## Abstract

We introduce a universal figure of merit to evaluate the processing speed (operation bandwidth) performance of arbitrary-order optical differentiators. In particular, we define the maximum-to-minimum bandwidth ratio (*MMBR*) as a main figure of merit of these devices, which essentially informs about the broadness of the acceptable input pulse bandwidth range. We derive and numerically confirm a general analytical expression for the *MMBR* of an arbitrary optical differentiator, showing that this can be expressed simply as a function of the differentiator’s amplitude resonance depth. The device *MMBR* can be improved by increasing the filter’s resonance depth, depending also on the differentiation order; in particular, the *MMBR* quickly deteriorates as the differentiator order is increased. In our analysis, photonic differentiators are considered in two main groups, namely (i) non-minimum phase and (ii) minimum phase optical filtering implementations. The derived analytical expression for the device *MMBR* is generalized for these two different solutions, and the validity of the obtained analytical estimates is verified through numerical simulations, including results for the cases of 1st-, 2nd-, and 3rd-order differentiators.

©2012 Optical Society of America

## 1. Introduction

Recently, with a considerably fast development in photonics technology, all optical circuits have been envisioned as a promising solution to overcome the signal processing speed limitations of present electronic technologies [1–3]. Some fundamental all-optical operators, such as a photonic temporal integrator [4,5], photonic temporal differentiator [6–16], photonic Hilbert transformer [17,18], photonic Fourier transformer [19,20] etc., have been designed and practically realized.

All-optical temporal differentiators are interesting as basic building blocks in ultrahigh-speed photonic analog/digital signal processing and computing circuits [6]. We refer to an *N*th-order optical differentiator (NOOD) as a device that provides the *N*th-time derivative of the temporal complex envelope of an arbitrary input optical signal. Besides their intrinsic interest in future ultrafast all-optical computing and information processing circuits, these devices have been envisioned for various other applications of more immediate interest. To give a few relevant examples, 1st-order optical differentiators have proved useful for measurement and characterization of optical signals (e.g. high-bit-rate data streams in optical telecommunication) and devices [21]. Arbitrary-order optical differentiators have been also successfully employed for ultra-short optical pulse shaping, including generation of 1st- and higher-order Hermite-Gaussian (HG) temporal waveforms from input Gaussian-like optical pulses [7,8,22,23]. These waveforms are particularly interesting in the context of optical telecommunications [24] and for advanced coding [22–24]. Some other important examples for applications of NOODs include: ultrashort flat-top pulse generation for use in the demultiplexing of 640 Gbit/s data in ultrahigh-speed transmission systems [25], ultrahigh bitrate (~Tbit/s) serial optical communication signals processing [26], optical dark-soliton detection [27] etc. In a more general framework, optical differentiators of different orders are necessary to create analog circuits capable of real-time solving differential equations at ultrahigh speeds [3].

We notice that optical differentiator designs have been also demonstrated for processing real-positive time-domain intensity waveforms: the so-called intensity differentiators [28–33]. Intensity differentiators have been mostly used for microwave photonics applications [33–35]. These solutions are, however, outside the scope of our present work. This paper reports a novel analytical performance analysis for arbitrary-order optical field differentiators (i.e. NOODs).

NOODs with different *device operation bandwidth*s (*DOB*s) in the range of 10s of GHz up to a few THz have been previously proposed and experimentally realized based on various optical device technologies (see Table 1
).

Experimental implementation of NOODs based on silicon microring resonator [11] and apodized fiber Bragg gratings (FBGs) working in reflection [12,13] has provided a *DOB* in the range of 10s of GHz. NOODs with a *DOB* in the range of 100s of GHz have been designed and experimentally demonstrated based on apodized chirped FBGs working in transmission [8]. Larger *DOB*s in the range of a few THz have also been targeted and experimentally achieved for NOODs based on long period fiber gratings (LPGs). It has been demonstrated that a uniform LPG operating in full-coupling condition implements a THz-bandwidth 1st-order differentiator (i.e. NOOD with *N* = 1) [7] and a uniform LPG incorporating *N*-1 π-phase shifts can serve as a THz-bandwidth NOOD (i.e. *N* = 2,3,4,…) [9,10]. It has also been proposed that apodized LPG-based NOODs could be designed to achieve *DOBs* > 10THz [16].

