## Extraction characteristics of a one dimensional Talbot cavity with stochastic propagation phase

Optics Express, Vol. 8, Issue 12, pp. 670-681 (2001)

http://dx.doi.org/10.1364/OE.8.000670

Acrobat PDF (249 KB)

### Abstract

We study the near- and far-fields of a linear array of fiber lasers in an external Talbot cavity. Each emitter has a random optical path difference (OPD)phase due to length and dispersion differences. The individual emitter fields are described by forward and reverse differential equations in the Rigrod approximation with the Talbot cavity coupling all emitters through boundary conditions. We analytically determine the effect of the rms phase on the increase in the threshold, the decrease in the emitter amplitude, and the decrease in the far-field intensity. These results are confirmed numerically by using a Monte Carlo technique for the phase. This leads to a locking probability, a coherence function, and the on-axis intensity as functions of the rms phase. Another issue which we investigate is the cavity performance for inter-cavity and external cavity phasing and find the latter preferable. We also determine the strong coupling limit for the fill factor.

© Optical Society of America

## 1 Introduction

_{2}laser arrays, but not yet extensively for solid state fiber laser arrays. In the following we study steady state diffractive coupling of fibers when each has a stochastic linear propagation phase. This model incorporates the entire range of gains, losses, and arbitrary fill factors, all in a statistical environment. Additionally, we assess the change on the far-field performance for internal and external phase corrections as a function of Talbot cavity length.

1. D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. **53**, 1165–1167(1988). [CrossRef]

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. Robert Waarts, David Mehuys, Derek Nam, David Welch, and William Streifer, “High-power, cw, diffraction-limited GaAlAs laser diode array in an external Talbot cavity,” Appl. Phys. Lett. **58**, 2586–2588(1991). [CrossRef]

4. James R. Leger, “Lateral mode control of an AlGaAs laser array in a Talbot cavity,” Appl. Phys. Lett. **55**,334–336(1989). [CrossRef]

5. James R. Leger, Miles L. Scott, and Wilfrid B. Veldkamp, “Coherent operation of AlGaAS lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. **52**, 1771–1773(1988). [CrossRef]

7. William J. Cassarly, John C. Ehlert, J. Michael Finlan, Kevin M. Flood, Robert Waarts, Davie Mehuys, Derek Nam, and Davie Welch, “Intercavity phase correction of an external Talbot cavity laser with the use of liquid crystals,” Opt. Lett. **17**,607–609(1992). [CrossRef] [PubMed]

9. V. P. Kandidov and A. V. Kondrat’ev,“Collective modes of laser arrays in Talbot cavities of various geometries,” Quantum Electronics **27**, 234–238(1997). [CrossRef]

10. V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Yu. P. Koval, V. V. Kuz’minov, D. A. Mashkovskii, and A. M. Prokhorov, “Investigations of linear and two-dimensional arrays of semiconductor laser diodes in an external cavity,” Quantum electronics **27**, 850–854(1997). [CrossRef]

11. V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Yu. P. Koval, V. V. Kuz’minov, D. A. Mashkovskii, and A. M. Prokhorov, “Spatial phase locking of linear arrays of 4 and 12 wide-aperture semiconductor laser diodes in an external cavity,” Quantum electronics **28**, 257–263(1998). [CrossRef]

12. V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Yu. P. Kocal, V. V. Kuz’minov, D. A. Mashkovskii, and A. M. Prokhorov, “Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity,” Quantum electronics **28**, 344–346(1998). [CrossRef]

13. V. P. Kandidov and A. V. Kondrat’ev,“Influence of the Talbot cavity selectivity on the evolution of collective operation of diffraction-coupled laser arrays,” Quantum Electronics **28**, 972–976(1998). [CrossRef]

14. V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, V. V. Kuzminov, D. A. Mashkovsky, and A,. M. Prokhorov,“Phase-locking of the 2D structures,” Opt. Express **4**, 972–976(1999). [CrossRef]

15. M. Wrange, P. Glas, D. Fischer, M. Leitner, D. V. Vysotsky, and A. P. Napartovich, “Phase locking in a nulticore fiber laser by means of a Talbot resonator,” Opt. Lett. **25**, 1436–1438(2000). [CrossRef]

