Arbitrary energy-preserving control of the line spacing of an optical frequency comb over six orders of magnitude through self-imaging

Hugues Guillet de Chatellus; Luis Romero Cortés; José Azaña

doi:10.1364/OE.26.021069

1. Introduction

Coherent optical frequency combs (OFCs) [1] are of fundamental interest for numerous applications, including frequency metrology [2], spectroscopy [3,4], and microwave photonics [5]. The capability of controlling the line spacing (LS) of an OFC is of critical importance. OFCs have been traditionally generated using passively mode-locked lasers, which have free spectral ranges (FSR) typically limited between a few tens to a few hundreds of MHz [6]. However, many applications of OFCs, e.g., astronomical spectrometer calibration or waveform synthesis, require frequency spacings beyond the GHz range [6,7]. Demonstrated successful solutions for GHz-LS comb generation include cavity miniaturization [8–10], frequency filtering [11] or optoelectronic generation [3,7,12–14]. On the other hand, OFCs with lower LSs provide high-intensity laser pulses, which can find applications in non-linear optics and spectroscopy [15]. However generating mode-locked OFCs with FSR in the MHz range would require the use of unpractical long laser cavities. In spite of all progress made towards development of OFC generation platforms over such a wide range of LSs, any of these platforms would still benefit from techniques that can provide additional capabilities to control further the LS of the generated OFCs, beyond the constraints imposed by the specific comb generation architecture. In this context, methods based on combined temporal and spectral self-imaging (SI), or Talbot effects, involving simple linear manipulations of the spectral and/or temporal phase profiles of the OFC, are particularly attractive: this set of techniques offers a practical solution to achieve nearly arbitrary LS control of any given OFC while preserving the main specifications of the original comb, namely its energy, spectral envelope, bandwidth, and reference frequency grid [16–26].

Temporal SI, or Talbot, effects are observed in the linear propagation of a periodic train of pulsed waveforms (e.g., a mode-locked pulse train) through a second-order dispersive medium [16,17]. In particular, temporal SI enables repetition rate multiplication of a mode-locked pulse train (corresponding to a phase-locked optical frequency comb, OFC, spectrum) by an integer factor $q$ , without affecting the individual pulse time duration and shape, through application of a proper quadratic phase profile along the pulse-train frequency spectrum, defined as:

φ_{k} = π \frac{p}{q} k^{2}

where

k

= 0, 1, 2, ... labels the OFC spectral line, and

p

and

q

are coprime integers. Since the temporal SI process ideally involves a pure spectral phase filtering transformation, the total energy of the incoming pulse train is ideally preserved (except for practical passive insertion losses in the dispersive filter). As a result, the energy of each individual pulse in the output pulse train is divided by a factor of

q

with respect to the input one. In practice, the required quadratic spectral phase profile can be achieved by second-order or group-velocity dispersion (GVD) in a transparent medium [16,17].

Similarly, in the frequency domain, spectral SI effects have been described that enable integer division of the frequency spacing of an input phase-locked OFC by an integer factor m, without affecting the shape and bandwidth of the comb spectral envelope, through application of a quadratic temporal phase modulation (PM) profile to the input train of pulses, defined by:

ψ_{n} = π \frac{s}{m} n^{2}

where

n

= 0, 1, 2, … labels the pulse, and

s

and

m

are coprime integers [18]. The required temporal phase mask (also referred to as a time-lens [19]) can be practically implemented by electro-optic phase modulation (EOPM) [18,20] or non-linear techniques [21]. Again, the energy of the output frequency lines is divided by a factor of

m

. It is noteworthy that contrary to mode-locked lasers, in both temporal and spectral SI, the repetition rate of the pulse train differs generally from the comb LS.

Recently, the extension of the capabilities of temporal SI beyond the simple case of integer division of the period was envisioned and demonstrated [22–24]. In particular, it has been shown that when the input pulse train is pre-conditioned through a suitable quadratic temporal phase modulation prior to propagation through the GVD medium, the period of the output pulse train can be set to be any arbitrary fractional multiple (higher or lower than 1) of the input period. This leads to the possibility of either repetition rate multiplication or division by any desired factor. Owing to the inherent energy-preserving property of this procedure, when the temporal period-multiplication factor is larger than unity, the remaining individual pulses in the output train are effectively amplified as compared to the input ones [22,24]. The method can be interpreted as a combination between spectral and temporal SI effects, respectively implemented by the PM and GVD processes. Through this generalized temporal SI method, the LS of the corresponding OFC is only affected (i.e., integer divided) by the first PM process, such as in conventional spectral SI.

