We demonstrate coherent spectral beam combining and femtosecond pulse spectral synthesis using three parallel fiber chirped pulse amplifiers, each amplifying different ultrashort-pulse spectra. This proof-of-concept experiment opens a path to simultaneously overcome individual-amplifier energy and power limitations, as well as limitations on amplified pulse spectra due to the gain narrowing in a single fiber amplifier.
©2013 Optical Society of America
Technological advantages of fiber lasers are associated with their practicality and their compatibility with high average power operation, due to high efficiency diode pumping, to compatibility with monolithic integration, which enables robust and compact laser systems, and due to fiber geometry with a large surface-to-volume ratio which facilitates efficient heat dissipation [1,2].
However, there are also significant limitations associated with individual fiber lasers. The primary one is a relatively low pulse energy achievable with an individual fiber laser due to a relatively small transverse modal area and a considerably long propagation length leading to nonlinear pulse distortions by stimulated Raman scattering (SRS), self-phase modulation (SPM), or four-wave mixing (FWM), or other constraints associated with energy extraction saturation and optical damage. For example, approximately ~1ns duration pulses in Yb-doped amplifiers (compatible with stretched ultrashort pulses) are limited to pulse energies of ~mJ . Another limitation is associated with gain spectral narrowing when amplifying broad band signals, such as ultrashort optical pulses. For example, although an Yb-doped fiber gain bandwidth can exceed 100 nm, chirped pulse amplification (CPA) of such a broad bandwidth pulse is not possible with a single Yb-doped fiber amplifier since gain narrowing limits the amplified spectrum to approximately 10-20 nm in high gain systems.
Individual-fiber pulse energy limitations can be overcome by combining multiple lasers or amplifiers. Pulse energy scaling of ultrashort pulses requires coherent phasing of multiple parallel fiber chirped-pulse amplifiers (FCPA), as has been recently demonstrated for up to 4 parallel channels  at low power, and up to two channels at high power [5,6]. On the other hand gain spectra limitations of individual lasers can be overcome by combining two different laser gain media, as has been demonstrated by locking and coherent phasing of two individual mode-locked oscillators , or seeding two different gain media with a single mode-locked oscillator [8,9]. Although very short durations of only a few optical cycles were produced, the coherent signal synthesis techniques involved in all these experiments are too cumbersome to be practical when combining more than two optical channels. Indeed, the approach in  requires repetition rate synchronization and phasing between individual mode-locked oscillators, the approach of  uses cross-correlation between the two channels to determine phasing and delay errors, and the approach in  relies on a passive optical-length matching between the two channels.
In this work we coherently spectrally combine multiple parallel fiber CPA channels, demonstrating that coherent phasing techniques used for beam combining of multiple parallel amplifiers can also be used for coherent spectral pulse synthesis. We developed a new phase tracking approach used in conjunction with LOCSET technique [10,11] which uses two-photon absorption (TPA) detector to measure phasing errors between spectrally non-overlapping combined signals. This approach allows for simultaneously increasing pulse energy and power from a fiber CPA array and to increase signal bandwidth to accommodate shorter pulses. For example, when applied to Yb-doped fiber amplifier arrays this technique allows amplification of individual spectrally-distinct broad-band signals, which when coherently spectrally combined at the amplifier array output can offset gain-narrowing effects of individual high-gain fiber amplification channels. In principle this technique could be extended to parallel array beam combining of different gain media, thus enabling a path towards energy scalable few-cycle optical pulses.
The structure of the paper is the following. First we describe the basic concept of coherent-spectral combining/signal synthesis using a multi-channel amplifier array. Then we discuss technical issues associated with phase-error detection and correction in this combining system, and show that TPA detection can be used to implement LOCSET technique [10,11] for coherent phasing of individual channels in a spectrally combined amplifier array. Further, we describe the experimental system used for the proof-of-the-principle demonstration and present experimental results, and finish the paper with a brief discussion and summary.
