Optical realization of Newton-cotes-based integrators for dark soliton generation

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 1, JANUARY 2006 563

Optical Realization of Newton–Cotes-BasedIntegrators for Dark Soliton Generation

Nam Quoc Ngo and Le Nguyen Binh

Abstract—An optical integrator is an analog optical signalprocessor that performs the time integral of an input optical signal.This paper presents a theory of Newton–Cotes optical integra-tors for high-speed optical signal processing. The Newton–Cotesoptical integrators are designed using the cascade of a finiteimpulse response (FIR) optical waveguide filter with an infiniteimpulse response (IIR) optical waveguide filter. To demonstratethe effectiveness of the proposed optical integrators, the authorsshow, by means of computer simulations, that a trapezoidal opticalintegrator can generate a fundamental dark-soliton pulse thatcan propagate stably over a large distance of single-mode opticalfiber. Although the analysis is directed at optical integrators im-plemented using waveguide technology, the theory, design method,and results are applicable to other physical systems such as opticalsystems based on free-space optics, fiber optics, and fiber gratings.

Index Terms—Optical filter, optical integrator, optical signalprocessor, optical waveguide, pulse shaping.

I. INTRODUCTION

U LTRAFAST optical signal processing at 1.55 µm, whichis difficult to achieve by traditional electronic techniques,

is important in the next generation of high-speed optical net-works operating at a data rate greater than 100 Gb/s. Ultrafastoptical signal processing that can perform pulse shaping andpulse pattern generation and recognition has recently beendemonstrated using a time-space-conversion technology basedon free-space optics in the visible wavelength region [1], [2].Pulse shaping using integrated optical devices based on arrayedwaveguide gratings [3], [4] and waveguide transversal filter [5]has also been demonstrated. In this paper, we propose a newtype of high-speed optical signal processor, namely, an opticaltemporal integrator.

An optical temporal integrator may be defined as an analogoptical signal processor that performs the time integral of acontinuous-time optical signal. An optical integrator approxi-mates the integral of a continuous-time optical signal x(t) bysampling or processing the optical signal at the discrete timet = nT , where n is an integer and T is the sampling period ofthe optical integrator. That is, the output of the optical integratorcan be defined as

y(t)|t=nT =

nT∫0

x(t) dt. (1)

Manuscript received June 23, 2005; revised August 15, 2005.N. Q. Ngo is with the Photonics Research Centre, School of Electrical and

Electronic Engineering, Nanyang Technological University, Singapore 639798,Singapore (e-mail: [email protected]).

L. N. Binh is with the Department of Electrical and Computer SystemsEngineering, Monash University, Clayton, Vic. 3168, Australia.

Digital Object Identifier 10.1109/JLT.2005.859842

The spectral response of an ideal optical integrator isgiven by

HI(ω) =

{1

jωT , 0 ≤ ωT(2π) ≤ 1

21

j(2π−ωT ) ,12 <

ωT(2π) ≤ 1

(2)

where j =√−1 and ω is the angular optical frequency of

the lightwave. It should be noted that, in the field of digitalsignal processing, digital integrators, which have been studiedin the last several decades, have been used in the design ofcompensators for control systems [6] and for measuring thecardiac output of the heart [7]. However, optical integrator isstill a new concept in the field of optics, especially in the areaof optical signal processing with many potential applications.An optical integrator can form an integral part of or can be usedas a basic building block of many ultrafast all-optical signalprocessing systems, because the time integral of signals maybe required for further use or analysis. An optical integratormay also be used for shaping of optical pulses or in an opticalfeedback control system. It is for this reason that this paperpresents a new theory of optical integrators for the processing ofultrafast optical pulses. As an example of the application of theproposed optical integrators, an optical integrator (trapezoidaltype) can modify the shape of a bright-soliton pulse withamplitude sech2(t) into a fundamental dark-soliton pulse withamplitude tanh(t).

