Ultrafast Response of Arrayed Waveguide Grating Pulse Shapers With

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 25, NO. 9, SEPTEMBER 2007

Ultrafast Response of Arrayed Waveguide Grating Pulse Shapers With Digital Amplitude and Phase Modulation Luis Grave de Peralta, Ayrton A. Bernussi, and Henryk Temkin, Fellow, IEEE

Abstract—We present a general quantitative description of the temporal and spectral responses of arrayed waveguide gratings pulse shapers to a single ultrashort optical pulse. Operation of the pulse shaper was described in terms of superposition of pulse replicas that are overlapping in the slab waveguide region of the device. We demonstrate that the analytical formulation requires only a combination of fast Fourier transform algorithm with waveguide-mode profile simulation to accurately describe the spectral and temporal responses of the pulse shaper. The proposed model was applied to pulse shapers based on digital amplitude and phase modulations. A supporting evidence of the effectiveness of the model described in this paper was obtained from the good agreement between simulations and experiments. Index Terms—Optical planar waveguide components, optical pulse shaping, ultrafast optics, waveguide arrays, wavelengthdivision multiplexing (WDM).

I. I NTRODUCTION

P

ULSE SHAPERS (optical encoders) are key devices for future optical code-division multiple-access communication networks [1]. Conventional pulse shaping transforms a single pulse of a mode-locked laser into a sequence of ultrafast pulses with very large repetition rates within the sequence. Different approaches have been used to generate fixed or arbitrary output optical waveform sequences [2], [3]. Among the existing pulse shaping techniques, direct space-to-time is of particular importance. This technique does not require Fourier transform computation of the mask element, making direct space-to-time a preferred approach for applications where a fast response of the pulse shaper is a critical requirement. This technique was initially implemented using a bulk-optic configuration [4]. More recently, integrated-optic pulse shapers that are based on arrayed waveguide grating (AWG) (de) multiplexers have been demonstrated [5], [6]. This approach offers a significant advantage in footprint when compared to bulk-optic implementation. Realization of the AWG pulse shapers (AWGManuscript received January 8, 2007; revised May 15, 2007. This work was supported in part by the Jack F. Maddox Foundation. L. Grave de Peralta is with the Department of Physics, Texas Tech University, Lubbock, TX 79409 USA (e-mail: luis.grave-de-peralta@ttu.edu). A. A. Bernussi is with the Department of Electrical and Computer Engineering, Texas Tech University, Lubbock, TX 79409 USA (e-mail: ayrton. bernussi@ttu.edu). H. Temkin is with the Department of Electrical and Computer Engineering, Texas Tech University, Lubbock, TX 79409 USA, and also with the Microsystems Technology Office, Defense Advanced Research Projects Agency, Arlington, VA 22203-1714 USA (e-mail: henryk.temkin@ttu.edu). Digital Object Identifier 10.1109/JLT.2007.901441

Fig. 1. Schematic representation of an AWG-PS layout. Represented planes X0 and X1 are arcs of circles with a radius of curvature f .

PS) is greatly simplified and practically eliminates the need of a sophisticated optical alignment. Another major advantage of the AWG-PS is the possibility of integration with other passive or active components in the same platform to perform multiple tasks. Although theoretical and numerical methods are being developed to predict the spectral response of AWGs that are designed for communication applications [7], [8], only recently, a comprehensive theoretical description of the temporal and spectral responses of an AWG-PS has been reported for the special case where the spectral width of the input laser pulse (∆ν) is larger than the free spectral range (FSR) of the AWG [9]. A previously published work in this field of research established an analogy between bulk-optic space-to-time pulse shaper and AWG-PS with the FSR < ∆ν [5]. Also, simple design formulas for the special case where FSR > ∆ν have been previously described for the AWG-PS [6], [10]–[12]. In this paper, we present for the first time a detailed analytical model for the simulation of AWG-PS with FSR > ∆ν. We present the results of the numerical simulations for relevant cases of digital amplitude and phase modulations. In addition, we compare the results of the numerical simulations with previously reported experiments [6], [10]–[12]. II. Q UANTITATIVE D ESCRIPTION The schematic of an AWG-PS based on a reflective AWG [13] is shown in Fig. 1. It consists of a set of input/output waveguides, followed by a slab or free propagation region (FPR) and an array of waveguides (grating) that terminates in a reflecting surface [6], [10]–[13]. Modulation of the electric field

