1 Introduction

In our previous paper[1] we studied extraction characteristics of one-dimensional Talbot arrays. In this model each emitter is described by a complex field differential equation in the Rigrod approximation. All emitters are coupled through the Talbot cavity which constitutes one-half of the two-point resonator boundary conditions. We showed that the far-field performance of linear arrays is a strong function of the Talbot cavity length displaying multiple maxima and minima as a function of Talbot cavity length. This is true for either internal or external cavity phasing. This complicated behavior is due primarily to the Talbot propagation phase embodied in the interference between emitters. Also, we showed that the best far-field performance was in the half-Talbot plane for external phasing; the quarter-Talbot plane gives almost the same performance[1]. When a linear stochastic propagation phase for each emitter is included we showed analytically that the standard threshold equation is replaced by a stochastic threshold equation which on the average predicts an increase in threshold gain. Further more, in the linear array development we introduced two measures of performance: a locking probability, and a coherence function. The former shows the number of times the integrator converges for a given rms phase σ ². This represents a measure of how often a lasing mode will be established for a certain rms optical path difference between the emitters. The coherence function is a normalized on-axis intensity.

In the following two-dimensional hexagonal case we will see entirely different behavior. There are four major differences. First, for a deterministic hexagonal Talbot cavity the far-field intensity as a function of the Talbot cavity length does not show any complicated structure, such a multiple maxima and minima as in the linear array case. Specifically, the far-field on-axis intensity shows a monotonic decrease as the Talbot cavity length increases. Next, when the stochastic propagation phase is included the coherence function shows an exp(-σ ²) behavior independent of the Talbot plane. In the linear array the coherence function changes dramatically depending on the Talbot cavity length. Third, again as the Talbot cavity length changes the coherence function, for small σ ² remains near 19, the number of emitters in a double-hexagonal array, where as the coherence function varied widely in the linear case[1]. Fourth, in our previous paper[1] we showed that the one-dimensional linear array starts in an threshold eigen-state, for example the out-of-phase supermode, and remains in this mode as the gain is increased. However, the hexagonal array threshold mode is not necessarily maintained above threshold. Instead, we find the lasing mode becomes uniform characterized by nearly equal amplitudes and phases for all emitters. These features indicate the dominance of an in-phase lasing mode which is due to rotational symmetry. Another difference between these two cases is in the losses. The losses for a hexagonal array are larger than that a linear array with an identical number of emitters. For example, in a linear arrays there are always just two end emitters, but in a single hexagonal array of say 7 emitters, six are peripheral and contribute to the losses. This leads to a larger threshold and smaller extraction. In contrast, many identical nearest neighbors in the hexagonal array makes it more stable. Specifically, the probability of locking does not decrease as rapidly as a function of the rms phase as that of the linear array. We clearly see this behavior in our simulations. Briefly then, the above-threshold hexagonal array lases in a mode which is almost in-phase and can, therefore, be considered to be naturally phased.

2 Theory

In our previous paper[1] we presented the theory of coupling between N emitters in a Talbot cavity when each emitter had a stochastic linear propagation phase due to different fiber lengths and dispersion. Rather than repeat than development we will just present a synopsis.

We consider N gain elements coupled through a Talbot cavity terminated by a monolithic mirror of field reflectivity r_t . The j^th gain element has outcoupling field reflectivity r_j and gain g_j . With these assumptions, and in the Rigrod approximation, the j^th gain element supports a forward $E_j^{+}$ (z) field and a reverse field $E_j^{¯}$(z) both described by

Since the gain is real the phases satisfy

and the constant C_j is given by

These four equations constitute our basic working equations.

 

Fig. 1. Double hexagonal array and coordinate nomenclature

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At the outcoupling end, z=0, we have $E_j^{+}$ (0)=r_j $E_j^{¯}$ (0). At the Talbot cavity end of the laser array, z=L, continuity of the electric fields requires that the reverse field is composed as $E_i^{¯}$ (L_i )=∑R_i,j$E_j^{+}$ (L_j ). R_i,j is the complex reflection matrix developed by evaluating the overlap integral between the Fresnel propagated electric field of the j^th emitter integrated over the aperture of the of the i^th laser, see the Appendix. These two restrictions constitute the two-point boundary conditions.