2. Coherent-spectral combining of fiber CPA arrays
2.1 Conceptual outline
Two basic conceptual-layout variations of coherently-spectrally combined fiber CPA arrays are shown in Figs. 1(a) and 1(b). The main concept here is that initial broad-band seed pulses from, for example, a mode-locked oscillator, are split into N channels spectrally so that each channel amplifies a relatively narrow-band and spectrally distinct signal. This is a major difference from a conventional coherent-combining scheme [4–6] where each channel amplifies identical signals. After amplification all signals are spectrally recombined into a broad-band signal again. The essential difference here from a conventional spectral-combining scheme  is that different spectral slices are combined coherently, i.e. phase-difference between all individual-channel signals is compensated using a suitable phase-error detection, tracking and phase-locking arrangements. This produces a bandwidth-limited broad-band ultrashort pulse consisting of all the different spectral “slices”. It also enables to combine partially overlapped spectra, so that a smooth pulse spectrum can be reconstituted without sacrificing combining efficiency or pulse shape fidelity, as described in more detail further in the text.
Configurations 1(a) and 1(b) are very different with respect to achievable pulse energy. In system 1(a) pulses are spectrally “sliced” after a common stretcher, thus reducing each-channel stretched-pulse duration N-times compared to that before spectral splitting (with N being a number of channels in an array). Since the maximum achievable energy is proportional to the stretched duration of pulses in an amplifier, achievable energies in this configuration are the same as would be from a single-channel CPA without a coherently combined array. Use of individual stretchers and compressors per each channel shown in 1(b) allows to use the same stretched pulse duration in each channel, thus leading to N-times increase in achievable pulse energy when simultaneously increasing the amplified-signal bandwidth by approximately N-times.
2.2 Spectral combining elements
There is a variety of spectral combining elements that can be used for incoherent spectral combining, such as diffraction gratings , volume Bragg gratings , and ultra-sharp edge spectral filters based on multi-layered dielectric coatings . Although in principle any of these combining elements could be used for coherent spectral combining, the broad-band spectrum of ultrashort optical pulses makes dielectric-coating filters a much preferred choice. Indeed, diffraction gratings and volume Bragg gratings rely on spatial spectral dispersion, which for broad-band spectrum is always associated with a spatial chirp of a beam. Dielectric-coating filters are spatial-dispersion free and, therefore, can accommodate any bandwidth without inducing spatial chirp.
Spectral splitters/combiners in systems depicted in Fig. 1 can be implemented as stacks of multilayer dielectric-film filters, consisting of either short-wave pass (SWP), long-wave pass (LWP), band-pass (BP), or band-reflection (BR) filters. Irrespective of a type of a dielectric filter used, the incident broad-band ultrashort-pulse spectrum after the spectral splitter will be “sliced” into separate partially-overlapping spectra for injecting into each parallel amplification channel. Note that this partial spectral overlap between adjacent channels can only be achieved with spatial-dispersion-free devices. The unique aspect of coherent spectral combining with such devices is that partially-overlapping spectra can be recombined (as shown in Fig. 1) without losing combining efficiency in the overlapping part of the spectra, thus enabling to reconstitute a smooth spectrum at the output, which cannot be achieved with incoherent spectral combining.
This can be shown using an illustration in Fig. 2(a) , which depicts wavelength-dependent normalized transmission T(λ) at a spectral edge of a filter. This edge can represent, for example, a short-wavelength edge of LWP or BP filters, or, alternatively, a long-wavelength edge of BR filters. Insert in this figure illustrates use of the filter as a beam combiner, when transmitted beam is overlapped spatially with the reflected beam thus producing an output consisting of both short-wavelength and long-wavelength portions of the input signals. Reversed propagation direction should be used for spectral beam splitting. For example, consider a beam incident from right-to-left with the wavelength-independent spectral intensity I(λ)I0, then, since within the spectral edge region this filter acts as a wavelength-dependent T(λ):(1-T(λ)) beam splitter, the transmitted and reflected wavelength-dependent beam intensities will be T(λ)∙I0 and (1- T(λ))∙I0, respectively. Now let’s consider an identical filter being used as a beam combiner for input beams with intensities I1 and I2 as shown in the insert. It is straightforward to find by direct calculation that, assuming both beams are in phase, the combining efficiency η = Icombined/(I1 + I2) isEq. (1) yields η = 100% for all wavelengths. In other words, all incident power can be coherently spectrally recombined into a single beam without any power loss provided that split-edge spectral shape is perfectly preserved between splitting and combining.
In practice preservation of such an ideally perfect spectral-edge-shape match between split and recombined signals might be challenging due to gain spectral reshaping in individual fiber amplifiers of a parallel array. Using relatively steep spectral-edge dielectric coating filters can mitigate this. Alternatively, it can be bypassed altogether by resorting to non-overlapping spectra, in which case spectrally smooth recombined spectra cannot be achieved.