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GRAVE DE PERALTA et al.: RESPONSE OF AWG-PS WITH DIGITAL AMPLITUDE AND PHASE MODULATION

of the light that is traversing each waveguide of the grating can be accomplished by placing an amplitude or phase mask at the reflecting surface that is terminating the AWG. The length of each consecutive waveguide in the grating is incremented by a fixed amount ∆L [13]. We assume here that the waveguides in the input/output array are transversally separated by the distance ∆xo = do at the FPR extreme (plane X0). The FPR has a radius of curvature f . Consecutive waveguides of the grating are transversally separated by distance ∆x1 = d1 at the FPR extreme (plane X1) and by distance ∆x2 = d2 at the reflecting surface that is terminating the grating (plane X2). Input, output, and grating waveguides at the FPR extreme are terminated with tapers having transversal spatial distribution profiles win (xo ), wout (xo ), and wg (x1 ), respectively. The key planes to be considered in our model are also indicated in Fig. 1. Consider an ultrafast optical pulse that is leaving the central waveguide (xo = 0) of the input/output array, with a temporal profile ein (t). The corresponding field eo (xo , t) at the FPR extreme can be described as [7]–[9]

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the spatial distribution of the illuminating field Wg (x1 ) and the waveguide transverse profile wg (x1 ) [7]–[9]. Therefore, the field coupled to the grating (e1 ) is given by e1 (x1 , t) ∝

r(x1 )Ein (ω)ejωt dω

(4)

with N

r(x1 ) =

2

ak wg (x1 − kd1 )

(5)

k=− N 2

ak ∝ Wg (kd1 ).

(6)

By using (5), (4) can be rewritten in the following way:

N

e1 (x1 , t) ∝

2

ak wg (x1 − kd1 )

Ein (ω)ejωt dω.

(7)

k=− N 2

eo (xo , t) = win (xo )ein (t) where ein (t) = u(t)ej2πνo t u(t) is the slowly varying amplitude of the field, and νo = ωo /2π is the central frequency of the pulse spectrum Ein (ω), which is defined as [5], [9] Ein (ω) ∝ ein (t)e−jωt dt. Here, we will assume transform-limited pulses of duration τp ∼ (1/∆ν), where ∆ν is the spectral width of the input pulse. Due to the small size of the AWGs and the single-mode character of the waveguides, we will neglect any pulse deformation due to chromatic dispersion. In addition, we consider an AWG with an ideal geometry, i.e., with a zero dispersion inside the device as a result of the phase errors [14], [15]. The field eo (x, t) is radiated from the central input waveguide to the FPR. The light spatial distribution (e1− ), which is arriving to the focal plane X1 in Fig. 1, can be obtained using the spatial Fourier transform of the input distribution [7]–[9], whereas the temporal profile at the slab–grating interface is identical to the input pulse, except for a phase factor; thus, the field arriving to the grating (e1− ) is described by e1− (x1 , ω) ∝ Wg (x1 )Ein (ω)

Equation (7) shows that the total field coupled to the grating can be described as N + 1 low-intensity pulse replicas that are simultaneously coupled to different waveguides of the grating, and each one has the same spectrum of the input pulse. The pulse replicas are reflected at the external mask that is used as the reflector. The reflecting surface of the external mask can be designed to provide different amplitude and phase modulations to each waveguide of the grating. In order to account for the modulation of light at each waveguide of the grating, the coefficient ak in (7) must be substituted with ck = ak δk eiϕk .

