Coupling of multiple lasers into a single coherent source as a method for achieving a high-power, high-brightness source remains a persistent problem in optics. Coherent coupling is particularly attractive for compact, rugged, high efficiency semiconductor- or fiber-based lasers which have output powers limited by the small transverse dimensions of the gain medium. Coherent combining methods include phase control, stimulated Brillouin scattering, four-wave mixing, and other nonlinear processes. Generally, these techniques require additional optical elements that are external to the gain element, such as phase correctors, scattering cells, and these these techniques usually include free space propagation. A distinct method of creating a coherent beam from multiple gain elements, and one that is easily adaptable to an all-fiber or other guided-wave laser configuration, is a compound cavity. In such a configuration, intracavity coupling is used to enforce coherence between gain elements; for example, semiconductor lasers are coherently combined using Talbot-effect cavities [1
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]
]. More recently fiber lasers are attracting considerable attention as a high-power, solid-state laser system, and there are efforts underway to combine multiple elements into a compact, spatially coherent source [2
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]
Fig. 1. Diagram and nomenclature for the compound cavity.
The utility of intracavity coupling of fiber lasers was recently demonstrated by studying their spectral combining properties [4
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]
]. This particular compound cavity [4
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]
] was formed with two single-mode Erbium (Er)-doped fiber amplifiers, each with a high reflectivity fiber Bragg grating (FBG) as a back mirror, coupled through one-half of a 2×2 fused taper single-mode coupler that acted as the compound cavity output coupler. The coupler was cleaved at the center of the interaction region. This configuration provided a straightforward method to couple multiple elements through a common aperture. In this experiment multiplexing was emphasized, thus the input fiber Bragg gratings for the two lasers had slightly different resonant wavelengths. Additionally, they looked at the criteria for injection locking. However, they did not assess extraction or the modal and coherence properties of this type of compound cavity.
Fig. 2. Schematic of the fiber compound cavity. Additional lengths of fiber can be inserted between the FBGs and the amplifiers, and polarization control elements between the amplifiers and the 50/50 coupler.
Here, we demonstrate that a coupled fiber-amplifier configuration can spontaneously generate a highly coherent output with longitudinal mode characteristics that differ from the modes of the component lasers. We show that when two identical fiber amplifiers are coupled through a standard 50/50 fiber coupler they spontaneously operate on modes that maximize the constructive interference of the fields within the coupler. These measurements are combined with our analytical modeling to give a consistent picture of the intracavity coupling. In the following we first describe the experimental configuration, apparatus and results. Then we develop an analytical model. Finally, we conclude by comparing the two and showing agreement.
Our experiment corresponding to fig. (1)
is an all-fiber arrangement shown in fig. (2)
. The fiber is single transverse mode, with a mode pattern compatible with industry standard SMF-28 fiber. Each gain leg is an independent Er-doped fiber terminated with a high reflector fiber Bragg grating. Cladding-pumped Er-doped fiber amplifiers provide the gain and have a length of approximately 10m. Separate pumping to each amplifier is provided by laser diodes under independent bias control. The small signal gain of the amplifiers is so high, up to 40 dB, that there is quite a bit of flexibility in the insertion of intracavity elements. Independent fiber components are connected using FC/APC connectors to minimize stray back-reflection while maintaining ease of reconfiguration. The FBGs are commercially available products specified to operate as a high reflector on one of the 100 GHz wide channels of the ITU grid. The gratings used in these experiments were fabricated for the grid channel at 1557nm. Approximately 2 meters of fiber pigtail and an FC/APC connector separate the gratings from the amplifier elements. Additional lengths of fiber can be inserted at the connector point. The mutual coupling is through the 50/50 fiber coupler; this corresponds to the beam splitter in fig. (1)
. Approximately 2 meters of fiber separates the coupler from the amplifier elements. The common cavity originates at port B and terminates on an uncoated fiber collimator that provides about 4% back reflection into the cavity and acts as the common output coupler. The length of the fiber in the common cavity is about 2m. The field exiting port D, termed EBS
, is lost from the cavity. Optical isolators prevent stray back reflections into the cavity.
