Abstract

A simple and general approach for designing practical all-optical (all-fiber) arbitrary-order time differentiators is introduced here for the first time. Specifically, we demonstrate that the N th time derivative of an input optical waveform can be obtained by reflection of this waveform in a single uniform fiber Bragg grating (FBG) incorporating N π-phase shifts properly located along its grating profile. The general design procedure of an arbitrary-order optical time differentiator based on a multiple-phase-shifted FBG is described and numerically demonstrated for up to fourth-order time differentiation. Our simulations show that the proposed approach can provide optical operation bandwidths in the tens-of-GHz regime using readily feasible FBG structures.

© 2007 Optical Society of America

1. Introduction

All-optical circuits for information processing are highly desired for a wide variety of applications, including computing and networking, mainly due to their strong potential for overcoming the speed limitations associated with conventional electronics implementations [1]. A promising alternative to implement all-optical signal processors is based on emulating the developments in the electronic domain using all-optical technologies. For this purpose, photonics equivalents of fundamental devices that form basic building blocks in electronic circuits would need to be designed and implemented. A temporal differentiator is one of these fundamental devices [2]–[6]. We refer to an N th-order optical temporal differentiator as a device that provides the N th time derivative of the complex envelope of an arbitrary input optical signal.

Besides the intrinsic interest in future ultrafast optical computing and information processing systems, these devices are also desired for other applications of immediate interest, including optical sensing and control, pulse shaping and ultra-high-speed coding [2]–[4]. As an example, first-order all-optical temporal differentiation can be applied for synthesizing an odd-symmetry Hermite-Gaussian (OS-HG) temporal waveform from an input Gaussian-like pulse [2]–[4]. An OS-HG pulse is orthogonal to any temporally symmetric waveform (including the input Gaussian pulse), which means that the two waveforms can interfere destructively and cancel each other out [4]. This interesting property can be exploited for advanced coding (where each of the two orthogonal waveforms may represent a different code in a transmission link). Even more interesting would be the implementation of arbitrary-order all-optical temporal differentiation of Gaussian pulses leading to the generation of higher-order HG waveforms [6]. HG polynomials constitute a complete family of orthogonal temporal functions [7], i.e. they could be potentially used as a complete family of orthogonal codes. Moreover, any desired temporal shape could be realized as a superposition of HG polynomials, e.g. generated by first and higher-order differentiation of a single Gaussian pulse [7].

Several schemes for performing real-time derivation of arbitrary signals in the optical domain have been previously proposed [2]–[6]. This includes the use of integrated-optic transversal filter topologies [2], fiber grating devices [3]–[5], and concatenated bulk-optics interferometers [6]. Among these alternatives, all-fiber solutions offer several distinctive advantages, including simplicity, relatively low cost, low losses, polarization independence and full compatibility with fiber optics systems. Two different all-fiber solutions have been previously demonstrated for all-optical time differentiation, namely long-period fiber gratings (LPGs) [3], [4] and phase-shifted FBGs [5]. Concerning the first solution, it has been demonstrated that a uniform LPG operating in full-coupling condition implements a first-order time differentiator over arbitrary optical signals. As an important practical advantage, the output signal from the LPG is directly obtained in the core mode (i.e. in the same mode through which the input signal is launched). Even though we have also shown that second-order differentiation can be achieved using a properly designed uniform LPG [3], the fact that the output signal needs to be recovered from the cladding mode poses a significant obstacle towards the practical implementation of this idea. We emphasize also that the LPG approach is specially suited for differentiating ultra-broadband optical waveforms (e.g. picosecond and sub-picosecond optical pulses) but it is extremely inefficient for processing signals with bandwidths narrower than ≈ 100 GHz.

As a complementary all-fiber solution, efficient first-order time differentiation of optical waveforms with bandwidths in the tens-of-GHz regime has been recently achieved using a simple and practical phase-shifted FBG (uniform fiber Bragg grating with a single π-phase shift in the middle of the grating) operating in reflection [5]. An important requirement in this approach is that the signals to be differentiated must be spectrally centered at the grating Bragg wavelength. An attractive feature of any FBG device is that the two coupling modes are confined in the core mode (i.e. the input and output signals are directly obtained in the fiber core).

Thus, all the previous fiber solutions only provide first-order time differentiation. In principle, an N th-order differentiator could be realized by concatenating N single FBG/LPG devices [6]; however, this would translate into an increased implementation complexity and a reduced energetic efficiency. For instance, in the FBG approach, an N th-order time differentiator could be implemented by concatenating N identical phase-shifted FBGs each incorporated in a different optical circulator. Moreover, the N concatenated FBGs should exhibit exactly the same Bragg wavelength, which is a particularly challenging practical requirement. An FBG filter could be designed using well-known synthesis tools (e.g. layer-peeling method or optimization algorithms) to achieve the spectral response corresponding to an N-order time differentiator; however, the application of these general-purpose methods to the problem of optical differentiation typically lead to complex grating profiles that may be difficult to fabricate in practice (see for instance the results in Ref. [8], where Bragg grating designs for synthesizing first and second-order HG optical pulses were presented).

In this communication, we report a new and simple technique for designing feasible FBG-based arbitrary-order optical time differentiators. This technique actually represents a generalization of the previously demonstrated approach for first-order time differentiation based on the use of a single phase-shifted FBG [5]. Specifically, we demonstrate that an Nth-order time differentiator can be implemented using a single FBG structure consisting of a uniform grating profile with N symmetrically located π phase shifts. The design procedure of an FBG-based arbitrary-order optical time differentiator is introduced here for the first time and it is confirmed by numerical simulations. In particular, FBG designs for up to fourth-order time differentiation are presented to illustrate the introduced technique. The resulting grating profiles from this general design method are remarkably simple and can be readily achieved with present FBG technology.