The second and third factors in (8) include the amplitude (δk ) and phase (ϕk ) modulations introduced at the reflecting surface, respectively. By design, there is a constant length difference ∆L between consecutive waveguides in the grating that can be expressed as [16], [17] 2∆L = m

∆τ = Wg (x1 ) ∝ Fxo [win (xo )]u= x1 α

α=

λo f , ns

λo =

c νo

(2) (3)

where Fx (u) is the spatial Fourier transform, c is the speed of the light, and ns is the effective refractive index of the slab region. The light coupled into each of the N + 1 waveguides of the grating is proportional to the overlap integral between

λo neff

(9)

where neff is the effective refractive index of the waveguides, and m is the diffraction order. The constant length increment can be converted into time delay using the relation

(1)

where

(8)

m 2∆L = . c/neff νo

(10)

The spectral component of the pulse replica that is traversing the kth waveguide (k = integer part of x1 /d1 ), with a frequency ν = ω/2π, experiences a relative phase shift with respect to the component with a frequency νo of the pulse replica that is traversing the central waveguide in the grating, and it is equal to [7]–[9] ω x1 d1

∆φ(x1 , ω) = e−jm νo

−jω ∆τ

=e

x1 d1

.

(11)

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Thus, the field that is leaving the grating back to the input/ output array (e1+ ) is given by the following expression: N

e1+ (x1 , t) ∝

2

x1 ck wg (x1 − kd1 ) e−jω∆τ d1 Ein (ω)ejωt dω.

k=− N 2

(12) In the absence of external modulation (δk ≡ 1, ϕk ≡ 0), (12) shows that the total field that is leaving the grating can be described as N + 1 low-intensity pulse replicas, where each one is coupled to a different waveguide of the grating. The pulse replicas have the same spectrum of the input pulse, but there is a delay ∆τ between pulse replicas that are traversing the consecutive waveguides of the grating. If the width of the input pulse τp is larger than the time delay ∆τ , pulse replicas overlap at the focal point of the FPR, resulting in a multiple-slit diffraction pattern. This occurs for AWG-PS with FSR > ∆ν. It is useful to rewrite (12) in terms of the mask function s(x) defined in the following way: N

2

s(x1 ) =

ck wg (x1 − kd1 ).

(13)

k=− N 2

Thus

jωt

E1 (x1 , ω)e

dω

(14)

with E1 (x1 , ω) ∝ s(x1 )e−jγωx1 Ein (ω),

γ=

∆τ . d1

(15)

The field (eo− ) arriving at the output waveguides is the spatial Fourier transform of (14), with the result [5], [9] 2πxo ∆τ − ω Ein (ω)ejωt dω (16) eo− (xo , t) ∝ S α d1 with S

2πxo ∆τ − ω α d1

∆ε =

ns do d1 FSR do = νo . SFSR neff 2∆Lf

∝ Fx1 s(x1 )e−jγωx1 u= xo .

(17)

α

Note that (16) shows that the complex spectrum of the field arriving at the output plane is given by 2πxo ∆τ − ω Ein (ω). (18) Eo− (xo , ω) ∝ S α d1 Without an external modulation, the mask function s(x) has a set of maxima that are separated by a distance d1 . Thus, for a given frequency, its spatial Fourier transform S[. . .] has maxima that are spatially separated by the amount α/d1 , which is the spatial FSR (SFSR) of the device [7]. With respect to the frequency variable, at a fixed position in the output plane, S[. . .] has a set of maxima that are separated by the amount 1/∆τ , which is the FSR of the AWG [7], [8]. The ratio FSR/SFSR gives the dispersion of the AWG-PS; thus, by being independent of the duration of the input pulse, light

(19)

The period of the mask function s(x) can be modified using an external periodic mask with a period (P d2 ) that is directly proportional to the separation (d2 ) between consecutive grating waveguides at the reflecting surface that is terminating the AWG. A periodic distribution (with a period P ) of ck values in (13) results in s(x), having a set of maxima that are separated by a distance P d1 ; thus, the FSR of the AWG-PS with a periodic external modulation (FSRM ) is reduced by a factor 1/P , and it is given by the following expression [6], [10]–[12]: FSRM =

c . 2neff P ∆L

(20)

For P = 1, (20) reduces to the FSR of the AWG-PS without external modulation [7]. Note that (16) also shows that the field arriving at the center of the central output waveguide is given by the following expression:

e1+ (x1 , t) ∝

leaves different output waveguides with different frequencies [9]. The frequency shift ∆ε between consecutive output channels that are separated by a distance do is given by the following expression:

eo− (xo = 0, t) ∝

∆τ S − ω Ein (ω)ejωt dω. d1

(21)

