## Abstract

A simple and efficient optical pulse re-shaper based on the concept of temporal coherence synthesization is proposed and analyzed in detail. Specifically, we demonstrate that an arbitrary chirp-free (transform-limited) optical pulse waveform can be synthesized from a given transform-limited Gaussian-like input optical pulse by coherently superposing a set of properly delayed replicas of this input pulse, e.g. using a conventional multi-arm interferometer. A practical implementation of this general concept based on the use of conventional concatenated two-arm interferometers is also suggested and demonstrated. This specific implementation allows the synthesis of any desired temporally-symmetric optical waveform with time features only limited by the input pulse bandwidth. A general optimization algorithm has been developed and applied for designing the system specifications (number of interferometers and relative time delays in these interferometers) that are required to achieve a desired optical pulse re-shaping operation. The required tolerances in this system have been also estimated and confirmed by numerical simulations. The proposed technique has been experimentally demonstrated by reshaping an ≈1-ps Gaussian-like optical pulse into various temporal shapes of practical interest, i.e. picosecond transform-limited flat-top, parabolic and triangular pulses (all centered at a wavelength of ≈ 1550nm), using a simple two-stage interferometer setup. A remarkable synthesis accuracy and high energetic efficiency have been achieved for all these pulse re-shaping operations.

© 2007 Optical Society of America

## 1. Introduction

Techniques for the precise synthesis and control of the temporal shape of optical pulses with durations in the picosecond and sub-picosecond regimes have become increasingly important for a wide range of applications in such diverse fields as ultrahigh-bit-rate optical communications [1]–[3], and nonlinear optics [4]. To give a few examples, (sub-)picosecond flat-top optical pulses are highly desired for nonlinear optical switching (e.g. for improving the timing-jitter tolerance in ultrahigh-speed optical time domain demultiplexing [1], [2]) as well as for a range of wavelength conversion applications [3]; high-quality picosecond parabolic pulse shapes are also of great interest, e.g. to achieve ultra-flat self-phase-modulation (SPM) - induced spectral broadening in supercontinuum generation experiments [4]. For all these applications, the shape of the synthesized pulse needs to be accurately controlled for achieving a minimum intensity error over the temporal region of interest.

The most commonly used technique for arbitrary optical pulse shaping is based on spectral amplitude and/or phase linear filtering of the original pulse in the spatial domain; this technique is usually referred to as ‘Fourier-domain pulse shaping’ and has allowed the programmable synthesis of arbitrary waveforms with resolutions better than 100fs [5]. Though extremely powerful and flexible, the inherent experimental complexity of this implementation, which requires the use of very high-quality bulk-optics components (high-quality diffraction gratings, high-resolution spatial light modulators etc.), has motivated research on alternate, simpler solutions for optical pulse shaping. This includes the use of integrated arrayed waveguide gratings (AWGs) [6], and fiber gratings (e.g. fiber Bragg gratings [7], or long-period fiber gratings [8]). However, AWG-based pulse shapers [6] are typically limited to time resolutions above 10ps. The main drawback of the fiber grating approach [7], [8] is the lack of programmability: a grating device is designed to realize a single pulse shaping operation over a specific input pulse (of prescribed wavelength and bandwidth) and once the grating is fabricated, these specifications cannot be later modified. Recently, a simple and practical pulse shaping technique using cascaded two-arm interferometers has been reported [9]. This technique can be implemented using widely accessible bulk-optics components and can be easily reconfigured to synthesize a variety of transform-limited temporal shapes of practical interest (e.g. flat-top and triangular pulses [9]) as well as to operate over a wide range of input bandwidths (in the sub-picosecond and picosecond regimes) and center wavelengths. Moreover, this novel technique has shown to provide a remarkably high energetic efficiency. We notice that a somehow similar concept has been also previously employed to synthesize flat-top optical pulses using a polarization-maintaining fiber interferometer, where two polarization modes were combined with a precisely controlled relative delay to achieve the required sinc-like function in the spectral domain [3].

In this paper, we provide a comprehensive study of this recently reported optical pulse shaping technique [9]. We refer to this new technique as temporal coherence synthesization method since it is essentially based on synthesizing the desired output pulse shape by coherently combining a set of input pulse replicas with different time delays. As a result, transform-limited output pulses can be obtained if a transform-limited input optical pulse is used. In fact, we demonstrate that the proposed optical pulse shaping technique can be programmed to synthesize any desired transform-limited optical pulse shape and we develop and test (both numerically and experimentally) a general design algorithm to obtain the system specifications, e.g. number of concatenated interferometers and relative time delays in the interferometers, that are required to achieve a desired arbitrary output pulse waveform (with the desired temporal shape and duration). In addition, we numerically investigate the effect of deviations of the system specifications (e.g. relative time delays in the concatenated interferometers) with respect to their optimal values on the system performance and estimate the acceptable tolerances for these specifications.