2. FBG-based high-order optical time differentiators: Design examples

Let us consider an optical signal with a carrier frequency ω 0 and a complex envelope e(t) - the spectrum corresponding to this envelope can be represented as E(ω-ω 0), where ω is the optical frequency variable. It can be proved that a signal with envelope of Ne(t)/∂tN (the temporally differentiated signal) has a frequency characteristic given by [j(ω-ω 0)]NE(ω-ω 0), where j = √-1 [9]. Thus, a N th-order temporal differentiator is essentially a linear filtering device providing a spectral transfer function of the form H(ω-ω 0) = [j(ω-ω 0)]N. We anticipate that the required spectral features of a N th-order time differentiator can be provided by the reflection response of a uniform FBG with multiple (N) symmetrically located π-phase shifts along its grating profile. The desired spectral response is achieved over a limited bandwidth around the grating Bragg wavelength, i.e. within the reflection resonance dip. The proposed approach will be first illustrated by designing and simulating a second-order and a third-order time differentiator.

2.1. Theoretical modeling of multiple-phase-shifted FBGs

An arbitrary multiple π-phase-shifted FBG can be modeled using the transfer matrix method combined with coupled-mode theory [11]. Specifically, we remind the reader that the transfer (2×2) matrix T(z0,L) relates the optical fields corresponding to the forward (transmission) EA(z0) and backward (reflection) EB(z0) propagating modes at the FBG input end (z = z 0) with the fields corresponding to these same modes at the FBG output end (z =z0+ L), i.e. EA(z0+L) and EB(z0+L), where L is the grating length:

[EA(z0+L)EB(z0+L)]=T(z0,L)[EA(z0)EB(z0)]=[T11(L)T12(z0,L)T21(z0,L)T22(L)][EA(z0)EB(z0)],

Assuming a uniform FBG, the elements of the corresponding transfer matrix can be obtained by solving the coupled mode equations, considering the following boundary conditions at the grating input and output ends: EA(z0)=1 and EB(z0+L)=0. The analytical expressions of the corresponding transfer matrix elements are as follows [10]:

T11=T22*=[cosh(γL)+jσγsinh(γL)]exp[j(πΛ)L],
T12=T21*=jκγsinh(γL)exp[j(πΛ)(2z0+L)],

where κ is the coupling coefficient, σ = β-π/Λ is the mismatch factor, β is the mode propagation constant, Λ is the grating period, and γ=(κ 2 -σ)1/2. The symbol * denotes complex conjugation.

It is also well known that the elements of the transfer matrix Φ corresponding to a phase shift φ in the grating perturbation are given by the following expressions [10]:

Φ11=exp(jφ2);
Φ22=exp(jφ2);
Φ12=Φ21=0

The total transfer matrix T of an arbitrary FBG profile (e.g. multiple-phase-shifted FBG) can be obtained by multiplying, in the appropriate order, the transfer matrices Tj corresponding to its compound uniform grating sections and the transfer matrices Φ corresponding to the discrete phase shifts along the grating profile. The complex field reflection coefficient, r, and the complex field transmission coefficient, τ of a Bragg grating structure can be found from the elements of its total transfer matrix T using the following expressions [10]:

r=T21T22=rexp(jϕr)
τ=1T22=τexp(jϕτ)

2.2. FBG-based second-order time differentiator

We have recently demonstrated [5] that a reflection phase-shifted FBG, consisting of a uniform Bragg grating with a single π-phase shift (φ = π) at its center (see Fig. 1), provides the spectral features that are required for first-order time differentiation of arbitrary optical signals (linear amplitude spectral response with a complex zero -including a π phase shift– at the filter’s central frequency). The desired spectral response is achieved over a relatively narrow bandwidth (up to a few tens of GHz, depending on the grating length, and the coupling coefficient) centered at the Bragg frequency of the uniform grating profile, i.e. within the grating reflection dip [5]. As a generalization of this idea, we demonstrate here that a FBG structure incorporating two π-phase shifts (see Fig. 2), operated in reflection, can provide the spectral response corresponding to a second-order time differentiator, r(ω-ω 0) ≈ [j(ω-ω 0)]2 (quadratic amplitude spectral response around the filter’s central frequency) over a certain optical bandwidth within the grating reflection resonance dip. As shown in Fig. 2, we have found out that in order to achieve the desired quadratic spectral response, the two required π-phase shifts must be symmetrically located with respect to the grating center.

 

Fig. 1. Structures of Bragg grating - based first-order and second-order differentiators

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This FBG structure can be described by the following matrix product:

T=T(L1+L2,L1)ΦT(L1,L2)ΦT(0,L1)

Using the results in Eq. (2) and Eq. (3), the elements of T can be easily derived as follows:

T21=[2jκγsinh(γL1)(cosh(γL1)cosh(γL2)σ2γ2sinh(γL1)sinh(γL2))jκγsinh(γL2)(κ2+σ2γ2sinh2(γL1)+cosh2(γL1))]exp(jπΛ(2L1+L2))
T22=[jσγsinh(γL2)(κ2γ2σ2γ2cosh(2γL1)jσγsinh(γL1)sinh(2γL1))cosh(γL2)(cosh(2γL1)jσγsinh(2γL1))]exp(jπΛ(2L1+L2))

The complex reflection coefficient can be obtained by introducing the matrix elements given by Eq. (7) into Eq. (4). Each of these matrix elements can be expanded in a Taylor series (as a function of σ), around the Bragg resonance condition (where σ → 0), resulting in the following expressions (in practice, one can use a computational general-purpose tool, such as Symbolic Solver from MathCad, to realize these relatively cumbersome calculations):

T21=jcosh(κL2)(cosh(2κL1)sinh(κL2)sinh(2κL1)cosh(κL2)cosh(κL1))+O(σ2)
T22=cosh(κL2)cosh(2κL1)+jκ(cosh(κL2)sinh(2κL1)+sinh(κL2))σ
+1κ(cosh(κL2)sinh(2κL1)L1+L22sinh(κL2)cosh(2κL1)+1κsinh(κL2)sinh(2κL1))σ2+O(σ3)