From (21), the temporal profile of the field arriving at xo = 0 is determined by the temporal profile of the input field that is convolved with a scaled version of the mask function [5], [9] t eout (t) ∝ ein (t) ∗ s −d1 . (22) ∆τ Without an external modulation, the scaled version of the mask function s(−d1 (t/∆τ )) has maxima at multiple values of ∆τ . If the spectral width of the input pulse spectrum is larger than the FSR (∆ν > FSR), the pulsewidth (τp ) is smaller than the period of the scaled version of the mask function; thus, the convolution (22) results in an output sequence of pulses with a repetition rate that is equal to ∆τ . This corresponds to a multiple-peaked spectrum with a peak-to-peak separation equal to the FSR of the AWG-PS without external modulation [9]. In the opposite case, where FSR > ∆ν(∆τ < τp ), the convolution (22) results in an elongated output pulse. This corresponds to an output spectrum that is containing a single narrow peak. However, as discussed above, the FSR of an AWG-PS can be reduced by a factor 1/P by using a reflective modulated mask with a period (P d2 ). Consequently, the output spectrum of an externally modulated AWG-PS, with FSRM < ∆ν [see (20)], is multiple peaked, with a peak-to-peak separation that is equal to FSRM , and the temporal output is a sequence of pulses with a repetition rate equal to P ∆τ . Finally, the light that is coupled into the kth output waveguide is proportional to the overlap integral between the spatial distribution of the impinging illuminating field and the output waveguide transverse profile wout (xo ) [7], [8].


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Therefore, the field coupled to the output waveguide (eout ) is given by the following expression: eout (xo , t) ∝ wout (xo − kdo ) ×S

2πxo ∆τ − ω Ein (ω)ejωt dω dxo . (23) α d1

The analytical expressions presented in this section could be further simplified for the special case of AWG-PS without external modulation, and simple Gaussian profiles for the transversal spatial distribution of all waveguide modes are assumed. However, a numerical evaluation of the analytic formulas of the model is required for more realistic simulations. A precise calculation of the waveguide modes can be carried out using beam propagation methods, while well-established fast Fourier transform (FFT) algorithms can be used to evaluate the analytical formulas, including arbitrary external modulation. In this paper, we used a mixed approach. A Gaussian approximation was adopted for the spatial waveguide-mode profiles, and all Fourier transforms involved in the formulation were numerically calculated using an FFT algorithm. First, single channel output spectra were numerically calculated. Then, the temporal response of the device was evaluated as the temporal Fourier transform of the spectral response. FFT algorithm was used repetitively in calculating both the spatial Fourier transforms involved in (2) and (17) and the temporal Fourier transform in (23). III. E XPERIMENTAL D ETAILS The experimental data that were used to validate the proposed model were obtained using AWG-PS that comprises silica-on-silicon reflective AWGs [13], which are originally designed for telecommunication applications, and external reflectors were used as phase or amplitude modulators. Details of the external masks can be found in [6], [10]–[12]. The reflective AWG multiplexers that were used in this paper consisted of 40 channels, with a channel-to-channel separation of 100 GHz, FSR = 8.4 THz, and the consecutive waveguides of the grating separated by d1 = 25(d2 = 31) µm at the FPR extreme (reflecting surface) [13], [17]. Spectral measurements were obtained using a continuous wave tunable laser as a light input source. A passive mode-locked pulsed laser was used to illuminate the AWG-PS through its input waveguide during the temporal measurements. The pulsed laser, with an average power of 5 mW, emits 0.5-ps pulses, with a repetition rate of 50 MHz. The laser has a time-limited spectrum with a spectral width of ∼7 nm. An optical spectrum analyzer with a 0.1-nm resolution was used to record the power spectra at the output waveguides (output channels) of the device. The temporal profile of the output signals was recorded via intensity cross correlation in a free-space apparatus. IV. A MPLITUDE M ODULATION The authors previously studied the effect of reconfigurable digital amplitude modulation in AWG-PS with ∆τ < τp [6].