*N*emitters in the Rigrod approximation. Each emitter has a random propagation phase originating from different cavity lengths, and dispersion effects. Integration of these coupled differential equations yields the emitter amplitudes, phases, as well as the far-field patterns all for a specific rms phase. Coupling between emitters is through a boundary condition utilizing a Talbot cavity reflectivity. The reflection coefficient is obtained from the overlap between an emitter and the sum of all the Fresnel propagated fields, as has been done before.[1

1. D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. **53**, 1165–1167(1988). [CrossRef]

1. D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. **53**, 1165–1167(1988). [CrossRef]

## 2 Theory

*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}. The period of the array is

*d*and the fiber emitting aperture diameter is

*a*

_{j}Further we allow each laser to run on a wavelength

*λ*

_{j}. With these assumptions the the j

^{th}gain element supports a forward

*z*is the coordinate along the fiber axis. Equation (1) can be brought into an integrable form by separating the amplitude and phase according to

*L*

_{j}. Later we incorporate the different

*λ*

_{j}’s, and

*L*

_{j}’s into a random variable. An important consequence of eq. (2) is a constant

*C*

_{j}that satisfies

*z*=0, we have that

*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. This has been calculated before[1

**53**, 1165–1167(1988). [CrossRef]

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]

*L*

_{j}and incorporate these effects in a random phase

*ϕ*

_{j}associated with the j

^{th}emitter. In the small-signal region the field grows, according to eq. (6), exponentially as

*L*)=

*g*

_{j}

*L*). Inserting this into eq. (7) gives

*i*=

*j*), is straight forward. However, the off-diagonal average<exp[

*i*(

*ϕ*

_{j}-

*ϕ*

_{i})]>requires more work. First, since

*i*≠

*j*and the emitters are independent this average becomes the product <exp(

*iϕ*

_{j})><exp(

*iϕ*

_{i})]>. The last step is to complete the ensemble average of just one of the exponentials. This process has been developed for the turblenece transfer function[16], and for atomic decay processes[17]. Without going into these details we just quote the result that <exp(

*iϕ*

_{j}) >=exp(-

*σ*

^{2}/2) when the phase satisfies Gaussian statistics, has zero mean, with a mean square phase of

*σ*

^{2}. Thus eq. (9) becomes

*σ*

^{2}to zero, see ref. (1). In eq. (10) the amplitude of

*R*

_{i,j}contains different losses for different supermodes and the phase condition determines the supermode frequency.

*A*

_{j}>. To this end we turn to the saturated gain region where eq. (6) has the approximate solution

^{2}) and as a consequence the amplitudes

*C*as

*N*exp(-σ

^{2}) for

*N*large. We will numerically verify this behavior in the next section.

## 3 Numerical Simulations

*z*=0,

*r*

_{j}

*z*=

*L*,

*L*)=∑

*R*

_{i,j}

*L*) through an iteration scheme. Specifically, at

*z*=0,

*C*

_{j}is formed through

*C*

_{j}=

^{2}/

*r*. Integration of eq. (1) gives

*L*). In this process the phases just follow eq. (3). Next these values determine the initial condition on the reverse wave through

*L*)=∑

*R*

_{i,j}

*L*) and a new constant

*C*

_{i}formed using

*C*

_{i}=

*L*)

*L*). Integrating back to

*z*=0 yields

*C*

_{i}(0). The entire process is repeated until convergence is achieved. This technique is far superior to a shooting method, especially for a large number of equations.

*g*

_{j}, reflectivity

*r*

_{j}, and length

*L*

_{j}. The latter is included as a random propagation phase added to the total phase after each forward and reverse propagation. This allows an assessment of the loaded cavity performance in the presence of random phase. For a specific ensemble of phases {

*ϕ*

_{j}}, convergence is achieved when the variables do not change between the

*N*and the

*N*-1 iteration. Thus, we introduce the probability of locking

*P*as the number of converged cases divided by the total number of attempts for a set of

*M*ensembles all of which have the same mean and rms phase. This is a measure of how easily an array with a given rms phase will lase. In the following, the number of emitters,

*N*, is 6, the number of ensembles

*M*=30, and each representation is iterated at least 60 times. The Monte Carlo technique is embodied in creating the

*M*=30 ensembles all with differentt phase distributions but each with the same average and rms phase, then applying these to the differential equation.