We note that in a first approximation, the variable σ is directly proportional to the base-band frequency variable (ω-ω B), where ω is the optical frequency variable and ωB ≈ cπ/neffΛ is the Bragg resonance frequency of the uniform grating (in this notation neff is the effective index of the propagating mode and c is the speed of light in vacuum). In particular, σ = β-π/Λ≈(neff/c)(ω-ωB). We recall that the FBG reflection coefficient can be calculated from Eq. (4) as r = -T 21(1/T 22). From the Taylor series expansions in Eq. (8) and Eq. (9), it can be easily inferred that the reflection coefficient will exhibit a predominantly quadratic dependence with frequency (σ 2), at least over a certain narrow bandwidth around the resonance frequency (i.e. within the reflection resonance dip), if the constant term in the Taylor series expansion of the transfer-matrix element T 21 (see Eq. (8)) is equal to zero. In this case, the first predominant term in the Taylor series expansion of T 21, and thus in the Taylor series expansion of the reflection coefficient r, will be the quadratic term proportional to σ 2 ∝ (ω-ω B)2. The reader can easily verify that this condition is satisfied when L 2 = 2L 1 (see Fig. 1). In this case, the reflection coefficient can be expressed as follows:

r=4jsinh2(κL1)sinh(2κL1)κ2cosh2(2κL1)σ2+O(σ3)

Figure 2 shows the amplitude, |r|, (solid, red curve) and the phase, arg(r), (dashed, blue curve) of the complex reflection coefficient of a FBG structure similar to that depicted in Fig.1 (bottom plot) with L 2 = 2L 1, where L 1 = 1 mm, i.e. total grating length 4L 1 = 4 mm, and coupling coefficient κ = 300π m-1. The Bragg gratings simulated in this paper were assumed to be written in conventional SMF-28 fiber, where the core mode propagation constant was approximated by the following accurate dispersion curve: β(λ)=2π(1.46409528-0.00829634λ-0.00184767λ2)/λ. In all the cases, the grating period was fixed to Λ = 0.5357 μm, which provides resonance Bragg reflection at 193.415 THz (1550 nm). For comparison, the ideal spectral response of a 2nd-order differentiator (parabolic function of frequency) is also shown in Fig. 2 (dashed, magenta curve). As theoretically predicted, the amplitude reflection spectrum of the simulated FBG is very close to the desired ideal quadratic response over a limited bandwidth of ≈12 GHz around the resonance frequency; moreover, the reflection phase is nearly linear over this operation bandwidth.

It is important to note that similarly to any FBG optical filter, the spectral bandwidth of the designed FBG-based optical differentiator (i.e. bandwidth of the resultant reflection resonance notch) can be tailored by properly fixing the total grating length and coupling coefficient. Roughly speaking, the differentiation spectral bandwidth can be increased by shortening the device length or by decreasing the coupling strength [11]. In particular, for a sufficiently low coupling coefficient, the differentiation bandwidth depends only on the grating length (i.e. the differentiation bandwidth is nearly inversely proportional to the device length). Since the proposed FBG designs provide the required spectral characteristics within a narrowband reflection resonance (induced by the grating phase shifts), the operation bandwidth of these differentiators is typically limited to a few GHz. More specifically, our numerical simulations demonstrate that operation bandwidths above 10–20 GHz can be achieved using readily feasible phase-shifted FBG structures (with total grating lengths of only a few millimeters). Such operation bandwidths would allow processing input optical waveforms with time features as fast as 50–100 ps. All the presented FBG designs in this paper have been numerically tested assuming an input 100-ps optical Gaussian pulse; these designs can be easily fabricated in practice.

 

Fig. 2. Amplitude (solid, red curve) and phase (dashed, blue curve) of the reflection spectral response of a FBG with two symmetrical π-phase shifts (parameters given in the text). For comparison, the amplitude spectrum of an ideal second-order differentiator is also represented (dashed, magenta curve).

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As anticipated, the grating structure simulated above (spectral characteristics shown in Fig. 2) could be used to calculate the second time derivative of an arbitrary optical signal launched at the FBG input by a simple reflection in the FBG. The signal to be processed should be spectrally centered at the grating Bragg frequency, ω 0 = ωB (193.415 THz) and its spectrum should lie within the differentiator operation bandwidth (≈ 12 GHz).

 

Fig. 3. Amplitude (solid, red curve) and phase (dashed, blue curve) temporal profiles of the output waveform from the FBG-based second-order differentiator assuming an input 100-ps Gaussian optical pulse. The dashed, magenta curve is the magnitude of the ideal (analytical) second time derivative of the input Gaussian.

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As an example, Fig. 3 shows the amplitude (solid, red curve) and phase (dashed, blue curve) profiles of the numerically simulated temporal response of the designed FBG to an input 100-ps (FWHM) Gaussian optical pulse spectrally centered at the grating Bragg frequency (solid, red curve). Notice that for the phase representation, we subtracted the linear phase term (βt) from the pulse temporal phase profile so that to be able to appreciate clearly the discrete phase jumps. For comparison, the magnitude of the ideal (analytical) second time derivative of the assumed Gaussian pulse is also shown in Fig. 3 (dashed, magenta curve). As expected, the output temporal waveform from the designed FBGs is very close to the second time derivative of the input Gaussian waveform. Specifically, the output time waveform consists of a central peak and two smaller side-lobes that are π-shifted with respect to the main peak. The estimated error or deviation between the output temporal waveform (in amplitude) and the ideal second-order time derivative of the input pulse was ≈ 0.9%. This deviation was estimated as the relative difference between the normalized optical intensities that correspond to the numerically obtained and to the ideal (analytical) temporal derivative evaluated over a temporal window where the signals exhibit nonzero intensity. All the time waveforms shown in this paper (e.g. Fig. 3) are represented as average optical intensities in normalized units. For completeness, we have also estimated the energetic efficiency of the optical differentiation process for each of the simulated cases; this energetic efficiency has been calculated as the ratio between the output signal total energy and the input signal total energy (where the total energy is obtained as the integral of the signal’s average optical intensity, |e(t)|2, over its total time duration). Specifically, the energetic efficiency of the differentiation process simulated above (Fig. 3) is ≈0.86%.

In practice, the FBG filtering response will essentially depend on the accuracy of the location of the π-phase shifts. Specifically, any deviation from the main design condition, L 2/L 1= 2, will translates into a distortion in the obtained output pulse shape. The relative deviation of the obtained output pulse shape from the ideal input pulse time derivative for different values of the ratio L 2/L 1 is quantitatively evaluated in Fig. 4, assuming the same input Gaussian pulse as in Fig. 3. We observed that for L 2/L 1 > 2, the side-lobes decreased very rapidly, practically disappearing at L 2/L 1 ≈ 2.5. For L2/L1 <2, the output pulse distortion was more pronounced and in particular, we observed that the central peak in the temporal waveform gradually decreased with respect to the side-lobes as the L 2/L 1 ratio was decreased until the reflected waveform was reduced to two identical optical pulses. Interestingly, this double-pulse waveform evolved into a nearly flat-top shape for even lower values of the L 2/L 1 ratio.