Fig. 2. (Filled curve) Simulated and (continuous line) experimental (a) spectra and (b) intensity cross correlation traces for a single output channel of an AWG-PS using a periodic digital amplitude mask as the external reflector, with NR = 13 and NT = 19.

However, the experimental data presented in this paper were obtained using fixed amplitude masks, which were fabricated on glass wafers using conventional optical photolithography. The reflecting pixels in the patterned mask were obtained by the deposition of a thin film of gold on selected spatial regions. The pattern consisted of a periodic array of NR consecutive reflecting pixels that are separated by NT transmitting pixels. In the experiments described here, an external mask with NR = 13 and NT = 19 was used. This corresponds to an amplitude mask with a spatial period defined as P d2 = (NR + NT )d2 = 32d2 . Fig. 2(a) shows a comparison between measured and simulated single channel spectra of the AWG-PS using the amplitude mask with NR = 13 and NT = 19. A good agreement between simulated and experimental data is evident from this figure. As expected, the periodic modulation of the light traversing the waveguides of the grating resulted in a reduction of the FSR of the AWG, and a multiple-peaked spectrum was observed at each output channel of the device [6]. A peak-to-peak separation of 2.1 nm that is obtained from the simulated spectrum [Fig. 2(a)] results from the evaluation of (8) and (13),

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but removing the central group of reflecting pixels. A good agreement between simulated and measured cross-correlation traces indicates the effectiveness of the proposed model that is discussed in this paper. V. P HASE M ODULATION

Fig. 3. (Filled curve) Simulated and (continuous line) measured intensity cross correlation traces of a single output channel from an AWG-PS using a digital amplitude periodic mask with NR = 13 and NT = 19. The central pulse in the sequence was eliminated by setting δk ≡ 0 for every k, corresponding to the central period of the mask function.

taking into account a constant phase (ϕk ≡ 0) for all grating waveguides and periodic amplitude modulation consisting of 13 consecutive pixels with δk = 1 (reflecting) that are separated by 19 consecutive pixels with δk = 0 (transmitting). This results in a mask function s(x) with a period equal to P d1 , i.e., P = 32 times larger than the period d1 of s(x) when δk ≡ 1. Consequently, the spatial and spectral periodicity of the function S[. . .] of the original AWG is reduced by a factor 1/P , and the corresponding peak-to-peak separation of the modified spectrum of the selected output channel is given by (20). Fig. 2(b) shows a comparison between simulated and measured intensity cross correlation traces of a selected output channel of the AWG-PS with the same amplitude mask, which is used to obtain the spectral data shown in Fig. 2(a). The observed sequence of pulses is in good correspondence with the multiple-peaked nature of the spectrum. A measured temporal separation of 4 ps between consecutive pulses in the sequence is identical to the one that is calculated from the reciprocal of the modified FSR of the AWG defined in (23). The good agreement between experimental and simulated cross-correlation temporal profiles is also evident from Fig. 2(b). As stated above, AWGs that are commonly used in telecommunication applications have ∆τ < τp . The simplest description of the device operation can be provided in terms of superposition of overlapping pulse replicas at the output channel of the device. Pulse replicas that are traversing consecutive waveguides of the grating with δk = 1 overlap in the FPR. This produces a constructive interference between them and results in a slightly elongated output pulse per period of the mask function. Each pulse in the sequence has a direct correspondence to a group of reflecting pixels in the amplitude mask. Consequently, it is possible to eliminate any pulse(s) from the output sequence by substituting a group(s) of reflecting pixels (δk ≡ 1) with transmitting ones (δk ≡ 0). This is demonstrated in Fig. 3, where we show the measured and simulated cross-correlation traces for an AWG-PS using an amplitude mask as the external reflector, with the same period (NR = 13 and NT = 19) as the one used to obtain the data in Fig. 2,