When the fields contain a random linear propagation phase the Talbot cavity boundary condition leads to the stochastic threshold equation[1]

which reduces to the deterministic form[2] for σ ²=0 and for σ ²≠0 this leads to an increased threshold gain.

As a measure of locking we define the coherence function C as

which is related to the far-field on-axis intensity I(0) by I(0)=C∑|A_m |². For completely coherent emitters this function has a value N equal to the number of emitters. If, however, the phases are random then the average of eq. (6) takes a simple form when the amplitude spectrum is also flat. With these restrictions using and the previous averaging procedure eq. (6) averages to

which approaches N exp(-σ ²) for large N.

3 Examples

Our example is a double hexagonal array with 19 elements as shown in fig. (1) along with our coordinate nomenclature. The period of a typical V-groove fiber holder is d=150µ; the emitting aperture a is 10µm. The monolithic Talbot mirror is 100% reflecting while outcoupling field reflectivities are r_j =r=.8, also we pick a value of g_j =g=4 for the gain.

We begin with a short discussion of previous results obtained for the hexagonal array Talbot planes[3]. These authors[3] consider Talbot cavity properties of oblique-angled arrays, in general. In particular, for a hexagonal array they show that the paraxial quadratic propagation phase factor is $i\frac{{\nu }_p^2+{\mu }_{p,q}^2}{2k}z$. The x-spatial frequency is ν_p =2πp/d, the y-spatial frequency is µ_p,q =4π(q + p/2)/√3d, p, q are integers, d is the x-period, and k=2π/λ. (Note that the linear array results are obtained for µ_p,q =0.) Thus, for the hexagonal array the self-replication Talbot plane is when $z_t\frac{4\pi }{3}\frac{⋋}{d^2}\left(p^2+q^2+pq\right)=2\pi$. This has the solution $z_t=\frac{3}{4}\left(\frac{2d^2}{⋋}\right)$ where the quantity in brackets is the Talbot distance for a linear array. The quadratic phase also shows that there is no precise out-of-phase plane similar to the linear case where the phase alternates between (0, π) for adjacent emitters. Since this phase factor analysis does not yield outstanding fractional Talbot cavity lengths we turn to our threshold results to indicate important intermediate planes, see eq. (8).

An analysis of the threshold equation, eq. (8), shows that there is a mode which is analogous to the linear array out-of-phase solution for z=2/3z_t . Here, however, the phases do not alternate between (0, π) but rather have the structure: (0, π, 0); (π, π, π, π); (0, π, 0, π, 0) corresponding to the rows shown in fig. (1). We refer to this plane as the out-of-phase Talbot plane. Similar to the linear case we look for an in-phase solution at 1/2 of the out-of-phase Talbot distance at 1/3z_t . Here, we find that the phases are small and have the structure :(.076, .078, .076); (.079, .086, .086, .079): (.076, .086, .098, .086, .076). Thus, this solution is almost in phase with a very small phase difference over the array. This solution acts in the far-field as an in-phase eigenvalue solution. We will consider these two planes in more detail in a moment.

Threshold is determined by solving

obtained from eq. (5) by setting σ ²=0. The threshold gain ḡ_i is related to the eigenvalue $\overline{\lambda }$ _i by ḡ_i =ln(1/r)+ln(1/|$\overline{\lambda }$ _i |) where r is the field outcoupling reflectivity. In fig. (2) we plot the three smallest threshold eigenvalue gains, ln(1/|$\overline{\lambda }$ _i |), along with the largest threshold eigenvalue gain all as functions of the normalized Talbot cavity length z/z_t for a field reflectivity of r=.8. This figure does not indicate the threshold supermode phase structure. Fig. (3) contains this information through the threshold eigenfunction coherence function C, see eq. (6), along with a repeat of the threshold eigenvalue gain for all 19 modes. Fig. (3a) is for in the in-phase plane (1/3z_t ) and fig, (3b) is for our out-of-phase plane (2/3z_t ). Now we return to fig. (2). This figure shows that the three lowest threshold eigenvalue gains are within 10% of one another; the only exception is the 1/3 plane which is the in-phase solution. With equal importance, figs. (3a,b) show that one of the lowest gains is an in-phase solution. In fig. (3a) the lowest threshold gain is the in-phase mode. However, in fig. (3b) the lowest threshold gain is the out-of phase mode with a gain of 2.02 while the next highest gain is the in-phase solution with a gain of 2.25 with C=.33 and 18.6, respectively.. Further more, when this coherence and threshold analysis is repeated for intermediate planes it again shows that one of the lowest two gains is an in-phase solution. Thus, because the threshold gains are nearly equal the lasing mode is not necessarily the threshold mode. In fact, the following simulations show that this is the case. This is not true for the linear array where the threshold eigenvalues are well separated and the lasing mode is the same as the threshold mode. Specifically, in the out-of phase plane the lowest threshold gain is, of course, the out-of phase mode, however, the next higher gain in this plane is about a factor of 4 greater[1].