Recently, we have shown how SI can be also generalized in the spectral domain to enable arbitrary control (division or multiplication by any desired fractional or integer factor) of the LS of an incoming OFC [25]. The method involves pre-conditioning the spectral phase of the incoming OFC, e.g., through GVD, prior to temporal PM, namely a combination of temporal and spectral Talbot effects. In this communication, we demonstrate experimentally some of the unique capabilities offered by this method, most prominently the possibility of controlling the comb LS over several orders of magnitude. As illustrated in Fig. 1, such an unprecedented comb-LS control is achieved while preserving the energy, spectral coverage (including bandwidth), and reference frequency grid of the original comb. We also report experimental results that show how the generalized spectral SI method provides additional important capabilities to set the absolute frequency location of the output comb lines.

Fig. 1 Illustration of energy-preserving control of an input OFC by integer/fractional division/multiplication of the input comb LS.

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Theoretically, the concept of generalized spectral SI applies to any optical frequency comb. In practice however, it relies on applying a suitable quadratic spectral phase to the OFC (temporal SI), prior to the temporal PM process (spectral SI) [25]. It turns out that the requirements in GVD can be challenging to meet with current technology. The first step (temporal SI) requires typical variations of the spectral phase between consecutive lines of the order of $π$ rad (Eq. (1)). This condition can be met using specially engineered optical dispersive media when the LS of the input OFC exceeds a few GHz [23,25], but would be extremely difficult to achieve with input LSs in the 10-100 MHz range, as found in conventional passively mode-locked lasers. Recall that these lasers remain the most widely available and used platform for OFC generation. Line-by-line pulse shaping techniques would also fail to provide the required spectral phase profile, due to the limited frequency resolution of conventional pulse shapers (typically in the order of tens of GHz) [26].

To demonstrate the full potential of the generalized spectral SI concept, we use here a platform based on a frequency-shifting loop (FSL) seeded with a CW single frequency laser [27,28]. This extremely simple system generates a comb of optical frequencies with a LS in the tens of MHz range. An interesting property of this system –key to the proposed LS control method– is that the generated combs present a built-in, and arbitrarily tunable, quadratic spectral phase. In other words, the CW-seeded FSL mimics a phase-locked frequency comb after propagation through an arbitrary amount of GVD [29]. This allows us to overcome the aforementioned limitations of available optical dispersive media. By exploiting the generalized spectral SI method in this laser platform, we report lossless redistribution of an input ~80 MHz-LS OFC into output OFCs with electronically-tunable LSs over six orders of magnitude, from ~8 kHz to ~8 GHz. Additionally, we illustrate application of the method for enhanced OFC detection and kHz-frequency point spacing comb-based spectral measurements. From a fundamental point of view, this work demonstrates the unique capabilities and flexibility offered by generalized spectral SI effects towards unprecedented LS control of frequency combs. From a practical perspective, this work paves the way to generation of arbitrary on-demand OFCs, currently out of reach of conventional systems.

Our article is organized as follows. In the first section, we describe in detail the theoretical background of the concept that enables full control of the LS of an incoming OFC by a combination of temporal and spectral SI steps. Second, we provide a short theoretical description of the CW injection seeded FSL, as a source of an OFC with a tunable built-in quadratic spectral phase, and report an experimental characterization of the system. Next, we prove the possibility of arbitrary tuning of the output LS over six orders of magnitude by generalized spectral SI, i.e., by use of an additional electro-optic PM stage at the output of the FSL system. Finally, we report two applications of generalized SI of optical frequency combs: detection of an OFC with a LS well below the spectral resolution of the spectrometer, and improvement of the frequency spacing of OFC-based spectral measurements by orders of magnitude.

2. Theoretical aspects of generalized self-imaging of OFCs

Figure 2 illustrates the proposed concept for arbitrary LS control of an input coherent frequency comb with an LS $f_{s}$ . We target multiplication of the original LS by a factor $q / m$ , to achieve an output frequency spacing equal to $(q / m) f_{s}$ . In a first step, a suitable spectral phase profile, Eq. (1), is imposed onto the input OFC to satisfy a temporal SI condition corresponding to pulse repetition-rate multiplication by a factor of $q$ . The LS of the frequency comb remains equal to $f_{s}$ , which implies that the output temporal train has acquired a pulse-to-pulse phase profile [30]. This is a deterministic temporal phase profile that can be obtained by analytical calculations [30,31], summarized here in an Appendix at the end of the paper.

Fig. 2 Principle of arbitrary LS control of an input OFC. a, scheme of the generalized spectral SI concept (involving a combination of temporal and spectral SI) for user-defined LS control of the OFC, with the notation in the text. In the plots, t stands for time and f for frequency. The intensity of the train of pulses is plotted in red, and the power spectral density (PSD) is represented in blue. The temporal and spectral phases ( $ψ$ and $φ$ respectively) are plotted in green and pink, respectively. The parabola-shaped dotted curves illustrate the intrinsic quadratic profile of the involved phases through the three steps of the SI process. In the shown example, $q = 3$ and $m = 4$ .

