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.

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).

 

Fig.1. Experimental setup. 1-laser diode array, 2-cylindrical lens, 3-cavity output mirror, R=50%, 4-lens (F=10 ñì), 5-videocamera, 6-monitor, 7-oscilloscope

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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.

 

Fig.2. Far field pattern of the “out-of-phase” supermode.

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The parameters of the pattern are as follows:

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.

 

Fig.3 (a). Far field pattern of the “in-phase” supermode for independent generation of the 2 laser diode arrays in a quarter-Talbot cavity.

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Fig.3 (b). Far field pattern of the “in-phase” supermode for 2D phase-locking of the 2 laser diode arrays in a quarter-Talbot cavity.

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

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 δλ₀ of the phase-locked wavelength λ_p from the center of gain profile λ₀ (δλ₀=λ₀-λ_p). Besides, correlated and non-correlated phase errors <δφ²> 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/N² 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=α₀l/I_s, G₀=α₀l-g₀, α₀-active medium gain, g₀-active medium losses. Eq.(4) is valid for δλ₀=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 δλ₀≠0 and suggest optimal value δ${\mathrm{\lambda }}_0^{\text{opt}}$ that maximize δλ_p. It is important to highlight that δλ₀ 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 <δφ²> 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_nΔ_m>=Δ²δ_nm, δ_nm is Kronecker data. Analysis [21] shows that if N(Δ/d)<<1 supermode eigenvalue experience decrease:

where γ₀ - 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/|γ₀|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:

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: