the theoretical estimations it follows that the
generation threshold for the “in-phase” supermode depends
slightly on the number N of phase-locked emitters if the array with relatively high
FF is employed. So, if the evidence for the “in-phase”
supermode selection is given for N=8 emitters, than there will be no problems with
N=16, 25⋯ and so on. Generation of the “in-phase”
supermode was demonstrated with the tilted mirror in our experiments on
phase-locking of two LDA’s (2D configuration).

Relatively large length of the external cavity (L_{c}=2.5÷5 cm)
forces to employ collimating optics (cylindrical lenses), that, in principle
complicates 2D phase-locking as it is necessary to decrease fast axis radiation
divergence to allow generation development in the Talbot cavity, that, in turn,
diminishes the efficiency of coupling between neighbor diode arrays.

Another circumstance that influence on the phase-locking consists of the uniform parameters of the laser diodes. Special attention was paid to selection of the LDA with slight deviation of the optical properties.

The experiments were carried out with two LDA’s of N=8 diodes separated by
1600 μm. At this stage the collective mode of the two LDA’s
existence in the quarter-Talbot cavity (L_{c}=Z_{T}/4) with the
tilted mirror (φ_{m}=λ/2d) was studied. The far field
pattern is presented on fig.3.

The patterns exhibit independent (a) and phase-locked (b) generation of the arrays. For the phase-locked regime:

- diffractional lobes angle corresponds to equation φ=mλ/d
_{x}, at λ=0.8 μm, d_{x}=1600 μm; - FWHM of the lobes is twice narrower the distance between the peaks in x axis;
- operating conditions: pulse duration - 400 ns, pulsed power - 3 W.

The features highlighted above point out 2D phase-locking for the system of two LDA’s involving N=8 wide aperture laser diodes. The chosen parameters allowed for fast axis and slow axis diffractional lobes FWHM δΨ=0.25 mrad and δΨ=0.5 mrad correspondingly.

## 3. Limitations in phase-locking

Successful phase-locking of a large synthesized aperture depends on simultaneous
matching of several conditions that can be divided into 3 groups. The first group
includes admissible range of cavity parameters variation: cavity length
δL_{c} and angle positioning of external mirror
φ_{m}. The second group consists of spectral parameters
variation: bandwidth of the spectral interval δλ_{p}
allowing phase-locking, detuning δλ_{0} of the
phase-locked wavelength λ_{p} from the center of gain profile
λ_{0}
(δλ_{0}=λ_{0}-λ_{p}).
Besides, correlated and non-correlated phase errors
<δφ^{2}> in the channels of
LDA should be included in this group. The third group unifies parameters that are
governed by technological processes of LDA manufacturing, mounting, AR-coatings
covering and so on. We summarize theoretical limitations and compare them with
experimental available data.

Consider the cavity parameters. Paper [3] presents estimation of the variation of the reproduction
coefficient δγ of the near field distribution with respect to
mirror displacement δL_{c}:

As it follows from eq.(1) FF influences intracavity losses primarily. In our
experiment δL_{c}=1 cm, Z_{T}=10 cm, FF=0.6 and eq. (1) results in δγ=0.008, that is 20%
displacement from Z_{T}/4 plane decreases reproduction coefficient only in
the third order. Theoretical estimation agrees experimentally observed weak
dependence of output radiation properties change with the mirror displacement in our
experiment that is a direct consequence of high FF.

The accuracy of external mirror angular positioning φ_{m} deduced
in [3] under assumption
φ_{m}<<λ/d,
δγ<<1 follows from equation:

Assuming δγ≈0.1, individual laser divergence
θ≈0.2, N=10, eq.(2) gives φ_{m}≈0.6 mrad, that
satisfies requirement
φ_{m}<<λ/d=4 mrad for our
experimental conditions λ=0.8 μm, d=200 μm. Eq. (2) elucidates the fact of strong dependence on the total size
of the array. The relationship φ_{m}~1/N^{2}
that follows from eq. (2) points out the necessity of precise cavity alignment for
large N.

If the angle mirror tilt is not small
(φ_{m}≥λ/d) than for a set of angles
φ_{m}=mλ/(2d) it is possible to obtain
phase-locking so that cavity supermodes coincide with that of
φ_{m}=0. Additional losses δγ resulted from
the tilt are estimated as:

Eq. (3) is valid if f<<1. The possibility of supermodes generation with significantly tilted mirror was confirmed in our experiments.

Summarizing estimations (1–3) we conclude that cavity parameters play important role but do not limit drastically phase-locking especially for the array with high FF.

The second group of parameters that unifies admissible range of spectral and phase
detunings is of great importance and, probably, plays the crucial role in
phase-locking. Deviation of lasers resonance wavelengths
δλ_{L} brings to slight dephasing first and than
breaks phase-locking if wavelengths interval extends over interval of phase-locked
bandwidth δλ_{p}. Existence of noise (spontaneous
emission, scattering on the coupling optical elements) also influence on
phase-locking negatively making δλ_{p} narrower.

