Abstract

In this manuscript, the multi-wavelength active coherent polarization beam combining (CPBC) system is investigated theoretically and experimentally. The relationship between the combining efficiency and the optical path difference (OPD), wavelength number, and the spectral density of power of the amplifier chains is analyzed and validated by establishing a two-channel multi-wavelength CPBC system. Further, the relationship between the combining efficiency and the voltage signal of the photo-detector is developed and validated experimentally. Finally, the feasibility of the active CPBC technique with complex spectral structures is verified and as high as 96% combining efficiency is obtained based on all fiber delay lines to compensate the OPD between different channels, which is crucial for further power scaling of the CPBC system. Our theoretical analysis offers a useful approach to estimate the influence of OPDs, wavelength number, and the spectral density of power of the amplifier chains on multi-wavelength active CPBC system.

© 2014 Optical Society of America

1. Introduction

High power fiber lasers/amplifiers have great potential in numerous scientific and industrial fields. However, the undesirable effects, such as fiber damage, nonlinear effects, and mode instability, will ultimately limit the power scaling of the monolithic amplifier/laser [1, 2]. Coherent beam combination is an alternative approach to overcome the limitations aforementioned. The fundamental issue in coherent beam combination is to achieve the mutual coherence among different channels. The most prevalent scheme is based on master oscillator power amplifier (MOPA) architecture, where a common seed is amplified by different gain elements, and then coherent combined by the servo phasing (active phasing) system [311]. Also, mutual coherence can be established by self-organization mechanism [1217]. Just considering the mutual coherence, a strictly single frequency seed laser is the optimal source for the CBC configurations [35]. However, due to the limitation of stimulated Brillouin scattering (SBS) effect, the power scaling of the individual channel in active CBC architecture was difficult, thus the maximal output power from a coherently combined single frequency laser array is at 1 kW power level up to now [4, 5]. In order to mitigate the influence of SBS effect, multi-wavelength or narrow linewidth (GHz) seed laser was incorporated into the active CBC configurations [610]. In this circumstance, by adjusting the optical differences (OPDs) to ensure the coherence, the combining effect was ensured successfully in the experiment. Nowadays, the 4 kW combined output power had been demonstrated by employing eight narrow linewidth (<10 GHz) fiber amplifiers [10] and the extendibility of active CBC had been verified [11]. Nevertheless, in the previous tiled array active CBC configurations, some power will be spread into the side-lobes, thus the beam quality and power concentration of the coherently combined beam will be degraded [18]. Recently, some active filled-aperture configurations have been proposed to overcome side-lobes, such as CBC based on diffractive optical element (DOE) [19], re-imaging waveguide [20], and coherent polarization beam combination [2125].

In the common active CBC configurations that without filled-aperture components, the influence of OPDs, spectral width, and wavelength number on the combining effect was analyzed theoretically [26]. As for the filled aperture active combining configurations, the influence of phase noise, pointing errors, divergence errors, beam size errors, and wave-front errors were also investigated theoretically [27, 28]. However, as for the discrete wavelength number, for example multi-seed active CBC system [6, 7] and/or active CBC system by phase modulating the seed laser [9, 29, 30], the spectral width is difficult to define and it is a plausible concept to depict the spectral structures. Furthermore, the combining effect of the CBC system is tightly related to the spectral density of power of the amplifiers. In this manuscript, we concentrate on the influence of OPDs, wavelength number, and the spectral density of power of the amplifiers on the combining efficiency of a new filled-aperture active CBC configuration (coherent polarization beam combination system) theoretically and experimentally.

2. Principle and theoretical analysis

The achievement of active CPBC can be simply illustrated as follows. When two orthogonally linear-polarized beams are combined by a polarization beam combiner (PBC), without phase controlling, the polarization state distribution of the combined beam is uncertain, as is the mixture of the two injected polarizations and related to the phase difference between the two injected beams. Thus, further combination is difficult because that remarkable power loss exists in the next PBC (as shown in Fig. 1(a)). However, when the phase difference (δ) between the two orthogonally linear-polarized beams is controlled and set to δ = nπ, where n is an integer, the polarization state of the combined beam is still linear-polarized, so it can be completely permeated in the next PBC by using a half wavelength plate (HWP) to rotate the polarization direction of the combined beam (see Fig. 1(b)). Thus it can be further combined with another linear-polarized beam, and multi-channel beams can be coherently polarization combined by phase controlling [21].

 figure: Fig. 1

Fig. 1 The schematic of the polarization beam combination process for (a) without phase controlling and (b) phase controlling.

Download Full Size | PPT Slide | PDF

In this section, we will establish analytical expressions between the combining efficiency of the unit of CPBC system and the influence factors of OPDs, wavelength number and spectral density of power of the amplifier chains. The analysis of the extendibility of the CPBC system can be achieved by repeating the recurrence process to multi-channel. The simplified schematic of the unit of CPBC system is shown in Fig. 2.

 figure: Fig. 2

Fig. 2 The simplified schematic of the unit of CPBC system.

Download Full Size | PPT Slide | PDF

Due to the influence of aforementioned factors, the polarization combined beam is not a pure linearly polarized one, so a small portion of power will still leak into the other port of PBC2. The leakage power in PBC2 cannot participate into next stage of polarization beam combination. The combining efficiency of the unit of CPBC system is depicted as

η=P/P0

In Eq. (1), P is the maximum output power in port1 of the PBC2 and P0 is the total power of the two injected laser beams.

Assuming that the single frequency seed laser is modulated into N discrete frequencies [9, 29, 30], the normalized complex amplitude of the optical field of the injected beam i (i = 1, 2) can be expressed by

Ei=n=1NEi,vnexp(j2πvnti)(i=1,2)
where
Ei,vn=2Pi,vnπ.1w.exp[(x2+y2)w2]exp(jϕi)
Pi,vn=|Ei,vn|2dxdy
ti=lic
where Φi is the phase noise of the ith amplifier chain, Pi,vnis the spectral density of power of beam i with frequency vn, w is the beam waist of each fiber laser beam, li is the optical path of beam i, and c is the light velocity in the vacuum.

Thus the Jones matrix of the combined beam in PBC1 can be expressed as

E(1)=[n=1NE1,vnexp(j2πvnt1)m=1NE2,vmexp(j2πvmt2)]

After PBC1, the combined intensity can be expressed by

I1=[n=1NE1,vnexp(j2πvnt1)][p=1NE*1,vpexp(j2πvpt1)]+[m=1NE2,vmexp(j2πvmt2)][q=1NE*2,vqexp(j2πvqt2)]=n=1Np=1NE1,vnE*1,vpexp[j2π(vnvp)t1]+m=1Nq=1NE2,vmE*2,vqexp[j2π(vmvq)t2]=n=1N|E1,vn|2+n=1Np=1pnNE1,vnE*1,vpexp[j2π(vnvp)t1]+m=1N|E2,vm|2+m=1Nq=1qmNE2,vmE*2,vqexp[j2π(vmvq)t2]

Considering that the spectral interval of the discrete spectrum is normally at MHz level for high power operations [29, 30], and Ei,vnis varied slowly along with time, so the long time integral of the cross items for different frequencies in Eq. (7) is approximately zero, thus the power of the combined beam in PBC1 can be calculated to be

PPBC1=n=1NP1,vn+n=1NP2,vn

Assuming that the polarization directions of beam 1 and beam 2 are parallel to the x-direction and y-direction, respectively, and the angle between the principal axis of the HWP and the x-direction isα, the Jones matrix of the HWP in front of PBC2 can be expressed by

JHWP=[cos(2α)sin(2α)sin(2α)cos(2α)]

By combination of Eq. (6) and Eq. (9), the Jones matrix of the combined beam that in front of the PBC2 can be expressed as

E'=[cos(2α)n=1NE1,vnexp(j2πvnt1)+sin(2α)m=1NE2,vmexp(j2πvmt2)sin(2α)n=1NE1,vnexp(j2πvnt1)cos(2α)m=1NE2,vmexp(j2πvmt2)]

Due to that the long time integrals of the cross items for different frequencies are approximately zeros, from Eq. (4) and Eq. (10), the laser power of the y-direction of PBC2 can be calculated to be

Py=12n=1NP1,vn[1cos(4α)]+12n=1NP2,vn[1+cos(4α)]sin(4α)n=1NP1,vnP2,vncos[ϕ1ϕ2+2πvn(l1l2)c]

