Abstract

Phase locking of two fiber lasers is obtained by mutual injection. The coherence properties of this composite laser are analyzed. In the most general case, the radiations at the two output mirrors of the laser are not mutually coherent. The cross-correlation of the two emitted beams shows that an extra-cavity delay line is required to get stable and high visibility interference fringes. Such a laser configuration is attractive for coherent beam combining in the far field provided the multiple output beams are properly time delayed.

©2009 Optical Society of America

1. Introduction

Over the past few years, fiber lasers were extensively studied in particular to achieve high powers with low beam divergence. Fiber based technology is now competitive in comparison with bulk technology for applications such as micro-machining, welding and other material processing. One of the present challenges consists in boosting the power far above that produced by a single laser by combining the radiations of several lasers. The coherent combining approaches are either active or passive depending on whether the phase locking is made after parallel amplifications or inside a common multi-arm laser cavity. In the second case, the radiations are synchronized either by intra-cavity spectral filtering [14], by intra-cavity spatial filtering [57] or by mutual injection [8,9]. The main advantage of the two last configurations is that the emission is made up of several elementary beams. The combining only occurs in the far field where the in-phase beams are superposed. With the spectral filtering method, the interferences are in the frequency domain and the emitted beam is single-mode which limits the combined power. The present paper deals only with laser configurations providing an array of laser beams (at least two beams).

If coupling is obtained by spatial filtering, the elementary beams are mutually coherent (and more precisely in-phase) in the plane where the coupling element is placed. Therefore, high visibility interference fringes were observed in the far field provided the elementary beams have identical paths from the reference plane [8,9]. In the case of mutual injection, a fraction of the radiation from one laser is injected into the adjacent one and the emissions are coupled step by step. This is the main advantage of such a configuration where no intra-cavity component has to withstand high power density. Because there are several coupling elements in this architecture, it seems not straightforward to anticipate the area where high visibility interference fringes are obtained. In this paper, we analyze the coherence of the emission of two continuous-wave fiber lasers coupled by mutual injection. The coherence properties of the radiations coming from the two elementary lasers are studied in particular by cross-correlation of the laser fields. We first describe the experimental device made of two fibered cavities coupled by mutual injection through unbalanced couplers. We analyze the coupling effect on the emitted spectrum. This gives the temporal structure and periodicities of the laser fields but this information is not sufficient to locate the areas of mutual coherence. Then we measured the fringe visibility according to a time delay introduced between the elementary laser outputs. Finally, the results were compared to the visibility calculated from the impulse response of the composite cavity.

2. Phase-locking in a mutual injection laser

We studied a mutual injection laser (MIL) made of two fiber amplifiers. The schematic setup is depicted in Fig. 1. Experimental results reported in this paper were obtained by using two elementary lasers made of Yb3+ doped double clad single mode fibers, pumped through fiber combiners (FC). These linear cavities were closed on the rear side by a maximum reflection fiber Bragg grating (FBG) and on the front side the cleaved output fiber facet, providing a reflection of 4%, served as output coupler. The elementary cavities numbered 1 and 2, characterized by the optical lengths L1 and L2 respectively were mutually injected through two identical unbalanced fiber couplers (UFC) with a 70:30 splitting ratio. The 70% ports were spliced to the two branches of the elementary lasers, whereas the 30% ports of both couplers were connected to ensure the radiation transfer between lasers 1 and 2. The unused coupler ports were angle cleaved to avoid any parasitic feedback. In this configuration, each elementary laser delivered about 500 mW. The laser scheme clearly shows a third mutual cavity of optical length LM ended by the FBG’s, crossing both amplifying fibers and couplers. Basically, at resonance, the length of a cavity is an integer number of half the wavelength. By applying such a condition to the three sub-cavities of the MIL (Li=qiλ/2; i≡1, 2 or M; q: integer number; λ: resonance wavelength), we deduce a fourth resonance condition connecting the three characteristic cavity lengths:

ΔL=LM(L1+L2)=q.λ2withqanintegernumber
 

Fig. 1. Experimental setup of the mutual injection laser (FBG, fiber Bragg grating; FC, fiber combiner; UFC, unbalanced fiber coupler; YDF, Ytterbium doped fiber; AC, angle cleave; L1, L2 and LM, lengths of the three sub-cavities; P, plane including the laser outputs) and of the fringe analysis device (extra-cavity delay line, reflective prism and CCD camera).