An example of spectral transmission characteristics of LWP filters (Semrock, LP02-1064RS-25) used as spectral combiners in the experiments reported in this work are shown in Fig. 2(b). The spectral-edge steepness of these filters is less than 1nm, and the spectral-edge wavelength can be adjusted by the input-beam angle of incidence as indicated in the figure.
2.2 Phase locking
Coherent phasing between all parallel amplification channels requires tracking and correcting phasing errors in each channel. In this work we are using LOCSET coherent phasing scheme in which each channel is “tagged” by individual harmonic-modulation frequency imprinted through a phase modulator [10, 11], the same modulator that is also used to correct the phase of the channel. Detection of individual-channel errors in this scheme is done with a single detector, in which each phase modulation frequency produces an identical-frequency electric current proportional to the magnitude of the corresponding-channel phase error. Electronic circuit connected to the detector recognizes each-channel phase error and applies the required-magnitude correction signal to the phase modulator in a corresponding channel.
LOCSET has been developed for conventional coherent phasing of spectrally identical parallel-channel signals, which are beam-combined at the system output interferometrically using some type of 50:50 beam splitters. This interference is what converts phase modulation in each channel into intensity modulation detected by the error-tracking detector. Main challenge in applying LOCSET to spectral-coherent combining is associated with the fact that the nature of spectral combining is non-interferometric, i.e. combined average power is independent of the relative phases between different-spectra signals, provided there is no spectral overlap between them. Interference can only occur in the regions of partial spectral overlap between the signals. Consequently, since in the coherent spectral combining all channel spectra either do not overlap or overlap only weakly, a linear detector (i.e. a detector with an output signal linearly proportional to the incident power) at the combined output will respectively detect either no interference or only a weak interference signal. Therefore, linear detector appears to be quite inefficient for phasing parallel channels in a coherently spectrally combined system.
As a result, it is necessary to devise a new phase-error detection approach, which is not based on combined-signal interference. Since the objective is to reconstitute from parallel-channel spectral slices a bandwidth-limited combined pulse, it is natural to seek a detection scheme that would sense combined pulse peak power. This can be done by replacing a linear detector with a quadratic one, i.e. a detector with an output signal quadratically proportional to the incident peak power. We have implemented a quadratic detector by using a standard semiconductor diode specially selected so that photon energy of ~1μm wavelength light falls well below its band gap edge. Consequently, output signal of this detector is produced only by two-photon absorption (TPA), and is, therefore, proportional to the square of the peak power. Compatibility of a TPA detector with frequency tagging to track phase errors in the LOCSET scheme is shown in the Appendix of this paper.
It is obvious that the highest TPA signal will occur for bandwidth-limited pulses, when all channels are in-phase, and the signal peak power is at its maximum. It is, however, far from obvious whether TPA signal always increases with decreasing phase mismatch between the different channels. Indeed, changing phases between different spectral slices from different channels is equivalent to phase shaping the combined pulse. In principle, in a N-channel system there are N degrees of freedom in controlling pulse shape. This leads to a very complicated “landscape” of possible combined-waveform shapes, and to a very complex change in these shapes with decreasing phase “mismatch” between the channels. If TPA signal would not always increase with decreasing phasing error then its use for phase correction feedback would be problematic.
In order to validate suitability of a quadratic detector for implementing coherent phasing of a fiber amplifier array we have performed a numerical statistical study of TPA signal dependence on inter-channel phasing error. This study shows that statistically the TPA detector output always increases with the decrease in the overall phasing error, and, therefore, confirms that the quadratic detector should work with the LOCSET scheme. To show that, it is necessary to express the combined signal spectrum as a coherent sum of individual-channel spectral slices:
Generation of carriers in a TPA detector is an instantaneous process, which is proportional to the square of the optical combined-signal irradiance. TPA detector electric response, however, is much slower than the optical pulse repetition period (but is selected to be much faster than LOCSET modulation period). Therefore, the TPA detector electric response is proportional to the time average of the optical combined-signal squared:
Random variation in optical path length due to external factors is accounted by assigning the probability with which each j-th channel phase magnitude ϕj occurs, and assuming that this probability distribution is described by a normal (Gaussian) distribution with its mean equal to zero and its variance equal to σ:
Statistical description of this normalized TPA response is achieved by running the calculations K times, each time with a different set of random phases , where k = 1, 2, …, K. All these random phase distributions Σk are calculated using the Gaussian probability distribution shown above, all with the same variance σ (i.e. the same overall inter-channel phasing error magnitude). Results of this statistical calculation can be cast as a histogram, with the horizontal axis representing TPA signaland subdivided into M slots with the width each, and the vertical axis representing the number of occurrences that the calculated values fall into each interval (m = 1, 2, …, M).