From Eq. (11), we see that, without phasing, the laser power of the y-direction of the PBC2 is fluctuating along with time due to that the phase noise difference between the two beams is changing along with thermal effect, experimental vibrations and so forth. Without losing generality, we assume that the laser power of the x-direction of the PBC2 is locked to be maximal and correspondingly the laser power of the y-direction is at minimal state (as shown in Fig. 2). Thus, the metric function of the active phasing algorithm is corresponding to the amplitude of the voltage signal transformed by the PD [2124]. The voltage signal transformed by the PD incorporates the information of phase difference between the two channels. Normally, this phase difference information can be extracted by modulation and demodulation technique [3, 5, 2124, 31]. In a practical system, when phase control is performed, the phase modulation and demodulation signals are continuously imposed and updated to the phase modulators of the two channels. Eventually, the phase difference between the two channels is compensated effectively and the laser power of the x-direction of the PBC2 can be locked to be at maximal state. For simplicity, we neglect the specific compensation process of the phase controller, and employ δ(t) to denote the phase compensation item for the multi-wavelength active CPBC system. With phase controlling, the laser power of the y-direction of PBC2 is expressed by the formula

Py=12n=1NP1,vn[1cos(4α)]+12n=1NP2,vn[1+cos(4α)]sin(4α)n=1NP1,vnP2,vncos[ϕ1ϕ2+2πvn(l1l2)c+δ(t)]

Besides, from Eq. (12), we also see that there exists an optimalαso that the combining efficiency of the active multi-wavelength CPBC system can be further at optimal state. By setting dPy/dα = 0, we obtains that

P'y=12n=1NP1,vn+12n=1NP2,vn124{n=1NP1,vnP2,vncos[ϕ1ϕ2+2πvn(l1l2)c+δ(t)]}2+(n=1NP1,vnn=1NP2,vn)2

Thus, the combining efficiency of the unit of CPBC system can be written by the formula

η=1P'ytn=1NP1,vn+n=1NP2,vn

In Eq. (14), < P′y>t is the averaged laser power of the y-direction of PBC2 when the system is at the optimal state.

In previous analysis, due to that we focused on the influence of OPD and spectral structures on the CPBC system, so we neglect the limited PERs of the two injected beams. The influence of limited PERs on the CPBC system will be discussed in the following section. Due to that the combining efficiency incorporates the influence of the OPD, wavelength number, and the spectral density of power of each amplifier chain, so their influence on the combining efficiency of the active multi-wavelength CPBC system can be analyzed specifically.

3. Theoretical and experimental results and discussions

The experimental architecture to implement two-channel active multi-wavelength CPBC system is shown in Fig. 3. The seed laser is a linear-polarized and single-frequency laser with a central wavelength of 1064.4 nm with ultra-short-cavity [32]. Firstly, the seed laser is modulated by a phase modulator to spread the single frequency spectrum into multi-wavelength spectrum. The maximal modulation frequency (f) and maximal modulation depth (d) of the signal generator imposed in the phase modulator are 100 MHz and 10 V, respectively. Then the modulated seed laser is amplified by a pre-amplifier, and the output power of the seed laser can be scaled to about 200 mW. Behind the pre-amplifier, the laser is split into two channels by a 50:50 fiber splitter. After the splitter, the laser of each channel is injected into a LiNbO3 phase modulator with 150 MHz electro-optical bandwidth and 2.5 V half wavelength voltages. After the phase modulator, the output power of each amplifier is about 40 mW. The loss of laser power is mainly induced by the insertion loss of the modulators and the fiber splitter. Then each fiber channel is amplified by a 2-stage all fiber polarization-maintained and single mode amplifier chain. In the first stage, 2.5 m single-clad Yb-doped fiber is employed, which had a core diameter of 6 um and a cladding diameter of 125 um. After pumping by a 480 mW single-mode fiber pigtailed laser diode with a central wavelength of 974 nm, the output power in each channel can be scaled to be 200 mW. The active fiber in the second stage is double-clad Yb-doped fiber, which had a 10 um core diameter and 125 um inner cladding diameter. In this stage, 3 m active fiber is employed in each amplifier. A laser diode with 975nm central wavelength, 105um pigtailed fiber are used to pump the active double-clad fiber in each channel via a (2 + 1) × 1 polarization maintained pump combiner. Finally, each output port of the active fiber is fused to a collimator with embedded isolator to prevent backscattering light and send the laser beam into free-space for polarization beam combination. After amplification, the output power of each channel can be boosted to 5 W power-level. The two collimated beams are firstly polarization combined in PBC1 by rotating the respective polarization directions through HWPs and coaxial adjusting the two laser beams. After polarization combination in the PBC1, the combined laser beam is injected in to a HWP and PBC2. The assembled HWP and PBC2 is utilized to measure the polarization extinction ratio (PER) of the combined beam, which is significant for further combining with another linear-polarized beam and extending the CPBC system. M1 is an all-reflectance mirror and M2 is a high- reflectance (99.9:0.1) mirror. After M2, a small part of the combined beam is collected by a photo detector (PD). The collected signal of the PD is transformed by the active phase controller for compensation the phase difference between the two channels. In our experiment, the phase controller is based on field programmable gate array (FPGA) and performed with single frequency dithering algorithm [5, 31]. P1 is a power meter that used for detect the leakage power in PBC1 result of the limited polarization extinction ratios (PERs) of the two injected beams. P2 and P3 are two power meters that employed to measure the power values of the two ports of PBC2.

 figure: Fig. 3

Fig. 3 Experimental setup of the two-channel multi-wavelength CPBC system. CO1-CO2: collimators; HWP: half wavelength plate; M1: all- reflected mirror; PBC1-PBC2: polarization beam combiners; M2: high-reflected mirror; PD-photo-detector; P1-P3: Power meters.

Download Full Size | PPT Slide | PDF

3.1 Experimental validation of the theoretical analysis of the multi-wavelength CPBC system

After two collimators, the output powers (Pi, i = 1, 2) of the two amplifier chains are all balanced to 5 W. The polarization extinction ratios (PERs) of the two channels are measured to be 26.2 dB and 27 dB. Firstly, we investigate the combining efficiency of the experimental system without modulating the seed laser. In this circumstance, the influence of spectrum-irrelative factors on the combining efficiency of the system can be confirmed. The influence of spectrum-irrelative factors should be considered in the following experimentally validating the theoretical analysis of the multi-wavelength CPBC system.

The active CPBC of the two amplifier chains is performed based on single-frequency dithering algorithm processor created by our group. In [31], Ma et al had presented the mathematical principle of single frequency dithering technique, and it is not repeated here to save place. As shown before, the voltage signal transformed by the PD incorporates the information of the phase difference between the two channels, which can be used to generate the phase control signal by modulation and demodulation technique. The active phasing process and phase noise suppressing results are shown in Fig. 4. Without phasing, the normalized energy in the PD fluctuates randomly because that the phase difference between the two amplifier chains continuously changes with the influence of thermal effects, environmental vibrations and so forth. However, with active phasing, the normalized energy in the PD is locked steady, which indicates that the phase difference between two channels is locked effectively. The comparison of the spectral density of power of the phase noise with and without active phasing is shown in Fig. 4(b). From Fig. 4(b), we see that the spectral density of power decreased sharply around~100 Hz without active phasing. Figure 4(b) also reveals that the phase noise of the amplifiers is suppressed efficiently in the closed loop.

 figure: Fig. 4

Fig. 4 The time dependent signals and spectral density of energy collected in the PD in open loop and closed loop. (a) Time dependent signals. (b) Spectral density of phase noise.

Download Full Size | PPT Slide | PDF

With active phasing, the power meter values of P1, P2 and P3 are measured to be 22 mW, 9.76 W and 0.22 W, respectively. The combining efficiency of the experimental system is calculated to be 97.6% by using the formula

ηa=Pp2Pp1+Pp2+Pp3

In Eq. (15), Pp1, Pp2, and Pp3 are the power values of the three power meters in Fig. 3, respectively.

In the present CPBC system, the spectrum-irrelative efficiency loss (Δη1) is 2.4% in the experiment. By calculation, the efficiency loss of the limited polarization is only 0.2% in the experiment. The residual efficiency loss is induced by some other imperfections, such as the residual phase error of the active controller, mode-mismatching of the two injected beams, coaxial errors and so forth.