The paper is organized as follows. Section II presents ageneralized theory of the Newton–Cotes digital integratorswhose derivation is given in the Appendix. Section III describesthe design of the Newton–Cotes optical integrators, whichconsist of the cascade of a finite impulse response (FIR) opticalwaveguide filter with an infinite impulse response (IIR) opticalwaveguide filter. Section IV presents the frequency responsesof the proposed Newton–Cotes optical integrators and describesthe application of the trapezoidal optical integrator as an opticaldark-soliton generator. Conclusion is given in Section V. Theproposed optical integrators are coherent optical signal proces-sors that process both the amplitude and phase of the inputoptical signal. Thus, the coherence time of the input opticalsignal must be much larger than the sampling period T ofthe optical integrators, and the optical source must be highlycoherent.

II. GENERALIZED THEORY OF THE NEWTON–COTES

DIGITAL INTEGRATORS

The derivation of a generalized theory of the Newton–Cotesdigital integrators presented here is given in the Appendix.

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564 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 1, JANUARY 2006

TABLE ICOMPUTED TAP COEFFICIENTS OF SEVERAL FAMILIES OF THE NEWTON–COTES DIGITAL INTEGRATORS USING (3)–(5).

WITHOUT LOSS OF GENERALITY, T = 1 HAS BEEN USED IN (3) AND (6) FOR THE SAKE OF SIMPLICITY

The transfer function of the pth-order Newton–Cotes digitalintegrator can be generally expressed as

Hmp(z)

=T[

11− z−p

]×

[m∑

i=0

Ci(p)∆iD(z)

]

=T[

11− z−p

]×

[C0(p)+C1(p)∆D(z)+C2(p)∆2D(z)

+ · · ·+ Cm(p)∆mD(z)] (3)

where the ith coefficient is given by

Ci(p) =

p∫0

(η

i

)dη (4a)

with the binomial coefficient being defined as(η

i

)=η(η − 1) · · · (η − (k − 1))

i!=

η!(η − i)!i!

(4b)

and the ith difference equation is given by

∆iD(z) = (−1)i(1− z−1)i (5)

where i = 0, 1, . . . ,m, 1 ≤ p ≤ m, and z = exp(jωT ) is thez-transform parameter. Equation (3) shows that a pth-orderNewton–Cotes digital integrator has m zeros and p poles(on the unit circle) in the z-plane. The derviation of (3)–(5)can be found in the Appendix, and (3)–(5) correspond to(A.32)–(A.34). Equation (3) can be generally expressed as theproduct of the transfer function of an FIR digital filter and thetransfer function of an IIR digital filter according to

Hmp(z) =T ·[

m∑i=0

biz−i

]×

[1

1− z−p

]

=T ·[

m∑i=0

|bi|ej∠biz−i

]×

[1

1− z−p

](6)

where the first term in brackets is the transfer function of theFIR digital filter (where bi is the ith tap coefficient, bi = bm−i

is positive for m = p, and bi is real (positive or negative)for m > p) and the second term in brackets is the transferfunction of the IIR digital filter. Furthermore, ∠bi = 0 (forbi > 0) or ∠bi = π (for bi < 0), where ∠bi represents theargument of bi. The transfer functions of several families ofthe Newton–Cotes digital integrators are tabulated in Table I.Note that H11(z), H22(z), H33(z), and H44(z) are, respec-tively, the well-known transfer functions of the trapezoidal,Simpson’s 1/3, Simpson’s 3/8, and Boole’s integrators [6]–[9].As the digital integrators are designed from the polynomialperspective (see the Appendix), the pth-order Newton–Cotesdigital integrator can accurately approximate the integral of apth-order polynomial continuous-time input signal, which isdescribed by x(t) = tp, where p ≥ 1. That is, the trapezoidalintegrator, the Simpson’s 1/3 integrator, the Simpson’s 3/8integrator, the Boole’s integrator, and the fifth-order integratorare extremely accurate if the function to be integrated is a first-order polynomial, second-order polynomial, third-order poly-nomial, fourth-order polynomial, and a fifth-order polynomial,respectively.

III. DESIGN OF NEWTON–COTES OPTICAL INTEGRATORS

This section describes the design of the Newton–Cotes opti-cal integrators using the cascade of an FIR optical waveguidefilter with an IIR optical waveguide filter.