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, thus obtaining complete information on the entire *ISB* range of operation of the device under analysis.

In this work, we define a new, universal figure of merit for any NOOD, namely the maximum to minimum bandwidth ratio, *MMBR*, which essentially determines the acceptable *ISB* range for a desired minimum processing accuracy. We obtain an analytical expression of general validity for this adimensional figure of merit, i.e. independent on the specific NOOD technological platform, 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. For this performance analysis, we consider the NOOD devices to be classified into two main broad groups, i.e. designs based on minimum phase (MP) [8,14] and on non-minimum phase (NMP) [7,10,12,13,16] filtering systems. We show 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 MP filters as compared to NMP filtering devices.

## 2. Operation principle of the optical differentiators

An NOOD is a linear filtering device that provides the *N*th-order time derivative of the input signal electric-field complex envelop. This device can be implemented using a linear optical filter with a spectral transfer function:

*ω*is the angular baseband frequency [6], defined by

*ω*=

*ω*

_{opt}-

*ω*

_{0}, where

*ω*

_{opt}is the optical angular frequency variable and

*ω*

_{0}is the carrier optical angular frequency of the input optical signal. The utilized variable

*Frequency*or

*f*hereafter, in the text or figures, is the baseband frequency defined as

*f = ω/2π*. Also the carrier optical frequency (i.e. the central or resonance frequency of the NOOD) is represented with

*f*

_{0}and is defined as

*f*

_{0}

*= ω*

_{0}

*/2π*. According to Eq. (1), the amplitude spectral response of an NOOD is proportional to |

*ω*|

*(*

^{N}*N*= 1, 2, 3,…). The spectral phase response for an even-order NOOD (i.e.

*N*= 2, 4, 6,…) has a linear profile (to account for the average propagation time through the device) and for an odd-order NOOD (i.e. N = 1, 3, 5, …) it has an additional discrete

*π*-phase shift at the NOOD’s central frequency (

*ω*= 0) [7,8]. Figure 1 shows a schematic for the operation principle of an NOOD. As evidenced by the frequency transfer function

*H*(

*ω*) in Eq. (1), an NOOD is a notch optical filter (i.e. a band-stop filter) which provides the

*N*th-time derivative of the temporal complex envelope of the input optical signal.

As mentioned above, NOODs can be realized based on various photonic device technologies, which can be generally classified into two main groups, i.e. MP and NMP linear systems. It is worth mentioning that the transfer function of an NOOD is a MP function [14] and MP functions can be realized based on either MP or NMP filtering systems: An NMP filter can be designed to implement any desired (physically realizable) spectral transfer function, including that of an MP system [36]. To be more concrete, in an MP linear filter, the phase (i.e. *Φ*(*ω*)) and the amplitude (i.e. |*H*(*ω*)|) profiles of the filter’s spectral response are related by means of the Hilbert transform, as follows [37,38]:

## 3. Basic definitions

The spectral amplitude of an ideal NOOD according to Eq. (1) can be expressed as |*H*(*ω*)| ∝ |*ω*|* ^{N}* which shows the need for a zero at the carrier optical frequency (i.e. at

*ω*= 0). Considering that a theoretical ideal zero (which corresponds to minus infinity in dB scale) in the transfer function,

*H*(

*ω*), of a physical filters is practically impossible, the depth of the spectral resonance dip of an NOOD can be anticipated to play a fundamental role in the processing performance of these devices. A schematic of the spectral amplitude response, |

*H*(

*f*)|, 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*=

*H*

_{max}/

*H*

_{min}in linear scale and

*d*= 20 × log(

*H*

_{max}/

*H*

_{min}) in dB scale, where log is the base-10 logarithm function.