To showcase the capabilities of the proposed approach, we report the experimental realization of three practically interesting optical pulse shaping operations, namely the generation of picosecond transform-limited (chirp-free) flat-top, parabolic, and triangular pulse waveforms (all centered at a wavelength of ≈ 1550nm). We illustrate the degree of control that we have on the pulse shaping operation by synthesizing these three waveforms with nearly the same full-width-half-maximum (FWHM) time duration (2-ps); moreover, a flat-top waveform of different duration (>3-ps) is also generated. All these pulse shapes have been synthesized from the same picosecond Gaussian-like input optical pulse by simply adjusting the relative time delays in a *two-stage* interferometric setup.

## 2. Operation principle

#### 2.1 Temporal coherence synthesization

The key concept of our proposed pulse shaping technique is that a Gaussian-like transform-limited (chirp-free) input pulse can be re-shaped into any desired arbitrary transform-limited (chirp-free) pulse waveform by coherently superposing a set of properly delayed replicas of this input pulse. The desired pulse re-shaping operation can be achieved by simply optimizing the relative time delays among the input pulse replicas. The word ‘coherent’ refers to the fact that the input pulse replicas need to interfere both in amplitude and phase. It is well known that a combination of interferometers and delay line structures can be used for producing customized ultrafast optical pulse sequences from a single input pulse [10]; similar architectures have been applied in the past for microwave arbitrary waveform generation by incoherent pulse interferometry followed by narrowband linear filtering [11]. In this work, we demonstrate for the first time to our knowledge, that similar interferometric structures can be used to coherently control and manipulate the temporal shape of individual optical pulses. A schematic of the proposed pulse shaping concept together with numerical examples of a few synthesized waveforms from an input Gaussian-like optical pulse are shown in Fig. 1. In the shown examples, eight replicas of the input optical pulse with an equally split power are coherently overlapped by independently controlling the individual relative time delays in a multi-arm interferometer. Specifically, the relative time delays have been optimized using a conventional least-square optimization algorithm to achieve a flat-top, a parabolic, a triangular and a trapezoidal intensity waveform (more details about the specific synthesis algorithm used for our designs are given below). Notice that as for any linear pulse shaping technique, the speed of the output pulse time features are ultimately limited by the input pulse bandwidth.

## 2.2 Practical implementation using concatenated two-arm interferometers

The temporal coherence synthesization concept suggested in Fig. 1 can be implemented using a set of two-arm (e.g. Michelson) interferometers connected in series; in practice, this system can be easily constructed using common (off-the-shelf) optical components. Figure 2 shows a schematic of this implementation. Here, in the Michelson interferometer configuration, it is assumed that the back-reflected light from each interferometer is completely blocked, e.g. by use of an appropriate aperture and tilt alignment. In this specific implementation, the relative time delays between the input pulse replicas are symmetrically located around the pulse center and this limits the applicability of this architecture for the synthesis of temporally symmetric optical pulse shapes. As an example, the flat-top, parabolic and triangular pulse re-shaping operations shown in Fig. 1 could be realized with a set of 3 cascaded two-arm interferometers with properly tailored relative delays between the two arms, giving rise to eight suitably delayed replicas of the input optical pulse. Notice that alternatively, the structure proposed in Fig. 2, or the more general one proposed in Fig. 1, could be implemented in integrated waveguide platforms; this would be a particularly interesting solution (e.g. offering a reduced size and increased stability) if a large number of relative time delays need to be realized.

## 3. General design algorithm

In what follows we focus on the pulse shaper implementation based on cascaded interferometers, see Fig. 2. Specifically, in this section we describe a general algorithm to design the system parameters (number of interferometers and relative time delays) that are required to achieve a desired pulse shaping operation. We anticipate that it is straightforward to extend this algorithm for application on the general synthesization problem shown in Fig. 1.

The original optical pulse and desired output optical pulse constitute the input information to the optimization algorithm. In order to proceed with the system design, the features of the desired output pulse need to be fully specified, including pulse shape, time duration, rising and falling times; additionally, a proper time weighting function can be used to emphasize the time regions of the synthesized pulse where the error should be minimized. We reiterate that the considered pulse shaper is a linear time-invariant (LTI) system and as a result, the output pulse bandwidth is limited by that of the input pulse.