 

Fig. 4. Relative deviation of the reflected waveform from its ideal shape (second time derivative of the input pulse) as a function of the ratio L2/L1. Optimal operation is achieved for L2/L1 =2.

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We emphasize that the results in Fig. 4 clearly indicate that in order to achieve an accurate operation of the designed optical differentiator, it is crucial to ensure that the π-phase shift is very precisely located along the device length according to the derived design specification. To give a reference, the maximum considered deviation from the main design specification (L 2/L 1= 2) in Fig. 4 is of ±5% and this may already induce very significant errors in the differentiator performance (deviation between the ideal and actually obtained output time waveforms >20%).

2.3. FBG-based third-order time differentiator

Based on our central hypothesis in this work, a third-order time differentiator can be realized using a uniform FBG profile with three π phase-shifts symmetrically located with respect to the grating center. A schematic of such FBG profile is shown in Fig. 5.

 

Fig. 5. Structure of a Bragg grating-based third-order optical differentiator.

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Figure 6 (cyan, solid curve) shows the reflection amplitude spectrum corresponding to the FBG structure of Fig. 5 assuming the same basic condition as for second-order time differentiation, namely L 2/L 1 =2. In these simulations we fixed κ=225πm-1 and L1 = 2/3 mm, which ensures the same total grating length as in the previous FBG design, i.e. 4 mm. It is obvious that the obtained amplitude spectrum considerably differs from the ideal cubic distribution (∝ (ω-ω 0)3) corresponding to a third-order optical differentiator (shown in Fig. 6 with the dashed, magenta curve). Thus the condition obtained above is only valid for the case of second-order differentiation but cannot be generalized for the design of higher-order differentiators. A similar strategy to that outlined above should be used to obtain the required design condition(s) (e.g. required L 2/L 1 ratio) to achieve a third-order optical differentiator using the FBG structure shown in Fig. 5.

The transfer matrix of the structure shown in Fig. 5 is given by the following expression:

T=T(L1+2L2,L1)ΦT(L1+L2,L2)ΦT(L1,L2)ΦT(0,L1)

The individual elements of this transfer matrix can be obtained using the procedure detailed above. The elements that are relevant for the calculation of the reflection coefficient, according to Eq. (4), can in turn be expanded in a Taylor series (as a function of σ) around the resonance Bragg condition (when σ → 0), resulting in the following expressions:

T21=jκ((cosh(κ(L1+L2))cosh(κ(L1L2)))2++(sinh(κ(L1+L2))sinh(κ(L1L2)))22sinh2(κ(L1+L2)))σ+O(σ3)
T22=j+[sinh(2κ(L2L1)+2sinh(2κL1)]σκ+
+j[κL1sinh(2κL1)+2sinh2(κL1)2sinh2(2κL2)++4sinh2(2κL1)sinh2(2κL2)sinh(2κL1)sinh(2κL2)]σ2κ2+O(σ3)

From these two expansions, it can be inferred that the reflection coefficient will be proportional to σ 3 ∝ (ω - ω 0)3 (over a certain narrow bandwidth around the resonance frequency) if the linear component of T21 is made equal to zero. The resultant equality can be reduced to the following expression:

12cosh(2κL1)+cosh(2κ(L2L1)=0

which is actually a condition for the L 2/L 1 ratio, depending on the coupling coefficient -length product κL 1:

α=L2L1=1+cosh1(2cosh(2κL1)1)2κL1

As an example, we assume a FBG structure such as that shown in Fig. 5 with κL 1 = 0.15π (where L 1 =2/3 mm). According to Eq. (15), this structure will behave as a third-order optical differentiator if we fix α = L 2/L 1 = 237. Fig. 6 shows the simulated amplitude (solid, red) and phase (dashed, blue) of the reflection spectrum of this FBG design; as expected, the FBG reflection spectrum approximates very precisely the spectral response of an ideal third-order optical differentiator (cubic dependence with the frequency variable, shown in Fig. 6 using a magenta, dashed curve) over a bandwidth of ≈ 23 GHz around the grating Bragg frequency (193.415 THz). This includes the necessary π phase shift at the filter’s central frequency. Notice that the designed FBG third-order differentiator is slightly longer (2L1+2αL1 = 4.493 mm) than the FBG second-order differentiator presented above. Although in principle, this would translate into a narrower operation bandwidth, this has been compensated by decreasing the coupling coefficient to κ=225π m-1 (as compared with κ =300π1 m-1 used for the second-order differentiator). In this way, the achieved operation bandwidth is almost twice than that of the second-order differentiator design presented above. In general, we have observed that the achievable operation bandwidth in our approach strongly depends on the differentiation order (i.e. different bandwidths can be achieved for different differentiation orders, assuming the same total grating length and coupling coefficient).

 

Fig. 6. Amplitude (solid, red curve) and phase (dashed, blue curve) of the reflection spectral response of a FBG with three π-phase shifts properly located to achieve third-order optical differentiation (parameters given in the text). The solid, cyan curve shows the amplitude reflection spectrum of a FBG with three symmetrically located π-phase shifts designed according to the conditions derived for second-order differentiation. For comparison, the amplitude spectrum of an ideal third-order differentiator is also represented (dashed, magenta curve).

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To confirm the proper operation of the designed FBG, we simulated the propagation of a 100-ps (FWHM) Gaussian optical pulse centered at the FBG Bragg frequency. Figure 7 shows the amplitude (solid, red curve) and phase (dashed, blue curve) temporal profiles of the output waveform after reflection of this input pulse in the designed FBG. The reflected waveform coincides very accurately (0.4% deviation) with the ideal (analytical) third time derivative of the input Gaussian waveform (amplitude shown in Fig. 6 with a dashed, magenta curve), including the expected π phase shifts between adjacent temporal lobes. The energetic efficiency of the signal processing operation shown in Fig. 7 is ≈5.2×10-3%.