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Step two involves compensation of this deterministic pulse-to-pulse phase variation by means of a temporal PM process: this operation leaves the multiplied pulse repetition rate ( $q f_{s}$ ) unchanged, but it locks in phase all pulses in the train. As a result, the LS of the resulting OFC is increased to $q f_{s}$ , implementing the target multiplication of the input LS by the factor q (Fig. 3). The described frequency and time-domain phase manipulation processes do not introduce any change on the comb offset frequency, so that the remaining spectral lines at the output are locked to those of the original OFC. Moreover, the specific set of input spectral lines that are transferred to the output OFC can be selected by simply shifting the temporal PM pattern with respect to the train of optical pulses: a temporal shift of the modulation pattern by the output pulse repetition period $1 / (q f_{s})$ translates into a shift of the selected set of frequency comb lines by a multiple of $f_{s}$ (Fig. 3).

Fig. 3 Numerical simulations illustrating LS control of a Gaussian mode-locked input OFC (input LS = $f_{S}$ ) through generalized spectral self-imaging, involving a combination of temporal SI (quadratic spectral phase filtering, left column) followed by standard spectral SI (quadratic temporal phase modulation, middle column) of the original OFC. The output and input OFCs for each of the simulated cases are shown in the right column.

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In a third step, the resulting temporal pulse train is again phase modulated according to Eq. (2) (spectral SI), achieving a division of the multiplied comb LS, $q f_{s}$ , by an integer m. In this way, an input OFC of LS $f_{s}$ is finally transformed into the target output OFC with an LS equal to $(q / m) f_{s}$ . In practice, the two described PM steps can be combined into a single phase sequence, delivered by a single PM device. Notice that the factor $q / m$ can be either larger or smaller than 1, so in principle, the original comb LS can be increased or decreased by any desired fractional or integer factor, while keeping the original frequency-grid reference.

The proposed concept relies on frequency and time-domain phase-only manipulations of the original OFC, so that the overall process is ideally lossless. Except for potential insertion losses in a practical setup, the entire energy of the input OFC is redistributed over the output OFC. Moreover, the spectral envelope and bandwidth of the original OFC are also preserved, while it is known that no noise is added by temporal and spectral SI processes [22,32]. Consistently, within our measurement capabilities, the experimental results shown below evidence that the OFC energy redistribution occurs without introducing any observable increase in the spectral noise floor and individual spectral line-width of the processed frequency combs.

Figure 3 shows numerical simulations illustrating LS control of a Gaussian mode-locked input OFC (input LS = $f_{S}$ ) using generalized spectral SI, as described above. In the temporal SI step (left column), a phase given by $φ_{k} = π (p / q) k^{2}$ (in pink) is imprinted into the comb spectrum ( $k$ is the index of the frequency line). Subsequently, we apply a temporal phase profile (green curve) equal to the sum of the opposite of the temporal phase resulting from the repetition-rate multiplication process (temporal SI step) and a temporal phase given by $ψ_{n} = π (s / m) n^{2}$ , as required to induce the desired spectral SI effect ( $n$ is the index of the pulse in the temporal sequence) . The resulting output OFCs (dark blue curves) are plotted in a logarithmic scale (right column) and compared to the input OFC (light blue curves) for each of the three cases considered here: integer LS multiplication ( $p = 1$ , $q = 25$ ; $s = 0$ ), fractional LS multiplication ( $p = 1$ , $q = 25$ ; $s = 1$ , $m = 2$ ) and fractional division of the LS ( $p = 1$ , $q = 2$ ; $s = 1$ or $2$ , $m = 7$ ). As predicted, in all cases, there is a lossless redistribution of the original comb energy into an OFC with an $F S R = (q / m) f_{S}$ over the entire input frequency bandwidth. Note that the three different simulation examples shown for the case of integer LS multiplication ( $p = 1$ , $q = 25$ ; $s = 0$ ) illustrate how the absolute location of the output frequency lines in the comb can be tuned by simply delaying the temporal PM sequence; in particular, a temporal shift of the modulation pattern by the output repetition period, $1 / (q f_{s})$ , translates into a shift of the selected set of comb lines by a multiple of the original LS ( $f_{s}$ ). The two different simulation examples for the case of fractional LS division ( $p = 1$ , $q = 2$ ; $s = 1$ or $2$ , $m = 7$ ) highlight the possibility of additionally shifting the output comb by multiples of $(q / m) f_{S}$ , when the product of the factors $m$ and s in the last PM step, $m \times s$ , is an odd number. The two shown plots of the optical spectra correspond to the same (central) region of the comb spectrum.

3. Implementation of the generalized spectral SI in a CW injection-seeded FSL

In practice, implementation of the required spectral phase masks for a frequency comb with a LS in the sub-GHz range would require the use of challenging high-resolution spectral line-by-line shaping methods [25], ultrahigh GVD techniques [33] or customized linear optical filtering devices [34]. For an experimental proof of the full potential of the concept, we use here an architecture based on a CW injection-seeded frequency shifting loop (FSL) (or frequency shifted feedback laser), Fig. 4. This simple system directly generates a discrete OFC with a tunable built-in quadratic spectral phase profile [27,28].