 

Fig. 2. The three lowest threshold eigenvalue gains (colored) and the bounding largest threshold eigenvalue gain (black) as functions of the normalized Talbot length.

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Fig. 3. (a) the coherence function (green) and the threshold eigenvalue gain (black) for the 1/3 Talbot plane;(b), the coherence function and the threshold eigenvalue gain for the 2/3 Talbot plane. Both are functions of mode number.

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We now turn to the above threshold extraction by solving eqs. (1) for the 19 coupled amplitude differential equations. Figs. (4a,b) shows the on-axis intensity, and the Talbot cavity effective reflectivity as functions of the Talbot cavity length normalized to z_t . The far-field on-axis intensity displays a smooth monotonic decrease as the cavity length increases and the coherence function remains at 19, but is not shown. Thus, the lasing modes are always in-phase affirming our earlier argument that the lasing mode is different from the threshold mode. This behavior is completely different from the on-axis intensity for a linear array which shows a very ragged dependence on the cavity length[1]. In fig. (4b) we also show the effective Talbot cavity reflectivity for an individual emitter defined by R_i =Ā _i (L)/$A_i^{+}$ (L). One can easily see that the decreasing behavior is given by a (z/z_t )^-1 dependence due to the decreasing energy of the propagated field E_p in the Talbot cavity, see the Appendix. This z/z_t dependence is the origin of the similar decrease seen in fig. (4a). In this example z ₀≈.01cm, z_t ≈2cm for d=150µm, ω ₀=a/√2=10/√2µm and λ=1.55µm. Note that 5%<R_i <56% for .2<z/z_t <.8, and that R_i is independent of i, as it is a ratio.

 

Fig. 4. (a), the far-field on-axis intensity; (b), the Talbot cavity effective reflectivity as functions of the normalized Talbot cavity length.

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Optimum operating conditions are always of interest. For z=1/3z_t and z=2/3z_t , fig. (5a) shows the outcoupled far-field on-axis intensity (1-r ²)I(0) as a function of the intensity reflectivity r ². Here, I(0)=|∑A_i exp(ϕ_i )|² where A_i and ϕ_i are solutions of the coupled differential equations, eqs. (1). For a gain of 4 and a fill factor of .047 fig. (5a) clearly shows an optimum outcoupling intensity reflectivity of $r_{\text{opt}}^2$=18% and 20%, for z/z_t =1/3 and z/z_t =2/3 respectively. Fig. (5b) shows the forward recirculating power for the central most emitter as a function of the outcoupling reflectivity r ². Here, the recirculating powers in the central emitter are similar except the 2/3 plane is reduced. Also, our simulations show that the effective reflectivity R_i , as defined above, is locked at 28% for all emitters over the entire range of r ² for the 1/3 plane and at R_i =7.2% for the out-of-phase plane where z/z_t =2/3. For all cases the coherence function is a constant 19 indicating that the lasing mode is an in-phase mode. Comparing the graphs in fig. (5a) we see that the optimum reflectivity is a mild function of Talbot cavity length. However, it is a stronger function of gain. For example, for g=3, $r_{\text{opt}}^2$=23%; and for g=8, then $r_{\text{opt}}^2$=10%.

 

Fig. 5. (a), the optimum far-field outcoupled on-axis intensity for the the 1/3 plane (red), and the 2/3 plane (black); (b) the forward recirculating power for the central emitter in the 1/3 plane (red), and in the 2/3 plane (black). All graphs are functions of the intensity reflectivity r ².