Photodiode #1 monitors the output transmitted through the fiber collimator after being collected at a second fiber collimator and is a measure of Eout
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 EBS
. The temporal characteristics of these signals were monitored using a digitizing oscilloscope.
We did not use polarization-maintaining fiber in the compound cavity but did manipulate the polarization. Initally, we simply twisted a fiber and clamped it down. Later we used commercial polarization control components inserted between the fiber amplifiers and the fiber coupler (not shown in fig. 2
). Polarization alignment at the coupling element was important for achieving a high degree of coherence between the output of the two gain legs of the coupled cavity. However, a single adjustment is sufficient for data taken over several hours. Using commercially available components in the configuration shown in fig. (2)
, the two component laser cavities had path lengths that could be matched within 1 cm. We also introduced length differences of up to approximately 33% by inserting lengths of passive fiber. The cavity mode spacing without additional fiber is approximately 6MHz corresponding to optical path length of about 16m × 1.53 = 25m in each leg. Because the mode spacing for the isolated lasers is approximately 6 MHz, this resolution does not permit the isolation of individual modes in the optical spectrum. However, we observed that the lasers operated on more than one mode simultaneously. Therefore, we investigated the individual mode characteristics by monitoring the power spectrum of the photodetected signal. Optical and power spectra of the solitary component lasers showed an output consists of only a sparse set of modes with significant optical power. Typically only 10 – 20 dominant modes appear out of the approximately 104
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.
shows the photodetector output from the uncoated fiber collimator for both of the two component lasers, and the compound configuration. A component laser is a result of turning off the pump laser diode to one of the fiber amplifiers. The Er-doped fiber of the unpumped amplifier is then a strong absorber so that, depending upon the component laser under test, port A or C of the fiber coupler becomes a second loss port. Coherence is evident in the compound cavity since its output is approximately four times that of one of the isolated amplifiers. We consider the origin of this factor of four in more detail following eq. (14
). Thus, almost all of the optical power exits the feedback surface and very little is lost through port D of the 50/50 beam coupler. A second feature of the output characteristics is that the time variation is relatively slow, corresponding to laboratory thermal changes, for example. Temporal changes corresponding to the time scale of the relaxation resonance (sub-millisecond) or the cavity mode spacing (sub-microsecond) were relatively small. However, there were significant power fluctuations on nanosecond and sub-nanosecond time scales, corresponding to the mode beating of the dominant modes.
Fig. 3. Time dependence of the compound (black) and component cavities (colored).
Fig. 4. (a) The extraction curves for the component cavities. (b) outcoupled power, left axis, and the power exiting the beam splitter, right axis for the compound cavity. The pump current is in amps.
Next we consider the steady state characteristics of the compound cavity. As a reference we begin with the component fiber amplifier. Fig. (4a)
shows the outputs of a single component cavity as monitored by both photodiodes as a function of the pump current. The threshold is near 0.3 amp and the extraction is linear; that is, there are no higher order processes like excited state absorption. In Fig (4b)
amplifier #2 is pumped until the signal from photodiode #1 reaches .1 Volt. Subsequently this component laser remains fixed and the ordinate in fig(4b)
shows the pumping to amplifier #1. As this pumping increases the output remains constant until threshold is reached at the component laser threshold of .3Amps. After this point the signal from the collimator grows while the photodiode signal from the loss port D goes through a minimum. The minimum occurs when the amplifiers are pumped to a similar output level. Note that the signal measured by photodiode number #1 is approximately four times, not two times, the single laser signal; see the discussion after eq. (14
). In this case almost all of the power is coupled to the fiber collimator and less than 10% is coupled through loss port D. Because the fused fiber coupler diverts the optical power based on the field characteristics in the interaction region, the direction of the optical power to one output arm requires a high degree of coherence between the two input beams. If we manipulate the polarization, we can achieve the situation where essentially equal output power to ports B and D is achieved in the coupled cavity configuration. This is expected for orthogonal polarizations of the input to the fiber coupler.