One representative example of this statistical calculation is shown in Fig. 3 . It shows calculated histograms of the statistics of the TPA responses for different overall inter-channel phasing-error magnitudes, as indicated by the different variance σ values shown in the figure. This particular calculation has been performed for 14-nm FWHM wide “flat top” spectrum, similar to the one used in actual experiments. The spectrum, however, was subdivided into 10 equal-width slots, which corresponds to a 10-channel array. To get reliable statistical distributions each calculation with each particular σ has been performed K = 10,000 times. Plotted results clearly show that the TPA electric-response signal increases with decreasing overall inter-channel phasing error σ.
3. Experimental setup
The experimental setup of 3-channel pulse synthesis is shown in Fig. 4 . A femtosecond seed source from the Nd:glass oscillator (central wavelength at 1059 nm, 72 MHz repetition rate, 12 nm spectral width) generates pulses that are stretched by a conventional diffraction-grating stretcher to around 500 ps in duration and coupled into a fiber CPA system. The stretched pulses are split into 3 channels and proceed to delay lines. The delay lines are composed of non-reciprocal fiber circulators, micro-optic mirrors, and 4-nm bandwidth bandpass spectral filters (Semrock, LL01-1064-25). The first two are used to fine tune the delay within a range of 1 cm whereas the last one is placed in a free space portion to divide the channel spectra into distinct parts. The schematic of the delay line with spectral filter and its 3D rendering is shown in Fig. 5 .
When using a linear detector (Thorlabs, DET10A) for partial spectral overlap, we slightly overlap the spectra of adjacent channels so that without phase locking some output power fluctuates due to random constructive and destructive interference from the common wavelength range, and thus can be used for feedback control. When using a TPA detector (Hamamatsu, G1735) for no spectral overlap, we just slice the spectra into three adjacent pieces with no spectral overlap.
After the delay line the pulses in two of three channels go to fiber piezo-stretchers (Optiphase, PZ1) for phase control. Following the piezo-stretchers the pulses are amplified with standard telecom grade single-mode pump diodes and PM fiber components. All three channels use identical in-fiber components to ensure the equal amount of linear and higher-order dispersion from these components in each channel. Moreover, each channel provides approximately 20-dB net gain in a fiber amplifier.
After amplification, the three separate channels are collimated out of the fibers into a free-space spectral combiner composed of two edge spectral filters (Semrock, LP02-1064RS-25). Note that each output from a single-mode fiber amplifier is diffraction-limited. Special care has been taken to ensure that a good spatial overlap between combined beams has been achieved. For this purpose we use a single-mode test fiber positioned after the beam combining. Spatial overlap is evaluated by measuring and comparing coupling efficiencies for each individual channel and for the combined beam. At 75% coupling, the difference in these coupling efficiencies is only 2%, indicating a very good spatial overlap between the beams. The combined pulse is then recompressed with a diffraction grating compressor. A small portion of the compressed output power is detected by either the linear or TPA detector and served as a feedback signal with phase error information for two of three channels with respect to the reference channel. Through the self-referenced LOCSET technique [10,11] and two sets of feedback signal processing units, the phase error signal is individually extracted and sent to piezo-stretchers for phase compensation. A feedback loop that maximizes the linear or TPA intensity reinforces the channels to interfere constructively and hence locks the phases of the separate channels. In our experiment, the combining efficiency is defined as the combined power at the output of spectral combiner divided by the sum of each individual input power of spectral combiner. Therefore, it refers to “absolute” combining efficiency.