*H*

_{max}and

*H*

_{min}are the maximum and minimum of the amplitude spectral response in the NOOD’s differentiation bandwidth. It is worth emphasizing that

*H*

_{max}is not necessarily the maximum of the amplitude spectral response in the entire resonance bandwidth of a notch filter used as an NOOD.

*H*

_{max}is defined according to the fraction of the filter’s resonance bandwidth that is useful for differentiation (i.e. device operation bandwidth,

*DOB*) and is typically quantified by limiting the maximum deviation (

*MD*) of the device amplitude spectral transfer function around its resonance frequency with respect to the ideal transfer function of an NOOD, i.e. Eq. (1). For instance in Refs. [7,9], where the operation bandwidth of the NOOD is just a fraction of the whole resonance bandwidth of an LPG,

*MD*= 9% was considered.

*H*

_{min}is referred to as the absolute depth of the optical filter (in some references it is simply called transmission depth). The experimental implementation of all-fiber LPG-based filters [43] shows that a transmission depth of over −60dB (i.e. 20 × log(1/

*H*

_{min}) = −60dB) for a single-unit optical filter is practically possible with current grating fabrication technologies.

The performance of an NOOD is evaluated by estimating the similarity between the output time-domain waveform and the corresponding ideal output (i.e. *N*th-order derivative of the input time-domain waveform). This is estimated by using the cross-correlation coefficient (*C*_{c}) between the ideal output and the actual NOOD’s output, defined as follows [38]:

*P*(

_{out}*t*) and

*P*(

_{ideal}*t*) are the time-domain intensity profiles of the NOOD’s output and the ideal output, respectively. The acceptable

*ISB*range is defined by considering a minimum limit in the cross-correlation coefficient, e.g. corresponding to

*C*

_{c}≥ 99%. In relation with these metrics, we use the

*MMBR*of the NOODs as a main figure of merit to evaluate their performance. The device’s

*MMBR*informs about the acceptable

*ISB*range and is expressed as the ratio between the upper (

*BW*

_{max}) and the lower (

*BW*

_{min}) limits of the acceptable

*ISB*range,

In this paper, for estimation of the *MMBR*, the input is assumed to be a Gaussian pulse. It should be noted that this shape exhibits a particularly poor performance in what concerns the cross-correlation coefficient value estimated from Eq. (3). This is due to the fact that one of the main practical deviations of a practical NOOD with respect to the ideal device’s response concerns the limited depth of its spectral resonance dip and the Gaussian pulse energy is heavily concentrated around this resonance frequency. In what follows, the input Gaussian pulse bandwidth (i.e. *ISB*) is considered to be the full width at 0.4% of the amplitude spectrum peak. This bandwidth definition has been fixed to ensure that the peak of the resulting *C*_{c} curves versus *ISB* is approximately reached at the *DOB*, see further discussions below.

As it is shown in this paper, the *MMBR* in a general NOOD filter essentially depends on the filter’s resonance depth (i.e. *d* in Fig. 2). The minimum acceptable performance of an NOOD corresponds to *MMBR* = 1, which reflects the case when a single, very specific *ISB* value can be accurately processed by the NOOD (to ensure a processing accuracy such that *C*_{c} = 99%). In this case, the depth of the device spectral response is defined as *d*_{min}. In other words, *d*_{min} is the minimum required depth for an NOOD’s amplitude spectral response to enable accurate processing of an input Gaussian pulse (i.e. with *C*_{c} = 99%) with a prescribed (single) bandwidth (corresponding to *BW*_{max} = *BW*_{min}). A pulse with a frequency bandwidth different to (narrower or broader than) the prescribed one will be differentiated with an unacceptable processing error (i.e. with *C*_{c} < 99%). It is obviously anticipated that the differentiator *MMBR* will be increased (> 1) as *d* is increased over the minimum value *d*_{min}.