From the theoretical analysis above-mentioned, we show that the combining efficiency of multi-wavelength CPBC system is related to OPD, wavelength number and spectral density of power of the amplifier chains. Firstly, we confirm the OPD between the two amplifier chains by replacing the seed laser in Fig. 3 into a mode-locked picosecond seed laser. The picosecond seed laser is a narrow-bandwidth mode-locked Yb-doped all fiber laser consisting of a fiber Bragg grating (FBG) and a LiNbO3 phase modulator (PM) in a linear cavity [33]. The time delay (Δτ) of the pulse trains from the two amplifier chains is measured to be 15.52 ns, so the OPD between the two channels is calculated by the formula c × Δτ, which is about 4.656 m in the experiment. Further, in order to theoretical analysis the combining efficiency of the experimental system, the spectral structures (wavelength number and spectral density of power) of the two amplified channels should also be obtained. Due to that the output power of the two amplifiers are equal, soP1,vnis approximately equal toP2,vnin the experiment. We select twelve representative spectral structures to validate the theoretical analysis above, and the modulation frequency (f) and modulation voltage (d) of each spectral structure is denoted in Fig. 5. The relative spectral density of power and discrete frequencies of the amplified chains are measured by using a Fabry-Pérot interferometer (FPI 100, FSR = 4 GHz, fineness~10 MHz, Toptica Inc.). In Fig. 5, Ps (%) is the relative spectral density of power of beam 1 and beam 2, and Pi,vn = Ps × Pi.

 figure: Fig. 5

Fig. 5 The relative spectral density of power and discrete frequencies with different modulation frequencies and voltages.

Download Full Size | PPT Slide | PDF

After acquiring the OPD and spectral structures of the system at different modulation frequencies and voltages, the theoretical combining efficiencies of the representative twelve different spectral structures are calculated and shown in Fig. 6. The only spectrum-related combining efficiencies (ηa + Δη1) are also measured by Eq. (15) in the experiment for validating the theoretical analysis of the multi-wavelength CBPC system, which is also revealed in Fig. 6. In Fig. 6, the abscissa represents the data number corresponding to the spectral structures in Fig. 5.

 figure: Fig. 6

Fig. 6 The theoretical and experimentally measured combining efficiencies of the representative twelve different spectral structures.

Download Full Size | PPT Slide | PDF

By comparing the theoretical results with experimental ones, we conclude that the theoretical analysis of the multi-wavelength CPBC system can be effectively employed to analyze the influence of OPD, wavelength number, and the spectral density of power of the amplifier chains on the combining efficiency. The maximal disparity between the theoretical and the experimental results is within 3.5%. By combination the combining efficiencies in Fig. 6 and the spectral structures in Fig. 5, we also conclude that the combining efficiency of the multi-wavelength CPBC system is sensitive to both the frequency intervals and the spectral density of power of the amplifier chains.

3.2 Experimental validation of the relationship between the combining efficiency and the amplitude of voltage in the PD

In the previous analysis, we experimentally validated the feasibility of the established relationship between the combining efficiency and the OPD, wavelength number, and the spectral density of power of the amplifier chains. We conclude that the influence of these factors on the combining efficiency can be obtained by acquiring the OPD and the spectral structures of the amplifiers. However, in a practical system with more complex spectral structures, the previous analysis is normally complicated or impractical. This is mainly attributed to the aspect that the spectral structures of the amplifiers are not always measurable by the Fabry-Pérot interferometer. Moreover, as for the system of complex spectral structures, higher precision of the OPD measurement is required for analyzing the system. In this circumstance, the method of OPD measurement afore-mentioned is possible not enough for the exact analysis, and normally high precision fiber delay lines should be utilized. In this section, we investigate the relationship between the combining efficiency and the amplitude of voltage signal in the PD. Thus the influence of the OPD and spectral structures on the combining efficiency can be analyzed more straightforwardly and the system with complex spectral structures can be also analyzed simply.

For a specific amplifier, we define that PSF and PMF are the SBS threshold power levels with single frequency operation and multi-frequency operation, respectively. Assuming that the discrepancy of the relative spectral structures of amplifiers change unobvious along with power scaling from PSF to PMF, thus the influence of OPD and spectral structures on the combining efficiency almost do not change in the process of power scaling from PSF to PMF. In this circumstance, the influence of OPD and spectral structures on the multi-wavelength CPBC system can be approximately estimated in the power level of PSF, which can be straightforwardly corresponding to the voltage signal collected by the PD in the practical experiment.

Specifically, from Eq. (14), we see that the combining efficiency of the system is related to the laser power (P′y) of the y-direction of PBC2. Due to that the collected energy (denoted by the amplitude of voltage signal) in the PD is proportional to P′y, so the combining efficiency of the multi-wavelength CPBC system can be also simply expressed by

η=11Ni=1NV(i)Vm

Where N is the sampling number of the voltage signal when the system is active phased, V(i) represents the amplitude of the voltage signal collected by the PD, Vm is the maximal amplitude of the voltage signal without the influence of OPD and spectral structures. In the following discussions, Vm is normalized to be 1 for simplicity.

From Eq. (13) and Eq. (14), a point should also be noted is that the combining efficiency of the multi-wavelength CPBC system calculated by Eq. (16) not only incorporates the influence of OPD and spectral structures, but also includes the influence of the residual phase errors of the phase controller. By using the normalized voltage signal without modulating the seed laser (shown in Fig. 4(a)), the influence of residual phase error on the combining efficiency is calculated to be 1.5%. From the analysis above, we conclude that the efficiency loss (Δη2) of other imperfections (such as mode-mismatching of the two injected beams and coaxial errors) is only 0.7% in the experimental setup.

In order to verify the feasibility of Eq. (16), the calculated combining efficiency (η) by the normalized voltage in the PD will be compared with the combining efficiency of (ηa + Δη2) in different cases.

Firstly, we set the modulation voltage of the signal generator to 10 V and investigate the comparing results by increasing the modulation frequency (f). The measured normalized voltage signals along with different modulation frequencies are shown in Fig. 7. The sample interval of the voltage signal is set to 0.002 s in the experiment.

 figure: Fig. 7

Fig. 7 The change of the normalized voltage signals along with the modulation frequencies with the modulation voltage of 10 V.

Download Full Size | PPT Slide | PDF

Figure 7 shows that the average amplitude of the normalized voltage signal in the active phasing state changes along with the increase of the modulation frequency. After acquiring the normalized voltage in the PD with different frequencies, the combining efficiencies of the experimental system are specifically estimated by Eq. (16), which is shown in Fig. 8. The experimentally measured combining efficiencies (ηa + Δη2) of the system are also shown in Fig. 8 for comparison.

 figure: Fig. 8

Fig. 8 The compared results of estimated combining efficiencies and the measured combining efficiencies with different modulation frequencies (modulation voltage-10V).

Download Full Size | PPT Slide | PDF

Further, we adjust the modulation frequency of the signal generator to 100 MHz and study the estimated combining efficiencies and measured combining efficiencies by increasing the modulation voltages (d). The measured normalized voltage signals along with different modulation voltages and the compared results are shown in Fig. 9 and Fig. 10, respectively.

 figure: Fig. 9

Fig. 9 The change of the normalized voltage signals along with the modulation voltages (d) with the modulation frequency of 100 MHz.

Download Full Size | PPT Slide | PDF

 figure: Fig. 10

Fig. 10 The compared results of estimated combining efficiencies and the measured combining efficiencies with different modulation voltages (modulation frequency-100 MHz).

Download Full Size | PPT Slide | PDF

According to Fig. 8 and Fig. 10, we see that the estimated combining efficiencies of the CPBC system matched precisely with the measured combining efficiencies by power meters. Specifically, the maximal matching errors in the process of increasing the modulation frequencies and voltages are just 1.1% and 2%, respectively.

By comparing the estimated combining efficiencies and the measured combining efficiencies in the circumstance of different modulation frequencies and voltages above, we validated the simple technique to evaluate the influence of OPD and spectral structures on the combining efficiency of the multi-wavelength CPBC system.