*R*

_{i,j}and illustrate some of its phase properties. Here, we consider the amplitudes. Fig. (1) shows the amplitude of the first, |

*R*

_{3,4}|

^{2}(black curve); second, |

*R*

_{2,4}|

^{2}(red curve); and third, |

*R*

_{1,4}|

^{2}(green curve), nearest neighbor coupling in the half-Talbot plane as a function of the fill factor. This figure shows that when the fill factor, defined by

*f*=

*a*/

*d*where

*a*is the emitting aperture width and

*d*the period, is greater than .16 coupling is dominated by just the nearest neighbors. However, for

*f*=.08 the coupling between first nearest neighbors and second nearest neighbors are comparable, .18 compared to .12, with non-negligible contribution from the third nearest neighbors at .08. For

*f*<.08 the coupling becomes even more uniform. Thus, the filling factor should not be greater than .08 in the half-Talbot plane for strong coupling. This amplitude behavior is manifested in the Talbot cavity operation. Specifically, in the region where the coupling is dominated by just the nearest neighbors

*f*>.16 the cavity experiences very little loss and the cavity supermodes become less distinct. Thus, as

*f*increases the loss decreases and so does the threshold gain[1

**53**, 1165–1167(1988). [CrossRef]

*f*< .08, more of the energy couples to the end emitters and exits the cavity so that the loss increases and consequently the threshold gain[1

**53**, 1165–1167(1988). [CrossRef]

*f*>.16 the probability of locking is smaller than the strong coupling case for a given rms phase. In passing we mention that the same functions for the quarter-Talbot plane shows that the form factor should be less than about .04.

*g*

_{j}

*L*

_{j}=

*gL*=4, outcoupling

*r*

_{j}=

*r*=.8, and a Talbot mirror reflectivity,

*r*

_{t}, of unity. The integration of

*dz*is from zero to one. The remaining parameters are the array fill factor

*f*=

*ω*

_{0}/

*d*, and the position of the Talbot mirror. For the latter we concentrate on the half-Talbot plane

*z*=

*z*

_{t}/2=

*d*

^{2}/λ. This is the out-of-phase plane where the return image is displaced by

*d*/2 and there is a null on-axis in the far-field. We choose a period of 150

*µ*m and an aperture of diameter 10/√(2)

*µ*m. This gives a fill factor of 0.47 and a Talbot distance of

*z*

_{t}=3.cm for λ=1.55µm.

*ϕ*

_{j}(

*L*) of eq.(3) equal to zero at each step of the integration. This case is instructive since it can drastically alter the performance of the resonator by driving it below threshold or by decreasing the near-field amplitudes, as we will show. Experimentally internal phasing is difficult to implement since changing an individual phase alters all other phases. The other type of phase correction is external to the outcoupling mirror. This is simulated by allowing the integrator to converge without any constraints and then afterwards setting the phase to zero. These two cases are compared with the uncorrected Talbot resonator. Figures (2a,2b) show the on-axis intensity as a function of

*z*/

*z*

_{t}. In order to show the above mentioned threshold behavior fig (2a)i s for a gain

*gL*=8 while fig. (2b)is for half that value. Both figures show that the uncorrected Talbot cavity displays several in-phase solutions at

*z*=

*z*

_{t}/4, .75

*z*

_{t},

*z*

_{t}, and at several other intermediate cavity lengths. Also, these figures show that .4<

*z*/

*z*

_{t}< .63 the on-axis intensity is near zero. This is a manifestation of the out-of-phase solution and this region narrows as the number of emitters increases or as the fill factor decreases. For external phasing, both figures show that the maximum on-axis intensity occurs for

*z*/

*z*

_{t}=.5 and that the on-axis intensity is slightly greater than in the quarter-Talbot plane. The major difference between these two figures is the behavior of the internally phased resonator (green curve). Fig. (2a) shows that the on-axis intensity is more erratic and is always less than or equal to the externally corrected resonator (red curve). In fact, in fig (2b), for

*gL*=4, the internally corrected resonator has dropped below threshold except for very small cavity lengths, see the truncated green line. In principle this behavior could be predicted by solving our threshold condition with the constraint that the phases

*ϕ*

_{j}=0.

*C*=

*N*=6; that of the internally corrected is erratic bouncing between 4 and 6, while the uncorrected

*C*looks like that in fig. (2a) Next, we turn to the transverse characteristics. In all cases the far-field is given by the diffractive envelope exp(-

*π*

^{2}

*x*

^{2}/

*A*

_{m}exp(

*iθ*

_{m})×exp(

*iπmx*/

*d*). The amplitude

*A*

_{m}is a solution of eq. (5) and

*θ*

_{m}is a random phase. Thus, the divergence and separation between main lobes for the coherent case goes as the inverse of the period, while the incoherent divergence goes as the inverse of the laser apertures, see the averaging process used in eq. (10).