Fig. 4 Simplified circuit diagram of a CW-seeded FSL with arbitrary LS control. The loop is seeded with a narrow-linewidth single-frequency (CW) laser and includes an Erbium Doped Fiber Amplifier (EDFA), an optical isolator, a tunable optical bandpass filter (OBPF) and an acousto-optic frequency shifter (AOFS) driven by an arbitrary function generator (AFG). The required temporal SI effect (step 1) is inherently realized by the FSL, while the desired spectral SI processes (steps 2 and 3) are achieved by temporal phase modulation of the pulse train from the FSL using a single external electro-optic phase modulator (EOPM) driven by an arbitrary waveform generator (AWG).

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As shown in Fig. 4, an FSL consists of an amplified loop, including a gain medium and a frequency shifter, typically an acousto-optics frequency shifter (AOFS). This system is characterized by $f_{s} = ω_{s} / 2 π$ , the frequency-shift per roundtrip, and $τ_{c} = 1 / f_{c}$ , the roundtrip time in the loop. A coupler enables to seed the loop and to extract a fraction of the light field circulating in the loop. If the FSL is seeded with a single-frequency laser (angular frequency: $ω_{0} = 2 π f_{0}$ ), a new spectral line is generated each roundtrip with a frequency spacing dictated by the frequency shifter, thus generating a periodic OFC with a LS equal to $f_{s}$ , where the offset optical frequency is controlled by the original seed laser. Each of the newly generated comb spectral lines is delayed by $τ_{c}$ with respect to its immediate frequency neighbor, so that a linear group delay profile –or the related quadratic phase variation- is intrinsically induced across the obtained frequency comb. More precisely, the electric field at the FSL output can be expressed as [27,28]:

E (t) = E_{0} e^{- i ω_{0} t} \sum_{k \geq 0} g (k) e^{- i k (ω_{s} t - ω_{0} τ_{c})} e^{i π k (k + 1) \frac{f_{s}}{f_{c}}}

where we recall that

τ_{c} = \frac{1}{f_{c}}

is the round-trip time in the loop and

g (k)

is a function describing the envelope of the comb (

g

is set by the profile of gain and losses in the loop). The spectral bandwidth of the resulting OFC can be adjusted by means of an optical bandpass filter (OBPF) inserted in the loop (Fig. 4).

When $f_{s}$ is a multiple of $f_{c}$ , neglecting a linear phase term that simply translates into an undistorted temporal shift, the electric field can be re-written as:

E (t) = E_{0} e^{- i ω_{0} t} \sum_{n \geq 0} g (n) e^{- i n ω_{s} t}

and the output field is equivalent to that of a mode-locked laser whose repetition rate is simply

f_{s}

.According to Eq. (3), in the general case, the output laser field is equivalent to a mode-locked laser having acquired a quadratic spectral phase, e.g. after propagation through a GVD medium. The quadratic spectral phase profile is mathematically defined by

π (f_{s} / f_{c}) k^{2}

, where k labels the frequency line number of the comb. This spectral phase can be tuned simply by changing the cavity round-trip frequency

f_{c}

, or the frequency shift

f_{s}

, enabling an agile implementation of the first temporal SI step. In practice, the spectral phase tuning was achieved by changing the acousto-optic frequency shift

f_{s}

(the length of the FSL being constant).

When $f_{s} / f_{c} = p / q$ ( $p$ and $q \neq 1$ being coprime integers), a fractional temporal Talbot condition, Eq. (1), is met and the output signal consists of a train of transform-limited pulses at a repetition rate equal to $q f_{s}$ (fractional temporal SI). The FSL shows the intrinsic capability of providing on-demand pulse trains, with reconfigurable repetition rates [27,28]. The maximum repetition rate is constrained by the spectral bandwidth of the frequency comb, and in particular, it is limited up to $N f_{s}$ , where $N$ is the total number of lines in the comb. The repetition rate is readily tunable by changing the length of the loop ( $f_{c}$ ), or more practically, by adjusting the frequency shift $f_{s}$ . Note that to be able to tune the repetition rate from $f_{s}$ to $N f_{s}$ by multiples of $f_{s}$ , the ratio $f_{s} / f$ needs to be varied by $1 / 2$ , which is relatively easy to achieve with current technologies: assuming a roundtrip time in the loop of 76.2 ns ( $f_{c}$ = 13.13 MHz) and an AOFS operating around 80 MHz, $f_{s}$ needs to be tuned between 77 and 83 MHz, which lies well within the bandwidth specifications of commercial AOFSs.