4. Experimental validation of CPBC with complex spectral structures

In this section, we will equalize the OPD between two laser channels and validate the feasibility of the active CPBC with complex spectral structures based on all fiber delay lines. Even that the adjustment precision of the all fiber delay lines is inferior to piezo-mounted mirrors, all fiber delay lines have advantageous of more economical, simple to adjust, and more stable compared with the piezo-mounted mirrors revealed in the previous publications [22, 25]. Thus, it is necessary to validate the feasibility of all fiber delay lines with complex spectral structures for remarkable power scaling of the compact and economical CPBC system. The experimental setup is similar to Fig. 3, and the complex spectral structures are generated by combining three single frequency seed lasers with different central wavelength (1064.4 nm, 1063.8 nm, and 1062.8 nm) [32], and then further modulated by the phase modulator. The modulation frequency and voltage are set to be 100 MHz and 10 V, respectively. The modulated spectral structures of each seed laser are shown in Figs. 11(a)11(c). The optical spectrum of the combined beam is also measured by a spectrum analyzer, which is shown in Fig. 11(d).

 figure: Fig. 11

Fig. 11 The modulated spectral structures of each seed laser ((a)-(c)) and the optical spectrum measured by a spectrum analyzer (d).

Download Full Size | PPT Slide | PDF

As for these complex spectral structures with different central wavelength, from Eq. (13) and Eq. (14), we see that the OPD between the amplifier chains should be normally adjusted precisely to obtain high combining efficiency. Firstly, we repeatedly replace the modulated seed lasers by the picosecond seed laser and compensate the time delay (15.52 ns) of the pulse trains from the two amplifier chains by fusing a section of passive fibers (about 3.1 m) in the second channel and adjusting the displacements of the two collimators. Secondly, we incorporate into a commercial and manual-controlled all fiber delay line in each channel to further precisely adjust the OPD between the two channels. The fiber delay line is adjusted by manually selecting and setting the delay values through computer and the resolution of the delay line is 0.496 um. When the system is active phased, by continuously setting the delay values of the fiber delay line, we confirmed the optimal state (namely the combining efficiency of the system is maximal) of the experimental setup. The active phasing process and results are shown in Fig. 12. At the optimal state, the combining efficiency of the system is measured to be as high as 96%, which is crucial for power scaling the CPBC system by suppressing SBS effect.

 figure: Fig. 12

Fig. 12 The voltage signals in the PD when the OPD is carefully compensated.

Download Full Size | PPT Slide | PDF

5. Conclusion

In conclusion, the multi-wavelength active CPBC system is analyzed theoretically and experimentally. We experimentally validated the established relationship between the combining efficiency of multi-wavelength CPBC system and the OPD, wavelength number, and the spectral density of power of the amplifiers. The feasibility of the amplitude of voltage signal collected by the photo-detector to evaluate the influence of OPD and spectral structures is also analyzed and validated, which is a more simple technique for a practical multi-wavelength CPBC system (especially with complex spectral structures). By employing modulated and multiple seeds with different central wavelength, the competence of the active CPBC system with complex spectral structures is validated and as high as 96% combining efficiency could be obtained based on all fiber delay lines to compensate the OPD between different channels, which is significant for further scaling the output power of the CPBC system by breaking through the limitation of SBS effect in fiber amplifiers.

Acknowledgments

This research is sponsored by the National Natural Science Foundation of China under NO. 11274386 and the innovation project of National University of Defense Technology for graduate student.

References and links

1. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008). [CrossRef]   [PubMed]  

2. C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013). [CrossRef]  

3. T. M. Shay, V. Benham, J. T. Baker, B. Ward, A. D. Sanchez, M. A. Culpepper, D. Pilkington, J. Spring, D. J. Nelson, and C. A. Lu, “First experimental demonstration of self-synchronous phase locking of an optical array,” Opt. Express 14(25), 12015–12021 (2006). [CrossRef]   [PubMed]  

4. A. Flores, T. M. Shay, C. A. Lu, C. Robin, B. Pulford, A. D. Sanchez, D. W. Hult, and K. B. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” in Conference on Laser and Electro-Optics (CLEO):Laser Applications to Photonic Applications, OSA Technical Digest (Optical Society of America, 2011), paper CFE3. [CrossRef]  

5. Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011). [CrossRef]   [PubMed]  

6. P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Coherent beam combination of three two-tone fiber amplifiers using stochastic parallel gradient descent algorithm,” Opt. Lett. 34(19), 2939–2941 (2009). [CrossRef]   [PubMed]  

7. X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011). [CrossRef]   [PubMed]  

8. G. D. Goodno, S. J. McNaught, J. E. Rothenberg, T. S. McComb, P. A. Thielen, M. G. Wickham, and M. E. Weber, “Active phase and polarization locking of a 1.4 kW fiber amplifier,” Opt. Lett. 35(10), 1542–1544 (2010). [CrossRef]   [PubMed]  

9. A. Flores, C. Lu, C. Robin, S. Naderi, C. Vergien, and I. Dajani, “Experimental and theoretical studies of phase modulation in Yb-doped fiber amplifiers,” Proc. SPIE 8381, 83811B (2012). [CrossRef]  

10. C. X. Yu, S. J. Augst, S. M. Redmond, K. C. Goldizen, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Coherent combining of a 4 kW, eight-element fiber amplifier array,” Opt. Lett. 36(14), 2686–2688 (2011). [CrossRef]   [PubMed]  

11. J. Bourderionnet, C. Bellanger, J. Primot, and A. Brignon, “Collective coherent phase combining of 64 fibers,” Opt. Express 19(18), 17053–17058 (2011). [CrossRef]   [PubMed]  

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

13. B. S. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50 W coherently combined passive laser array,” Opt. Lett. 34(7), 863–865 (2009). [CrossRef]   [PubMed]  

14. A. Shirakawa, T. Saitou, T. Sekiguchi, and K. Ueda, “Coherent addition of fiber lasers by use of a fiber coupler,” Opt. Express 10(21), 1167–1172 (2002). [CrossRef]   [PubMed]  

15. M. Khajavikhan and J. R. Leger, “Modal Analysis of path length sensitivity in superposition architectures for coherent laser beam combining,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).

16. J. Lhermite, A. Desfarges-Berthelemot, V. Kermene, and A. Barthelemy, “Passive phase locking of an array of four fiber amplifiers by an all-optical feedback loop,” Opt. Lett. 32(13), 1842–1844 (2007). [CrossRef]   [PubMed]  

17. Y. F. Yang, M. Hu, B. He, J. Zhou, H. K. Liu, S. J. Dai, Y. R. Wei, and Q. H. Lou, “Passive coherent beam combining of four Yb-doped fiber amplifier chains with injection-locked seed source,” Opt. Lett. 38(6), 854–856 (2013). [CrossRef]   [PubMed]  

18. M. A. Vorontsov and S. L. Lachinova, “Laser beam projection with adaptive array of fiber collimators. I. Basic considerations for analysis,” J. Opt. Soc. Am. A 25(8), 1949–1959 (2008). [CrossRef]   [PubMed]  

19. S. M. Redmond, D. J. Ripin, C. X. Yu, S. J. Augst, T. Y. Fan, P. A. Thielen, J. E. Rothenberg, and G. D. Goodno, “Diffractive coherent combining of a 2.5 kW fiber laser array into a 1.9 kW Gaussian beam,” Opt. Lett. 37(14), 2832–2834 (2012). [CrossRef]   [PubMed]  

20. R. Uberna, A. Bratcher, T. G. Alley, A. D. Sanchez, A. S. Flores, and B. Pulford, “Coherent combination of high power fiber amplifiers in a two-dimensional re-imaging waveguide,” Opt. Express 18(13), 13547–13553 (2010). [CrossRef]   [PubMed]  

21. R. Uberna, A. Bratcher, and B. G. Tiemann, “Coherent polarization beam combination,” IEEE J. Quantum Electron. 46, 1191–1196 (2010). [CrossRef]   [PubMed]  

22. L. A. Siiman, W. Z. Chang, T. Zhou, and A. Galvanauskas, “Coherent femtosecond pulse combining of multiple parallel chirped pulse fiber amplifiers,” Opt. Express 20(16), 18097–18116 (2012). [CrossRef]   [PubMed]  

23. P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012). [CrossRef]  

24. P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013). [CrossRef]  

25. A. Klenke, S. Breitkopf, M. Kienel, T. Gottschall, T. Eidam, S. Hädrich, J. Rothhardt, J. Limpert, and A. Tünnermann, “530 W, 1.3 mJ, four-channel coherently combined femtosecond fiber chirped-pulse amplification system,” Opt. Lett. 38(13), 2283–2285 (2013). [CrossRef]   [PubMed]  

26. H. Kai, X. Xiaojun, and L. Zejin, “Analysis of multiwavelength coherent beam combining effect,” Appl. Opt. 51(34), 8132–8140 (2012). [CrossRef]   [PubMed]  

27. G. D. Goodno, C. C. Shih, and J. E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express 18(24), 25403–25414 (2010). [CrossRef]   [PubMed]  

28. P. Ma, P. Zhou, X. Wang, Y. Ma, R. Su, and Z. Liu, “Influence of perturbative phase noise on active coherent polarization beam combining system,” Opt. Express 21(24), 29666–29678 (2013). [CrossRef]   [PubMed]  

29. V. Khitrov, K. Farley, R. Leveille, J. Galipeau, I. Majid, S. Christensen, B. Samson, and K. Tankala, “kW level narrow linewidth Yb fiber amplifiers for beam combining,” Proc. SPIE 7686, 76860A (2010). [CrossRef]  

30. D. Engin, W. Lu, M. Akbulut, B. McIntosh, H. Verdun, and S. Gupta, “1 kW cw Yb-fiber-amplifier with <0.5 GHz linewidth and near-diffraction limited beam quality, for coherent combining application,” Proc. SPIE 7, 1407 (2011).

