## Extraction characteristics of a dual fiber compound cavity

Optics Express, Vol. 10, Issue 20, pp. 1060-1073 (2002)

http://dx.doi.org/10.1364/OE.10.001060

Acrobat PDF (599 KB)

### Abstract

We study experimentally the time dependence, steady state behavior and spectra of a dual fiber-laser compound cavity. Theoretically we confirm the CW and spectral characteristics. This particular cavity is formed with two Er-doped fiber amplifiers, each terminated with a fiber Bragg grating, and coupled through a 50/50 coupler to a common feedback and output coupling element. The experiment and theory show that a low Q, high gain symmetric compound cavity extracts nearly 4 times the power of a component resonator. This extraction is maintained even when there is significant difference in the optical pathlengths of the two component elements. Further, our measurements and theory show that the longitudinal modes of the coupled cavity are distinct from the modes of the component cavities and that the coherence is formed on a mode-by-mode basis using these coupled-cavity modes. The time behavior of the compound cavity shows slow fluctuations, on the order of seconds, consistent with perturbations in the laboratory environment.

© 2002 Optical Society of America

## 1 Introduction

1. See, for example,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**, 19 (1999), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-1-19. [CrossRef] [PubMed]

2. P. K. Cheo, A. Liu, and G. G. King, “A High-Brightness Laser Beam from a Phase-Locked Multicore Yb-doped Fiber Laser Array,” IEEE Photon. Technol. Lett. **13**, 439 (2001). [CrossRef]

3. E. J. Bochove, “Theory of Spectral Beam Combining of Fiber Lasers,” IEEE J. Quantum Electron. **38**, 432 (2002). [CrossRef]

4. V. A. Kozlov, J. Hernandez-Cordoero, and T. F. Morse, “All-fiber Coherent Beam Combining of Fiber Lasers,” Opt. Lett. **24**, 1814 (1999). [CrossRef]

4. V. A. Kozlov, J. Hernandez-Cordoero, and T. F. Morse, “All-fiber Coherent Beam Combining of Fiber Lasers,” Opt. Lett. **24**, 1814 (1999). [CrossRef]

## 2 Experiment

*E*

_{BS}, is lost from the cavity. Optical isolators prevent stray back reflections into the cavity.

*E*

_{out}of figure (1). This photodiode has a bandwidth of greater than 10GHz allowing broadband power spectra to be generated using a microwave spectrum analyzer. Optical spectra also are generated using a heterodyne technique with a distributed feedback laser diode acting as the local oscillator. By ramping the bias current of the laser diode the optical frequency of the local oscillator is swept over a range of approximately 25GHz with a spectral resolution of approximately 100 MHz. This detector also provides a signal proportional to the average output power, with a bandwidth of a few kHz. This lower bandwidth signal is compared with the output of a second photodiode, photodiode #2 in figure (2), that monitors the optical power lost from the cavity at port D to measure

*E*

_{BS}. The temporal characteristics of these signals were monitored using a digitizing oscilloscope.

^{4}available modes within the high-reflection band of the fiber Bragg grating. Similar optical spectra occur under coherent coupling, except that isolated modes become mode clusters of multiple peaks within the 100 MHz resolution of the optical spectra.

## 3 Theory

*G*

_{i}, beam splitter reflectivity

*r*

^{2}, output reflectivity

*R*

^{2}, and the three different cavity lengths

*l*

_{i},

*L*. We begin with the complex gain equation in the Rigrod approximation [5] which we write as

*g*

_{i}is the single pass small signal gain.

*E*

_{i}is either the forward going fields

*F*

_{i}for the plus sign, or the reverse fields

*B*

_{i}for the negative sign. These in turn decompose as

*F*(0) =

*RB*(0).

_{i},

*i*= 0, ±1, ±2,….. The amplitude of the spectra is found by inserting these roots back into eqs.(8). It is clear that the solutions exhibit a periodicity in Ψ, and the intensities show a modulation given by

*π*/

*δl*. In the case of no coupling eq. (8) shows that the losses are determined by the logarithm of the outcoupling term

*R*. However, for the compound cavity the losses depend on Ψ, the propagation constant of each longitudinal mode. Therefore, threshold for a specific longitudinal mode depends critically on the value of the propagation constant.

*I*

_{incoh}= ∑

*I*

_{i}where

*I*

_{i}=

*G*

_{i}+ ln (

*R*),

*i*= 1,2. The conditions for maximum coherent output power clearly requires that the resonator must be symmetric with

*t*

^{2}=

*r*

^{2}= 1/2 and

*G*

_{1}=

*G*

_{2}=

*G*. Further, eq. (12) and the following equations show that as both

*δl*and

*R*

^{2}become small the ratio

*I*

_{out}/

*I*

_{incoh}is less than but approaches unity for arbitrary gain. That is, under these two conditions the total available incoherent power from the two isolated lasers is extracted as a coherent beam. However, if

*δl*or

*R*

^{2}is not small, say near 10% then this ratio is less than unity. Similarly, under the two above conditions of smallness the ratio of

*I*

_{out}/

*I*

_{component}is greater than 4 and only approaches 4 as the gain becomes large. These are our experimental conditions as shown in fig. (3). Again, if

*δl*or

*R*

^{2}is greater than about 10% or the gain is small this ration deviates from 4; a large gain would be greater than about 15. Thus, in order to extract the maximum coherent power the compound resonator must have a low Q and be symmetric.

*r*= 0 or

*t*= 0 in eq. (6). Picking the former condition gives Ψ(1 -

*δl*) =

*nπ*, for

*n*= 0, ±1, ±2,…. However, eq.(9) requires that

*β*

_{1}= (-1)

^{n}

*R*> 0. Therefore, only even values of

*n*contribute to non trivial longitudinal modes in agreement with expectations. Thus, eq. (7) leads to

*t*= 0. The second case is identical amplifiers characterized by

*G*

_{1}=

*G*

_{2},

*r*=

*t*= 1/√2, however

*l*

_{1}≠

*l*

_{2}. From eq. (6) we find that

*δl*=

*π*+2

*nπ*which leads to the nonlasing solution

*β*

_{1}=

*β*

_{2}= 0. The second solution is Ψ =

*nπ*,

*n*= 0, ±1,±2,… which leads to

*β*

_{1}=

*β*

_{2}=

*R*(-1)

^{n}cos(

*nπδl*). Thus, since

*β*

_{i}must be positive, cos(

*nπδl*) is positive for even

*n*, and cos(

*nπδl*) is negative for

*n*odd. Therefore, the mode are separated by exactly 2

*π*except at the transition from negative to positive where the separation of the modes is 3

*π*. We note that our simulations show this behavior. In this case the mode separation is

*n*even to

*n*odd. Thus, for the case of a nearly identical gain regions,

*l*

_{1}≈

*l*

_{2}in eq. (7), the mode spacing is, again, determined by the component round trip length. Finally, we note that the modes are situated at

*ν*

_{m}=

*mc*/(2(

*l*

_{1}+

*l*

_{2}+ 2

*L*)),

*m*= 1, 2, 3…. However, the complexities of the spectra

*δl*)) in eq. (12). In both cases the allowable modes are restriced by

*β*

_{1}=

*β*

_{2}=

*R*(-1)

^{n}cos(

*nπδl*) > 0.

## 4 Comparison

*l*

_{1}= 14m,

*l*

_{2}=

*l*

_{1}+ .01

*m*,

*L*= 2

*m*; the index-of-refraction is 1.53. This gives

*δl*= .03%. The reflectivities are

*R*

^{2}= 4%,

*r*

^{2}=

*t*

^{2}= 50%. The gains

*G*

_{1}and

*G*

_{2}are in the neighborhood of 8 for a small signal gain of about 40dB. Threshold gain

*G*

_{th}= 2.3 as can be seen from eq. (14) for

*I*

_{component}= 0. Our first result comes from eq. (17) which gives a mode spacing of 6MHz which agrees with our experiment.

*I*

_{out}, and the exiting beam splitter power

*I*

_{BS}as a function of the gain. This figure is obtained by solving the eigenvalue equation, eq(6) for Ψ

_{n}and then inserting these values into the spectrum equation, eq. (8), and the exiting intensity equations eqs. (12,13). To conform with the experimental data, see fig. (4b), we set

*G*

_{1}= 3.5 which is about 1.5 times above threshold. Then

*G*

_{2}is varied from just above threshold,

*G*

_{2}= 2.5, to a value of

*G*

_{2}= 5.3. This interval corresponds to the experiment. Note, in our simulations we allow 3 modes to run which is again consistent with the experiment. Both figs. (4b,8) clearly show a minimum in the beam splitter power when the two gains are equal; that is the resonator is symmetric as the theory predicts. Further, for a gain of 3.5

*I*

_{component}, see eq.(14), has a value of 1.8. Thus, from fig. (8) the ratio

*I*

_{out}/

*I*

_{component}is 6.4 which is greater than the value of 4 shown in fig. (3). The difference is attributable to more losses in the experiment than have been accounted for in the model.

_{n}/2

*π*. Figure (9a) is near threshold for

*G*

_{1}=

*G*

_{2}= 3.5 and fig. (9b) is well above threshold for

*G*

_{1}=

*G*

_{2}= 8. Both graphs are normalized to the available incoherent power

*I*

_{incoh}= ∑

*I*

_{i}where

*I*

_{i}=

*G*

_{i}+ ln (

*R*),

*i*= 1,2. Both graphs clearly show the modulation Ψ

_{mod}=

*π*/2

*δl*= 1, 600 which gives a frequency of

*ν*

_{mod}=

*c*/2

*n*Δ

*l*= 10Ghz for Δ

*l*= .01m. The distinguishing feature is that modes near

*ν*

_{mod}/2 experience destructive interference and do not lase for small gains, but appear at larger gains. Although this figure does not resolve the mode spacing a closer look show that ΔΨ

_{n}/2

*π*is an integer and gives a Δ

*ν*=

*c*/(

*l*

_{1}+

*l*

_{2}+ 2

*L*) = 6Mhz spacing.

*δl*= (

*l*

_{1}-

*l*

_{2})/(

*l*

_{1}+

*l*

_{2}+ 2

*L*). Fig. (10) shows the output power

*I*

_{out}given by eq. (12) as a function of

*δl*% for

*r*

^{2}= .5,

*G*

_{1}=

*G*

_{2}= 8 and

*r*

^{2}= .04; note that we are still allowing only 3 modes as before. The graph is symmetric about the maximum output of 115 which occurs at

*δl*= 0. The maximum output drops by 15% for a relative length difference of

*δl*= .03%. If the outcoupling is increased to

*R*

^{2}= .5 the maximum is increased to a value of 138, but the remainder of the graph stays as in fig. (10). Also, we mention that our simulations show that for a change in

*r*

^{2}of 20% results in a decrease of the

*δl*= 0 peak of about 2%. Thus, the reflectivity as well as the path lengths do not have to be accurately specified.

## 5 Summary

## Appendix

*f*

_{i}(

*z*)

*b*

_{i}(

*z*) = constant = f

_{i}(L)b

_{i}(L) = f

_{i}(L + l

_{i})b

_{i}(L + l

_{i}). The third property comes from the imaginary part of eq. (1) and gives

*ϕ*

_{i}(

*L*) -

*ϕ*

_{i}(

*L*+

*l*

_{i}) =

*k*

_{i}

*l*

_{i}and

*β*

_{i}(

*L*+

*l*

_{i}) -

*β*

_{i}(

*L*) =

*k*

_{i}

*l*

_{i}. However, at

*z*=

*L*+

*l*

_{i}the phase is preserved, that is

*ϕ*(

*L*+

*l*

_{i}) =

*β*(

*L*+

*l*

_{i}) since the imposed field boundary condition is

*F*

_{i}(

*L*+

*l*

_{i}) =

*B*

_{i}(

*L*+

*l*

_{i}). Thus,

*F*

_{i}(

*L*+

*l*

_{i}) =

*B*

_{i}(

*L*+

*l*

_{i}), to eq. (A1). This gives

*δl*is the normalized gain length mismatch, and

*δ*is the OPD mismatch.

## References and links

1. | See, for example,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 |

2. | P. K. Cheo, A. Liu, and G. G. King, “A High-Brightness Laser Beam from a Phase-Locked Multicore Yb-doped Fiber Laser Array,” IEEE Photon. Technol. Lett. |

3. | E. J. Bochove, “Theory of Spectral Beam Combining of Fiber Lasers,” IEEE J. Quantum Electron. |

4. | V. A. Kozlov, J. Hernandez-Cordoero, and T. F. Morse, “All-fiber Coherent Beam Combining of Fiber Lasers,” Opt. Lett. |

5. | A. E. Siegman, |

**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(140.3410) Lasers and laser optics : Laser resonators

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 30, 2002

Revised Manuscript: September 20, 2002

Published: October 7, 2002

**Citation**

T. Simpson, A. Gavrielides, and Phillip Peterson, "Extraction characteristics of a dual fiber compound cavity," Opt. Express **10**, 1060-1073 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-20-1060

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### References

- See, for example, V. V. Apollonov, S. I. Derzhavin, V. I. Kislov, V. V. Kuzminov, D. A. Mashkovsky andA . M. Prokhorov, �??Phase-locking of the 2D Structures,�?? Opt. Express 4, 19 (1999), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-1-19">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-1-19</a>. [CrossRef] [PubMed]
- P. K. Cheo, A. Liu, and G. G. King, �??A High-Brightness Laser Beam from a Phase-Locked Multicore Yb-doped Fiber Laser Array,�?? IEEE Photon. Technol. Lett. 13, 439 (2001). [CrossRef]
- E. J. Bochove, �??Theory of Spectral Beam Combining of Fiber Lasers,�?? IEEE J. Quantum Electron. 38, 432 (2002). [CrossRef]
- V. A. Kozlov, J. Hernandez-Cordoero, and T. F. Morse , �??All-fiber Coherent Beam Combining of Fiber Lasers,�?? Opt. Lett. 24, 1814 (1999). [CrossRef]
- A. E. Siegman, Lasers, (University Science Books, Mill Valley CA, 1986.)

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