Let us assume that our system consists of *M* concatenated two-arm interferometers. As detailed below, *M* actually represents the maximum number of available interferometers to be used in the pulse shaping system. The parameters to be optimized are the *M* relative time delays, i.e. *τ*
* _{j}* with

*j*=1 …

*M*; each one corresponding to each of the concatenated interferometers. Assuming also that an input pulse

*e*(

*t*) (corresponding optical spectrum

*E*(

*ω*)) is launched at the system input, the output pulse spectrum

*A*(

*ω*) can be expressed as follows [12]:

In this notation, *ω*=*ω*
^{′}+*ω*
* _{c}*, where

*ω*

^{′}is the base-band frequency, and

*ω*

*is the central (carrier) optical frequency of the input pulse. The corresponding relationship in the temporal domain can be written as:*

_{c}$$\otimes \dots \otimes \left(\delta \left(t\right)+\delta \left(t-{\tau}_{M}\right)\mathrm{exp}\left[-i{\omega}_{c}{\tau}_{M}\right]\right)$$

where “⊗” denotes the convolution operation and *a*(*t*) and *e*(*t*) are the complex envelopes of the output and input optical pulses, respectively. In what follows, we assume that *e*(*t*) is a real function, representing a transform-limited pulse. The design is further simplified if the relative time delays in the interferometers are fixed to satisfy the following ‘phase-matching’ condition: *τ*
* _{j}*=2

*n*

_{j}*π*/

*ω*

*, where*

_{c}*n*

*is an arbitrary integer. In this case, Eq. (2) can be reduced to:*

_{j}As anticipated above, Eq. (3) indicates that under the given conditions, the temporal envelope of the output optical signal *a*(*t*) is the result of overlapping various (2* ^{M}*) delayed replicas of the input optical pulse envelope

*e*(

*t*). This overlapping process can be properly designed to construct the desired output envelope shape. Specifically, in order to determine the system parameters, namely the relative time delays, that are required to obtain a target output pulse shape

*a*

*(*

_{targ}*t*), we use a weighted root-mean-square error function, defined as follows:

where *I*
* _{targ}*=|

*a*

*|*

_{targ}^{2}and

*I*

*=|*

_{calc}*a*

*|*

_{calc}^{2}are the target intensity (desired output pulse shape) and calculated (actually obtained) intensity, respectively,

*a*

*(*

_{calc}*t*)=

*a*(

*t*) (which is obtained from Eq. (3)),

*t*

*is the discrete set of times where the output pulse envelope is evaluated,*

_{n}*N*being the total number of samples over a predefined time window, and

*W*

*=*

_{n}*W*(

*t*

*) is the weight for each sampling point. We use a general algorithm to search for the optimum set of parameters (relative time delays) that minimizes this error function. As stated before, the purpose of the weighting function is to emphasize the local regions of the output waveform over which a higher re-shaping accuracy is desired (as an example, this procedure can be used to minimize the oscillations in the flat-top region of a rectangular-like pulse waveform by simply allocating a higher weighting value over this more critical region).*

_{n}The developed optimization algorithm is similar to the conventional least-square search algorithm, except for the fact that an additional mechanism is incorporated to determine the minimum number of interferometers that need to be concatenated in order to achieve a prescribed accuracy. The algorithm of this optimization procedure is as follows:

## Procedure make(shape, weight, M, T,D)

{

*shape* ⇒ *I*
* _{targ}* (

*t*

_{1}…

*t*

*)*

_{N}*weight* ⇒ *W* (*t*
_{1}… *t*
* _{N}*)

*set*: *Err*=∞

*for*
*τ*
_{1}=*τ*
_{1.1}…*τ*
_{1,S}

…

*for*
*τ*
* _{M}*=

*τ*

_{M,1}…

*τ*

_{M,S}

*calculate the output intensity based on the*

*relativetime delays vector* (*τ*
_{1},…*τ*
* _{M}*)⇒

*I*

*(*

_{calc}*t*

_{1}…

*t*

*);*

_{N}*calculate the absolute error*: *Error*=$\sum _{n=1}^{N}W\left({t}_{n}\right){\mid {I}_{\mathrm{targ}}\left({t}_{n}\right)-{I}_{\mathrm{calc}}\left({t}_{n}\right)\mid}^{2}$;

*if*(*Error*<*D*×*Err*)

{

*Err*←*Error*;