31. Y. Ma, P. Zhou, X. Wang, H. Ma, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination with single frequency dithering technique,” Opt. Lett. 35(9), 1308–1310 (2010). [CrossRef]   [PubMed]  

32. S. H. Xu, Z. M. Yang, W. N. Zhang, X. M. Wei, Q. Qian, D. D. Chen, Q. Y. Zhang, S. X. Shen, M. Y. Peng, and J. R. Qiu, “400 mW ultrashort cavity low-noise single-frequency Yb3+-doped phosphate fiber laser,” Opt. Lett. 36(18), 3708–3710 (2011). [CrossRef]   [PubMed]  

33. R. Tao, X. Wang, P. Zhou, L. Si, and Z. Liu, “Widely tunable mode-locked all-fiberized Yb-doped fiber laser with near-transform-limited spectrum linewidth,” Appl. Phys. B 116, 115–119 (2014).

References

  • View by:
  • |
  • |
  • |

  1. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008).
    [Crossref] [PubMed]
  2. C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013).
    [Crossref]
  3. T. M. Shay, V. Benham, J. T. Baker, B. Ward, A. D. Sanchez, M. A. Culpepper, D. Pilkington, J. Spring, D. J. Nelson, and C. A. Lu, “First experimental demonstration of self-synchronous phase locking of an optical array,” Opt. Express 14(25), 12015–12021 (2006).
    [Crossref] [PubMed]
  4. A. Flores, T. M. Shay, C. A. Lu, C. Robin, B. Pulford, A. D. Sanchez, D. W. Hult, and K. B. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” in Conference on Laser and Electro-Optics (CLEO):Laser Applications to Photonic Applications, OSA Technical Digest (Optical Society of America, 2011), paper CFE3.
    [Crossref]
  5. Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
    [Crossref] [PubMed]
  6. P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Coherent beam combination of three two-tone fiber amplifiers using stochastic parallel gradient descent algorithm,” Opt. Lett. 34(19), 2939–2941 (2009).
    [Crossref] [PubMed]
  7. X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011).
    [Crossref] [PubMed]
  8. G. D. Goodno, S. J. McNaught, J. E. Rothenberg, T. S. McComb, P. A. Thielen, M. G. Wickham, and M. E. Weber, “Active phase and polarization locking of a 1.4 kW fiber amplifier,” Opt. Lett. 35(10), 1542–1544 (2010).
    [Crossref] [PubMed]
  9. A. Flores, C. Lu, C. Robin, S. Naderi, C. Vergien, and I. Dajani, “Experimental and theoretical studies of phase modulation in Yb-doped fiber amplifiers,” Proc. SPIE 8381, 83811B (2012).
    [Crossref]
  10. C. X. Yu, S. J. Augst, S. M. Redmond, K. C. Goldizen, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Coherent combining of a 4 kW, eight-element fiber amplifier array,” Opt. Lett. 36(14), 2686–2688 (2011).
    [Crossref] [PubMed]
  11. J. Bourderionnet, C. Bellanger, J. Primot, and A. Brignon, “Collective coherent phase combining of 64 fibers,” Opt. Express 19(18), 17053–17058 (2011).
    [Crossref] [PubMed]
  12. C. J. Corcoran and F. Durville, “Experimental demonstration of a phase-locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118 (2005).
  13. B. S. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50 W coherently combined passive laser array,” Opt. Lett. 34(7), 863–865 (2009).
    [Crossref] [PubMed]
  14. A. Shirakawa, T. Saitou, T. Sekiguchi, and K. Ueda, “Coherent addition of fiber lasers by use of a fiber coupler,” Opt. Express 10(21), 1167–1172 (2002).
    [Crossref] [PubMed]
  15. M. Khajavikhan and J. R. Leger, “Modal Analysis of path length sensitivity in superposition architectures for coherent laser beam combining,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).
  16. J. Lhermite, A. Desfarges-Berthelemot, V. Kermene, and A. Barthelemy, “Passive phase locking of an array of four fiber amplifiers by an all-optical feedback loop,” Opt. Lett. 32(13), 1842–1844 (2007).
    [Crossref] [PubMed]
  17. Y. F. Yang, M. Hu, B. He, J. Zhou, H. K. Liu, S. J. Dai, Y. R. Wei, and Q. H. Lou, “Passive coherent beam combining of four Yb-doped fiber amplifier chains with injection-locked seed source,” Opt. Lett. 38(6), 854–856 (2013).
    [Crossref] [PubMed]
  18. M. A. Vorontsov and S. L. Lachinova, “Laser beam projection with adaptive array of fiber collimators. I. Basic considerations for analysis,” J. Opt. Soc. Am. A 25(8), 1949–1959 (2008).
    [Crossref] [PubMed]
  19. S. M. Redmond, D. J. Ripin, C. X. Yu, S. J. Augst, T. Y. Fan, P. A. Thielen, J. E. Rothenberg, and G. D. Goodno, “Diffractive coherent combining of a 2.5 kW fiber laser array into a 1.9 kW Gaussian beam,” Opt. Lett. 37(14), 2832–2834 (2012).
    [Crossref] [PubMed]
  20. R. Uberna, A. Bratcher, T. G. Alley, A. D. Sanchez, A. S. Flores, and B. Pulford, “Coherent combination of high power fiber amplifiers in a two-dimensional re-imaging waveguide,” Opt. Express 18(13), 13547–13553 (2010).
    [Crossref] [PubMed]
  21. R. Uberna, A. Bratcher, and B. G. Tiemann, “Coherent polarization beam combination,” IEEE J. Quantum Electron. 46, 1191–1196 (2010).
    [Crossref] [PubMed]
  22. L. A. Siiman, W. Z. Chang, T. Zhou, and A. Galvanauskas, “Coherent femtosecond pulse combining of multiple parallel chirped pulse fiber amplifiers,” Opt. Express 20(16), 18097–18116 (2012).
    [Crossref] [PubMed]
  23. P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
    [Crossref]
  24. P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
    [Crossref]
  25. A. Klenke, S. Breitkopf, M. Kienel, T. Gottschall, T. Eidam, S. Hädrich, J. Rothhardt, J. Limpert, and A. Tünnermann, “530 W, 1.3 mJ, four-channel coherently combined femtosecond fiber chirped-pulse amplification system,” Opt. Lett. 38(13), 2283–2285 (2013).
    [Crossref] [PubMed]
  26. H. Kai, X. Xiaojun, and L. Zejin, “Analysis of multiwavelength coherent beam combining effect,” Appl. Opt. 51(34), 8132–8140 (2012).
    [Crossref] [PubMed]
  27. G. D. Goodno, C. C. Shih, and J. E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express 18(24), 25403–25414 (2010).
    [Crossref] [PubMed]
  28. P. Ma, P. Zhou, X. Wang, Y. Ma, R. Su, and Z. Liu, “Influence of perturbative phase noise on active coherent polarization beam combining system,” Opt. Express 21(24), 29666–29678 (2013).
    [Crossref] [PubMed]
  29. V. Khitrov, K. Farley, R. Leveille, J. Galipeau, I. Majid, S. Christensen, B. Samson, and K. Tankala, “kW level narrow linewidth Yb fiber amplifiers for beam combining,” Proc. SPIE 7686, 76860A (2010).
    [Crossref]
  30. D. Engin, W. Lu, M. Akbulut, B. McIntosh, H. Verdun, and S. Gupta, “1 kW cw Yb-fiber-amplifier with <0.5 GHz linewidth and near-diffraction limited beam quality, for coherent combining application,” Proc. SPIE 7, 1407 (2011).
  31. Y. Ma, P. Zhou, X. Wang, H. Ma, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination with single frequency dithering technique,” Opt. Lett. 35(9), 1308–1310 (2010).
    [Crossref] [PubMed]
  32. S. H. Xu, Z. M. Yang, W. N. Zhang, X. M. Wei, Q. Qian, D. D. Chen, Q. Y. Zhang, S. X. Shen, M. Y. Peng, and J. R. Qiu, “400 mW ultrashort cavity low-noise single-frequency Yb3+-doped phosphate fiber laser,” Opt. Lett. 36(18), 3708–3710 (2011).
    [Crossref] [PubMed]
  33. R. Tao, X. Wang, P. Zhou, L. Si, and Z. Liu, “Widely tunable mode-locked all-fiberized Yb-doped fiber laser with near-transform-limited spectrum linewidth,” Appl. Phys. B 116, 115–119 (2014).