 

Fig. 7. Amplitude (solid, red curve) and phase (dashed, blue curve) temporal profiles of the output waveform from the FBG-based third-order differentiator together with the ideal third time derivative (dashed, magenta curve) assuming an input 100-ps Gaussian optical pulse

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As we have just demonstrated, for the FBG-based third order optical differentiator, the ratio α = L 2/L 1 depends on the coupling coefficient – length product κL 1, according to the expression given in Eq. (15). This dependence is plotted in Fig. 8. It is observed that for a relatively weak Bragg grating, the L 2/L 1 ratio is always larger than 2 and asymptotically approaches to 2 as the grating gets stronger.

 

Fig. 8. Graphical representation of the main design condition (Eq. (15)) of a FBG-based third-order optical differentiator.

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3. General design approach for FBG-based high-order optical differentiation.

3.1. General design strategy

From the specific design examples considered above (second-order and third-order optical differentiation), one can easily formulate a general design principle for FBG-based arbitrary-order optical differentiators. In what follows, we outline the strategy to be followed to design an N th-order all-optical time differentiator based on a multiple-phase-shifted FBG structure. As anticipated, such differentiator can be implemented using a FBG structure based on a uniform grating profile incorporating N π-phase shifts, which should be symmetrically located with respect to the FBG center. In all the cases, the FBG consists of (N+1) uniform grating sections of identical coupling strength and period but of different lengths; each two consecutive uniform sections need to be π phase shifted and the structure should be symmetric with respect to the grating center. Our goal is to determine the lengths of the different uniform grating sections so that to achieve the desired reflection spectral response r(ω-ω 0) ∝ [j(ω-ω 0)]N. Due to the symmetry of the FBG structure, the number of length variables to be fixed is reduced to (N/2)+1 when N is an even number, whereas only (N+1)/2 different length variables need to be determined when N is an odd number.

As detailed above, the field reflection coefficient of the considered multiple-phase-shifted FBG structure can be calculated from the elements T 21 and T 22 of the corresponding transfer-matrix, i.e. r = -T 21(1/T 22). Each of these elements can be expressed as a function of the mismatch frequency factor (σ) in the form of a Tailor series expansion around the grating resonance condition (σ → 0). In general, the element T 22 has contributions from all the Taylor series terms, i.e. in general, (1T22)n=0Anσn, where A n is the Taylor series coefficient of order n and where the constant term is always present, A 0 ≠ 0. In contrast, the Taylor series expansion of the element T 21 strongly depends on the parity of N. Specifically, in the case of an even-order differentiator, i.e. FBG with an even number N of π phase shifts, only the even terms of the corresponding Taylor series (including the constant term, of order n = 0) are present (the odd terms are zero):

T21n=0(N2)1F2nL1L(N2)+1σ2n+O(σN)forevenN

Similarly, in the case of a FBG-based odd-order differentiator (N odd), only the odd terms of the Taylor series expansion are present:

T21n=0(N3)2F2n+1L1L(N+1)2σ2n+1+O(σN)foroddN

In the previous equations, Fm are the Taylor series coefficients given by Fm = [m(T 21)/∂σm]σ→0. As indicated by Eq. (16) and Eq. (17), each of these Taylor series coefficients depends on the grating length variables to be fixed in the design problem. The condition(s) that these length variables need to satisfy in order to ensure that the FBG structure provides the spectral features corresponding to an N th-order optical differentiator can be derived by solving the system of trigonometric equations that result from application of the following set of equalities: F0 = 0, F2 = 0 ,…, F N-2 = 0 (for N even) or F1 = 0, F3 = 0 ,…, F N-2 = 0 (for N odd). These set of equalities ensure that the first sum term in Eq. (16) [or in Eq. (17)] is equal to zero, i.e. T 21σN + O(σ N+2), which means that as required for N th-order optical differentiation, the FBG reflection transfer function depends predominantly on the N th power of the optical frequency variable over a relatively narrow bandwidth around the grating Bragg frequency (where σ → 0), i.e. r = -T 21(1/T 22)∝ σN + O(σ N+1).

It is easy to observe that the number of resultant independent equations from application of the set of equalities introduced above is always equal to the number of length variables to be determined minus one. Specifically, for N even, the number of independent equations is equal to N/2 while the number of length variables to be determined is equal to (N/2)+1 (e.g. for N = 2, we have 1 independent equation and 2 length variables, see example above); similarly, when N is odd, the number of independent equations and length variables are given by (N- 1)/2 and (N+1)/2, respectively (e.g. for N = 3, we have 1 independent equation and 2 length variables, see example above). Thus, the resultant set of equations can be always solved as a function of one length variable (e.g. L 1 in the examples shown above); obviously, this offers an important additional degree of flexibility in the design stage as this length variable can be freely fixed according to the desired filter features (e.g. desired operation bandwidth). Finally, it should be also mentioned that the resultant set of equations may become more complicated as the differentiation order N increases, thus eventually requiring a numerical (instead of a fully analytical) solution.

3.2. Design example: FBG-based fourth-order optical differentiator.

To illustrate further the general FBG design approach sketched above, this method is now applied to the design of a fourth-order time differentiator. The required FBG structure consists of five uniform grating sections of identical coupling strength and period, separated by four π phase shifts. Thus, in this structure there are 3 different grating lengths to be determined. The total transfer matrix of this structure can be obtained using the general procedure described above:

T=T(L1+2L2+L3,L1)ΦT(L1+L2+L3,L2)ΦT(L1+L2,L3)ΦT(L1,L2)ΦT(0,L1)

The transfer matrix element T 21 can be obtained from the solution of Eq. (18). As anticipated above, the Taylor coefficients F 1 and F 3 of this transfer-matrix element are equal to zero. Thus in order to ensure that the FBG reflection spectral response is proportional to the factor σ 4, the grating lengths should be fixed to ensure that the constant (F 0) and quadratic terms (F 2) of the Taylor series expansion of T 21 are also equal to zero. As expected, this translates into a system of 2 independent equations with 3 length variables. Specifically, the reader can verify that the equation F 0(L 1, L 2, L 3) = 0 leads to the following simplified condition:

η=L3L1=2(α1)

where we recall that α = L 2/L 1. A second condition governing the relationship between L 2 and L 1 can be found from the second equation F 2(L 1, L 2, L 3) = 0. In practice, this equation is too cumbersome to be solved analytically. However, the solution (i.e. value of α that satisfies this second equation) can be easily found using numerical simulations. Specifically, we have found out that if we fix κL 1 = 0.15π, then the optimal L 2/L 1 ratio is α=2.522. The reflection spectrum of the FBG that satisfies Eq. (19) with L 1 =0.5 mm and α=2.522 (total length L 1(2+2α+η)=5.044 mm) is shown in Fig. 9 (amplitude spectrum shown with the solid, red curve, and phase spectrum shown with the dashed, blue curve); the obtained response approximates very precisely the ideal spectral response of a fourth-order optical differentiator (ideal amplitude shown in Fig. 9 with the dashed, magenta curve), including a nearly ideal linear phase profile, over a bandwidth of ≈25 GHz. This was confirmed by simulating the grating reflection response to an input 100-ps (FWHM) Gaussian optical pulse, centered at the FBG resonance frequency. The resultant amplitude temporal waveform is shown in Fig. 10 (solid, red curve); there is an excellent agreement between the obtained temporal shape at the FBG output and the ideal (analytical) fourth time derivative of the input Gaussian waveform (dashed, magenta curve). The energetic efficiency of the differentiation process shown in Fig. 10 is ≈5×10-4%. As in the two previously shown examples, the dashed, blue curve in Fig. 10 represents the phase temporal profile of the output pulse; as expected, the obtained temporal lobes in the output waveform are phase shifted by π with respect to each other.

 

Fig. 9. Amplitude (solid, red curve) and phase (dashed, blue curve) of the reflection spectral response of a FBG with four symmetrical π-phase shifts (parameters given in the text). For comparison, the amplitude spectrum of an ideal fourth-order differentiator is also represented (dashed, magenta curve).

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Fig. 10. Amplitude (solid, red curve) and phase (dashed, blue curve) temporal profiles of the output waveform from the FBG-based fourth-order differentiator assuming an input 100-ps Gaussian optical pulse. The dashed, magenta curve represents the magnitude of the ideal (analytical) fourth time derivative of the input Gaussian pulses.

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3. Conclusion

We have proposed and numerically demonstrated a simple and general approach for implementing arbitrary-order time differentiation of optical waveforms. The basis of this general approach can be found in our previous experimental realization of a first-order optical differentiator based on a single π phase-shifted FBG [5]. Specifically, in this paper, we have shown that the N th time derivative of an input optical waveform can be obtained by simple reflection of this waveform in a single uniform FBG incorporating N symmetrically located π phase shifts. We have described a general design strategy to determine the proper location of the N phase shifts to be introduced in a FBG structure so as to implement a N th-order time differentiator. This general strategy has been illustrated by designing second-order, third-order and fourth-order optical differentiators, which have been numerically tested using input Gaussian optical pulses. We have shown that this approach can provide optical operation bandwidths in the tens-of-GHz regime, well beyond the reach of present electronic technologies, using readily feasible FBG devices.

We emphasize that the proposed approach is based on FBG technology and thus provides the inherent advantages of an all-fiber method. Moreover, the required FBG devices, i.e. multiple-phase-shifted gratings, are remarkably simple and can be practically implemented using well-established FBG fabrication methods. Specifically, these devices can be fabricated using the conventional photo-inscription method based on the use of a uniform phase-mask, where the required π-phase shifts can be accurately introduced at the desired locations by displacement of the phase mask relative to the photosensitive fiber using a piezoelectric actuator (this was the specific method employed for fabrication of the recently reported FBG-based first-order differentiator [5]). Alternatively, the π-phase shifts could be directly imprinted in the phase-mask [12]. We also anticipate that it should be relatively straightforward to transfer this same concept for implementation in integrated optics platforms. Finally, it should be also noted that as discussed above for the second-order differentiator case, the proper performance of a practically fabricated device depends critically on the precise location of the π phase shifts along the FBG length (i.e. the ratios between the lengths of the different uniform grating sections in the FBG structure must be fixed with a very high accuracy), according to the derived design specifications. Obviously, this requirement becomes more challenging as the differentiation order increases since a larger number of π phase shifts need to be accurately introduced in the grating profile.

All-optical time differentiators are intrinsically interesting as basic building blocks in ultrafast information processors; other applications of more immediate interest for first and higher-order all-optical time differentiators include optical sensing and control, ultrafast coding, optical pulse shaping and ultra-wideband microwave signal generation and processing.

Acknowledgments

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).

References and links

1. J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti, edts., special issue on “Optical Signal Processing,” in J. Lightwave Technol. 24, 2484–2767 (2006). [CrossRef]  

2. N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230, 115–129 (2004). [CrossRef]  

3. M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. 30, 2700–2702 (2005). [CrossRef]   [PubMed]  

4. R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express 14, 10699–10707 (2006). [CrossRef]   [PubMed]  

5. N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Express 15, 371–381 (2007). [CrossRef]   [PubMed]  

6. Y. Park, J. Azaña, and R. Slavík, “Ultrafast all-optical first and higher-order differentiators based on interferometers,” Opt. Lett. 32, 710–712 (2007). [CrossRef]   [PubMed]  

7. H. J. A. Da Silva and J. J. O’Reilly, “Optical pulse modeling with Hermite - Gaussian functions,” Opt. Lett. 14, 526–528 (1989). [CrossRef]   [PubMed]  

8. C. Wu and M. G. Raymer, “Efficient picosecond pulse shaping by programmable Bragg gratings,” IEEE J. Quantum Electron. 42, 871–882 (2006). [CrossRef]  

9. A. Papoulis, Fourier Integral and its Applications, (McGraw-Hill, New York, 1987).

10. R. Kashyap, Fiber Bragg Gratings, (Academic Press, San Diego, 1999).

11. T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15, 1277–1294 (1997). [CrossRef]  

12. J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Damman fiber Bragg gratings and phase-only sampling for high-channel counts,” IEEE Photon. Technol. Lett. 14, 1309 – 1311 (2002). [CrossRef]  