*Optimum* (*τ*
_{1},…*τ*
* _{M}*)←(

*τ*

_{1},…

*τ*

*);*

_{M}}

}

Each relative time delay, *τ*
* _{j}* (to be optimized), is assigned S different possible values uniformly increasing from

*τ*

_{j,1}=0 to a pre-defined maximum value

*τ*

_{j,S}(ignoring the parameters permutation). Thus, the total number of possible combinations that are evaluated for searching the optimal set of time delays (with minimum error) would be equal to

*M*×

*S*, where

*M*is the maximum number of available interferometers. Notice that in our algorithm we have assumed that permutations between the different interferometers are not important and we can allocate the relative time delays to arbitrarily placed interferometers. For each set of time delays, the resultant output pulse waveform,

*a*

*(*

_{calc}*t*) is calculated according to Eq. (3) and subsequently, the new weighted mean squared error is obtained from Eq. (4). Notice that since all the time delays,

*τ*

*(1, 2,…,*

_{j}*M*) are initialized at ‘0’, (which is equivalent to assume that there is no interferometer), the number of “required” interferometers in the system increases to (

*j*+1) after every

*j*×

*S*loops (

*j*=1, 2,…,

*M*). In principle, the new set of design parameters will be accepted as the optimal one (

*Optimum*) only if their associated weighted squared error (

*Error*) is lower than the present minimum weighted squared error (

*Err*) multiplied by a parameter

*D*(<1). The parameter

*D*(margin factor) defines an additional mechanism to determine the minimum number of interferometers that are required to achieve the target pulse re-shaping operation with the desired accuracy. Basically, a new set of parameters will be accepted as the optimal solution

*only*if they provide a certain improvement over the presently selected solution; the required “improvement” is quantitatively evaluated in terms of the associated mean squared error and is controlled through the parameter

*D*. To be more concrete, the margin factor

*D*sets the minimum relative error difference (

*δ*=1-

*D*) that is necessary to accept the new solution as the optimal one, i.e. the solution under evaluation (with associated mean squared error

*Error*) will be accepted as the optimal solution only if (

*Err*-

*Error*)/

*Err*>

*δ*(where we reiterate that

*Err*is the mean squared error of the presently selected solution). Thus, if the use of more than

*L*(<

*M*) interferometers does not offer an improvement over the already selected solution, according to the criterion fixed through the parameter

*D*, then the rest of the time delays, i.e. variables from

*τ*

_{1}~

*τ*

_{M-L}will be set to zero in the optimal solution. In this way, the minimum number of required interferometers (

*L*) will be automatically provided by the algorithm, according to the minimum relative error improvement fixed “a priori” by an experienced designer.

It is worth noting that in the proposed design procedure, the phase variation associated with the carrier optical frequency is first obviated [i.e. the output complex envelope is calculated using Eq. (3)]; this is equivalent to assume that the relative time delays in the concatenated interferometers satisfy the following condition: *τ*
* _{j}*=

*2πn*

*/*

_{j}*ω*

*. In this way, the time sampling that is used in the design algorithm for each time-delay variable can be made much longer than the ultrashort time resolution associated with the used carrier frequency. In practice, the above condition can be achieved by very fine tuning of each of the relative time delays around its nominal design value (see more discussions on this point in Section 5).*

_{c}## 4. Design examples

In what follows we provide some design examples of our proposed pulse re-shaping architecture to synthesize different time waveforms of practical interest; specifically, we consider the problem of synthesization of square (flat-top), triangular and parabolic optical pulses. Notice that these three ideal waveforms exhibit an infinite bandwidth; however, in practice, the output pulse bandwidth is limited by that of the input pulse. Thus, the target output pulse waveform in each of these cases will be a bandwidth-limited (“softened”) version of the ideal waveform.

For design purposes, it is convenient to have available mathematical expressions that can approximate the target optical pulse shapes. For example, in our designs, the square and triangular target shapes have been mathematically described using a general raised cosine function, which can represent very precisely the desired bandwidth-limited versions of the square and triangular ideal pulse shapes, i.e. obtained as a convolution between an input Gaussian pulse and the ideal time waveforms. The raised cosine function can be mathematically described in the time domain as follows [13]:

and is characterized by two main parameters, namely the roll-off factor *α* (0≤*α* ≤1), and the Full-Width Half-Maximum (FWHM) pulse time-width *T*. The corresponding frequency-domain representation of the raised cosine function in Eq. (5) is:

Figures 3(a) and 3(b) show the time and frequency domain representations of the general raised cosine function, for different values of the roll-off factor *α*. The figure evidences the role of the two parameters (roll-off factor *α* and pulse time width *T*) in the general function. As the parameter *α* varies from zero towards one, the shape of the raised cosine function is softened, changing from an ideal square shape (*α*=0) to a “minimally smoothed” triangular shape (*α*=1).

