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Optics Express

  • Editor: J. H. Eberly
  • Vol. 9, Iss. 8 — Oct. 8, 2001
  • pp: 373–385
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Extraction characteristics of a cw double-hexagonal Talbot cavity with stochastic propagation phase

P. Peterson, A. Gavrielides, and M. P. Sharma  »View Author Affiliations


Optics Express, Vol. 9, Issue 8, pp. 373-385 (2001)
http://dx.doi.org/10.1364/OE.9.000373


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Abstract

We solve the coupled cw electric field differential equations for an hexagonal array of fiber gain elements all sharing a common monolithic Talbot cavity mirror. A threshold analysis shows that the lowest gains are nearly equal, within 10% of one another, and that one of these corresponds to an in-phase supermode. Above threshold we study the extraction characteristics as a function of the Talbot cavity length, and we also determine the optimum outcoupling reflectivity. These simulations show that the lasing mode is an in-phase solution. Lastly, we study extraction when random linear propagation phases are present by using Monte Carlo techniques. This shows that the coherence function decreases as exp(-σ2), and that the near-field intensity decreases faster as the rms phase σ2 increases. All of the above behaviors are strongly influenced by the hexagonal array rotational symmetry.

© Optical Society of America

1 Introduction

2 Theory

We consider N gain elements coupled through a Talbot cavity terminated by a monolithic mirror of field reflectivity rt . The jth gain element has outcoupling field reflectivity rj and gain gj . With these assumptions, and in the Rigrod approximation, the jth gain element supports a forward Ej+ (z) field and a reverse field Ej¯(z) both described by

dAj±dz=±gj1+Aj±2+CjAj±2Aj±
(1)
Ej±(z)=Aj±(z)exp(ϕj±(z)).
(2)

Since the gain is real the phases satisfy

ϕj±(Lj)ϕj±(0)=±kjLj
(3)

and the constant Cj is given by

Cj=Aj+(z)Aj(z)=Aj+(Lj)Aj(Lj)=Aj+(0)Aj(0).
(4)

These four equations constitute our basic working equations.

Fig. 1. Double hexagonal array and coordinate nomenclature

At the outcoupling end, z=0, we have Ej+ (0)=rj Ej¯ (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 Ei¯ (Li )=∑Ri,jEj+ (Lj ). Ri,j is the complex reflection matrix developed by evaluating the overlap integral between the Fresnel propagated electric field of the jth emitter integrated over the aperture of the of the ith 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

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]

]

<Ei+(L)>=rexp(2gL)[Ri,i<Ei>+exp(σ2)jiNRi,j<Ej+(L)>].
(5)

which reduces to the deterministic form[2

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]

] 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

c=ΣEj(0)2ΣEj(0)2
(6)

<c>=1N<i,jNexp[i(ϕjϕi)>=1N[N+N(N1)exp(σ2)],
(7)

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 rj =r=.8, also we pick a value of gj =g=4 for the gain.

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=2/3zt . 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 1/3zt . 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

Ei+(L)rexp(2gL)jNRi,jEj+(L)=0
(8)

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

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 zt . 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

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]

]. In fig. (4b) we also show the effective Talbot cavity reflectivity for an individual emitter defined by Ri =Ā i (L)/Ai+ (L). One can easily see that the decreasing behavior is given by a (z/zt )-1 dependence due to the decreasing energy of the propagated field Ep in the Talbot cavity, see the Appendix. This z/zt dependence is the origin of the similar decrease seen in fig. (4a). In this example z 0≈.01cm, zt ≈2cm for d=150µm, ω 0=a/√2=10/√2µm and λ=1.55µm. Note that 5%<Ri <56% for .2<z/zt <.8, and that Ri 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.

Optimum operating conditions are always of interest. For z=1/3zt and z=2/3zt , 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)=|∑Ai exp(ϕi )|2 where Ai and ϕi 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 ropt2=18% and 20%, for z/zt =1/3 and z/zt =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 Ri , as defined above, is locked at 28% for all emitters over the entire range of r 2 for the 1/3 plane and at Ri =7.2% for the out-of-phase plane where z/zt =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, ropt2=23%; and for g=8, then ropt2=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.

Our last topic is the decreased performance when the emitters have a linear stochastic phase[1

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]

]. 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/zt =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 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.

Appendix

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

E0(x,y,z)=j=0NEjexp[i(xxj)2+(yyj)2wo2].
(A1)

For clarity, we list the 19 hexagonal elements with the coordinates: (xj =j, yj =-√3), 1≤j≤3; (xj =1/2+(j-4), yj =-√3/2), 4≤j≤7; (xj =j-8, yj =0), 8≤j≤12; with these coordinates flipped for positive yj . 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

EP(x,z)=iexp(ikz)zE0(x,y)exp(ik2z[(xx')2+(yy')2]dx'dy'.
(A2)

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

EP(x,y,z)=exp(ikz)11+izz0j=0NEjexp[ik2(ziz0)((xxj)2+(yyj)2)]
(A3)

where j labels a laser with coordinates (xj, yj ) as given above, z 0=πω02/λ, and k=2π/λ.

Now we move to the reflection coefficient Ri,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,

Ri,j=Gi(x,y,0)*Gj(x,y,z)dxdyGi(x,y,0)*Gi(x,y,0)dxdy
(A4)

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

Gj(x,y,z)=exp(ikz)11+izz0exp[ik2(ziz0)((xxj)2+(yyj)2)].
(A5)

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

Ri,j=exp(ikz)11+iz2z0exp[ik2(zi2z0)[(xixj)2+(yiyj)2]].
(A6)

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

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]

OCIS Codes
(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects
(140.3410) Lasers and laser optics : Laser resonators

ToC Category:
Research Papers

History
Original Manuscript: August 8, 2001
Published: October 8, 2001

Citation
Phillip Peterson, Athanasios Gavrielides, and M. Sharma, "Extraction characteristics of a cw double-hexagonal Talbot cavity with stochastic propagation phase," Opt. Express 9, 373-385 (2001)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-8-373


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References

  1. P. Peterson, A. Gavrielides, 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, M. B. Surovitskii, "Collective modes of two-dimensional laser arrays in a Talbot cavity," Quant. Elect. 28, 692-696 (1998). [CrossRef]

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