2014 (1)

R. Tao, X. Wang, P. Zhou, L. Si, and Z. Liu, “Widely tunable mode-locked all-fiberized Yb-doped fiber laser with near-transform-limited spectrum linewidth,” Appl. Phys. B 116, 115–119 (2014).

2013 (5)

2012 (5)

2011 (6)

2010 (6)

2009 (2)

2008 (2)

2007 (2)

M. Khajavikhan and J. R. Leger, “Modal Analysis of path length sensitivity in superposition architectures for coherent laser beam combining,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).

J. Lhermite, A. Desfarges-Berthelemot, V. Kermene, and A. Barthelemy, “Passive phase locking of an array of four fiber amplifiers by an all-optical feedback loop,” Opt. Lett. 32(13), 1842–1844 (2007).
[Crossref] [PubMed]

2006 (1)

2005 (1)

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

2002 (1)

Akbulut, M.

D. Engin, W. Lu, M. Akbulut, B. McIntosh, H. Verdun, and S. Gupta, “1 kW cw Yb-fiber-amplifier with <0.5 GHz linewidth and near-diffraction limited beam quality, for coherent combining application,” Proc. SPIE 7, 1407 (2011).

Alley, T. G.

Augst, S. J.

Baker, J. T.

Barthelemy, A.

Barty, C. P. J.

Beach, R. J.

Bellanger, C.

Benham, V.

Bourderionnet, J.

Bratcher, A.

Breitkopf, S.

Brignon, A.

Chang, W. Z.

Chen, D. D.

Christensen, S.

V. Khitrov, K. Farley, R. Leveille, J. Galipeau, I. Majid, S. Christensen, B. Samson, and K. Tankala, “kW level narrow linewidth Yb fiber amplifiers for beam combining,” Proc. SPIE 7686, 76860A (2010).
[Crossref]

Corcoran, C. J.

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

Culpepper, M. A.

Dai, S. J.

Dajani, I.

A. Flores, C. Lu, C. Robin, S. Naderi, C. Vergien, and I. Dajani, “Experimental and theoretical studies of phase modulation in Yb-doped fiber amplifiers,” Proc. SPIE 8381, 83811B (2012).
[Crossref]

Dawson, J. W.

Desfarges-Berthelemot, A.

Dong, X.

Du, W.

Durville, F.

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

Eidam, T.

Engin, D.

D. Engin, W. Lu, M. Akbulut, B. McIntosh, H. Verdun, and S. Gupta, “1 kW cw Yb-fiber-amplifier with <0.5 GHz linewidth and near-diffraction limited beam quality, for coherent combining application,” Proc. SPIE 7, 1407 (2011).

Fan, T. Y.

Farley, K.

V. Khitrov, K. Farley, R. Leveille, J. Galipeau, I. Majid, S. Christensen, B. Samson, and K. Tankala, “kW level narrow linewidth Yb fiber amplifiers for beam combining,” Proc. SPIE 7686, 76860A (2010).
[Crossref]

Flores, A.

A. Flores, C. Lu, C. Robin, S. Naderi, C. Vergien, and I. Dajani, “Experimental and theoretical studies of phase modulation in Yb-doped fiber amplifiers,” Proc. SPIE 8381, 83811B (2012).
[Crossref]

Flores, A. S.

Galipeau, J.

V. Khitrov, K. Farley, R. Leveille, J. Galipeau, I. Majid, S. Christensen, B. Samson, and K. Tankala, “kW level narrow linewidth Yb fiber amplifiers for beam combining,” Proc. SPIE 7686, 76860A (2010).
[Crossref]

Galvanauskas, A.

Goldizen, K. C.

Goodno, G. D.

Gottschall, T.

Gupta, S.

D. Engin, W. Lu, M. Akbulut, B. McIntosh, H. Verdun, and S. Gupta, “1 kW cw Yb-fiber-amplifier with <0.5 GHz linewidth and near-diffraction limited beam quality, for coherent combining application,” Proc. SPIE 7, 1407 (2011).

Hädrich, S.

He, B.

Heebner, J. E.

Hu, M.

Jauregui, C.

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013).
[Crossref]

Kai, H.

Kermene, V.

Khajavikhan, M.

M. Khajavikhan and J. R. Leger, “Modal Analysis of path length sensitivity in superposition architectures for coherent laser beam combining,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).

Khitrov, V.

V. Khitrov, K. Farley, R. Leveille, J. Galipeau, I. Majid, S. Christensen, B. Samson, and K. Tankala, “kW level narrow linewidth Yb fiber amplifiers for beam combining,” Proc. SPIE 7686, 76860A (2010).
[Crossref]

Kienel, M.

Klenke, A.

Lachinova, S. L.

Leger, J. R.

M. Khajavikhan and J. R. Leger, “Modal Analysis of path length sensitivity in superposition architectures for coherent laser beam combining,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).

Leng, J.

Leng, J. Y.

Leveille, R.

V. Khitrov, K. Farley, R. Leveille, J. Galipeau, I. Majid, S. Christensen, B. Samson, and K. Tankala, “kW level narrow linewidth Yb fiber amplifiers for beam combining,” Proc. SPIE 7686, 76860A (2010).
[Crossref]

Lhermite, J.

Limpert, J.

Liu, H. K.

Liu, Z.

Liu, Z. J.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011).
[Crossref] [PubMed]

Lou, Q. H.

Lu, C.

A. Flores, C. Lu, C. Robin, S. Naderi, C. Vergien, and I. Dajani, “Experimental and theoretical studies of phase modulation in Yb-doped fiber amplifiers,” Proc. SPIE 8381, 83811B (2012).
[Crossref]

Lu, C. A.

Lu, W.

D. Engin, W. Lu, M. Akbulut, B. McIntosh, H. Verdun, and S. Gupta, “1 kW cw Yb-fiber-amplifier with <0.5 GHz linewidth and near-diffraction limited beam quality, for coherent combining application,” Proc. SPIE 7, 1407 (2011).

Ma, H.

Ma, P.

Ma, P. F.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

Ma, Y.

Ma, Y. X.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011).
[Crossref] [PubMed]

Majid, I.

V. Khitrov, K. Farley, R. Leveille, J. Galipeau, I. Majid, S. Christensen, B. Samson, and K. Tankala, “kW level narrow linewidth Yb fiber amplifiers for beam combining,” Proc. SPIE 7686, 76860A (2010).
[Crossref]

McComb, T. S.

McIntosh, B.

D. Engin, W. Lu, M. Akbulut, B. McIntosh, H. Verdun, and S. Gupta, “1 kW cw Yb-fiber-amplifier with <0.5 GHz linewidth and near-diffraction limited beam quality, for coherent combining application,” Proc. SPIE 7, 1407 (2011).

McNaught, S. J.

Messerly, M. J.

Mies, E.

Minden, M.

Murphy, D. V.

Naderi, S.

A. Flores, C. Lu, C. Robin, S. Naderi, C. Vergien, and I. Dajani, “Experimental and theoretical studies of phase modulation in Yb-doped fiber amplifiers,” Proc. SPIE 8381, 83811B (2012).
[Crossref]

Nelson, D. J.