Fig. 5. Spectra of the component (colored) and the compound cavity (black).
Typical optical spectra for the two component amplifiers (colored) and the coupled configuration (black) are shown in fig. (5)
. Because the high reflection bandwidth of the FBGs is approximately 100GHz only part of the optical spectra is shown. The laser oscillates on several modes simultaneously but these modes come and go continuously on a time scale on the order of seconds. This time scale is consistent with small environmental changes in the laboratory, and the time scale of power fluctuations of the coupled cavity.
Finally, mode beating spectra that give information about individual modes are shown in Figures 6
. Figure 6
shows one mode beating frequency for the case when the two component cavities are nearly equal. It shows the mode-beating spectrum of the 79th harmonic of the fundamental mode frequency. To generate this spectra, the microwave spectrum analyzer is set on its “peak hold” setting that saves the highest power level at each frequency and the curves are generated over time as the laser modes randomly changed. Note that the frequencies of the peaks for the two component lasers are offset and that the peak of the compound cavity lies between them. The offset between the two component-laser frequencies corresponds to a fractional cavity-length mismatch of approximately 0.0008, or approximately 1 cm. With larger cavity mismatches a high degree of coherence is still maintained in the compound cavity. Figure 7
shows the mode beating spectra for the case when an additional 5 meters of fiber, corresponding to approximately a 0.3 fractional change in length, is added to one of the component cavities. Again, the “peak hold” setting of the spectrum analyzer is used to generate the spectra. Now the individual mode spectra differ and the compound-cavity modes are quite distinct from the component cavity modes. However, high output power is observed only on those compound cavity modes that are also simultaneously near resonant with a cavity mode from each of the two component cavities.
Fig. 6. Frequency offset of the 79
harmonic for both component lasers and the compound laser.
Fig. 7. Mode beating spectra of the two component cavities and the compound cavity. One of the two components cavities has approximately 5 meters of additional fiber inserted. The spectrum analyzer is operating on its “peak hold” setting and the spectra shown here are accumulated over a period of a couple of minutes with the beating frequencies filling in as the laser randomly changed oscillation modes. Note that the compound cavity only showed strong output on those modes that are simultaneously nearly resonant with the two component cavity modes.
In summary, the two-laser compound cavity is robust, and over 90% of the output is coherently coupled from the cavity through the common output coupler. The mutual locking of the two lasers occurs on a mode-by-mode basis and the cavity mode frequencies of the coupled cavity differ from those of the component cavities. In passing, we mention that we also built a four-element version of figure (2)
using a tree-type configuration of fiber couplers. Preliminary results with this configuration showed a much higher sensitivity to fluctuations in the laboratory environment, and typically only around 75% of the power was outcoupled and the power fluctuations were large.
In this section, with the details relegated to the appendix, we derive the steady state resonator eigenvalue equation, the outcoupled power, and the power exiting the beam splitter as functions of the gains Gi
, beam splitter reflectivity r
, output reflectivity R
, and the three different cavity lengths li
. We begin with the complex gain equation in the Rigrod approximation [5
5. A. E. Siegman, Lasers, (University Science Books, Mill Valley CA, 1986.)
] which we write as
is the single pass small signal gain. Ei
is either the forward going fields Fi
for the plus sign, or the reverse fields Bi
for the negative sign. These in turn decompose as
The boundary condtions at the beam splitter are
which is supplemented with the outcoupling boundary condition, F(0) = RB(0).
The eignevalue equation which satifies eqs. (1 –5
) is derived in the appendix and is
where the trignometric arguments are
Again, refering to the appendix eq. (A11
), the amplitudes are given by
A numerical solution of eq.(6
) gives the periodic roots Ψ
= 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
. 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.