The acceptable *ISB* range of an NOOD can be expressed as *BW*_{min}<*ISB*<*BW*_{max}. Figures 3
and 4
show an example of calculated *ISB* ranges corresponding to different 3rd-order differentiator designs (i.e. NOOD with *N* = 3). Figure 3 shows the amplitude spectral response profiles of the considered NOOD designs, all having the same resonance depth of *d* = 70dB. In all cases, a purely linear phase spectral response with an ideal pi phase shift at the central frequency is assumed (NMP filter implementations). The considered NOOD designs just differ on their amplitude spectral response outside the differentiation frequency band (*DOB*), as it is typically the case in practical designs based on different device technologies. For instance, the amplitude spectral response of a reflection FBG-based NOOD decays to zero outside the *DOB* (similar to H_{2}-H_{5} in Fig. 3) [12,13], whereas the amplitude response of an LPG-based NOOD does not decay to zero outside its operation bandwidth (similar to H_{1} in Fig. 3) [7,9,10]. Figure 4 shows the simulated *C*_{c} curves, from Eq. (3), as a function of the input Gaussian pulse bandwidth (i.e. *ISB*) for each considered spectral amplitude profile in Fig. 3 (H_{1}-H_{5}). The minimum acceptable *ISB* (i.e. *BW*_{min}) in Fig. 4 corresponds to the left intersection of the curves with *C*_{c} = 99%. As it can be seen in Fig. 4, the *BW*_{min} is the same for all the considered spectral phase profiles (H_{1}-H_{5}) as this essentially depends on the filters’ resonance depth (*d*), see discussions below. However, the maximum acceptable *ISB* (i.e. *BW*_{max}), which corresponds to the right intersections of the curves with *C*_{c} = 99% in Fig. 4, strongly depends on the shape of the amplitude spectral response outside the differentiation frequency band (i.e. |*Frequency|*>*DOB/2* in Fig. 4).

As it can be seen in Fig. 4, the value of the parameter *BW*_{max} (maximum *ISB* that can be processed with the prescribed accuracy) strongly depends on the shape of the filter’s amplitude spectral responses outside the differentiation bandwidth. In any case, the parameter *BW*_{max} is necessarily higher than the *DOB*, regardless of the specific filter’s spectral shape (see Fig. 4). This suggests a more general definition for the minimum value of the *MMBR* parameter, fully independent on the chosen filter device technology (i.e. independent on the filter’s spectral shape outside the differentiation bandwidth): The minimum acceptable *ISB* range can be estimated by defining the *MMBR* within the *DOB* (*MMBR*_{WDOB}) as below:

## 4. Analytical expression for *MMBR*_{WDOB}

In this section, we derive an analytical, general expression for the *MMBR*_{WDOB} of NOODs based on MP and NMP systems.

The spectral amplitude response, |*H*(*f*)|, of a physically realizable NOOD according to Fig. 2, can be expressed as:

*H*

_{min},

*H*

_{max}and

*DOB*are previously defined and also shown in Fig. 2. Evaluation of Eq. (6) at

*f*=

*BW*

_{min}/2 (see Fig. 2) leads to:

*H*

_{0}, as shown in Fig. 2, is the amplitude spectral response of the NOOD evaluated at

*f*=

*BW*

_{min}/2. Equation (7) can be rewritten as:

According to the definitions introduced in Section 3 (see Fig. 2 and Eq. (5)), *H*_{0}/*H*_{min}, *H*_{max}/*H*_{min} and *BW*_{min}/*DOB* are *d*_{min}, *d* and 1/*MMBR*_{WDOB}, respectively. Therefore, the *MMBR*_{WDOB} for a general NOOD can be expressed as follows:

Equation (9) quantitatively shows how the *MMBR*_{WDOB} is increased by increasing the resonance depth in the NOOD’s spectral response (i.e. *d*). As expected, the NOOD’s minimum acceptable performance, which is expressed by *MMBR*_{WDOB} = 1, corresponds to *d* = *d*_{min}. On the other hand Eq. (9) shows that in order to obtain the same performance for NOODs with higher-orders (i.e. higher *N*), higher depth in the spectral response (i.e. larger *d*) is required. In the other word, assuming the same depth in the spectral responses of NOODs with different orders (*N* = 1, 2, 3, etc.), we expect to have a shorter acceptable *ISB* range for higher-order differentiators.

*d*_{min} can be estimated by using the defined minimum limit in the cross-correlation coefficient of the acceptable outputs, e.g. *C*_{c} = 99%. Table 2
shows the numerically estimated *d*_{min} of NOODs (*N* = 1,2,3) based on MP and NMP filtering systems, considering a Gaussian input pulse. For the odd-order NMP-based NOODs, an ideal π-phase jump has been considered at the NOOD’s central frequency (*ω* = 0). However, we recall that in MP-based NOODs, the spectral phase response is dependent on the spectral amplitude response, see discussions in Section 2 above. This translates into the fact that for an MP-based NOOD, *d*_{min} in Eq. (9) depends on the value of the filter’s resonance depth, *d*. The estimated relations for *d*_{min} in the case of MP-based NOODs are obtained by numerical curve fitting and are given in Table 2.

## 5. Numerical simulations

The obtained analytical expression for *MMBR*_{WDOB} (Eq. (9)) have been fully verified through numerical simulations of NMP filters-based and MP filters-based NOODs with different orders (*N* = 1, 2, 3) and different resonance depths (*d* = 50dB and 70dB).

#### 5.1. MMBR_{WDOB} as a function of the resonance depth

Equation (9) predicts that the *MMBR*_{WDOB} increases by increasing the resonance depth. To verify this prediction, three NMP-based 3rd-order differentiators, with the spectral amplitude profiles shown in Fig. 5
, are considered. Their spectral amplitude responses are assumed to have three different depths of *d* = 50dB, *d* = 70dB and *d* = infinity dB (i.e. ideal case). Also the differentiator is assumed to have an ideal linear phase profile including an exact π-phase-jump at the NOOD’s central frequency.

Figure 6
shows the simulated *C*_{c} curves, using the definition in Eq. (3), versus the input Gaussian pulse bandwidth (i.e. *ISB*). The *MMBR*_{WDOB} for the three studied cases of *d* = 50dB, *d* = 70dB and *d* = infinity dB can be numerically estimated from the simulations in Fig. 6 using Eq. (5). The *MMBR*_{WDOB} can be also evaluated from the derived analytical equation Eq. (9) with the *d*_{min} value defined in Table 2 (38.57dB for the three considered filters). Table 3
verifies the validity of Eq. (9) to obtain a precise estimate of the *MMBR*_{WDOB}. According to the investigated examples (results in Fig. 5 and Fig. 6), it can be easily inferred that not reaching an exact zero at the resonance wavelength implies a deviation in the filter's spectral response with respect to the ideal response of an NOOD (ideal curve corresponding to d = infinity dB in Fig. 5) over a certain bandwidth around this resonance wavelength. The closer the response gets to zero at the resonance (i.e. the deeper the resonance dip), the narrower the 'deviation' frequency bandwidth (i.e. smaller *BW*_{min}) would be. Therefore by considering the fact that the resonance depth (*d*) of any physical filter cannot reach infinity, there will be a minimum operation bandwidth (i.e. *BW*_{min}) in practical field differentiators (i.e. NOODs) and as a result, the corresponding *MMBR*_{WDOB} will be a finite number.