Pax, P. H.

Peng, M. Y.

Pilkington, D.

Primot, J.

Pulford, B.

Qian, Q.

Qiu, J. R.

Redmond, S. M.

Ripin, D. J.

Robin, C.

A. Flores, C. Lu, C. Robin, S. Naderi, C. Vergien, and I. Dajani, “Experimental and theoretical studies of phase modulation in Yb-doped fiber amplifiers,” Proc. SPIE 8381, 83811B (2012).
[Crossref]

Rothenberg, J. E.

Rothhardt, J.

Saitou, T.

Samson, B.

V. Khitrov, K. Farley, R. Leveille, J. Galipeau, I. Majid, S. Christensen, B. Samson, and K. Tankala, “kW level narrow linewidth Yb fiber amplifiers for beam combining,” Proc. SPIE 7686, 76860A (2010).
[Crossref]

Sanchez, A.

Sanchez, A. D.

Sekiguchi, T.

Shay, T. M.

Shen, S. X.

Shih, C. C.

Shirakawa, A.

Shverdin, M. Y.

Si, L.

Siders, C. W.

Siiman, L. A.

Spring, J.

Sridharan, A. K.

Stappaerts, E. A.

Su, R.

Su, R. T.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

Tankala, K.

V. Khitrov, K. Farley, R. Leveille, J. Galipeau, I. Majid, S. Christensen, B. Samson, and K. Tankala, “kW level narrow linewidth Yb fiber amplifiers for beam combining,” Proc. SPIE 7686, 76860A (2010).
[Crossref]

Tao, R.

R. Tao, X. Wang, P. Zhou, L. Si, and Z. Liu, “Widely tunable mode-locked all-fiberized Yb-doped fiber laser with near-transform-limited spectrum linewidth,” Appl. Phys. B 116, 115–119 (2014).

Thielen, P. A.

Tiemann, B. G.

R. Uberna, A. Bratcher, and B. G. Tiemann, “Coherent polarization beam combination,” IEEE J. Quantum Electron. 46, 1191–1196 (2010).
[Crossref] [PubMed]

Tünnermann, A.

Uberna, R.

Ueda, K.

Verdun, H.

D. Engin, W. Lu, M. Akbulut, B. McIntosh, H. Verdun, and S. Gupta, “1 kW cw Yb-fiber-amplifier with <0.5 GHz linewidth and near-diffraction limited beam quality, for coherent combining application,” Proc. SPIE 7, 1407 (2011).

Vergien, C.

A. Flores, C. Lu, C. Robin, S. Naderi, C. Vergien, and I. Dajani, “Experimental and theoretical studies of phase modulation in Yb-doped fiber amplifiers,” Proc. SPIE 8381, 83811B (2012).
[Crossref]

Vorontsov, M. A.

Wang, B. S.

Wang, X.

Wang, X. L.

Ward, B.

Weber, M. E.

Wei, X. M.

Wei, Y. R.

Wickham, M. G.

Xiao, H.

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

Xiaojun, X.

Xu, S. H.

Xu, X.

Xu, X. J.

Yang, Y. F.

Yang, Z. M.

Yu, C. X.

Zejin, L.

Zhang, Q. Y.

Zhang, W. N.

Zhao, Y.

Zhou, J.

Zhou, P.

R. Tao, X. Wang, P. Zhou, L. Si, and Z. Liu, “Widely tunable mode-locked all-fiberized Yb-doped fiber laser with near-transform-limited spectrum linewidth,” Appl. Phys. B 116, 115–119 (2014).

P. Ma, P. Zhou, X. Wang, Y. Ma, R. Su, and Z. Liu, “Influence of perturbative phase noise on active coherent polarization beam combining system,” Opt. Express 21(24), 29666–29678 (2013).
[Crossref] [PubMed]

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011).
[Crossref] [PubMed]

Y. Ma, P. Zhou, X. Wang, H. Ma, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination with single frequency dithering technique,” Opt. Lett. 35(9), 1308–1310 (2010).
[Crossref] [PubMed]

P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Coherent beam combination of three two-tone fiber amplifiers using stochastic parallel gradient descent algorithm,” Opt. Lett. 34(19), 2939–2941 (2009).
[Crossref] [PubMed]

Zhou, T.

Zhu, J.

Appl. Opt. (1)

Appl. Phys. B (1)

R. Tao, X. Wang, P. Zhou, L. Si, and Z. Liu, “Widely tunable mode-locked all-fiberized Yb-doped fiber laser with near-transform-limited spectrum linewidth,” Appl. Phys. B 116, 115–119 (2014).

Appl. Phys. Lett. (1)

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

IEEE J. Quantum Electron. (1)

R. Uberna, A. Bratcher, and B. G. Tiemann, “Coherent polarization beam combination,” IEEE J. Quantum Electron. 46, 1191–1196 (2010).
[Crossref] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

M. Khajavikhan and J. R. Leger, “Modal Analysis of path length sensitivity in superposition architectures for coherent laser beam combining,” IEEE J. Sel. Top. Quantum Electron. 13, 460–472 (2007).

IEEE Photon. Technol. Lett. (1)

P. F. Ma, P. Zhou, H. Xiao, Y. X. Ma, R. T. Su, and Z. J. Liu, “Generation of a 481 W single frequency and linearly polarized beam by coherent polarization locking,” IEEE Photon. Technol. Lett. 25(19), 1936–1938 (2013).
[Crossref]

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

Laser Phys. Lett. (1)

P. F. Ma, P. Zhou, R. T. Su, Y. X. Ma, and Z. J. Liu, “Coherent polarization beam combining of eight fiber lasers using single-frequency dithering technique,” Laser Phys. Lett. 9(6), 456–458 (2012).
[Crossref]

Nat. Photonics (1)

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013).
[Crossref]

Opt. Express (8)

T. M. Shay, V. Benham, J. T. Baker, B. Ward, A. D. Sanchez, M. A. Culpepper, D. Pilkington, J. Spring, D. J. Nelson, and C. A. Lu, “First experimental demonstration of self-synchronous phase locking of an optical array,” Opt. Express 14(25), 12015–12021 (2006).
[Crossref] [PubMed]

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008).
[Crossref] [PubMed]

J. Bourderionnet, C. Bellanger, J. Primot, and A. Brignon, “Collective coherent phase combining of 64 fibers,” Opt. Express 19(18), 17053–17058 (2011).
[Crossref] [PubMed]

A. Shirakawa, T. Saitou, T. Sekiguchi, and K. Ueda, “Coherent addition of fiber lasers by use of a fiber coupler,” Opt. Express 10(21), 1167–1172 (2002).
[Crossref] [PubMed]

R. Uberna, A. Bratcher, T. G. Alley, A. D. Sanchez, A. S. Flores, and B. Pulford, “Coherent combination of high power fiber amplifiers in a two-dimensional re-imaging waveguide,” Opt. Express 18(13), 13547–13553 (2010).
[Crossref] [PubMed]

L. A. Siiman, W. Z. Chang, T. Zhou, and A. Galvanauskas, “Coherent femtosecond pulse combining of multiple parallel chirped pulse fiber amplifiers,” Opt. Express 20(16), 18097–18116 (2012).
[Crossref] [PubMed]

G. D. Goodno, C. C. Shih, and J. E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express 18(24), 25403–25414 (2010).
[Crossref] [PubMed]

P. Ma, P. Zhou, X. Wang, Y. Ma, R. Su, and Z. Liu, “Influence of perturbative phase noise on active coherent polarization beam combining system,” Opt. Express 21(24), 29666–29678 (2013).
[Crossref] [PubMed]

Opt. Lett. (12)

A. Klenke, S. Breitkopf, M. Kienel, T. Gottschall, T. Eidam, S. Hädrich, J. Rothhardt, J. Limpert, and A. Tünnermann, “530 W, 1.3 mJ, four-channel coherently combined femtosecond fiber chirped-pulse amplification system,” Opt. Lett. 38(13), 2283–2285 (2013).
[Crossref] [PubMed]

Y. Ma, P. Zhou, X. Wang, H. Ma, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination with single frequency dithering technique,” Opt. Lett. 35(9), 1308–1310 (2010).
[Crossref] [PubMed]

