## Extraction characteristics of a cw double-hexagonal Talbot cavity with stochastic propagation phase

Optics Express, Vol. 9, Issue 8, pp. 373-385 (2001)

http://dx.doi.org/10.1364/OE.9.000373

Acrobat PDF (6107 KB)

### 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(-*σ*^{2}), and that the near-field intensity decreases faster as the rms phase *σ*^{2} increases. All of the above behaviors are strongly influenced by the hexagonal array rotational symmetry.

© Optical Society of America

## 1 Introduction

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]

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]

*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.

*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

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]

**8**, 670–682(2001), http://www.opticsexpress.org/oearchive/source/32913.htm [CrossRef] [PubMed]

## 2 Theory

**8**, 670–682(2001), http://www.opticsexpress.org/oearchive/source/32913.htm [CrossRef] [PubMed]

*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.

*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

*z*) field and a reverse field

*z*) both described by

*C*

_{j}is given by

*z*=0, we have

*r*

_{j}

*z*=

*L*, continuity of the electric fields requires that the reverse field is composed as

*L*

_{i})=∑

*R*

_{i,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.

**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]

*σ*

^{2}=0 and for

*σ*

^{2}≠0 this leads to an increased threshold gain.

*C*as

*N*exp(-

*σ*

^{2}) for large

*N*.

## 3 Examples

*d*=150

*µ*; the emitting aperture

*a*is 10

*µ*m. 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.

*z*=2/3

*z*

_{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 1/3

*z*

_{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.

*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

**8**, 670–682(2001), http://www.opticsexpress.org/oearchive/source/32913.htm [CrossRef] [PubMed]

*R*

_{i}=

*Ā*

_{i}(

*L*)/

*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*=150

*µ*m,

*ω*

_{0}=

*a*/√2=10/√2

*µ*m and λ=1.55

*µ*m. 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.

*z*=1/3

*z*

_{t}and

*z*=2/3

*z*

_{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

*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,

*g*=8, then

**8**, 670–682(2001), http://www.opticsexpress.org/oearchive/source/32913.htm [CrossRef] [PubMed]

*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 19

^{2}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.

*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 (2/3

*z*

_{t}) plane and less than .08 for the in-phase (1/3

*z*

_{t}) plane. We do not show these graphs since they are similar to those in our previous work[1

**8**, 670–682(2001), http://www.opticsexpress.org/oearchive/source/32913.htm [CrossRef] [PubMed]

## 4 Conclusion

*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.

## Appendix

*R*

_{i.j}for a 2-dimensional array of symmetric Gaussians. Thus, the aperture function is

*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*=2

*ω*

_{0}/√2.

*z*gives

*j*labels a laser with coordinates (

*x*

_{j}

*, y*

_{j}) as given above,

*z*

_{0}=

*k*=2

*π*/λ.

*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,

*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.

## 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 |

2. | David Mehuys, William Streifer, Robert G. Waarts, and Davie F. Welsh, “Modal analysis of linear Talbot-cavity semiconductor lasers,” Opt. Lett. |

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. |

**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

- 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]
- 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]
- 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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