Universal Figure of Merit for Arbitrary-Order Optical Differentiators

Reza Ashrafi and José Azaña Institut National de la Recherche Scientifique - Énergie, Matériaux et Télécommunications,

Varennes, Québec, Canada, J3X 1S2 ashrafi@emt.inrs.ca

Abstract—A universal analytical figure of merit for performance evaluation of optical differentiators is introduced, which allows one to estimate the acceptable input pulse bandwidth-range to achieve a desired processing accuracy from the filter’s physical parameters.

Keywords- ultrafast optical signal processing; differentiation; waveguide/fiber gratings; performance evaluation.

I. INTRODUCTION All-optical temporal differentiators are basic building

blocks in ultrahigh-speed photonic analog/digital signal processing and computing circuits [1]. We refer to an Nth-order optical differentiator (NOOD) as a linear filtering device that provides the Nth-time derivative of the temporal complex envelope of an arbitrary input optical signal. This device can be implemented using a linear optical filter with a transfer function:

( )( )H f j f Ν∝ − (1) where f is the baseband frequency [1], defined by f=fopt-f0, where fopt is the optical frequency variable and f0 is the carrier optical frequency of the input optical signal. Fig. 1 shows a schematic for the operation principle of an NOOD.

( ) ( )out in

N

Nddt

e t e t⎡ ⎤∝ ⎣ ⎦

0Input @ f 0Output @ f

( )ine t01 f

( ) ( )NH f j f∝ −

NOOD

Fig. 1. A schematic of an Nth-order optical differentiator (output curve corresponds to N=2).

Besides their intrinsic interest in future ultrafast all-optical computing and information processing circuits [2], NOODs have been already demonstrated for a broad range of applications, including measurement and characterization of ultrafast optical signals and devices [3] and ultra-short optical pulse shaping [4]. NOODs with different device operation bandwidths (DOBs), in the range of 10s of GHz up to THz bandwidths, have been previously proposed and experimentally realized based on various optical device technologies (see Table 1). Previous studies have used the DOB, i.e. approximately the maximum input signal bandwidth (ISB) that can be accurately processed, as the main performance parameter of a differentiator. However, in evaluating the

processing-speed performance of these devices one should also consider the minimum ISB that can be processed with a prescribed accuracy. For this purpose, we define here a new figure of merit for any NOOD, namely the maximum to minimum bandwidth ratio, MMBR, which essentially determines the broadness of the acceptable ISB range for a desired minimum processing accuracy. We obtain an analytical expression of general validity for this new figure of merit and demonstrate that the MMBR of an optical differentiator essentially increases with the resonance depth of the NOOD’s amplitude spectral response. This suggests that the operation range, i.e. ISB range, of an optical differentiator can be improved only by increasing the depth of the device resonance notch. Moreover, it is also shown that the needed resonance depth for a prescribed processing accuracy increases with the differentiation order and it is also significantly higher for solutions based on the use of minimum-phase filters (as compared to non-minimum-phase filtering devices). Table 1. Previously demonstrated optical differentiators with different device operation bandwidths (DOBs) based on various photonic devices.

Photonic devices for optical differentiation DOB

Silicon microring resonator [5],Apodized FBGs working in reflection [6] 10s of GHz

Apodized chirped FBG working in transmission [7] 100s of GHz

Uniform long period gratings [8] A few THz

Apodized long period gratings [9] > 10 THz

II. ANALYTICAL PERFORMANCE EVALUATION NOODs can be realized based on various photonic linear-

filtering devices, which can be generally classified into two main groups, minimum phase (MP) and non-minimum phase (NMP) filters [10]. For instance NOODs based on fiber Bragg gratings (FBGs) working in transmission and reflection are MP and NMP solutions, respectively [10]. A schematic of the spectral amplitude response, |H(f)| (according to Eq. 1), of a practical NOOD is shown in Fig. 2. As it is shown in Fig. 2, the depth of the spectral response is defined as d=Hmax/Hmin , where Hmax and Hmin are the maximum and minimum of the amplitude spectral response along the NOOD’s differentiation bandwidth. For evaluation of the processing accuracy of an NOOD we consider the degree of similarity between the time-domain intensity profiles of the NOOD’s output Pout(t) and the

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ideal output Pideal(t), which is estimated by using normalized cross-correlation coefficient, defined as below:

( )( )2 2

( ) ( )100%

( ) ( )

out idealc

out ideal

P t P t dtC

P t dt P t dt

+∞

−∞

+∞ +∞

−∞ −∞

= ×∫

∫ ∫ (2)

As illustrated in Fig. 2, the acceptable ISB range is defined by considering a minimum limit in the cross-correlation coefficient of the acceptable outputs, e.g. corresponding to Cc > 99%. In particular, this acceptable ISB range is quantified using a new figure of merit, namely maximum-to-minimum bandwidth ratio (MMBR) of the NOOD, which is defined as the ratio between the DOB and the lower limit of the acceptable ISB range (BWmin):

min

DOBMMBRBW

= (3)