In our following designs, we assume a transform-limited Gaussian-like input optical pulse with a FWHM time width of 0.95 ps (corresponding FWHM spectral bandwidth Δ*λ*=3.2 nm). The considered input pulse time shape [intensity shown in the inset of Fig. 4(a)] was obtained by taking the inverse Fourier transform of the square root of the measured input pulse spectrum, assuming a constant spectral phase (see descriptions in Section 6 below). For our numerical designs, the time variable has been quantized in sampling steps of 0.01 ps.

## 4.1 Pulse shaping based on two cascaded interferometers

In this section, we present three design examples of pulse re-shaping operations that can be accurately achieved with only two cascaded interferometers. In all these pulse shaping examples, the use of a number of interferometers larger than two does not represent any significant advantage (i.e. the improvement in the operation error is almost negligible). From expression (3), it can be easily inferred that in the case of two cascaded interferometers, the output pulse is formed by the interference of four delayed copies of the input optical pulse (with relative time delays given by 0, *τ*
_{1}, *τ*
_{2} and *τ*
_{3}=*τ*
_{1}+*τ*
_{2}). Specifically, we have targeted the synthesis of flat-top, triangular and parabolic optical pulse shapes, all of them with the same FWHM time duration of 2 ps. The corresponding numerically simulated results are shown in Figs. 4–6. Additionally, this same system has been designed to synthesize a longer flat-top pulse, see results in Fig. 4(b).

## 4.1.1 Flat-top optical pulses

The optimization algorithm described above is first used to design a cascaded interferometer pulse shaping system aimed to generate a 2-ps (FWHM) flat-top optical pulse from the input 0.95-ps Gaussian pulse. The target pulse shape was a raised cosine function (as given by Eq. (5)) with parameters *α*=0.75 and *T*=2 ps. A proper weighting function was used to minimize the error along the flat-top region (weight factor of 1000) and along the waist region (weight factor of 100). The used weighting function (normalized units) is plotted in Fig. 4(a) with a black dotted line. The use of this weighting function allowed us to minimize the fluctuations in the flat-top region while obtaining the desired time width and suitable rising and falling slopes. As anticipated, the desired pulse re-shaping algorithm could be accurately implemented using only two concatenated interferometers; the optimum relative delays for this design were 0.37 ps and 1.22 ps, respectively. Figure 4(a) (blue, dotted curve) shows the intensity of the numerically simulated output pulse after propagation through the optimized pulse shaper. The numerically obtained output pulse shape was almost identical to the target square-like time waveform (not shown here).

In order to illustrate further the flexibility provided by the proposed pulse shaping technique, the system has been designed to achieve a flat-top pulse of longer duration (3.1 ps). In this case, the target was a raised cosine function with parameters *α*=0.75 and *T*=3.1 ps. A similar weighting function to that of the previous flat-top pulse was employed (the normalized weighting function is shown in Fig. 4(b) with a black dotted curve). The optimum relative time delays in this case were 0.93 ps and 1.90 ps. The numerically obtained output pulse shape (shown in Fig. 4(b) with a blue, dotted curve) was almost identical to the target square-like shape.

## 4.1.2 Triangular optical pulse

In a triangular pulse re-shaping operation, the input pulse bandwidth essentially limits the sharpness of the pulse peak and the straightness of the triangular slopes. The optimization algorithm is now applied to the design of a cascaded interferometer system aimed to generate a 2-ps (FWHM) triangular optical pulse from the input 0.95-ps Gaussian pulse. The target shape is defined as a raised cosine function with parameters *α*=1 and *T*=2 ps. No weighting function is required in this case. As anticipated, this pulse re-shaping operation can be accurately achieved using only two concatenated interferometers; the optimum relative time delays in these interferometers are 0.76 ps and 1.40 ps. Simulation results are shown in Fig. 5, where the blue, dotted line is the numerically simulated output pulse waveform. We emphasize again that the numerically obtained pulse waveform was almost undistinguishable from the target waveform.