## 4 Conclusion

## Appendix

*N*Gaussians of period

*d*. Each Gaussian has a radius

*ω*

_{0}, where

*ω*

_{0}is the

*e*

^{-1}of the electric field. The initial electric field distribution is then

*a*=2

*ω*

_{0}/√2.

*z*the Fresnel propagated electric field is

*z*

_{0}=

*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,

*x*

_{m}=

*md*. This equation can be cast in a form which clearly shows the uniqueness of the Talbot distance

*z*

_{t}=2

*d*

^{2}/λ. In the usual configuration

*z*,

*z*

_{t}>>

*z*

_{0}, in our case

*z*

_{t}/

*z*

_{0}≈10

^{3}, and eq. (A5) becomes

*z*=

*z*

_{t}/2,

*z*

_{t}/4 which are determined by the phase of

*R*

_{i,j}. In the half-Talbot plane the phase factor of the coupling coefficient is an alternating ±1, which is the out-of-phase solution and in the quarter-Talbot plane the phase is always +1 which is the in-phase solution. The former displays a half-period shift in the half-Talbot plane which results in a far-field null on axis.

## References and links

1. | D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. |

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

3. | Robert Waarts, David Mehuys, Derek Nam, David Welch, and William Streifer, “High-power, cw, diffraction-limited GaAlAs laser diode array in an external Talbot cavity,” Appl. Phys. Lett. |

4. | James R. Leger, “Lateral mode control of an AlGaAs laser array in a Talbot cavity,” Appl. Phys. Lett. |

5. | James R. Leger, Miles L. Scott, and Wilfrid B. Veldkamp, “Coherent operation of AlGaAS lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. |

6. | R. G. Waarts, D. W. Nam, D. F. Welch, D. Mehuys, W. Cassarly, J. C. Ehlert, J. M. Finlan, and K. M. Flood, “Semiconductor laser array in an external Talbot cavity,” |

7. | William J. Cassarly, John C. Ehlert, J. Michael Finlan, Kevin M. Flood, Robert Waarts, Davie Mehuys, Derek Nam, and Davie Welch, “Intercavity phase correction of an external Talbot cavity laser with the use of liquid crystals,” Opt. Lett. |

8. | W. J. Cassarly, J. C. Ehlert, S. H. Chakmakjian, D. Harnesberger, J. M. Finlan, K. M. Flood, R. Waarts, D. Nam, and D. Welch “Automatec two-dimensional phase sensing and control using phase contrast imaging,” |

9. | V. P. Kandidov and A. V. Kondrat’ev,“Collective modes of laser arrays in Talbot cavities of various geometries,” Quantum Electronics |

10. | V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Yu. P. Koval, V. V. Kuz’minov, D. A. Mashkovskii, and A. M. Prokhorov, “Investigations of linear and two-dimensional arrays of semiconductor laser diodes in an external cavity,” Quantum electronics |

11. | V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Yu. P. Koval, V. V. Kuz’minov, D. A. Mashkovskii, and A. M. Prokhorov, “Spatial phase locking of linear arrays of 4 and 12 wide-aperture semiconductor laser diodes in an external cavity,” Quantum electronics |

12. | V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Yu. P. Kocal, V. V. Kuz’minov, D. A. Mashkovskii, and A. M. Prokhorov, “Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity,” Quantum electronics |

13. | V. P. Kandidov and A. V. Kondrat’ev,“Influence of the Talbot cavity selectivity on the evolution of collective operation of diffraction-coupled laser arrays,” Quantum Electronics |

14. | V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, V. V. Kuzminov, D. A. Mashkovsky, and A,. M. Prokhorov,“Phase-locking of the 2D structures,” Opt. Express |

15. | M. Wrange, P. Glas, D. Fischer, M. Leitner, D. V. Vysotsky, and A. P. Napartovich, “Phase locking in a nulticore fiber laser by means of a Talbot resonator,” Opt. Lett. |

16. | B. R. Frieden, it Probability, Statistical Optics, and Testing, (Springer-Verlag, New York, 1983), pp.186. |

17. | Murray Sargent III, Marlan O. Scully, and Willis E. Lamb, it Laser Physics, (Addisonn-Wesley Reading Mass., 1977), pp. 86. |