A. FIR Optical Filter

This section presents the characteristics of an FIR (or op-tical hybrid) waveguide filter, which is a building block ofthe Newton–Cotes optical integrators. Fig. 1(a) shows theschematic of the proposed (m+ 1)-tap FIR (or optical hy-brid) optical waveguide filter, which has the unique advantagethat the filter tap coefficients can be independently controlled(and can also be programmed) to yield arbitrary filtering

NGO AND BINH: NEWTON–COTES-BASED INTEGRATORS FOR DARK SOLITON GENERATION 565

Fig. 1. (a) Schematic diagram of the proposed (m + 1)-tap FIR optical waveguide filter. (b) Schematic diagram of a waveguide-based splittter. (c) Schematicdiagram of a waveguide-based combiner.

characteristics [10]. The (m+ 1)-tap optical hybrid filter sim-ply consists of a 1× (m+ 1) waveguide-based splitter forsignal splitting, m+ 1 waveguide arms with a constant timedelay difference of T (or sampling period of the filter) for delay-ing the signals, and tunable couplers (TCs) and phase shifters(PSs), with a phase shift of φi each, for arbitrarily weightingthe signals, and (m+ 1)× 1 waveguide-based combiner forcoherently combining the tapped signals at the output. Fig. 1(b)and (c) shows the schematic diagrams of the waveguide-basedsplitter and combiner, respectively, and the splitter and com-biner (and, hence, the optical hybrid) have low polarizationdependence [10]. The PS is a thin-film heater loaded on thewaveguide and utilizes the thermooptic effect to change thephase of the optical carrier. When an electric voltage is appliedto the thin-film heater, the optical path length of the heatedwaveguide will change because of the temperature dependenceof the refractive index. For instance, a change in the optical pathlength of the heated waveguide by 1.55 µm will correspondto a change in the phase of a 1.55-µm optical carrier by 2π.The TC is a symmetrical Mach–Zehnder interferometer [see theinset in Fig. 1(a)] that consists of two identical 3-dB directionalcouplers (DCs), two waveguide arms of equal length, and athin-film heater, with a phase change of ϕi, deposited on oneof the arms for controlling the output amplitude of the TC.The TC is stable against temperature variation, because it is thetemperature difference between the two waveguide arms, butnot the absolute temperature of each arm, which is importantfor tuning or switching operation. The electric-field transferfunction of the TC is given by

κi = |κi|ej∠κi = 0.5[ejϕi − 1], i = 0, 1, . . . ,m (7)

where ∠κi denotes the argument of κi. A desired output ampli-tude of the TC is described by

|κi| =√

0.5− 0.5 cos ϕi (8)

which can be obtained by controlling the phase shift of PS,which is given by

ϕi = cos−1[1− 2|κi|2

](9)

and this results in the output phase of the TC as given by

∠κi = tan−1

[sin ϕi

(cos ϕi − 1)

]. (10)

The output amplitude |κi| and output phase ∠κi of the TC canbe changed from 0 to 1 and from −π/2 to +π/2, respectively,when ϕi varies from 0 to 2π. The z-transform transfer functionof the (m+ 1)-tap FIR optical waveguide filter is given by

HFIR(z) =1

m+ 1

m∑i=0

κiejφiz−i

=1

m+ 1

m∑i=0

|κi|ej∠κiejφiz−i. (11)

B. IIR Optical Filter

This section presents the characteristics of an IIR opticalwaveguide filter, which is a building block of the Newton–Cotesoptical integrators. Fig. 2 shows the schematic diagram of theproposed IIR optical waveguide filter, which is basically anall-pole optical filter [11]. The IIR optical waveguide filter


Fig. 2. Schematic diagram of the proposed IIR optical waveguide filter, whichis also an all-pole optical filter.