4. Experimental results

4.1 Architecture and characterization of the CW injection-seeded FSL

As shown in Fig. 4, the FSL developed for demonstration of the concept is based on an optical fiber loop (single mode fiber-SMF 28) including an optical isolator to eliminate backward propagation, an Erbium doped fiber amplifier (EDFA) driven at a moderate current (<100 mA), a tunable optical bandpass filter (OBPF) and a polarization controller (PC). The total roundtrip time in the loop is about 76.2 ns ( $f_{c}$ = 13.13 MHz). A free-space acousto-optic frequency shifter (AOFS) driven around $f_{s}$ ~80 MHz is inserted in the loop, introducing an insertion loss of approximately 6 dB. The AOFS is set to provide a downshift of the incoming light, and is driven around 80 MHz by an arbitrary function generator (AFG). The driving voltage is optimized to minimize the AOFS losses. The spectral bandwidth of the FSL is adjustable through the OBPF between a few GHz to a few hundreds of GHz. A 2% Y-coupler enables to seed the loop with a CW narrow-linewidth (< 0.1 kHz) single-frequency laser delivering 1 mW at 1550.0 nm. The polarization of the seed laser is controlled by a fiber-pigtailed PC, and is linear at the input of the FSL. The PC inserted in the FSL is adjusted, in order to maintain the polarization over the successive roundtrips in the loop. In practice, we recombine the signal at the output of the FSL with a fraction of the seed laser, and tune the PC, so as to maximize the beat signal. The system is stable at a typical time scale of 30 minutes. A 1% Y-coupler enables to extract a fraction of the intra-cavity field. The output signal from the loop is amplified with a second EDFA and sent through a 40-GHz bandwidth electro-optic phase modulator (EOPM, V_π = 4.5 V), driven by a high-speed arbitrary waveform generator (AWG). The output signal from the AWG is amplified with a fast RF amplifier to match the half-wave voltage of the EOPM.

The output train of pulses is detected with a fast photodiode and a fast real-time oscilloscope (28-GHz bandwidth), while the optical spectrum is measured with a 5-MHz spectral resolution Optical Spectrum Analyzer. In Fig. 5(a), we plot the optical spectrum measured recorded by the high-resolution OSA. As expected, the comb consists of ~1200 lines starting from $f_{0}$ , the frequency of the seed laser, and separated by an LS of ~80 MHz. The comb spectral bandwidth is limited by the OBPF.

Fig. 5 Spectral and temporal properties of the experimental CW injection-seeded FSL. a, experimental 100 GHz-bandwidth OFC generated from the implemented FSL. The AOFS operates at $f_{s}$ ~80 MHz in a frequency down-shifting mode, i.e., the frequency of the CW seed laser ( $f_{0}$ ) corresponds to the rightmost line. b, generation of a built-in tunable quadratic phase, and the related temporal SI effects, in a 30 GHz-bandwidth seeded FSL ( $f_{c}$ = 13.13 MHz). The spectral phase is measured by heterodyning the output of the FSL with a fraction of the seed laser, and the temporal trace is detected with a fast photo-detector (see text). According to the theory of temporal SI, when $f_{s} / f_{c} = p / q$ the repetition rate of the train of pulses at the FSL output is equal to q $f_{s}$ , as in b.1 (integer SI, q = 1) and b.3, b.4 (fractional SI, q = 8, and q = 5, respectively). The results in b.2 illustrate the influence of the quadratic spectral phase on the pulse shape (or equivalent GVD) in the vicinity of the integer SI condition, leading to the observed severe temporal broadening of the individual pulses.

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For some measurements, including characterization of the output spectra after phase modulation (with reduced LS) as well as monitoring of the spectral phase of the FSL comb, a self-heterodyne measurement setup is used [35]. For this purpose, a fraction of the CW seed laser is mixed, by means of a 50% Y-coupler, with the direct output from the FSL (Fig. 4). The RF spectrum is then recorded by the fast photodiode and the oscilloscope, providing a direct measurement of the power and phase of the optical spectrum of the FSL in the RF domain. Beyond its very high resolution, this scheme is particularly well suited to our setup: the seed frequency corresponds to the highest one in the FSL spectrum, which prevents any aliasing of the optical frequencies in the measured RF spectrum. This technique enables an experimental observation of the spectral phase of the OFCs at the output of the FSL system, including the predicted quadratic dependence of this spectral phase, see details in Fig. 5(b). Notice that in all measurements reported in the following, the estimated width of the OFC individual lines is limited by the integration time of the measurement. However, it is expected that a broadening of the lines would be observed by increasing the integration time. This phenomenon is mainly due to vibrations and thermal drifts of the system, so that it could be strongly reduced by mechanical isolation or active stabilization of the length of the loop.