Likewise the intensity exciting the cavity via the common beam splitter is
) all the fields are expanded in terms of longitudinal modes identified by the solution to the eigenvalue equation, eq. (6
). It is assumed that the detectors are slow and the frequency components corresponding to beating between longitudinal modes are not resolved. Thus, upon averaging the above equations only the terms of the same frequency are retained. Consequently, the diagonal terms give
If a single amplifier is disconnected, see fig.(2)
, the output intensity per mode of the remaining component fiber is
as can be seen from eqs. (8
These equations are the foundation for the power extraction analysis. The maximum extractable incoherent intensity from both isolated resonators is Iincoh
+ ln (R
= 1,2. The conditions for maximum coherent output power clearly requires that the resonator must be symmetric with t
= 1/2 and G
. Further, eq. (12
) and the following equations show that as both δl
become small the ratio Iout
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
is not small, say near 10% then this ratio is less than unity. Similarly, under the two above conditions of smallness the ratio of Iout
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
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.
Before leaving the theoretical development we consider two special cases of the eigenvalue equation. The first case is for the single cavity specified by either 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
> 0. Therefore, only even values of n
contribute to non trivial longitudinal modes in agreement with expectations. Thus, eq. (7
) leads to
which is in agreement with the usual single resonator conditions. Similar results hold if t
= 0. The second case is identical amplifiers characterized by G
= 1/√2, however l
. From eq. (6
) we find that
This equation has two solutions. The first is 2Ψδl = π+2nπ 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)
cos(nπδl). Thus, since βi
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 2π except at the transition from negative to positive where the separation of the modes is 3π. We note that our simulations show this behavior. In this case the mode separation is
except at the transition from n
even to n
odd. Thus, for the case of a nearly identical gain regions, l
in eq. (7
), the mode spacing is, again, determined by the component round trip length. Finally, we note that the modes are situated at νm
= 1, 2, 3…. However, the complexities of the spectra
are largely determined by the interference term (1 + cos(2Ψδl
)) in eq. (12
). In both cases the allowable modes are restriced by β
) > 0.
Fig. 8. Simulated extraction curves for the compound cavity. The red line is the outcoupled power and the black curve is power exiting the beam splitter.
In this section we first examine our experiment by simulating fig. (4b)
and by illustrating the spectra near and above threshold. Then, we investigate the more general changes in extraction as different parameters are altered. Prior to presenting these results we list the experimental parameters. The cavities lengths are l
= 14m, l
; the index-of-refraction is 1.53. This gives δl
= .03%. The reflectivities are R
= 4%, r
= 50%. The gains G
are in the neighborhood of 8 for a small signal gain of about 40dB. Threshold gain Gth
= 2.3 as can be seen from eq. (14
) for Icomponent
= 0. Our first result comes from eq. (17
) which gives a mode spacing of 6MHz which agrees with our experiment.
shows the steady state outcoupled power, Iout
, and the exiting beam splitter power IBS
as a function of the gain. This figure is obtained by solving the eigenvalue equation, eq(6
) for Ψ
and then inserting these values into the spectrum equation, eq. (8
), and the exiting intensity equations eqs. (12
). To conform with the experimental data, see fig. (4b)
, we set G
= 3.5 which is about 1.5 times above threshold. Then G
is varied from just above threshold, G
= 2.5, to a value of G
= 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 Icomponent
, see eq.(14
), has a value of 1.8. Thus, from fig. (8)
the ratio Iout
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.
Fig. 9. (a) Output spectra just above threshold. (b) Output spectra far above above threshold. The power is normalized to the total incoherent power.
Next we consider the spectra in more detail. Figs. (9a,9b)
shows the outcoupled power, the summand in eq. (12
), as a function of Ψ
. Figure (9a)
is near threshold for G
= 3.5 and fig. (9b)
is well above threshold for G
= 8. Both graphs are normalized to the available incoherent power Iincoh
+ ln (R
= 1,2. Both graphs clearly show the modulation Ψ
= 1, 600 which gives a frequency of νmod
= 10Ghz for Δl
= .01m. The distinguishing feature is that modes near νmod
/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
is an integer and gives a Δν
) = 6Mhz spacing.
Fig. 10. Power output as a function of δl
Our last subject is the outcoupled power as a function of detuning δl
). Fig. (10)
shows the output power Iout
given by eq. (12
) as a function of δl
% for r
= .5, G
= 8 and r
= .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
= .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
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.