We have analytically derived and verified by simulation that the MMBR of any NOOD can be obtained from the following expression:

min

11

NdMMBR

d−=

− (4)

where dmin is the minimum required resonance depth to obtain the desired processing accuracy (target CC) when the ISB is equal to the DOB. Assuming that the desired performance is defined by the limit Cc = 99%, the values of dmin for MP and NMP NOODs of different differentiation orders are given in Table 2.

minH

maxH( )H f

min

2BW

2DOB (Hz)f

mind

max

min

Hd

H=

Fig. 2. A schematic of the spectral amplitude response of an NOOD.

1 2 3 4 5 6 7 8 9 1096

97

98

99

100

Input pulse bandwidth (THz)

Cc

%

1st-order (N=1), d=50dB2nd-order (N=2), d=50dB3rd-order (N=3), d=50dB3rd-order (N=3), d=70dB

DOB

Cc=99%

Fig. 3. An example of simulation results showing the cross correlation coefficient versus input Gaussian pulse bandwidth (full width at 0.3981% of the amplitude spectrum peak) for 1st-, 2nd- and 3rd-order differentiators.

To verify the obtained analytical expression (i.e. Eq. 4) for MMBR we have simulated NOODs with different orders (N=1,2,3) and different resonance depths (d=50dB and 70dB, reported here). Fig. 3 shows some examples of the simulation results. Fig. 4 presents a comparison between the estimated MMBR results from simulation according to Eq. 3 (i.e. circle points) and from the analytical expression in Eq. 4 (i.e. solid curves). The observed excellent agreement for the mentioned

comparison in Fig. 4, clearly proves the validity of the obtained analytical performance expression for the main figure of merit of arbitrary-order optical differentiators.

Table 2. Estimation of dmin for the NOODs by considering the limit of Cc = 99% for the minimum processing-accuracy performance. For the even-order NPM-NOODs a linear spectral phase profile and for odd-order NPM-NOODs an additional ideal π-phase-jump at the NOOD’s central frequency (i.e. f=0) have been-considered.

NOOD Based on NMP systems Based on MP systems

1st-order (N=1) dmin=20.24dB dmin ≥40.70dB

dmin≈40.70+0.25tanh[(d-42)/4)] dB

2nd-order (N=2) dmin=34.68dB dmin ≥54.67dB

dmin≈54.671+1.041tanh[(d-55)/10)] dB

3rd-order (N=3) dmin=38.57dB dmin ≥81.45dB

dmin≈83.535+3.535tanh[(d-89.5)/11)] dB

20 40 60 80 100 120 1400

2

4

6

8

10

d (dB)

MM

BR

1st-order, NMP2rd-order, NMP3nd-order, NMP

(a)20 40 60 80 100 120 1400

2

4

6

8

10

d (dB)

MM

BR

1st-order, MP2nd-order, MP3rd-order, MP

(b)

Fig. 4. MMBR versus d for the NOODs based on (a) NMP and (b) MP systems. The solid curves (a) and (b) are the plots of the obtained theoretical expression Eq. 4, and the circle points are obtained from the simulation.

III. CONCLUSION We have introduced and numerically confirmed a universal

analytical figure of merit to evaluate the performance of arbitrary-order optical differentiators, i.e. MMBR (Eq. 4), which allows one to estimate the acceptable input pulse bandwidth range to achieve a desired processing accuracy from the filter’s physical parameters (notably its resonance depth).

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differentiators”, Opt. Commun., vol. 230, pp. 115–129, 2004. [2] J. Azaña, "Ultrafast analog all-optical signal processors based on fiber-

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[4] L. K. Oxenlwe et al., "640 Gb/s timing jitter-tolerant data processing using a long period fiber-grating-based flat-top pulse shaper," IEEE J. Sel. Topics Quantum Electron., vol. 14, pp. 566–572, 2008.

[5] F. Liu et al., "Compact optical temporal differentiator based on silicon microring resonator," Opt. Express, vol. 16, pp. 15880–15886, 2008.

[6] D. Gatti et al. "Temporal differentiators based on highly-structured fibre Bragg gratings," Electron. Lett., vol. 46, pp. 943–945, 2010.

[7] L. M. Rivas et al., "Experimental demonstration of ultrafast all-fiber high-order photonic temporal differentiators," Opt. Lett., vol. 34, pp. 1792–1794, 2009.

[8] R. Slavík at al. "Terahertz-bandwidth high-order temporal differentiators based on phase-shifted long-period fiber gratings," Opt. Lett., vol. 34, pp. 3116–3118, 2009.

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[10] J. Skaar, “Synthesis of fiber Bragg gratings for use in transmission,” J. Opt. Soc. Am. A, vol. 18, pp. 557-564, 2001.

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