For the case of Talbot like phase-locking δλ_{p} is
given [18]:

where M-is coupling constant, P-noise power,
β=α_{0}l/I_{s},
G_{0}=α_{0}l-g_{0},
α_{0}-active medium gain, g_{0}-active medium losses. Eq.(4) is valid for δλ_{0}=0 and
reveals that δλ_{p} increases with N and diminishes
with P_{n}. Theoretical simulations predict significant increase
(>10 times) of allowed for phase-locking bandwidth
δλ_{p} if
δλ_{0}≠0 and suggest optimal value
δ${\mathrm{\lambda}}_{0}^{\text{opt}}$ that maximize
δλ_{p}. It is important to highlight that
δλ_{0} can be, in principle, arbitrary within the
limits of active medium gain profile. This circumstance is truly important and,
probably, comprise the only feasibility to obtain phase-locking of powerful arrays
in which element-to-element wavelength shift exists due to heat release.

In the case of nearest neighbor emitters coupling and neglecting spontaneous emission term

where *v* - is supermode number.

For either “in-phase” or “out-of-phase” supermode and N>>1 eq.(5) gives:

Comparison of eq.(4,6) shows that spectral conditions in the case of nearest emitters coupling are N/2 times severe suggesting the preference of Talbot like coupling.

In the description above, laser diodes of the array were assumed identical so that
path-length errors are constant for all of the diodes. If such assumption is
excluded and non correlated phase deviation between the channels
<δφ^{2}> is taken into
consideration than additional cavity losses can be written as [3]:

Experiments with liquid crystal array (LCA) [19,20] inserted in the Talbot cavity displayed the necessity to control both correlated (tilt of a focusing lens) and non-correlated phase errors. For the non-correlated errors it is shown that a 0.10 π element-to-element route-mean square (rms) phase difference produces 9.5% reduction in far field lobes intensity while a 0.26 π rms phase difference results in 50 % lobes intensity decrease. Hence, λ/8 phase distortions should be taken into account.

The third group of technological parameters, doubtlessly, is very important. To achieve coupling with minimal losses in Talbot cavity lasers should provide equal intensities of the output radiation. Otherwise, interference of N beams inside the cavity will not result in zero field intensity exactly between the emitters and maximum of field intensity falling exactly on the emitters.

Another technological limitation arises from the admissible variation of LDA period
d. Consider Δ_{n} as a random deviation of the n-th diode from
nominal position, so that <Δ_{n}>=0,
<Δ_{n}Δ_{m}>=Δ^{2}δ_{nm},
δ_{nm} is Kronecker data. Analysis [21] shows that if N(Δ/d)<<1
supermode eigenvalue experience decrease:

where γ_{0} - is supermode eigenvalue in the absence of
displacement, p=L_{c}/Z_{T}. To estimate the role of displacement
suppose the relative reduction in the square of
|γ|^{2}/|γ_{0}|^{2} should not
exceed 20%, that gives:

Standard powerful LDA has d=400 mkm and N=25 that results in Δ<1.6 mkm that is quite “allowed” accuracy of manufacturing.

In conclusion of phase-locking limitations consideration one may deduce that non-correlated errors and spectral detunings are of prime importance and special care should be taken to obtain powerful lobes of the output radiation.

## 4. Conclusion

In the presented paper we analyzed both 1D and 2D phase-locking of laser diode arrays. Special attention was given to the consideration of wide aperture laser diodes employment for phase-locking of the arrays with high FF that is required for high power output radiation. Emphasis is given to FF=0.5÷5-0.65 because of:

- standard powerful arrays have S=200 μm, d=400 μm, so that FF=0.5;
- FF=0.5÷5-0.65 still provides cavity supermodes selection;
- FF=0.65 is optimal for power conversion from side lobes to the central one (85%) [17];
- high FF provides stability of the output power at significant distortion of the cavity length;
- high FF allows for weak dependence of the “in-phase” supermode losses on the number of emitters N in a cavity with the tilted mirror.

Our experiments both confirm actual feasibility of 1D and 2D phase-locking of LDA’s with specified parameters and illustrate theoretical predictions of the system stability and selectivity, so that:

- phase-locking is obtained in the external quarter Talbot cavity (L
_{c}=Z_{T}/4) for N=8 wide aperture (S=120 μm) laser diodes in the array of high FF=0.6. It allowed diffraction lobes FWHM δΨ=0.5 mrad; - high value of FF provided stability of the output radiation parameters for cavity mirror displacement of 20% from Z
_{T}/4 plane; - “In-phase” supermode generation was selected in the cavity with tilted mirror (φ
_{m}=λ/2d) placed at Z_{T}/4 plane; - 2D phase-locking is demonstrated for the two LDA’s. The chosen parameters allowed for fast axis and slow axis diffractional lobes FWHM δΨ=0.25 mrad and δΨ=0.5 mrad correspondingly.

## Acknowledgments

This work was supported by CILAS, France (Marcoussis), contract #135454

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