4.2 Demonstration of arbitrary LS control by SI

In a first example (Fig. 6), we report LS increase by a factor of $q = 100$ . In particular, $f_{s}$ is set to 78.91 MHz so that Eq. (1) is satisfied with $p / q = f_{s} / f_{c} = 601 / 100$ . As expected, the optical pulse train at the output of the FSL exhibits a repetition rate of 7.891 GHz, while the comb LS remains at 78.91 MHz. Subsequently, a temporal phase sequence, opposite of the output pulse train phase profile, is applied by means of an EOPM driven by an electronic arbitrary waveform generator, AWG (Fig. 6(a)).

Fig. 6 Integer multiplication of the comb LS by a factor of 100. a, driving voltage signal applied to the EOPM (green) and theoretical phase sequence (grey). b, output optical power spectrum (dark blue, “PM on”). The input optical spectrum (generated in the FSL) is plotted in red (“PM off”) for comparison. The width of the individual frequency lines (~2 MHz at −3 dB) is unchanged by the SI process and is limited by the measurement integration time of the OSA (inset). c, input (red, “PM off”) and output (blue, “PM on”) baseband spectra acquired through self-heterodyne measurements. Inset: illustration of the possibility of selecting the specific set of output comb lines, within the original frequency grid, by temporal shift of the PM sequence. The relative temporal delay between the PM sequences corresponding to the blue and the purple curves in the plot is equal to $2 / (100 f_{s})$ (twice the period of the rate-multiplied temporal pulse train). Such a delay in the PM profile produces a shift in the absolute frequency location of the generated OFC of twice the input LS (shifting two lines in the original frequency grid). The width of the input and output frequency lines (~1 MHz at −3dB) is unchanged, and limited by the measurement integration time.

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Figure 6(b) shows direct measurements of the input and output comb optical spectra, whereas Fig. 6(c) shows measurements of the corresponding base-band spectra, obtained by heterodyning with the input seed CW light, as described above. The measured results in Fig. 6(b) and 6(c) confirm multiplication of the comb LS by a factor of 100, leading to generation of a 7.891 GHz-spacing OFC. As predicted, the overall OFC energy is nearly preserved: the power ratio between the measured output and input comb lines is ~17 dB, approaching the expected ideal value of 20 dB (100 in logarithmic scale). Residual energy remains in the output spectrum at discrete frequencies corresponding to the input OFC lines. This imperfect realization of the ideal process is mainly attributed to practical deviations in the temporal PM profile imposed on the FSL output, as compared with the theoretical prescription (Fig. 6(a)). For the rest, the spectral envelope and bandwidth of the OFC are not affected by the SI process and the output individual lines show no noticeable broadening (within the limit of the measurement integration time) (insets of Fig. 6(b) and Fig. 6(c)). As predicted, the output frequency lines are locked to those of the input comb.

Additionally, the results shown in the inset of Fig. 6(c) illustrate the capability of the method to select the specific set of input frequency lines that form the output OFC by simply delaying the temporal PM sequence with respect to the incoming pulse train. A relative temporal delay in the PM sequence of $2 / (100 f_{s})$ (twice the period of the rate-multiplied pulse train), corresponding to the blue and purple curves, shifts the absolute frequency location of the generated OFC over two lines of the original frequency grid. Finally, it is also observed that the spectral noise floor level of the OFC (mainly due to amplified spontaneous emission in the FSL) is not affected by the spectral SI process (inset of Fig. 6(c)).

In a second example, we demonstrate fractional multiplication of the input LS by a factor of $q / m$ = 25/2. For this purpose, we select $f_{s}$ = 82.46 MHz, to satisfy $p / q = f_{s} / f_{c} = 157 / 25$ , inducing pulse repetition rate multiplication by $q = 25$ . Subsequently, the temporal phase modulation is designed to compensate for the temporal phase profile of the $25 \times f_{s}$ rate-multiplied pulse train, combined with the phase variation that is needed for spectral SI (Eq. (2)) with $s = 1$ and $m = 2$ (LS division by 2).

The results in Fig. 7(a) demonstrate the predicted LS multiplication by a factor of 12.5, with full control of the selected frequency lines (see Fig. 7(a)-inset for an example). The OFC energy is again nearly preserved, with a power ratio between the output and input lines of ~9 dB, slightly short from the ideal 11 dB value. This is mainly due to practical deviations in the temporal PM profile with respect to the theoretical one, as can be seen from the imperfect removal of the input lines in Fig. 7(a). Figure 7(b) shows fractional LS reduction (division) by a factor of $q / m$ = 3/11; in this case, $f_{s}$ = 83.16 MHz $(p / q = f_{s} / f_{c} = 19 / 3)$ , and the temporal phase sequence is defined by $s = 2$ and $m = 11$ . As expected, the measured output LS is 22.68 MHz, energy is closely preserved from the input to the output OFC, and the offset frequency of the comb is similarly determined by the relative time delay of the PM sequence (see Fig. 7(b)-inset).