In the feedback system, the piezo-stretcher can provide a π phase shift per 2.6 V driving voltage and can be driven up to ± 500 V equivalent to 384π continuous phase control. Our feedback electronics is limited to an output of ± 5 V (3.8π). To attain long-time stable locking, we use the high-voltage amplifiers to reach at least ± 100 V (76π). Moreover, the noise regime of our environment associated with acoustic vibrations and temperature drift is below 1 kHz . We choose RF modulation/demodulation frequencies at 5 kHz and 6 kHz to be fast enough to detect the phase drift as well as to be slower than the repetition rate (72 MHz) of the oscillator to maintain the validity of self-referenced LOCSET technique. The integration time of the feedback electronics is set to 50 msec to cancel phase disturbances up to 20 Hz in frequency and the phase modulation amplitude β is set to 0.25 to provide stable locking.
Owing to two different off-the-shelf types of spectral filters for splitting and combining the spectra, it is inevitable to consider their steepness mismatch. To achieve the best combining efficiency for partial spectral overlap, we adjust bandpass and edge filters to let any two adjacent spectra overlap the minimal detectable range so that phase locking can still be fulfilled as well as minimizing the power loss.
4. Experimental results
The objective of the proof-of-the-principle experiment was to demonstrate synthesis of coherently spectrally combined pulses with the combined-pulse durations much shorter than that from each individual channel, and to characterize combined and synthesized pulse temporal quality. We explored two different coherent spectral combining cases: one with partially overlapping spectra, and another with separated non-overlapping spectra.
Experimental results for the partially overlapping case are shown in Fig. 6 . As is apparent from Fig. 6(a), each individual channel spectra are roughly triangular with approximately 3-nm bandwidth, as determined by the spectral-edge steepness of the filters used in the experiment. Overlap between any two adjacent individual spectra was set to be approximately 1nm. The full-width-half-maximum (FWHM) of the combined spectrum is 8 nm. Due to the partial spectral overlap it was possible to use in this case both linear and TPA detectors for tracking inter-channel phasing errors. The measured background-free SHG autocorrelation traces of individual-channel and combined signals are shown as solid lines in Fig. 6(b). For reference we also show in this figure the calculated autocorrelation trace (dash line) of the bandwidth-limited pulse obtained from the measured combined spectrum (red line) from Fig. 6(a). The measured autocorrelation trace durations of individual-channel and combined pulses are 1328 fsec (829 fsec), 1368 fsec (777 fsec), 1501 fsec (871 fsec), and 547 fsec (403 fsec) respectively, with the the corresponding deconvolved pulse durations given in parentheses. These results clearly show that the combined pulse is approximately 2 to 3 times shorter than each individual-channel pulse. Comparison between the measured and calculated autocorrelation traces indicates that the combined pulse duration of 403 fsec is slightly longer than the calculated transform-limited 363 fsec duration, and the measured trace has long wings, which are absent in the calculated trace. We identified that this difference is caused by some residual third-order dispersion uncompensated in the system. The absolute combining efficiency and combined power are measured to be 76.3% and 257 mW respectively. Since the spectral combiner has intrinsic loss from each edge filter with 95% transmission and 99.7% reflection, it gives 94.9% spectral filter efficiency in a 3-channel setup. The additional 18.6% efficiency loss in this experiment was from the slope mismatch between the bandpass and edge filters in each channel.
Experimental results for non-overlapping spectra are shown in Fig. 7 . In this case phasing between the channels was achievable only with the TPA detector. Figure 7(a) shows individual-channel and combined spectra. The FWHM of the combined spectrum spans approximately 10 nm and is comprised from 3 nearly identical triangular-shaped individual spectra. The measured autocorrelation traces are shown in Fig. 7(b) by solid lines, and the dashed line shows the calculated transform-limited trace of the combined spectrum. Corresponding autocorrelation-trace and deconvolved pulse (shown in parentheses) durations for individual-channel and combined pulses are 1700 fsec (1071 fsec), 1420 fsec (761 fsec), 1327 fsec (834 fsec), and 474 fsec (356 fsec). The combined pulse width in this case is also approximately 2 to 3 times shorter than each individual-channel ones, and longer than the calculated transform-limited 278 fsec duration. Note, however, that in this case the satellite structure appears in the wings of both the measured as well as the calculated bandwidth-limited traces of the combined pulses, with wings in the measured trace still being noticeably larger than in the calculated trace. The latter is still caused by the uncompensated third-order dispersion in the system. The satellites in the combined pulse appear due to the strongly modulated profile of the combined spectrum, produced by the triangular shape and large separation between individual channel spectra. The absolute combining efficiency and combined power are measured to be 85.8% and 273 mW respectively. With 94.9% spectral filter efficiency, the 9.1% additional loss corresponds to efficiency loss from the some spectral mismatch between signal splitting and combining arrangements. Note that the small overlap in Fig. 7 does not capture the fact that there was a much more significant mismatch between spectral characteristics of the spectral splitters and combiners used in this experiment. In both partial overlapping and non-overlapping cases, the autocorrelation traces were very stable in time, as indicated by the fact the autocorrelation trace from a phase-locked array observed on an oscilloscope screen was not changing in time.