#### 5.2. MMBR_{WDOB} as a function of the differentiation order

Equation (9) predicts that the *MMBR*_{WDOB} decreases by increasing the differentiation order (for the same resonance depth). To verify this prediction, we consider three NMP-based differentiators (i.e. a 1st-order, a 2nd-order and a 3rd-order differentiator) with the spectral amplitude profiles shown in Fig. 7
. Ideal spectral phase profiles are assumed for the three evaluated NOOD filters. The same resonance depth of *d* = 50dB is assumed for all these differentiators. The simulated *C*_{c} curves for the three considered differentiators are shown in Fig. 8
. As detailed in Table 4
, in all cases, Eq. (9) provides a very precise estimate of the *MMBR*_{WDOB}, using the corresponding *d*_{min} values defined in Table 2, in agreement with the numerically obtained value (from the simulation results in Fig. 8).

#### 5.3. Comparison between NMP and MP implementations

Equation (9) predicts that the needed resonance depth for a prescribed processing accuracy is significantly higher for an MP filter - based NOOD as compared to the NMP filter -based implementation. To verify this prediction, two 1st-order differentiators respectively implemented using an MP filter and an NMP filter are considered. Their spectral amplitude responses are the same as for the 1st-order case in Fig. 7 and the two filters are also assumed to have the same resonance depth of *d* = 50dB. The spectral phase response of the MP filter is calculated from its spectral amplitude response using the HT relationship defined in Eq. (2). In the NMP implementation example, an ideal linear spectral phase variation with an exact π phase shift at the differentiator’s central frequency is assumed. The simulated *C*_{c} curves for these differentiators are shown in Fig. 9
. For the analytical evaluation of the *MMBR*_{WDOB}, we use the corresponding values of the parameter *d*_{min} according to the definitions in Table 2, i.e. 21.31dB and 40.70 + 0.25tanh[(50-42)/4)] = 40.941dB, respectively. As detailed in Table 5
, the analytically obtained values for the *MMBR*_{WDOB} provide again a precise estimate of this figure of merit, in agreement with the numerically estimated values (from the results in Fig. 9).

#### 5.4. Full confirmation of the analytical estimates for the MMBR_{WDOB} of NOODs.

To conclude our work, Fig. 10
presents a comparison between the *MMBR*_{WDOB} vs. resonance depth (*d*) curves obtained from numerical simulations (circle points) and from the analytical expression in Eq. (9), using the *d*_{min} values in Table 2 (solid curves). The curves are calculated for different differentiation orders (i.e. 1st-, 2nd- and 3rd-order) and for MP and NMP filtering implementations of the NOODs.

The results in Fig. 10, together with the partial results detailed in Table 2, Table 3 and Table 4, clearly proves the validity of the obtained analytical performance expression for the defined figure of merit (*MMBR*_{WDOB}) of arbitrary-order optical differentiators.

## 6. Conclusion

We have introduced a universal figure of merit to evaluate the performance of arbitrary-order optical differentiators, i.e. maximum-to-minimum bandwidth ratio (*MMBR*). We have derived and numerically confirmed a general analytical expression for this figure of merit, namely Eq. (9), which allows one to estimate the acceptable input signal bandwidth range to achieve a desired processing accuracy from the filter’s physical parameters. Our analysis shows that the *MMBR* of any given optical differentiator device can be increased by increasing the filter’s resonance depth, regardless of the maximum operation bandwidth enabled by the specifically used device technology. A smaller *MMBR* is obtained for the same resonance depth as the differentiation order is increased, or in other words, to obtain the same *MMBR* performance, higher-order differentiators require an increased resonance depth. Moreover, our study also reveals that implementations based on minimum-phase linear filters require a higher resonance depth than those based on non-minimum-phase filters to achieve the same *MMBR* performance.

## Acknowledgments

This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), le Fonds Québécois de la Recherche sur la Nature et les Technologies (FQRNT), and Institut National de la Recherche Scientifique (INRS).

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