## Design of high-order all-optical temporal differentiators based on multiple-phase-shifted fiber Bragg gratings

Optics Express, Vol. 15, Issue 10, pp. 6152-6166 (2007)

http://dx.doi.org/10.1364/OE.15.006152

Acrobat PDF (263 KB)

### 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

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]

*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.

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]

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]

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

*complete*family of orthogonal temporal functions [7

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]

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]

**230**, 115–129 (2004). [CrossRef]

**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]

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]

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]

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

*N*

^{th}-order differentiator could be realized by concatenating

*N*single FBG/LPG devices [6

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]

*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

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

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]

*N*th-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

*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

*∂*(

^{N}e*t*)/

*∂t*(the

^{N}*temporally differentiated signal*) has a frequency characteristic given by [

*j*(

*ω*-

*ω*

_{0})]

*(*

^{N}E*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

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

**T**(

*z*) relates the optical fields corresponding to the forward (transmission)

_{0},L*E*and backward (reflection)

_{A}(z_{0})*E*propagating modes at the FBG input end (

_{B}(z_{0})*z*=

*z*

_{0}) with the fields corresponding to these same modes at the FBG output end (

*z*=

*z*+

_{0}*L*), i.e.

*E*and

_{A}(z_{0}+L)*E*, where

_{B}(z_{0}+L)*L*is the grating length:

*E*=

_{A}(z_{0})*1*and

*E*=

_{B}(z_{0}+L)*0*. The analytical expressions of the corresponding transfer matrix elements are as follows [10]:

*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.

**Φ**corresponding to a phase shift

*φ*in the grating perturbation are given by the following expressions [10]:

**T**of an arbitrary FBG profile (e.g. multiple-phase-shifted FBG) can be obtained by multiplying, in the appropriate order, the transfer matrices

_{∑}**T**corresponding to its compound uniform grating sections and the transfer matrices

_{j}**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]:

_{∑}### 2.2. FBG-based second-order time differentiator

**15**, 371–381 (2007). [CrossRef] [PubMed]

*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

**15**, 371–381 (2007). [CrossRef] [PubMed]

*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.

*σ*is directly proportional to the base-band frequency variable (

*ω*-

*ω*

_{B}), where

*ω*is the optical frequency variable and

*ω*≈ cπ/

_{B}*n*Λ is the Bragg resonance frequency of the uniform grating (in this notation

_{eff}*n*is the effective index of the propagating mode and

_{eff}*c*is the speed of light in vacuum). In particular,

*σ*=

*β*-

*π*/Λ≈(

*n*/

_{eff}*c*)(

*ω*-

*ω*). We recall that the FBG reflection coefficient can be calculated from Eq. (4) as

_{B}*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}= 2

*L*

_{1}(see Fig. 1). In this case, the reflection coefficient can be expressed as follows:

*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}= 2

*L*

_{1}, where

*L*

_{1}= 1 mm, i.e. total grating length 4

*L*

_{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-0.00184767λ

^{2})/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 2

^{nd}-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.

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

*ω*

_{0}=

*ω*(193.415 THz) and its spectrum should lie within the differentiator operation bandwidth (≈ 12 GHz).

_{B}*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%.

*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

*L*/

_{2}*L*<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

_{1}*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.

*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

*L*

_{2}/

*L*

_{1}=2. In these simulations we fixed

*κ*=

*225π*m

^{-1}and

*L*= 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 (∝ (

_{1}*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.

**σ**) around the resonance Bragg condition (when

**σ**→ 0), resulting in the following expressions:

**σ**

^{3}∝ (

*ω*-

*ω*

_{0})

^{3}(over a certain narrow bandwidth around the resonance frequency) if the linear component of

*T*is made equal to zero. The resultant equality can be reduced to the following expression:

^{∑}_{21}*L*

_{2}/

*L*

_{1}ratio, depending on the coupling coefficient -length product

*κL*

_{1}:

*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 (2

*L*+2

_{1}*αL*= 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

_{1}**=225π m**

*κ*^{-1}(as compared with

**=300π1 m**

*κ*^{-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).

^{-3}%.

*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.

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

### 3.1. General design strategy

*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.

*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 (

**→ 0). In general, the element**

*σ**T*

^{∑}

_{22}has contributions from all the Taylor series terms, i.e. in general,

*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):

*N*odd), only the odd terms of the Taylor series expansion are present:

*F*are the Taylor series coefficients given by

_{m}*F*= [

_{m}*∂*(

^{m}*T*

^{∑}

_{21})/

*∂σ*]

^{m}_{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:

*F*= 0,

_{0}*F*= 0 ,…,

_{2}*F*

_{N-2}= 0 (for

*N*even) or

*F*= 0,

_{1}*F*= 0 ,…,

_{3}*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*

^{∑}

_{21}∝

*σ*+

^{N}*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})∝

*σ*+

^{N}*O*(

*σ*

^{N+1}).

*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.

*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:

*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.

## 3. Conclusion

*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

**15**, 371–381 (2007). [CrossRef] [PubMed]

*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.

**15**, 371–381 (2007). [CrossRef] [PubMed]

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]

## Acknowledgments

## 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. |

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. |

3. | M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. |

4. | R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express |

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 |

6. | Y. Park, J. Azaña, and R. Slavík, “Ultrafast all-optical first and higher-order differentiators based on interferometers,” Opt. Lett. |

7. | H. J. A. Da Silva and J. J. O’Reilly, “Optical pulse modeling with Hermite - Gaussian functions,” Opt. Lett. |

8. | C. Wu and M. G. Raymer, “Efficient picosecond pulse shaping by programmable Bragg gratings,” IEEE J. Quantum Electron. |

9. | A. Papoulis, |

10. | R. Kashyap, |

11. | T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. |

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. |

**OCIS Codes**

(060.2340) Fiber optics and optical communications : Fiber optics components

(200.3050) Optics in computing : Information processing

(230.1150) Optical devices : All-optical devices

(320.5540) Ultrafast optics : Pulse shaping

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: February 16, 2007

Revised Manuscript: April 29, 2007

Manuscript Accepted: May 2, 2007

Published: May 3, 2007

**Citation**

Mykola Kulishov and José Azaña, "Design of high-order all-optical temporal differentiators based on multiple-phase-shifted fiber Bragg gratings," Opt. Express **15**, 6152-6166 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6152

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### References

- 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]
- 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]
- M. Kulishov and J. Azaña, "Long-period fiber gratings as ultrafast optical differentiators," Opt. Lett. 30, 2700-2702 (2005). [CrossRef] [PubMed]
- R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, "Ultrafast all-optical differentiators," Opt. Express 14, 10699-10707 (2006). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- C. Wu and M. G. Raymer, "Efficient picosecond pulse shaping by programmable Bragg gratings," IEEE J. Quantum Electron. 42, 871-882 (2006). [CrossRef]
- A. Papoulis, Fourier Integral and its Applications, (McGraw-Hill, New York, 1987).
- R. Kashyap, Fiber Bragg Gratings, (Academic Press, San Diego, 1999).
- T. Erdogan, "Fiber Grating Spectra," J. Lightwave Technol. 15, 1277-1294 (1997). [CrossRef]
- 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]

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