Fig. 7 Fractional multiplication/division of the comb LS. Experimental comparison of the baseband spectra of the input frequency comb directly generated from the FSL (red, “PM off”) and the output frequency comb after the SI process (blue/purple, “PM on”). All spectra are measured by heterodyning with the seed laser. a, fractional LS multiplication ( $q = 25, m = 2$ ). Inset: the relative temporal shift between the PM sequences corresponding to the blue and the purple curves is $2 / (25 f_{s})$ . b, fractional LS division ( $q = 3, m = 11)$ . Inset: the temporal shift between the PM sequences corresponding to the blue and the purple curves is $2 / (3 f_{s})$ .

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In a last set of experiments, we demonstrate the ability of the technique for generating sub-MHz LSs through controlled division of the original comb LS. In particular, we set $f_{s} / f_{c} = 6$ ( $q = 1$ ), which leads to direct generation from the FSL of a uniform-phase mode-locked OFC with an LS (and pulse repetition rate) of 78.8 MHz. Subsequent temporal modulation of this input OFC, according to Eq. (2), enables lossless division of the original comb LS by the desired integer factor, m. Experimental results for four different cases are reported in Fig. 8 ( $s = 1, m = 10, 100, 1000, 10000$ ), corresponding to output OFCs with a measured LS of 7.88 MHz, 788 kHz and 78.8 kHz and 7.88 kHz respectively. Notice that in the output spectra, the frequency lines that were initially present in the input OFC are more intense than the new lines created by the SI process. Similarly to the case of integer/fractional multiplication of the comb LS, this effect is mainly due to the imperfect PM signal applied by the EOM on the pulse train.

Fig. 8 Integer division of the comb LS. Comparison of the baseband spectra of the OFC directly generated from the FSL (red, “PM off”) and at the output after the SI process (blue, “PM on”). All spectra are measured by heterodyning with the seed laser. In each case, the output OFC exhibits an LS that is decreased (divided) by a factor of m with respect to the original one with a, $q = 1, m = 10$ ; b, $q = 1, m = 100$ ; c, $q = 1, m = 1000$ ; and d, $q = 1, m = 10000$ . Notice that the noise floor level and width of the spectral lines (~2 kHz at −3dB, limited by the measurement integration time) remain unchanged.

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4.3 Applications of enhanced spectral SI of OFCs

Finally, we report results on two different practical applications of the proposed comb LS control method in order to illustrate the unique capabilities enabled by the concept. First, Fig. 9(a) shows results on the use of LS multiplication for precise characterization of an OFC with an original LS (~80 MHz) that is well below the frequency resolution of the detection system, e.g., conventional spectrometers. The LS multiplication is produced without affecting any of the other features of the comb, thus allowing us to precisely resolve the individual frequency lines and to obtain key information of the original comb (e.g., spectral intensity envelope, noise floor etc.) On the other hand, Fig. 9(b) illustrates the use of LS division to enhance the frequency point spacing, and related sensitivity, of OFC-based spectral measurements, e.g., for spectroscopy applications. Here, only integer (or plain) LS division is shown, but fractional division could also have been used. The method enables adapting the measurement frequency sampling rate -as dictated by the comb LS- for ultra-high resolution, while featuring the key advantages of comb-based techniques (e.g., increased measurement speed and precision). The possibility of controlling the frequency point spacing brings significant simplification as compared to techniques that make use of successive interleaving of frequency-shifted frequency combs [35].

Fig. 9 Application of LS control for enhanced OFC detection and kHz frequency spacing spectroscopy. a.1-2, demonstration of LS control for improved OFC detection. The input comb (orange curve) generated by the FSL ( $f_{s}$ = 79.91 MHz) cannot be resolved using spectrometers having a larger frequency resolution (a1: 140 MHz and a.2: 1 GHz). Self-imaging (SI) of the input OFC produces OFCs (in blue) identical to the input one but with LS exceeding the resolution limit of the detection, revealing the detailed comb structure. The output LS is multiplied by 7 and 100 in a.1 and a.2 resp. b.1-4, adaptive frequency sampling rate by SI for spectroscopy with kHz frequency point spacing. The output OFC is sent through a Fabry-Perot (FP) resonator (LS: 16 GHz, finesse: 250) for characterization of the resonator’s spectral response. The FP theoretical spectral transmission function is plotted in red (dash). The transmitted optical spectrum is measured by heterodyning the output of the FP with the monochromatic (CW) seed laser (acquisition time: 100 µs) (b.1). b.2, the input comb generated by the FSL (LS = 80.93 MHz, orange curve) cannot resolve the transmission peaks of the FP (red curve) nor the additional FP transverse modes at ~1,4 and ~1,7 GHz, contrary to self-imaged OFCs displayed on b.3 (blue comb: LS = 40.46 MHz, and cyan comb: LS = 8.093 MHz). b.4, LS division through SI by factors of 100 (top, comb LS = 809.3 kHz) and 1000 (bottom, comb LS = 80.93 kHz) enables simultaneous, precise sampling of the FP transmission peaks with kHz point spacing.