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Our last topic is the decreased performance when the emitters have a linear stochastic phase[1]. Briefly, this is modeled by adding the same propagation phase to each emitter before and after each propagation. However, between all emitters the phase ensemble is random specified by some rms phase. Eqs. (1) are then integrated until convergence or non-convergence for many such ensembles. Thus, we introduce the locking probability as the ratio of the number of times the integrator converges to the total number of tries for a specific rms phase. This then allows an assessment of the lasing performance in a stochastic environment.

Our simulations have shown that the hexagonal array lases in an in-phase mode. Thus, consideration of just one Talbot plane is sufficient and for this we pick our threshold out-of-phase plane located at z/z_t =2/3. As a function of rms phase, Fig. (6a) shows the coherence function; fig. (6b) displays the outcoupled near-field intensity of the central emitter; fig. (6c) shows the far-field on-axis intensity of the array; and fig. (6d) shows the probability of locking. The average of the coherence function in fig. (6a) satisfies eq. (6) to within 10%. The decrease in the near-field emitter intensity, fig. (6b) is due to a decrease in extracted energy as the rms phase increases. Thus the on-axis intensity, fig. (6c), decreases much more rapidly than the coherence function due to the added decrease in the near-field extraction. Finally, the probability of locking remains near unity which means that the Talbot phase dominates. These figures are consistent because for a symmetric phased mode the central emitter near-field intensity of 1.3, shown in fig. (6b), times 19² gives the far-field on-axis intensity of approximately 450, shown in fig. (6c), all for a zero rms phase. This, again, conforms to an in-phase solution. These results are typical for any Talbot cavity length.

We conclude our analysis with a consideration of the most favorable fill factor by just plotting the amplitude of R_i,j for nearest, second-nearest, and third-nearest neighbors. The strong coupling region occurs when these three amplitudes are nearly equal. For the hexagonal array coupling to many neighbors beyond just the nearest neighbors requires the fill factor f be less than .13 for the out-of phase (2/3z_t ) plane and less than .08 for the in-phase (1/3z_t ) plane. We do not show these graphs since they are similar to those in our previous work[1].

We have only considered a double hexagonal array of 19 elements and did not present any results for a single array of just 7 elements. The reason is that the 7 element array solutions do not show the symmetry evident in the 19 element array and are more like the linear array results. In other words, the emitters of the 7 element array do not have as many identical neighbors as the 19 element array and thus the solutions are not as appealing.

4 Conclusion

We have simulated the coherence function, far-field on-axis intensity, near-field intensity of a 19 element hexagonal array in an Talbot cavity when each emitter has a linear stochastic propagation phase. We do this by solving 19 coupled differential equations by an iteration technique for the amplitude and the phase. This is done using a Rigrod model for the gain. Unique to the hexagonal array is its rotational symmetry which forces several of the lower threshold gains to be nearly equal. Additionally, one of these lower gain solutions is an in-phase solution. As a consequence the lasing solution is an in-phase mode for all Talbot cavity lengths. That is, the coherence function remains near 19 as the Talbot cavity length is changed.

We also identified the optimum reflectivity and found that it is near 20% for fill factors near .05 and a large small-signal gain of 4. The optimum reflectivity is a mild functions of outcoupling reflectivity. Finally, we investigated the decrease in far-field performance when the emitters have random linear propagation phase. We showed that the coherence function obeyed an exp(-σ ²) behavior to within 10% while the far-field on-axis intensity decreased much faster that this. Another manifestation of the rotational symmetry is that the locking probability remains near unity which means that the array will lase for rather large rms phase difference.

 

Fig. 6. (a), coherence function; (b) central emitter near-field intensity; (c), far-field on-axis intensity; (d) the locking probability. All are functions of the rms phase for the 2/3 Talbot plane.

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Appendix

In the following we develop the overlap integral R_i.j for a 2-dimensional array of symmetric Gaussians. Thus, the aperture function is

(A1)$$E_0(x,y,z)=\sum _{j=0}^N{E}_j\exp \left[i\frac{{\left(x-x_j\right)}^2+{\left(y-y_j\right)}^2}{w_o^2}\right].$$

For clarity, we list the 19 hexagonal elements with the coordinates: (x_j =j, y_j =-√3), 1≤j≤3; (x_j =1/2+(j-4), y_j =-√3/2), 4≤j≤7; (x_j =j-8, y_j =0), 8≤j≤12; with these coordinates flipped for positive y_j . The period in the x-direction is d and that in the y-direction is d√3/2. Also, if the laser aperture is a then a=2ω ₀/√2.