References

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  1. J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti, eds., Special issue on "Optical Signal Processing," in J. Lightwave Technol. 24, 2484-2767 (2006).
    [CrossRef]
  2. N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, "A new theoretical basis of higher-derivative optical differentiators," Opt. Commun. 230, 115-129 (2004).
    [CrossRef]
  3. M. Kulishov and J. Azaña, "Long-period fiber gratings as ultrafast optical differentiators," Opt. Lett. 30, 2700-2702 (2005).
    [CrossRef] [PubMed]
  4. R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, "Ultrafast all-optical differentiators," Opt. Express 14, 10699-10707 (2006).
    [CrossRef] [PubMed]
  5. N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, "Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating," Opt. Express 15, 371-381 (2007).
    [CrossRef] [PubMed]
  6. Y. Park, J. Azaña, and R. Slavík, "Ultrafast all-optical first and higher-order differentiators based on interferometers," Opt. Lett. 32, 710-712 (2007).
    [CrossRef] [PubMed]
  7. H. J. A. Da Silva, and J. J. O’Reilly, "Optical pulse modeling with Hermite - Gaussian functions," Opt. Lett. 14, 526-528 (1989).
    [CrossRef] [PubMed]
  8. C. Wu and M. G. Raymer, "Efficient picosecond pulse shaping by programmable Bragg gratings," IEEE J. Quantum Electron. 42, 871-882 (2006).
    [CrossRef]
  9. A. Papoulis, Fourier Integral and its Applications, (McGraw-Hill, New York, 1987).
  10. R. Kashyap, Fiber Bragg Gratings, (Academic Press, San Diego, 1999).
  11. T. Erdogan, "Fiber Grating Spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
    [CrossRef]
  12. J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
    [CrossRef]

2007 (2)

2006 (2)

C. Wu and M. G. Raymer, "Efficient picosecond pulse shaping by programmable Bragg gratings," IEEE J. Quantum Electron. 42, 871-882 (2006).
[CrossRef]

R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, "Ultrafast all-optical differentiators," Opt. Express 14, 10699-10707 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (1)

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, "A new theoretical basis of higher-derivative optical differentiators," Opt. Commun. 230, 115-129 (2004).
[CrossRef]

2002 (1)

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

1997 (1)

T. Erdogan, "Fiber Grating Spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

1989 (1)

Azaña, J.

Berger, N. K.

Da Silva, H. J. A.

Erdogan, T.

T. Erdogan, "Fiber Grating Spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

Fischer, B.

Kam, C.H.

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, "A new theoretical basis of higher-derivative optical differentiators," Opt. Commun. 230, 115-129 (2004).
[CrossRef]

Kulishov, M.

Levit, B.

Li, H.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

Li, Y.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

Morandotti, R.

Ngo, N. Q.

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, "A new theoretical basis of higher-derivative optical differentiators," Opt. Commun. 230, 115-129 (2004).
[CrossRef]

O’Reilly, J. J.

Park, Y.

Plant, D. V.

Popelek, J.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

Raymer, M. G.

C. Wu and M. G. Raymer, "Efficient picosecond pulse shaping by programmable Bragg gratings," IEEE J. Quantum Electron. 42, 871-882 (2006).
[CrossRef]

Rothenberg, J. E.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

Sheng, Y

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

Slavík, R.

Tjin, S. C.

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, "A new theoretical basis of higher-derivative optical differentiators," Opt. Commun. 230, 115-129 (2004).
[CrossRef]

Wang, Y.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

Wilcox, R. B.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

Wu, C.

C. Wu and M. G. Raymer, "Efficient picosecond pulse shaping by programmable Bragg gratings," IEEE J. Quantum Electron. 42, 871-882 (2006).
[CrossRef]

Yu, S. F.

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, "A new theoretical basis of higher-derivative optical differentiators," Opt. Commun. 230, 115-129 (2004).
[CrossRef]

Zweiback, J.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. Wu and M. G. Raymer, "Efficient picosecond pulse shaping by programmable Bragg gratings," IEEE J. Quantum Electron. 42, 871-882 (2006).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, "Damman fiber Bragg gratings and phase-only sampling for high-channel counts," IEEE Photon. Technol. Lett. 14, 1309 - 1311 (2002).
[CrossRef]

J. Lightwave Technol. (1)

T. Erdogan, "Fiber Grating Spectra," J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

Opt. Commun. (1)

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, "A new theoretical basis of higher-derivative optical differentiators," Opt. Commun. 230, 115-129 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Other (3)

A. Papoulis, Fourier Integral and its Applications, (McGraw-Hill, New York, 1987).

R. Kashyap, Fiber Bragg Gratings, (Academic Press, San Diego, 1999).

J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti, eds., Special issue on "Optical Signal Processing," in J. Lightwave Technol. 24, 2484-2767 (2006).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

Structures of Bragg grating - based first-order and second-order differentiators

Fig. 2.
Fig. 2.

Amplitude (solid, red curve) and phase (dashed, blue curve) of the reflection spectral response of a FBG with two symmetrical π-phase shifts (parameters given in the text). For comparison, the amplitude spectrum of an ideal second-order differentiator is also represented (dashed, magenta curve).

Fig. 3.
Fig. 3.

Amplitude (solid, red curve) and phase (dashed, blue curve) temporal profiles of the output waveform from the FBG-based second-order differentiator assuming an input 100-ps Gaussian optical pulse. The dashed, magenta curve is the magnitude of the ideal (analytical) second time derivative of the input Gaussian.

Fig. 4.
Fig. 4.

Relative deviation of the reflected waveform from its ideal shape (second time derivative of the input pulse) as a function of the ratio L2 /L1 . Optimal operation is achieved for L2 /L1 =2.

Fig. 5.
Fig. 5.

Structure of a Bragg grating-based third-order optical differentiator.

Fig. 6.
Fig. 6.

Amplitude (solid, red curve) and phase (dashed, blue curve) of the reflection spectral response of a FBG with three π-phase shifts properly located to achieve third-order optical differentiation (parameters given in the text). The solid, cyan curve shows the amplitude reflection spectrum of a FBG with three symmetrically located π-phase shifts designed according to the conditions derived for second-order differentiation. For comparison, the amplitude spectrum of an ideal third-order differentiator is also represented (dashed, magenta curve).