Generation of periodic sequences of pulses using AWGPS with phase masks has been recently demonstrated by the authors [10]. In these experiments, amplitude masks were substituted with digital phase masks. The phase masks were fabricated on a silicon wafer using conventional optical photolithography and etching procedures. To produce a phase change of π radians, the pixel regions were etched down to ∼269 nm. After etching, a thin film of gold was deposited on the entire wafer. The pattern consisted of a periodic array of corrugated reflective pixels with a period equal to P d2 . However, simple phase masks only allow for generation of periodic sequences of pulses [10], which is not enough for practical applications of pulse shapers as the optical encoding element in future code-division multiple-access communications networks. Optical encoders must generate arbitrary sequences of pulses with very high repetition rates. This can be achieved using AWG-PS with more complex phase masks, which consist of groups of consecutive pixels with the same phase that is separated by sets of pixels with alternating zero and π phases [12]. A periodic sequence of pulses is observed in Fig. 4(a), where we show simulated and measured crosscorrelation traces for a selected output channel of the AWGPS using a periodic phase mask as the external reflector, which comprises N0 = 13 consecutive pixels with a constant phase ϕk = 0 that is separated by sets of Nπ /N0 = 19 pixels with alternating phases ϕk = 0 and ϕk = π. The simulation was obtained using (8) and (13), and δk ≡ 1. The selected pattern period is chosen to be identical to the one that is used in the amplitude experiments shown in Figs. 2 and 3. This allows us to perform a direct comparison between both modulation mechanisms. The patterned phase mask contains two distinct periods: a large one corresponding to P d2 (where P = 32) and a short period corresponding to 2d2 as a result of the presence of the alternating phase pixels in each large period of the phase mask. The temporal sequence that is observed in Fig. 4(a) is nearly identical to the one that is observed in Fig. 2(b), where the same pulse-to-pulse separation value of 4 ps was determined. This is in good agreement with the value 1/FSRM that is calculated using (23) with P = 32. The effect of the additional short period (2d2 ) within the patterned phase mask is shown in Fig. 4(b), where simulated and measured spectral responses of an output channel of the device are shown. The simulated spectrum consists of three sets of multiple peaks: The maximum of the central set is located at ∼1561 nm, with the peak-to-peak wavelength separation within the set of 2.1 nm. This result is similar to the one that is shown in Fig. 2(a), and it corresponds to the modification of the FSR of the R-AWG by the external phase mask with a pattern period of P = 32. The maxima of the other two sets of peaks are located symmetrically at ±33.5 nm from the central maximum (at ∼1561 nm). This corresponds to half of the original FSR of


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Fig. 5. (Filled curve) Simulated and (continuous line) measured intensity cross correlation traces of a single output channel from an AWG-PS using a phase mask with N0 = 13 and N0 /Nπ = 19. The central pulse in the sequence was eliminated by substituting the corresponding group of pixels with constant phase with pixels with alternating zero and π radians phases.

Fig. 4. (Filled curve) Simulated and (continuous line) experimental (a) intensity cross correlation traces and (b) spectra of a single output channel from an AWG-PS using a phase mask as the external reflector, with N0 = 13 and N0 /Nπ = 19.

the R-AWG. The larger spectral separation (33.5 nm) between the sets of peaks in the phase-modulated response spectrum of the R-AWG is attributed to a short period with P = 2, corresponding to the spatial regions with alternating phases in the patterned mask. This interpretation is supported by the data that are also shown in Fig. 4(b), where a phase mask with the same period (N0 = 13 and N π/N0 = 19) was used as the external reflector, but the spectrum is now measured over a wider wavelength span. The spectrum consists of two sets of multiple peaks, where the signature of each set is similar to the simulated one. The absence of a set of peaks at longer wavelengths in Fig. 4(b) is due to the wavelength span limitations of our experimental setup. Pulses propagating through different waveguides in the grating are reflected back at the patterned mask with modified or preserved phases, depending on the patterned feature. A constructive interference in the AWG-PS occurs only between pulse replicas that traversed the grating with the same phase. Each group of pixels with a constant phase contributes to a pulse in the output sequence, which is in analogy to the amplitude modulation. Pulse replicas that are traversing the grating waveguides with alternating phases zero and π radians