## 4.1.3 Parabolic optical pulse

The generalized expression for an ideal transform-limited parabolic pulse is given by [4]:

where *P*
* _{p}* is the peak power of the pulse, and

*T*

*is the FWHM temporal duration. The envelope shape (intensity) of an ideal 2-ps (FWHM) parabolic pulse is shown in Fig. 6 (orange, solid curve). As for an ideal square or triangular waveform, an ideal parabolic pulse exhibits an infinite bandwidth and consequently, cannot be synthesized in practice. A practical, bandwidth-limited version of this waveform could be obtained by simply convolving the ideal parabolic function [given by Eq. (7)] with a bandwidth-limited (e.g. Gaussian) pulse. In our design, we have chosen a 2-ps ideal parabolic pulse as the target time waveform but have used a proper weighting function (shown in Fig. 6 with a black, dotted curve) to fully optimize the synthesization process over the central region of the optical pulse. Specifically, this was achieved by fixing the weighting factor to 1,000 over the region of interest. Notice that the central part of a parabolic waveform is indeed the most important pulse region from a practical viewpoint [4]. In this way, we were able to synthesize a practically realizable parabolic waveform, where the sharp edges of the ideal waveform were softened according to the available input bandwidth. Based on our design algorithm, the desired pulse shape could be again synthesized using two concatenated interferometers with the relative time delays fixed to 0.61 ps and 1.29 ps. The numerically simulated output pulse shape from this designed system is shown in Fig. 6 with a blue, dotted curve. There is again an excellent agreement between the obtained pulse shape and the target parabolic pulse along the central region of interest.*

_{P}## 4.2 Design example with more than two concatenated interferometers

In this section, we illustrate through a numerical design example how the parameter *D* in the proposed optimization algorithm can be used to determine the minimum number of interferometers that are required to achieve a desired pulse shaping operation with a sufficiently high accuracy. Specifically, we assumed the same input optical pulse as in the examples shown above (0.95-ps Gaussian-like pulse) and we targeted the generation of a triangular pulse waveform with a FWHM time duration of 3 ps (i.e. the target function was the raised cosine with parameters *α*=1 and *T*=3 ps). The target pulse waveform is shown in Fig. 7 (blue, dotted curves). A suitable weighting function was used to optimize the temporal response along the waveform slopes (a weight of 1,000 was employed over these critical regions). Figure 7(a) (solid, red curve) shows the optimized output pulse waveform when only two interferometers were considered in the design (*M*=2); in this case, the optimal relative delays were 1.17 ps and 2.11 ps. A significant deviation is observed between the numerically calculated output waveform and the target shape, particularly in the slopes of the triangular pulse. Following this optimization process, the relative deviation between the mean squared errors of the two best solutions (solutions with the two lowest mean squared errors) was estimated to be 1.5 × 10^{-15}. In a second trial, the number of interferometers was left undetermined (*M* > 3) and the parameter *D* was fixed to 1-1.5×10^{-16}. This value was set to ensure that the algorithm could escape from the relative minimum-error solution found for the case of two interferometers (*δ*=1-*D*<1.5×10^{-15}). The optimization algorithm provided a solution consisting in 3 concatenated interferometers (*L*=3) with relative time delays fixed to 0.9 ps, 1.2 ps, and 2 ps, respectively. In this case, we estimated a relative deviation between the two lowest mean squared errors of 1.4×10^{-16} (as expected, this relative deviation is lower than the minimum acceptable relative error difference fixed in the algorithm, *δ*=1-*D*=1.5×10^{-16}). Figure 7(b) (solid, red line) shows the numerically simulated output pulse shape obtained with this optimal configuration; the synthesized waveform is visibly much closer to the target pulse than in the case of two interferometers [Fig. 8(a)], having synthesized the pulse slopes with a very high precision.

## 5. Error analysis

In this section, we analyze the effect of deviations in the system’s specifications, particularly in the relative time delays of the concatenated interferometers, with respect to their optimal values and provide an estimation of the acceptable tolerances for these specifications. As discussed above, in order to ensure the proper operation of the pulse shaping system, the relative time delays in the interferometers are assumed to satisfy the following condition: *τ*
* _{j}*=2

*n*

_{j}*π*/

*ω*

*, where*

_{c}*n*

*is an arbitrary integer an*

_{j}*ω*

*is the central (carrier) optical frequency of the input pulse. This condition guarantees that the delayed replicas of the input optical pulses are summed up with no phase variations among them. In practice, the time delays may deviate from this condition and this is expected to affect critically the pulse shaping result. One should bear in mind that given the high value of the central optical frequency, even a small change in each relative time delay may translate into a significant phase variation in the interfering pulses. Mathematically, it can be inferred from Eq. (2) and Eq. (3) above that deviations in the relative time delays (Δ*