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As a result of phase locking, the MIL delivers two beams with an identical and structured spectrum. The MIL resonant frequencies are the modal congruencies of the sub-cavities. The composite laser only amplifies these common longitudinal modes to reach a steady state operating regime. One of the spectrum modulations is periodic and inversely proportional to the length difference ΔL. An illustration is given in Fig. 2(a). This curve is the result of the product of three frequency combs in connection with the characteristic lengths (L1, L2, LM) of the MIL in the manner of a generalized Vernier effect. This spectral signature depending on ΔL is characteristic of the phase locking: the elementary beams carry the same optical frequencies. We experimentally verified this spectral behavior with a laser configuration identical to that presented in Fig. 1. Because of the long fiber lengths (L1=28.80 m, L2=36.56 m, LM=64.94 m), we measured the beat modes, by means of a 1GHz bandwidth photodiode connected to an electrical spectrum analyzer (Rohde&Schwarz Spectrum Analyzer 9kHz/7GHz). On Fig. 2(b), the modulation c/2ΔL (c speed of light) due to the three coupled cavities is clearly visible in the spectrum. As the mutual injection laser provides several output beams, a spectral analysis alone is not sufficient to check that these beams are mutually coherent. Nevertheless, this last situation becomes crucial for achieving the coherent combination of the whole output beams, in the far field for instance. Thus, we completed this study by an analysis of the mutual coherence of the two elementary beams.

 

Fig. 2. (a) Calculated product of the three resonance frequency combs of the MIL sub-cavities. (b) Measured electrical spectrum with the same sub-cavity lengths (L1=28.8 m, L2=36.56 m, LM=64.94 m).

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3. Analysis of mutual coherence of two mutually injected fiber lasers

The difference in the statistical properties of the fields U1 and U2 exiting the laser from its two separate outputs can be deduced from a measure of the degree to which their fluctuations are correlated [10]. The degree of coherence g12 is described by:

g12(τ)=G12(τ)I0

with

G12(τ)=+U1(t)U2*(t+τ)dt

U* being the complex conjugate of U.

G12 (τ) denotes the cross-correlation of U1 and U2 and τ stands for a time delay between the two laser outputs. I0 is the average intensity of the two beams supposed of equal amplitude. g12 (τ) is connected to the intensity of the interference pattern between these beams by the following relationship:

I(τ)=2I0[1+Re{g12(τ)}]

The mutual coherence can be estimated from the visibility V of the interference fringes according to the time delay:

V(τ)=Imax(τ)Imin(τ)Imax(τ)+Imin(τ)=g12(τ)

|g12| being the modulus of g12.

 

Fig. 3. Visibility of the interference fringes between the MIL outputs as function of their relative delay Δz=c.τ, (a) calculated from the impulse response of the MIL, (b) measured from the cross-correlation of the output beams. The red and green stars mark the Δz values for which the fringe pattern of Fig. 4(a) and Fig. 4(b) were respectively recorded.

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A high level of mutual coherence should lead to a visibility close to one. Firstly, we calculated the visibility from the impulse response of the composite cavity. A pulse of duration δt, inversely proportional to the laser bandwidth BW (BW=0.2 nm), was used as an input and launched through one of the two outputs of the mutual injection cavity. Then, we calculated the two pulse trains available from the outputs after several iterations across the MIL. A constant gain was considered to compensate for the cavity losses. The fringe visibility V(τ) was deduced from the calculated pulse trains according to the relation (5). A typical calculated visibility function is reported Fig. 3(a). The MIL characteristic lengths were L1=33.85 m, L2=33.76 m, LM=67.64 m. The curve exhibits several coherence peaks about 6 cm away from each other corresponding to the length difference 2ΔL in agreement with the beatings already noticed in the electrical spectrum of Fig. 2(b). In a second experimental step, the interference fringes between the output beams of the MIL were recorded by a CCD camera, and the visibility was deduced from the data (Fig. 1). Experimentally, a free space variable delay line gave the delay τ required to get the evolution of the fringes contrast versus the time lag (Fig. 3(b)). This contrast was calculated from snap shots of interference patterns. The experimental plot is consistent and in good qualitative agreement with the theoretical approach. We first note that the visibility function is dissymmetric because of the geometry of the cavity: dissymmetric in length of elementary lasers and dissymmetric in the position of the couplers. Secondly, it is clear, in the present experimental situation, that the two output beams are not mutually coherent in the plane P defined by the position of the two front mirrors of the cavity. It is therefore not possible to get the coherent combination of the elementary beams directly in the far field. The positions of the experimental coherence peaks confirm the significance of the length difference 2ΔL. At such positions, the fringe visibility can reach a high value close to the maximum of unity as shown on Fig. 4(a). The difference of visibility in these areas between the experimental and theoretical curves of Fig. 3 is due to the way the fringes are recorded (snapshots). Averaged measurements would have significantly decreased the value of the minimum of visibility to get closer to the numerical one. Moreover, the fringes are extremely stable in time in spite of the perturbations withstood by the laser due to environmental changes in the laboratory (temperature, vibrations…). The various peaks indicate the domains of mutual coherence between the MIL’s outputs. The peak width Lp corresponds to the coherence length of the laser given by its bandwidth BW (Lp=λ02/BW; λ0 laser average wavelength). Out of these peaks, the interference fringes became unstable; they shifted continuously back and forth and the poor contrast changed in time (Fig. 4(b)). Moreover, the figure 3 shows that coherence peaks are distributed symmetrically on both sides of the zero path length difference Δz. This general feature was confirmed numerically and experimentally whatever the characteristic lengths L1, L2, LM of the MIL, except for a particular configuration where ΔL=0. That can be considered as the most convenient case, because one can imagine the possibility to get coherent interaction without addition of a delay line. Nevertheless, in such a case, the spectrum modulations in connection with ΔL become of comparable magnitude to the laser bandwidth. As the absolute position of these large spectral modulations shift in the laser bandwidth depending on environmental perturbations, frequently there are only few modal congruencies. Consequently, the elementary lasers are only partially coupled, and deliver a combination of free oscillating frequencies with few common resonant frequencies. As observed experimentally, it leaded to unstable contrast and unstable transverse position of the interference fringes whatever the value of the delay Δz. The influence of such spectrum modulations was already observed in other coherent combining techniques [3,4] and was extensively studied in Ref. [3]. In a MIL system, because the ΔL=0 configuration appears unsuited to practical situation, it is essential to implement an extra-cavity delay line. In addition, an adjustment of the delay-line down to a fraction of wavelength is necessary to ensure in-phase emission and efficient coherent combining.

 

Fig. 4. Experimental fringe patterns between MIL outputs recorded for the two Δz values marked on Fig. 3(b): (a) into a mutual coherence area, (b) out of the mutual coherence areas.

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4. Conclusion

To conclude, we have analyzed the coherence properties of a pair of fiber lasers coupled by mutual injection. The observed modulated spectrum proves the phase-locking of the emitted beams. We have shown that on the contrary to techniques relying on spatial filtering for phase-locking of laser array, mutual coupling leads to more complex coherence features. In particular, in the most general case, the two laser beams are not mutually coherent right at the laser output. Cross-correlation between the laser beams provides a visibility function with multiple peaks that depends on the lengths of the MIL sub-cavities and on the laser bandwidth. Experiments have confirmed the theoretical expectations derived from a simple impulse response approach. The coherence peaks are distributed symmetrically on both sides of a null delay between the MIL fields. Therefore a high degree of mutual coherence between the two outputs of such a laser can be obtained at the expense of a proper extra-cavity delay line or by a proper positioning of the laser exits. Then, coherent combination can be obtained for instance in the far field. More complex behaviors can be expected from laser with a larger number of elementary cavities.

Acknowledgments

The authors thank ASTRIUM (B. Esmiller, ASTRIUM Space Transportation, Les Mureaux, France, Bruno.esmiller@astrium.eads.net) and CILAS (D. Sabourdy, sabourdy@cilas.com) for their support in the reported study.

References and links

1. T. B. Simpson, A. Gavrielides, and P. Peterson, “Extraction characteristics of a dual fiber compound cavity,” Opt. Express 10, 1060–1073 (2002). [PubMed]  

2. A. Shirakawa, T. Saitou, T. Sekiguchi, and K. Ueda, “Coherent addition of fiber lasers by use of a fiber coupler,” Opt. Express 10, 1167–1172 (2002). [PubMed]  

3. D. Sabourdy, V. Kermène, A. Desfarges-Berthelemot, L. Lefort, A. Barthélémy, P. Even, and D. Pureur, “Efficient coherent combining of widely tunable fiber lasers,” Opt. Express 11, 87–97 (2003). [CrossRef]   [PubMed]  

4. M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunning, D. L. Hammon, A. J. Solis, and L. Vaughan, “Self-organized coherence in fiber laser arrays,” Proc. SPIE 5335, 89–97 (2004). [CrossRef]  

5. J.C. Corcoran and F. Durville, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118–201118-3 (2005). [CrossRef]  

6. J. Lhermite, A. Desfarges-Berthelemot, V. Kermène, and A. Barthélémy, “Passive phase locking of an array of four fiber amplifiers by an all-optical feedback loop,” Opt. Lett. 32, 1842–1844 (2007). [CrossRef]   [PubMed]  

7. L. Li, A. Schülzgen, H. Li, V. L. Temyanko, J. V. Moloney, and N. Peyghambarian, “Phase-locked multicore all-fiber lasers: modeling and experimental investigation,” J. Opt. Soc. Am. B 24, 1721–1728 (2007). [CrossRef]  

8. Z. Chen, J. Hou, P. Zhou, and Z. Jiang, “Mutual injection-locking and coherent combining of two individual fiber lasers,” IEEE J. Quantum Electron. 44, 515–519 (2008). [CrossRef]  

9. B. Lei and Y. Feng, “Phase locking of an array of three fiber lasers by an all-fiber coupling loop,” Opt. Express 15, 17114–17119 (2007). [CrossRef]   [PubMed]  

10. B.E.A. Saleh and M.C. Teich, Fundamental of photonics (Wiley Series in Pure and Applied Optics, 1991). [CrossRef]  

References

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  1. T. B. Simpson, A. Gavrielides, and P. Peterson, “Extraction characteristics of a dual fiber compound cavity,” Opt. Express 10, 1060–1073 (2002).
    [PubMed]
  2. A. Shirakawa, T. Saitou, T. Sekiguchi, and K. Ueda, “Coherent addition of fiber lasers by use of a fiber coupler,” Opt. Express 10, 1167–1172 (2002).
    [PubMed]
  3. D. Sabourdy, V. Kermène, A. Desfarges-Berthelemot, L. Lefort, A. Barthélémy, P. Even, and D. Pureur, “Efficient coherent combining of widely tunable fiber lasers,” Opt. Express 11, 87–97 (2003).
    [Crossref] [PubMed]
  4. M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunning, D. L. Hammon, A. J. Solis, and L. Vaughan, “Self-organized coherence in fiber laser arrays,” Proc. SPIE 5335, 89–97 (2004).
    [Crossref]
  5. J.C. Corcoran and F. Durville, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118–201118-3 (2005).
    [Crossref]
  6. J. Lhermite, A. Desfarges-Berthelemot, V. Kermène, and A. Barthélémy, “Passive phase locking of an array of four fiber amplifiers by an all-optical feedback loop,” Opt. Lett. 32, 1842–1844 (2007).
    [Crossref] [PubMed]
  7. L. Li, A. Schülzgen, H. Li, V. L. Temyanko, J. V. Moloney, and N. Peyghambarian, “Phase-locked multicore all-fiber lasers: modeling and experimental investigation,” J. Opt. Soc. Am. B 24, 1721–1728 (2007).
    [Crossref]
  8. Z. Chen, J. Hou, P. Zhou, and Z. Jiang, “Mutual injection-locking and coherent combining of two individual fiber lasers,” IEEE J. Quantum Electron. 44, 515–519 (2008).
    [Crossref]
  9. B. Lei and Y. Feng, “Phase locking of an array of three fiber lasers by an all-fiber coupling loop,” Opt. Express 15, 17114–17119 (2007).
    [Crossref] [PubMed]
  10. B.E.A. Saleh and M.C. Teich, Fundamental of photonics (Wiley Series in Pure and Applied Optics, 1991).
    [Crossref]

2008 (1)

Z. Chen, J. Hou, P. Zhou, and Z. Jiang, “Mutual injection-locking and coherent combining of two individual fiber lasers,” IEEE J. Quantum Electron. 44, 515–519 (2008).
[Crossref]

2007 (3)

2005 (1)

J.C. Corcoran and F. Durville, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118–201118-3 (2005).
[Crossref]

2004 (1)

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunning, D. L. Hammon, A. J. Solis, and L. Vaughan, “Self-organized coherence in fiber laser arrays,” Proc. SPIE 5335, 89–97 (2004).
[Crossref]

2003 (1)

2002 (2)

Barthélémy, A.

Bruesselbach, H.

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunning, D. L. Hammon, A. J. Solis, and L. Vaughan, “Self-organized coherence in fiber laser arrays,” Proc. SPIE 5335, 89–97 (2004).
[Crossref]

Chen, Z.

Z. Chen, J. Hou, P. Zhou, and Z. Jiang, “Mutual injection-locking and coherent combining of two individual fiber lasers,” IEEE J. Quantum Electron. 44, 515–519 (2008).
[Crossref]

Corcoran, J.C.

J.C. Corcoran and F. Durville, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118–201118-3 (2005).
[Crossref]

Desfarges-Berthelemot, A.

Dunning, G. J.

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunning, D. L. Hammon, A. J. Solis, and L. Vaughan, “Self-organized coherence in fiber laser arrays,” Proc. SPIE 5335, 89–97 (2004).
[Crossref]

Durville, F.

J.C. Corcoran and F. Durville, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118–201118-3 (2005).
[Crossref]

Even, P.

Feng, Y.

Gavrielides, A.

Hammon, D. L.

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunning, D. L. Hammon, A. J. Solis, and L. Vaughan, “Self-organized coherence in fiber laser arrays,” Proc. SPIE 5335, 89–97 (2004).
[Crossref]

Hou, J.

Z. Chen, J. Hou, P. Zhou, and Z. Jiang, “Mutual injection-locking and coherent combining of two individual fiber lasers,” IEEE J. Quantum Electron. 44, 515–519 (2008).
[Crossref]

Jiang, Z.

Z. Chen, J. Hou, P. Zhou, and Z. Jiang, “Mutual injection-locking and coherent combining of two individual fiber lasers,” IEEE J. Quantum Electron. 44, 515–519 (2008).
[Crossref]

Jones, D. C.

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunning, D. L. Hammon, A. J. Solis, and L. Vaughan, “Self-organized coherence in fiber laser arrays,” Proc. SPIE 5335, 89–97 (2004).
[Crossref]

Kermène, V.

Lefort, L.

Lei, B.

Lhermite, J.

Li, H.

Li, L.

Mangir, M. S.

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunning, D. L. Hammon, A. J. Solis, and L. Vaughan, “Self-organized coherence in fiber laser arrays,” Proc. SPIE 5335, 89–97 (2004).
[Crossref]

Minden, M. L.

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunning, D. L. Hammon, A. J. Solis, and L. Vaughan, “Self-organized coherence in fiber laser arrays,” Proc. SPIE 5335, 89–97 (2004).
[Crossref]

Moloney, J. V.

Peterson, P.

Peyghambarian, N.

Pureur, D.

Rogers, J. L.

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunning, D. L. Hammon, A. J. Solis, and L. Vaughan, “Self-organized coherence in fiber laser arrays,” Proc. SPIE 5335, 89–97 (2004).
[Crossref]

Sabourdy, D.

Saitou, T.

Saleh, B.E.A.

B.E.A. Saleh and M.C. Teich, Fundamental of photonics (Wiley Series in Pure and Applied Optics, 1991).
[Crossref]

Schülzgen, A.

Sekiguchi, T.

Shirakawa, A.

Simpson, T. B.

Solis, A. J.

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunning, D. L. Hammon, A. J. Solis, and L. Vaughan, “Self-organized coherence in fiber laser arrays,” Proc. SPIE 5335, 89–97 (2004).
[Crossref]

Teich, M.C.

B.E.A. Saleh and M.C. Teich, Fundamental of photonics (Wiley Series in Pure and Applied Optics, 1991).
[Crossref]

Temyanko, V. L.

Ueda, K.

Vaughan, L.

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunning, D. L. Hammon, A. J. Solis, and L. Vaughan, “Self-organized coherence in fiber laser arrays,” Proc. SPIE 5335, 89–97 (2004).
[Crossref]

Zhou, P.

Z. Chen, J. Hou, P. Zhou, and Z. Jiang, “Mutual injection-locking and coherent combining of two individual fiber lasers,” IEEE J. Quantum Electron. 44, 515–519 (2008).
[Crossref]

Appl. Phys. Lett. (1)

J.C. Corcoran and F. Durville, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118–201118-3 (2005).
[Crossref]

IEEE J. Quantum Electron. (1)

Z. Chen, J. Hou, P. Zhou, and Z. Jiang, “Mutual injection-locking and coherent combining of two individual fiber lasers,” IEEE J. Quantum Electron. 44, 515–519 (2008).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Express (4)

Opt. Lett. (1)

Proc. SPIE (1)

M. L. Minden, H. Bruesselbach, J. L. Rogers, M. S. Mangir, D. C. Jones, G. J. Dunning, D. L. Hammon, A. J. Solis, and L. Vaughan, “Self-organized coherence in fiber laser arrays,” Proc. SPIE 5335, 89–97 (2004).
[Crossref]

Other (1)

B.E.A. Saleh and M.C. Teich, Fundamental of photonics (Wiley Series in Pure and Applied Optics, 1991).
[Crossref]

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Figures (4)

Fig. 1.
Fig. 1. Experimental setup of the mutual injection laser (FBG, fiber Bragg grating; FC, fiber combiner; UFC, unbalanced fiber coupler; YDF, Ytterbium doped fiber; AC, angle cleave; L1, L2 and LM, lengths of the three sub-cavities; P, plane including the laser outputs) and of the fringe analysis device (extra-cavity delay line, reflective prism and CCD camera).
Fig. 2.
Fig. 2. (a) Calculated product of the three resonance frequency combs of the MIL sub-cavities. (b) Measured electrical spectrum with the same sub-cavity lengths (L1=28.8 m, L2=36.56 m, LM=64.94 m).
Fig. 3.
Fig. 3. Visibility of the interference fringes between the MIL outputs as function of their relative delay Δz=c.τ, (a) calculated from the impulse response of the MIL, (b) measured from the cross-correlation of the output beams. The red and green stars mark the Δz values for which the fringe pattern of Fig. 4(a) and Fig. 4(b) were respectively recorded.
Fig. 4.
Fig. 4. Experimental fringe patterns between MIL outputs recorded for the two Δz values marked on Fig. 3(b): (a) into a mutual coherence area, (b) out of the mutual coherence areas.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

ΔL=LM(L1+L2)=q.λ2withqanintegernumber
g12(τ)=G12(τ)I0
G12(τ)=+U1(t)U2*(t+τ)dt
I(τ)=2I0[1+Re{g12(τ)}]
V(τ)=Imax(τ)Imin(τ)Imax(τ)+Imin(τ)=g12(τ)

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