interfere destructively at the output channel of the device. Thus, substituting a group of pixels with constant phase with a set of pixels with alternating zero and π radians results in the elimination of the corresponding output pulse from the sequence. Fig. 5 shows the calculated and measured cross-correlation traces for an AWG-PS by using a patterned periodic phase mask as the external reflector, with N0 = 13 and Nπ /N0 = 19, but with the central group of pixels with a constant phase replaced by a set of pixels with phases corresponding to alternating zero and π radians. Elimination of the central pulse from the output sequence is evident. This result is similar to the one that is shown in Fig. 3, where an amplitude mask was used. Again, a good agreement between the simulation and the experiment was obtained for the AWG-PS with phase modulation. Our results demonstrate the efficacy of the proposed approach in producing arbitrary sequences of ultrafast pulses using a combination of AWGs and external phase masks. VI. C ONCLUSION A general quantitative description of the temporal and spectral responses of AWG-PS to a single ultrashort optical input pulse was presented and applied to relevant cases of digital amplitude and phase modulations. The operation of AWG-PS based on AWGs with FSR > ∆ν was described in terms of superposition of overlapping pulse replicas at the FPR of the device. It was shown that a precise numerical evaluation of the presented analytic formulation only requires the use of the wellestablished FFT algorithm in combination with the calculation of the waveguide modes. A comparison between experimental results and numerical simulations by assuming simple Gaussian spatial waveguide-mode profiles was presented. A good agreement between the simulation and the experiments for both cases of amplitude and phase modulations was demonstrated. Further improvements in the simulations can be obtained using spatial waveguide-mode profiles that are obtained from beam propagation methods. This is necessary for the simulation of the fine details of the experimental data.

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Luis Grave de Peralta received the M.S. degree in physics from Oriente University, Santiago de Cuba, Cuba, in 1982 and the Ph.D. degree in electrical engineering from Texas Tech University (TTU), Lubbock, in 2000. He was a Professor with the Department of Experimental and Theoretical Physics, Oriente University, until 1989. His earlier work focused on Vertical Cavity Surface Emitting Lasers (VCSEL) design and material characterization (luminescence and X-ray reflectivity). Since 2000, he has been working in planar lightwave circuit design and development in Lubbock at Applied WDM Inc., TTU, and Multipass Corporation. He is currently a Professor with the Department of Physics, TTU.

Ayrton A. Bernussi received the B.S., M.S., and Ph.D. degrees in physics from the State University of Campinas, Campinas, Brazil, in 1981, 1984, and 1990, respectively. In 1988, he joined the Optoelectronic group at the Brazilian Telecommunication Company (Telebras), where he was involved in the development of semiconductor lasers and materials. From 1994 to 1995, he was a Postdoctoral Researcher with the Electrical Engineering Department, Colorado State University, Fort Collins. His research during this period was on hightemperature properties of strained quantum well lasers. In 2000–2001, he was with the National Synchrotron Light Laboratory, Brazil, where he was involved in the development of semiconductor high-power lasers and strained nanostructures. From 2001 to 2004, he was a Research Associate with the Electrical Engineering Department, Texas Tech University (TTU), Lubbock, where he was involved in the development and characterization of planar lightwave circuits. In 2004, he joined the faculty of TTU as an Assistant Professor within the Electrical and Computer Engineering Department. His current research interests include generation of arbitrary sequences of ultrafast pulses using integrated-optic devices, design, fabrication, and characterization of nanocavities and plasmonic waveguides.

Henryk Temkin (SM’87–F’93) has a broad record of research in semiconductors. He was a Distinguished Member of Staff with the Physics Division, Bell Laboratories, Murray Hill, NJ, from 1977 to 1992, where he was involved in research on optoelectronic devices. In 1992, he joined the faculty of Colorado State University, Fort Collins, as a Professor of electrical and computer engineering. He moved to Texas Tech University, Lubbock, in 1996, where he currently serves as a Maddox Chair in Engineering. He is also currently a Program Manager in the Microsystems Technology Office, DARPA, Arlington, VA. Recent research interests include wide bandgap devices, devices for wavelength division multiplexing, and optical properties of semiconductors. He is the author and coauthor of over 400 publications and is the holder of 27 U.S. patents.