**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: April 11, 2001

Published: June 2, 2001

**Citation**

Phillip Peterson, Athanasios Gavrielides, and M. Sharma, "Extraction characteristics of a one dimensional Talbot cavity with stochastic propagation phase," Opt. Express **8**, 670-681 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-12-670

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### References

- D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, and A. Yariv, "Supermode control in diffraction-coupled semiconductor laser arrays," Appl. Phys. Lett. 53, 1165-1167 (1988). [CrossRef]
- 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]
- Robert Waarts, David Mehuys, Derek Nam, David Welch, William Streifer, "High-power, cw, diffraction-limited GaAlAs laser diode array in an external Talbot cavity," Appl. Phys. Lett. 58, 2586-2588 (1991). [CrossRef]
- James R. Leger, "Lateral mode control of an AlGaAs laser array in a Talbot cavity," Appl. Phys. Lett. 55, 334-336 (1989). [CrossRef]
- James R. Leger, Miles L. Scott, and Wilfrid B. Veldkamp, "Coherent operation of AlGaAS lasers using microlenses and diffractive coupling," Appl. Phys. Lett. 52, 1771-1773 (1988). [CrossRef]
- R. G. Waarts, D. W. Nam, D. F. Welch, D. Mehuys, W. Cassarly, J. C. Ehlert, J. M. Finlan, K. M. Flood, "Semiconductor laser array in an external Talbot cavity," Laser Doiode Technology and Applications, SPIE 1634, 288-298 (1992).
- William J. Cassarly, John C. Ehlert, J. Michael Finlan, Kevin M. Flood, Robert Waarts, Davie Mehuys, Derek Nam, and Davie Welch, "Intercavity phase correction of an external Talbot cavity laser with the use of liquid crystals," Opt. Lett. 17, 607-609 (1992). [CrossRef] [PubMed]
- W. J. Cassarly, J. C. Ehlert, S. H. Chakmakjian, D. Harnesberger, J. M. Finlan, K. M. Flood, R. Waarts, D. Nam, D. Welch "Automatec two-dimensional phase sensing and control using phase contrast imaging," Laser Diode Technology and Applications, SPIE 1634, 299-3091992).
- V. P. Kandidov, A. V. Kondrat'ev,"Collective modes of laser arrays in Talbot cavities of various geometries," Quantum Electronics 27, 234-238 (1997). [CrossRef]
- V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Yu. P. Koval. V. V. Kuz'minov, D. A. Mashkovskii, A. M. Prokhorov, "Investigations of linear and two-dimensional arrays of semiconductor laser diodes in an external cavity," Quantum electronics 27, 850-854 (1997). [CrossRef]
- V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Yu. P. Koval. V. V. Kuz'minov, D. A. Mashkovskii, A. M. Prokhorov, "Spatial phase locking of linear arrays of 4 and 12 wide-aperture semiconductor laser diodes in an external cavity," Quantum electronics 28, 257-263 (1998). [CrossRef]
- V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, A. A. Kazakov, Yu. P. Kocal. V. V. Kuz'minov, D. A. Mashkovskii, A. M. Prokhorov, "Phase locking of eight wide-aperture semiconductor laser diodes in one-dimensional and two-dimensional configurations in an external Talbot cavity," Quantum electronics 28, 344-346 (1998). [CrossRef]
- V. P. Kandidov, A. V. Kondrat'ev,"Influence of the Talbot cavity selectivity on the evolution of collective operation of diffraction-coupled laser arrays," Quantum Electronics 28, 972-976 (1998). [CrossRef]
- V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, V. V. Kuzminov, D. A. Mashkovsky, A,. M. Prokhorov,"Phase-locking of the 2D structures," Opt. Express 4, 19-26 (1999), http://www.opticsexpress.org/oearchive/source/8312.htm [CrossRef]
- M. Wrange, P. Glas, D. Fischer, M. Leitner, D. V. Vysotsky, A. P. Napartovich, "Phase locking in a nulticore fiber laser by means of a Talbot resonator," Opt. Lett. 25, 1436-1438 (2000). [CrossRef]
- B. R. Frieden, Probability, Statistical Optics, and Testing, (Springer-Verlag, New York, 1983), pp.186.
- Murray Sargent III, Marlan O. Scully, Willis E. Lamb, Laser Physics, (Addisonn-Wesley Reading Mass., 1977), pp. 86.

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