_{c}*τ*

*) with respect to its optimal values (*

_{j}*τ*

*) will result into the following distorted output waveform:*

_{j}$$\otimes \left(\delta \left(t\right)+\delta \left(t-\left({\tau}_{2}+\Delta {\tau}_{2}\right)\right)\mathrm{exp}\left[-i{\omega}_{c}\left({\tau}_{2}+\Delta {\tau}_{2}\right)\right]\right)$$

$$\otimes \dots \otimes \left(\delta \left(t\right)+\delta (t-({\tau}_{M}+\Delta {\tau}_{M})\right)\mathrm{exp}\left[-i{\omega}_{c}\right({\tau}_{M}+\Delta {\tau}_{M}\left)\right])$$

Based on Eq. (8), in order to ensure that the phase fluctuations induced by the variations in the relative time delays remain negligible (i.e. Δ*φ*
* _{j}*=

*ω*

*Δ*

_{c}*τ*

*≪2π), these delay variations should satisfy the following condition: Δ*

_{j}*τ*

*≪2π/*

_{j}*ω*

*. Assuming a typical central optical frequency*

_{c}*ω*

*=2π×192THz (corresponding wavelength around 1.55µm), this translates into very tight acceptable tolerances for the relative time delays in the system, i.e. Δ*

_{c}*τ*

*≪5.2-fs; in other words, the path-length differences in the used interferometers need to be controlled with sub-micron resolutions. Obviously, this condition is much more critical than the one resulting from deviations in the time delays among the overlapped pulse replicas (assuming no phase variations), since the tolerances associated with these deviations are given by the following condition Δ*

_{j}*τ*

*≪*

_{j}*τ*

*, typically resulting in tolerances in the sub-picosecond range (assuming (sub-)picosecond pulse re-shaping operations such as those investigated in this paper).*

_{j}Our previous estimations have been numerically confirmed. As an example, we consider the interferometer design given above for synthesizing a 2-ps flat-top pulse from a 0.95-ps Gaussian-like input pulse [results shown in Fig. 4(a)]. In our following simulations, we assume a central optical pulse frequency *ω*
* _{c}*=2π×192THz. We remind the reader that the optimum relative time delays are 0.369 ps and 1.218 ps. Notice that in order to evaluate the effect of phase variations, which was obviated in the above system design procedure, the sampling time is now fixed to 1-fs, corresponding to phase changes of 1.2 rad. Two different cases are separately evaluated, namely the effect of deviations in the longer time delay [results shown in Fig. 8(a)] and the effect of deviations in the shorter time delay [results shown in,. 8(b)], assuming in each case that the other time delay is set to its optimal value. In both cases, the time delay under analysis was deviated between -1fs to 1fs with respect to its optimal value which corresponds to ±0.3 µm optical path-length instability in air. As expected, these small time delay variations were sufficient to induce a visible distortion in the synthesized output waveforms. Concerning the variations in the longer time delay [Fig. 8(a)], we observed that as we deviated further from the optimum relative time delay, the synthesized pulse was broadened and a deeper undershoot was introduced over the flat-top region. The system appeared to be more sensitive to a decrease in the time delay (with respect to its optimal value) rather than to a delay increasing. In the case of variations in the shorter time delay [Fig. 8(b)], the synthesized pulse was slightly broadened with a significant central undershoot when the delay was increased with respect to its optimal value; in contrast, a delay decreasing translated into a narrowing of the pulse width.

## 6. Experimental results and discussions

The three pulse re-shaping operations (‘flat-top’, ‘triangular’, and ‘parabolic’) described in Section 4.1 have been experimentally realized. The input pulse source in our experiments was a passively mode-locked wavelength-tunable fiber laser (Pritel Inc.) operating at a repetition rate of 20MHz. This source generated nearly transform-limited optical pulses with a FWHM temporal duration of 0.7~1.2 ps depending on the center wavelength. The inset in Fig. 9 shows the input pulse spectrum directly generated from the fiber laser. Specifically, the pulse exhibited a FWHM spectral bandwidth of 3.2-nm, centered at 1549.5 nm, corresponding to a 0.95-ps (FWHM) transform-limited Gaussian-like input pulse [see inset in Fig. 4(a)]. In each pulse shaping experiment, the required phase matching condition (*τ*
* _{j}*=

*2πn*

*/*

_{j}*ω*

*) was precisely adjusted by monitoring the output pulse spectrum (using an OSA); specifically, the path-length difference in each interferometer needed to be tuned with sub-micrometer resolution to ensure that the periodic spectrum modulation corresponding to the assigned time delay,*