S. H. Xu, Z. M. Yang, W. N. Zhang, X. M. Wei, Q. Qian, D. D. Chen, Q. Y. Zhang, S. X. Shen, M. Y. Peng, and J. R. Qiu, “400 mW ultrashort cavity low-noise single-frequency Yb3+-doped phosphate fiber laser,” Opt. Lett. 36(18), 3708–3710 (2011).
[Crossref] [PubMed]

B. S. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50 W coherently combined passive laser array,” Opt. Lett. 34(7), 863–865 (2009).
[Crossref] [PubMed]

S. M. Redmond, D. J. Ripin, C. X. Yu, S. J. Augst, T. Y. Fan, P. A. Thielen, J. E. Rothenberg, and G. D. Goodno, “Diffractive coherent combining of a 2.5 kW fiber laser array into a 1.9 kW Gaussian beam,” Opt. Lett. 37(14), 2832–2834 (2012).
[Crossref] [PubMed]

J. Lhermite, A. Desfarges-Berthelemot, V. Kermene, and A. Barthelemy, “Passive phase locking of an array of four fiber amplifiers by an all-optical feedback loop,” Opt. Lett. 32(13), 1842–1844 (2007).
[Crossref] [PubMed]

Y. F. Yang, M. Hu, B. He, J. Zhou, H. K. Liu, S. J. Dai, Y. R. Wei, and Q. H. Lou, “Passive coherent beam combining of four Yb-doped fiber amplifier chains with injection-locked seed source,” Opt. Lett. 38(6), 854–856 (2013).
[Crossref] [PubMed]

C. X. Yu, S. J. Augst, S. M. Redmond, K. C. Goldizen, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Coherent combining of a 4 kW, eight-element fiber amplifier array,” Opt. Lett. 36(14), 2686–2688 (2011).
[Crossref] [PubMed]

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Coherent beam combination of three two-tone fiber amplifiers using stochastic parallel gradient descent algorithm,” Opt. Lett. 34(19), 2939–2941 (2009).
[Crossref] [PubMed]

X. L. Wang, P. Zhou, Y. X. Ma, J. Y. Leng, X. J. Xu, and Z. J. Liu, “Active phasing a nine-element 1.14 kW all-fiber two-tone MOPA array using SPGD algorithm,” Opt. Lett. 36(16), 3121–3123 (2011).
[Crossref] [PubMed]

G. D. Goodno, S. J. McNaught, J. E. Rothenberg, T. S. McComb, P. A. Thielen, M. G. Wickham, and M. E. Weber, “Active phase and polarization locking of a 1.4 kW fiber amplifier,” Opt. Lett. 35(10), 1542–1544 (2010).
[Crossref] [PubMed]

Proc. SPIE (3)

A. Flores, C. Lu, C. Robin, S. Naderi, C. Vergien, and I. Dajani, “Experimental and theoretical studies of phase modulation in Yb-doped fiber amplifiers,” Proc. SPIE 8381, 83811B (2012).
[Crossref]

V. Khitrov, K. Farley, R. Leveille, J. Galipeau, I. Majid, S. Christensen, B. Samson, and K. Tankala, “kW level narrow linewidth Yb fiber amplifiers for beam combining,” Proc. SPIE 7686, 76860A (2010).
[Crossref]

D. Engin, W. Lu, M. Akbulut, B. McIntosh, H. Verdun, and S. Gupta, “1 kW cw Yb-fiber-amplifier with <0.5 GHz linewidth and near-diffraction limited beam quality, for coherent combining application,” Proc. SPIE 7, 1407 (2011).

Other (1)

A. Flores, T. M. Shay, C. A. Lu, C. Robin, B. Pulford, A. D. Sanchez, D. W. Hult, and K. B. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” in Conference on Laser and Electro-Optics (CLEO):Laser Applications to Photonic Applications, OSA Technical Digest (Optical Society of America, 2011), paper CFE3.
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1 The schematic of the polarization beam combination process for (a) without phase controlling and (b) phase controlling.
Fig. 2
Fig. 2 The simplified schematic of the unit of CPBC system.
Fig. 3
Fig. 3 Experimental setup of the two-channel multi-wavelength CPBC system. CO1-CO2: collimators; HWP: half wavelength plate; M1: all- reflected mirror; PBC1-PBC2: polarization beam combiners; M2: high-reflected mirror; PD-photo-detector; P1-P3: Power meters.
Fig. 4
Fig. 4 The time dependent signals and spectral density of energy collected in the PD in open loop and closed loop. (a) Time dependent signals. (b) Spectral density of phase noise.
Fig. 5
Fig. 5 The relative spectral density of power and discrete frequencies with different modulation frequencies and voltages.
Fig. 6
Fig. 6 The theoretical and experimentally measured combining efficiencies of the representative twelve different spectral structures.
Fig. 7
Fig. 7 The change of the normalized voltage signals along with the modulation frequencies with the modulation voltage of 10 V.
Fig. 8
Fig. 8 The compared results of estimated combining efficiencies and the measured combining efficiencies with different modulation frequencies (modulation voltage-10V).
Fig. 9
Fig. 9 The change of the normalized voltage signals along with the modulation voltages (d) with the modulation frequency of 100 MHz.
Fig. 10
Fig. 10 The compared results of estimated combining efficiencies and the measured combining efficiencies with different modulation voltages (modulation frequency-100 MHz).
Fig. 11
Fig. 11 The modulated spectral structures of each seed laser ((a)-(c)) and the optical spectrum measured by a spectrum analyzer (d).
Fig. 12
Fig. 12 The voltage signals in the PD when the OPD is carefully compensated.

Equations (16)

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

η=P/ P 0
E i = n=1 N E i, v n exp(j2π v n t i ) ( i=1, 2)
E i, v n = 2 P i, v n π . 1 w .exp[ ( x 2 + y 2 ) w 2 ]exp(j ϕ i )
P i, v n = | E i, v n | 2 dxdy
t i = l i c
E (1) =[ n=1 N E 1, v n exp(j2π v n t 1 ) m=1 N E 2, v m exp(j2π v m t 2 ) ]
I 1 =[ n=1 N E 1, v n exp(j2π v n t 1 ) ][ p=1 N E * 1, v p exp(j2π v p t 1 ) ]+[ m=1 N E 2, v m exp(j2π v m t 2 ) ][ q=1 N E * 2, v q exp(j2π v q t 2 ) ] = n=1 N p=1 N E 1, v n E * 1, v p exp[j2π( v n v p ) t 1 ] + m=1 N q=1 N E 2, v m E * 2, v q exp[j2π( v m v q ) t 2 ] = n=1 N | E 1, v n | 2 + n=1 N p=1 pn N E 1, v n E * 1, v p exp[j2π( v n v p ) t 1 ] + m=1 N | E 2, v m | 2 + m=1 N q=1 qm N E 2, v m E * 2, v q exp[j2π( v m v q ) t 2 ]
P PBC1 = n=1 N P 1, v n + n=1 N P 2, v n
J HWP =[ cos(2α) sin(2α) sin(2α) cos(2α) ]
E ' =[ cos(2α) n=1 N E 1, v n exp(j2π v n t 1 )+sin(2α) m=1 N E 2, v m exp(j2π v m t 2 ) sin(2α) n=1 N E 1, v n exp(j2π v n t 1 )cos(2α) m=1 N E 2, v m exp(j2π v m t 2 ) ]
P y = 1 2 n=1 N P 1, v n [1cos(4α)]+ 1 2 n=1 N P 2, v n [1+cos(4α)] sin(4α) n=1 N P 1, v n P 2, v n cos[ ϕ 1 ϕ 2 + 2π v n ( l 1 l 2 ) c ]
P y = 1 2 n=1 N P 1, v n [1cos(4α)]+ 1 2 n=1 N P 2, v n [1+cos(4α)] sin(4α) n=1 N P 1, v n P 2, v n cos[ ϕ 1 ϕ 2 + 2π v n ( l 1 l 2 ) c +δ(t)]
P ' y = 1 2 n=1 N P 1, v n + 1 2 n=1 N P 2, v n 1 2 4 { n=1 N P 1, v n P 2, v n cos[ ϕ 1 ϕ 2 + 2π v n ( l 1 l 2 ) c +δ(t)] } 2 + ( n=1 N P 1, v n n=1 N P 2, v n ) 2
η=1 P ' y t n=1 N P 1, v n + n=1 N P 2, v n
η a = P p2 P p1 + P p2 + P p3
η=1 1 N i=1 N V(i) V m

Metrics