Fresnel propagation of the electric field a distance of z gives

(A2)$$E_P(x,z)=\frac{i\exp \left(ikz\right)}{⋋z}\int \int E_0(x,y)\exp (\frac{ik}{2z}\left[{\left(x-x\text{'}\right)}^2+{\left(y-y\text{'}\right)}^2\right]dx\text{'}dy\text{'}.$$

Integrating eq. (A2) along with eq. (A1) gives the propagated field

(A3)$$E_P(x,y,z)=\exp (ikz)\frac{1}{1+iz⁄z_0}\sum _{j=0}^N{E}_j\exp \left[i\frac{k}{2\left(z-i{z}_0\right)}\left({\left(x-x_j\right)}^2+{\left(y-y_j\right)}^2\right)\right]$$

where j labels a laser with coordinates (x_j, y_j ) as given above, z ₀=${\pi \omega }_0^2$/λ, and k=2π/λ.

Now we move to the reflection coefficient R_i,j . This is given by the overlap integral between the electric field propagated through a distance z and the initial electric field distribution at z=0. Thus,

(A4)$$R_{i,j}=\frac{\int G_i(x,y,0)*G_j(x,y,z)dxdy}{\int G_i(x,y,0)*G_i(x,y,0)dxdy}$$

where eq. (A3) identifies G_j (x, y, z) as

(A5)$$G_j(x,y,z)=\exp (ikz)\frac{1}{1+iz⁄z_0}\exp \left[i\frac{k}{2\left(z-i{z}_0\right)}\left({\left(x-x_j\right)}^2+{\left(y-y_j\right)}^2\right)\right].$$

Completing the integrations and forming the ratio gives the two-dimensional overlap integral as

(A6)$$R_{i,j}=\exp (ikz)\frac{1}{1+iz⁄2z_0}\exp \left[i\frac{k}{2\left(z-i2z_0\right)}\left[{\left(x_i-x_j\right)}^2+{\left(y_i-y_j\right)}^2\right]\right].$$

We close with contour plots of the Talbot intensity using eq. (A3) for a fill factor of .047. We only show the upper half of the pattern. Fig. (7) is in the Talbot plane z/z_t =1, the z/z_t =1/3 plane, and the z/z_t =2/3 plane. In the Talbot plane the 5 on-axis emitters, the next four emitters at y=√3/2, and the last 3 emitters at y=√3 are all positioned as in the original aperture. Thus, the original pattern is duplicated with some spillage at the aperture edge. In the z/z_t =1/3 plane the pattern becomes more complex with extra maxima inserted into the original hexagonal pattern. Here the peaks are narrower and more intense that in the Talbot plane. For z/z_t =2/3, the bottom figure, the pattern is still more complicated than the in-phase plane with many subsiderary maxima. There is no clear shifting of the original pattern in the out-of-phase plane as there is for the linear array.

 

Fig. 7. I(x, y) contour plot for z=z_t , z=1/3z_t , z=2/3z_t , from top to bottom. The maximum contour intervals are .0875–.1, .36–.4, .1225–.14, respectively.

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References and links

1. P. Peterson, A. Gavrielides, and M. P. Sharma, “Extraction characteristics of a one dimensional Talbot cavity with stochastic propagation phase,” Opt. Express 8, 670–682(2001), http://www.opticsexpress.org/oearchive/source/32913.htm [CrossRef]   [PubMed]  

2. David Mehuys, William Streifer, Robert G. Waarts, and Davie F. Welsh, “Modal analysis of linear Talbot-cavity semiconductor lasers,” Opt. Lett. 16, 823–825(1991). [CrossRef]   [PubMed]  

3. V. P. Kandidov, A. V. Kondrat’ev, and M. B. Surovitskii, “Collective modes of two-dimensional laser arrays in a Talbot cavity,” Quant. Elect. 28, 692–696(1998). [CrossRef]