_{c}*τ*

*, was properly shifted to be exactly symmetric with respect to the pulse center wavelength. Examples of output spectra obtained after these fine adjustments are shown in the inset of Fig. 4(b) (results corresponding to the two flat-top pulse synthesis experiments described above).*

_{j}We used a Fourier-transform spectral interferometry (FTSI) setup [14], [15] to retrieve the complex temporal waveform of the pulses synthesized at the system output, which allowed us to directly monitor and optimize the temporal pulse re-shaping operations. Figure 9 shows the schematic of the conducted experiments, including the FTSI setup. The secondary optical output from the fiber laser was used as the reference pulse for the SI measurements. The time delay between the secondary (reference) and primary arms in the FTSI setup was adjusted to ≈45 ps by use of a conventional free-space delay line. The spectral interference between the two arms of the FTSI setup was measured with an OSA (wavelength resolution=0.02 nm) after properly adjusting their polarization state with a polarization controller. The complex temporal and spectral profiles of the output pulse were recovered from the recorded spectral interference pattern using the well-known FTSI algorithm [14].

Figures 4–6 show the temporal intensity (red, solid curves) and phase profiles (greed, dot-dashed curves) recovered from FTSI measurements of the re-shaped optical pulses, according to the designs described in Section 4.1, leading to the synthesis of (i) two ‘flat-top’ pulses with FWHM’s of 2 ps and 3.1 ps, respectively (Fig. 4), (ii) a 2-ps triangular pulse (Fig. 5), and (iii) a 2-ps parabolic pulse (Fig. 6). In each case, there is a remarkable agreement between the numerical pulse designs and the experimental synthesizations. FTSI measurements also confirmed that the synthesized pulses were nearly transform-limited as these pulses did not exhibit significant phase variations along their temporal profile. The energetic efficiencies (ratio between output and input average pulse powers) of the reported pulse shaping operations were measured to be 26% and 15% for the 2-ps and 3.1-ps flat-top pulses, respectively, 21% for the 2-ps triangular pulse, and 23% for the 2-ps parabolic pulse. These efficiencies compare very favorably with those typically achieved with previous pulse shaping techniques, e.g. Fourier-based processing methods [5].

## 7. Conclusions

In conclusion, a simple and efficient optical pulse shaping method based on the concept of temporal coherent synthesization has been proposed and demonstrated. In this technique, the desired optical pulse shape is synthesized by coherently overlapping various properly delayed replicas of an input optical pulse. This method allows the synthesis of any arbitrary transform-limited pulse shape from a transform-limited Gaussian-like input optical pulse e.g. using a multi-arm interferometer. In practice, temporal coherent synthesization can be implemented using a multi-stage interferometric setup, which can be easily reconfigured to generate any desired temporally-symmetric arbitrary pulse waveform, only limited by the input pulse bandwidth, by simply varying the relative time delays in the interferometer arms. An advanced ‘least-square’ optimization algorithm has been developed to determine the system specifications (number of interferometer stages and relative time delays) that are required to achieve a desired target waveform from a given input optical pulse. The practical limitations of the proposed system have been investigated; for this purpose, the pulse shape distortion caused by practical time delay tolerances has been numerically evaluated, confirming that picosecond pulse shaping can be implemented using an interferometric setup with sub-micron accuracies in the optical-path length of the interferometer arms.

The developed optimization algorithm has been tested by designing a multi-stage interferometer for re-shaping a 0.95-ps Gaussian-like optical pulse (centered at 1550 nm) into three practically interesting pulse waveforms, namely flat-top, triangular, and parabolic optical pulses, all of them with a FWHM duration of 2 ps; the design flexibility offered by the proposed method has been further illustrated by synthesizing a longer (3.1-ps) flat-top optical pulse. All these pulse re-shaping operations have been experimentally realized by simply adjusting the relative time delays in a two-stage interferometric apparatus; a remarkable agreement between the theoretical (target) and experimentally obtained pulse shapes has been demonstrated. Moreover, high energetic efficiencies above 15% have been measured for all these re-shaping operations. Given the simplicity, demonstrated efficiency and flexibility of the proposed general optical pulse shaper, we believe that this method should prove very useful for the practical synthesis of a wide range of (sub-)picosecond temporal shapes of fundamental and practical interest.

## Acknowledgments

The authors would like to thank Dr. Radan Slavík and Dr. Mykola Kulishov for invaluable discussions and comments. This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), and by the Fonds Québécois de la Recherche sur la Nature et des Technologies (FQRNT).

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