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We demonstrate here spectral characterization of the transmission function of a narrow-linewidth (~5 MHz) Fabry-Perot resonator using a single OFC probe with kHz frequency spacing. Not only the Fabry-Perot spectral transmission peaks are precisely resolved, but the enhanced resolution also reveals high-finesse transverse spectral modes at very specific frequencies. To our knowledge, this result represents the first practical application of spectral self-imaging to spectroscopy.

5. Conclusion

We have demonstrated the unique capabilities of combined temporal and spectral SI effects, so-called generalized spectral SI [25], for arbitrary control of the LS of a frequency comb while preserving the OFC features, including its overall energy, spectral envelope and original frequency grid reference. In particular, we have demonstrated unprecedented LS tunability, over six orders of magnitude, in a simple dedicated fiber-optics OFC generation platform, involving a frequency-shifting recirculating loop seeded by a CW laser and an electro-optic temporal phase modulator. Our reported results provide demonstration of lossless LS division and multiplication by both fractional and integer factors, including LS multiplication of a MHz-rate OFC by factors as high as 100.

We have also shown experimental demonstration of passive frequency-line amplification of the OFC with a gain as high as 17 dB, a functionality that could be useful for detection of coherent OFCs from a strong noise background [25]. Besides the fundamental interest of the experiments reported here to confirm central capabilities and key features of generalized spectral SI, the developed platform for OFC generation and their enhanced control should prove useful for practical applications. Specifically, the demonstrated system combines several advantages that, beyond the afore-mentioned applications to spectroscopy, could make it attractive for other uses in the context of frequency metrology [2] or optical waveform synthesis [7]. Similarly to electro-optic OFCs, the optical frequencies in the developed platform derive from a single CW laser that could be externally stabilized [14]. Moreover, the LS is directly set by an electronic function generator, and not by an optical cavity as in mode-locked OFCs. Finally, our system can be easily programmed to provide a user-defined arbitrary number of frequency lines over a given spectral envelope with a very large degree of flexibility (from ~10 to ~10⁷ lines demonstrated here), a feature that should be especially valuable for multi-heterodyne interferometry, microwave photonic applications and others [15,36–39].

Appendix

The temporal phase profile of a rate-multiplied optical pulse train by fractional temporal SI (step 1) will need to be compensated for through a suitable temporal PM process in order to achieve integer multiplication of the LS of the input OFC by the pulse-rate multiplication factor, $q$ (step 2). This temporal phase profile can be determined by calculations that have been summarized in [30,31], and are here recalled for completeness. Assume an infinite flat-envelope comb of optical frequencies with a frequency spacing (LS) of $f_{s}$ , showing a quadratic spectral phase defined by $π \frac{p}{q} k^{2}$ (Eq. (1)), so as to generate a fractional temporal SI effect, inducing repetition-rate multiplication by a factor $q$ . Recall that $p$ and $q$ are coprime integers whereas is the frequency line index (k = 0, 1, 2, …) The output pulses are then emitted at times $t_{n} = n / (q f_{s}) - p q / (2 f_{s})$ , with relative phases equal to $- π \frac{p'}{q} n^{2}$ (plus an unimportant constant term), where n (=0, 1, 2, …) labels the pulse number in the rate-multiplied temporal sequence and $p'$ depends directly on $p$ and $q$ [30,31]:when both $p$ and $q$ are odd,

p' = 8 p {[\frac{1}{2}]}_{q} {({[\frac{1}{2 p}]}_{q})}^{2},

when

p

or

q

is even,

p' = p {({[\frac{1}{p}]}_{q})}^{2},

where

{[\frac{1}{a}]}_{b}

is the inverse of

a

modulo

b

, i.e., the only integer smaller than

b

satisfying

a {[\frac{1}{a}]}_{b} = 1 mod b .

Funding

Natural Sciences and Engineering Research Council (NSERC) of Canada, Région Rhône-Alpes (C Mira Explora Pro 12986), Agence Nationale de la Recherche (Grant ANR-14-CE32-0022).

Acknowledgments

The authors thank S. Costrel and O. Jacquin for technical help at the beginning of the project.

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Arbitrary energy-preserving control of the line spacing of an optical frequency comb over six orders of magnitude through self-imaging

Abstract

1. Introduction

2. Theoretical aspects of generalized self-imaging of OFCs

3. Implementation of the generalized spectral SI in a CW injection-seeded FSL

4. Experimental results

4.1 Architecture and characterization of the CW injection-seeded FSL

4.2 Demonstration of arbitrary LS control by SI

4.3 Applications of enhanced spectral SI of OFCs

5. Conclusion

Appendix

Funding

Acknowledgments

References and links

Cited By

Figures (9)

Equations (6)

Optics Express

Hugues Guillet de Chatellus	https://orcid.org/0000-0003-2075-4056
Luis Romero Cortés	https://orcid.org/0000-0002-9534-4476