Phase locking results for the two cases are shown in Fig. 8 . Figure 8(a) shows phase–error tracking detector output for the configuration with partial spectral overlap. For this particular measurement a linear detector has been used. Figure 8(b) shows TPA detector output for the configuration with non-overlapping spectra. Both figures compare results for locked and free running operation. As one can see from these figures stable phase locking has been achieved in both cases and with both detectors (as indicated by red lines in the figures). Note that for free running operation (blue lines) in both cases phase tracking detector output fluctuates, as expected. However, the span of these fluctuations does not reach zero for reasons that are very different for each of the two cases. In Fig. 8(a) fluctuations measured with linear detector are due to the interference in the spectrally overlapped regions. Since the fraction of the power in these overlapped regions is small compared to the total power, fluctuation amplitude is only a small fraction of the detector output voltage. In Fig. 8(b) fluctuations measured with TPA detector are due to peak intensity variation in the combined signal. In this case, although the detector operates at frequencies below its absorption bandgap and in principle should only detect the two-photon signal, there is still a small residual linear absorption which produces a background signal comparable to the TPA signal magnitude. Moreover, the TPA signal in a free-running system should not go to zero even in the absence of residual linear absorption, when the background signal would be comprised only of the sum of the individual-channel TPA signals. This background signal is constant both for locked and free running operation.
5. Discussion and summary
These experimental results show that, since coherent spectral combining of ultrashort pulses essentially constitutes a signal synthesis, combined-pulse temporal shape very strongly depends on individual-channel spectra. Generation of background-free pulses requires smooth combined spectrum, which can be achieved using partially overlapping individual-channel spectra. Combining of separate, non-overlapping individual-channel spectra leads to significantly structured combined-pulse background. In addition, controlling the phase between the channels can be used to shape the temporal profile of the combined signal.
To summarize, we had demonstrated coherent spectral combining of femtosecond optical pulses from multiple parallel fiber CPA channels. This technique enables ultrashort pulse amplification with an aggregate spectrum significantly exceeding amplification bandwidths of each individual amplifier, as well as allows scaling average power and pulse energy beyond single fiber limitations. Potentially, this technique could lead to fiber laser sources of tens of femtosecond duration multi-mJ pulses with high average powers.
Here we evaluate the two-photon-absorption (TPA) phase error signal under the LOCSET scheme. The total electric field in time domain from a system using self-referenced LOCSET technique [10,11] can be written as
Since the optical fields from the unmodulated element and all of the phase modulated elements are superimposed on the photodetector, the TPA photodetector current is
The phase error signal is extracted from the TPA photocurrent using coherent demodulation in the RF domain. The TPA photocurrent is multiplied by sin(ωit) and integrated over a time T that is larger than the optical period (2π/ωL) but smaller than any of the modulation periods (2π/ωi). Then, the phase error signal for the ith element is given by
Using the Fourier series expansionsEq. (6), then we can substitute Eq. (6) into Eq. (7) and calculate the phase error signal from Eq. (8) to give
From the complicated Eq. (10), one can conclude specific phase error signal is extracted when specific demodulation frequency is implemented so that each modulated element has its own specific phase error signal without confusion with other elements. Moreover, Eq. (10) is nonzero under the free-running operation except when the system is phase-locked, ϕi=ϕj (i, j=1, 2,...N), then cos(ϕi-ϕj)=1, sin(ϕi-ϕj)=0, and thus Si=0.
If TPA photocurrent is demodulated by sin(ωkt), where ωk is different from the tagging modulation frequency ωi, then
This is due to the orthogonality of Fourier series for different frequencies over an integration of a time period T. With Si≠0 and Sk,i=0, one can summarize TPA provides with the correct “frequency tagged” signal for the LOCSET-scheme phase error signal detection.
The authors acknowledge the financial support from Office of Naval Research (Grant No. N00014-07-1-1155).
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