Fig. 7.
Fig. 7.

Amplitude (solid, red curve) and phase (dashed, blue curve) temporal profiles of the output waveform from the FBG-based third-order differentiator together with the ideal third time derivative (dashed, magenta curve) assuming an input 100-ps Gaussian optical pulse

Fig. 8.
Fig. 8.

Graphical representation of the main design condition (Eq. (15)) of a FBG-based third-order optical differentiator.

Fig. 9.
Fig. 9.

Amplitude (solid, red curve) and phase (dashed, blue curve) of the reflection spectral response of a FBG with four symmetrical π-phase shifts (parameters given in the text). For comparison, the amplitude spectrum of an ideal fourth-order differentiator is also represented (dashed, magenta curve).

Fig. 10.
Fig. 10.

Amplitude (solid, red curve) and phase (dashed, blue curve) temporal profiles of the output waveform from the FBG-based fourth-order differentiator assuming an input 100-ps Gaussian optical pulse. The dashed, magenta curve represents the magnitude of the ideal (analytical) fourth time derivative of the input Gaussian pulses.

Equations (25)

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[ E A ( z 0 + L ) E B ( z 0 + L ) ] = T ( z 0 , L ) [ E A ( z 0 ) E B ( z 0 ) ] = [ T 11 ( L ) T 12 ( z 0 , L ) T 21 ( z 0 , L ) T 22 ( L ) ] [ E A ( z 0 ) E B ( z 0 ) ] ,
T 11 = T 22 * = [ cosh ( γ L ) + j σ γ sinh ( γ L ) ] exp [ j ( π Λ ) L ] ,
T 12 = T 21 * = j κ γ sinh ( γ L ) exp [ j ( π Λ ) ( 2 z 0 + L ) ] ,
Φ 11 = exp ( j φ 2 ) ;
Φ 22 = exp ( j φ 2 ) ;
Φ 12 = Φ 21 = 0
r = T 21 T 22 = r exp ( j ϕ r )
τ = 1 T 22 = τ exp ( j ϕ τ )
T = T ( L 1 + L 2 , L 1 ) ΦT ( L 1 , L 2 ) ΦT ( 0 , L 1 )
T 21 = [ 2 j κ γ sinh ( γ L 1 ) ( cosh ( γ L 1 ) cosh ( γ L 2 ) σ 2 γ 2 sinh ( γ L 1 ) sinh ( γ L 2 ) ) j κ γ sinh ( γ L 2 ) ( κ 2 + σ 2 γ 2 sinh 2 ( γ L 1 ) + cosh 2 ( γ L 1 ) ) ] exp ( j π Λ ( 2 L 1 + L 2 ) )
T 22 = [ j σ γ sinh ( γ L 2 ) ( κ 2 γ 2 σ 2 γ 2 cosh ( 2 γ L 1 ) j σ γ sinh ( γ L 1 ) sinh ( 2 γ L 1 ) ) cosh ( γ L 2 ) ( cosh ( 2 γ L 1 ) j σ γ sinh ( 2 γ L 1 ) ) ] exp ( j π Λ ( 2 L 1 + L 2 ) )
T 21 = j cosh ( κ L 2 ) ( cosh ( 2 κ L 1 ) sinh ( κ L 2 ) sinh ( 2 κ L 1 ) cosh ( κ L 2 ) cosh ( κ L 1 ) ) + O ( σ 2 )
T 22 = cosh ( κ L 2 ) cosh ( 2 κ L 1 ) + j κ ( cosh ( κ L 2 ) sinh ( 2 κ L 1 ) + sinh ( κ L 2 ) ) σ
+ 1 κ ( cosh ( κ L 2 ) sinh ( 2 κ L 1 ) L 1 + L 2 2 sinh ( κ L 2 ) cosh ( 2 κ L 1 ) + 1 κ sinh ( κ L 2 ) sinh ( 2 κ L 1 ) ) σ 2 + O ( σ 3 )
r = 4 j sinh 2 ( κ L 1 ) sinh ( 2 κ L 1 ) κ 2 cosh 2 ( 2 κ L 1 ) σ 2 + O ( σ 3 )
T = T ( L 1 + 2 L 2 , L 1 ) Φ T ( L 1 + L 2 , L 2 ) Φ T ( L 1 , L 2 ) Φ T ( 0 , L 1 )
T 21 = j κ ( ( cosh ( κ ( L 1 + L 2 ) ) cosh ( κ ( L 1 L 2 ) ) ) 2 + + ( sinh ( κ ( L 1 + L 2 ) ) sinh ( κ ( L 1 L 2 ) ) ) 2 2 sinh 2 ( κ ( L 1 + L 2 ) ) ) σ + O ( σ 3 )
T 22 = j + [ sinh ( 2 κ ( L 2 L 1 ) + 2 sinh ( 2 κ L 1 ) ] σ κ +
+ j [ κ L 1 sinh ( 2 κ L 1 ) + 2 sinh 2 ( κ L 1 ) 2 sinh 2 ( 2 κ L 2 ) + + 4 sinh 2 ( 2 κ L 1 ) sinh 2 ( 2 κ L 2 ) sinh ( 2 κ L 1 ) sinh ( 2 κ L 2 ) ] σ 2 κ 2 + O ( σ 3 )
1 2 cosh ( 2 κ L 1 ) + cosh ( 2 κ ( L 2 L 1 ) = 0
α = L 2 L 1 = 1 + cosh 1 ( 2 cosh ( 2 κ L 1 ) 1 ) 2 κ L 1
T 21 n = 0 ( N 2 ) 1 F 2 n L 1 L ( N 2 ) + 1 σ 2 n + O ( σ N ) for even N
T 21 n = 0 ( N 3 ) 2 F 2 n + 1 L 1 L ( N + 1 ) 2 σ 2 n + 1 + O ( σ N ) for odd N
T = T ( L 1 + 2 L 2 + L 3 , L 1 ) Φ T ( L 1 + L 2 + L 3 , L 2 ) Φ T ( L 1 + L 2 , L 3 ) Φ T ( L 1 , L 2 ) Φ T ( 0 , L 1 )
η = L 3 L 1 = 2 ( α 1 )

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