consists of a waveguide loop interconnected by two identicalDCs, with each having an intensity coupling coefficient of d.In the waveguide loop, an optical waveguide amplifier with anintensity gain ofG (e.g., an erbium-doped waveguide amplifier;see, e.g., [12]) is used to compensate for the losses in the loop.An amorphous-silicon (a-Si) film loaded on the upper section ofthe waveguide loop can be used to eliminate the stress-inducedwaveguide birefringence by laser trimming the a-Si film [13].The waveguide loop has a loop delay of Tp which is p timesthe sampling period T of the FIR optical waveguide filter (seeSection III-A), that is, Tp = pT for p ≥ 1. Using the signal-flow graph method [11], the z-transform transfer function of theIIR optical waveguide filter (from the input port to the outputport) is simply given by

HIIR(z) =−√Gd2

1−√G(1− d)2e−2αLpz−p

. (12)

Here, e−2αLp is the intensity loss of the waveguide loopdue to the propagation loss of the waveguide, where α is theamplitude waveguide loss and Lp is the length of the waveguideloop. From (12), the IIR optical filter is, in fact, an all-poleoptical filter, which has p nonzero poles and p zeros at the originin the z-plane [11].

C. Design of Newton–Cotes Optical Integrators

This section describes the design of the Newton–Cotes opti-cal integrators using the cascade of an FIR optical waveguidefilter (see Section III-A) with an IIR optical waveguide filter(see Section III-B). The z-transform transfer function of theNewton–Cotes optical integrators, which is simply the productof the transfer function of the FIR optical filter [see (11)] andthe transfer function of the IIR optical filter [see (12)], is givenby

Hmp(z) =

[−√Gd2

m+ 1

][m∑

i=0

|κi|ej∠κiejφiz−i

]

×[

11−

√G(1− d)2e−2αLpz−p

]. (13)

The synthesis of a Newton–Cotes optical integrator [asdescribed by Hmp(z) in (13)] from the characteristics of aNewton–Cotes digital integrator [as described by Hmp(z) in(6)] requires Hmp(z) = Hmp(z), from which the followingequations are obtained:

|bi|ej∠bi = |κi|ej∠κiejφi (14)

1 =√G(1− d)2e−2αLp . (15)

From (14), the required output amplitude |κi| of the TC [asdefined in (8)] is given by

|κi| = |bi| (16)

and the required phase shift φi of the PS on each delay line ofthe FIR optical filter is given by

φi = ∠bi − ∠κi. (17)

From (15), the required intensity optical gain G is given by

G =1

(1− d)2e−2αLp(18)

where G decreases with a decrease in d (e.g., use d = 0.05).

IV. RESULTS AND DISCUSSION

Section IV-A presents the magnitude responses of the pro-posed Newton–Cotes optical integrators. Section IV-B de-scribes the application of the proposed trapezoidal opticalintegrator H11(z) as an optical dark-soliton generator.

A. Characteristics of the Newton–Cotes Optical Integrators

Fig. 3 shows the magnitude responses of the trapezoidal fam-ily,Hm1(z; p = 1; m = 1, 2, 3, 4, 5), of the Newton–Cotes op-tical integrators, where the normalized frequency correspondsto ωT/(2π) and Hmp means Hmp(z). It can be seen thatthe magnitude response of the trapezoidal optical integratorH31(z) approximates that of the ideal optical integrator muchbetter than other optical integrators of the same trapezoidalfamily. These results demonstrate the effectiveness of the pro-posed theory of the Newton–Cotes optical integrators. It canalso be seen that the ideal optical integrator is effectively anoptical bandstop filter within one free spectral range (FSR).The magnitude responses are plotted over one normalized FSRof ωT/(2π) = 1, and it should be noted that the magnituderesponses are periodic with a frequency period equal to theFSR. It should be noted that the actual value of the samplingperiod T (and, hence, the size of the device) will depend onthe application requirement. The sampling period T relates tothe FSR of the optical integrator by T = 1/FSR. For example,FSR = 517 GHz gives T = 1.93 ps, which corresponds to awaveguide length of about 400 µm (for silica waveguide).Furthermore, it should be noted that the normalized frequencyin the x-axis of Fig. 3 is given by ωT/(2π) = fT = cT/λ,where f and λ are the operating frequency and wavelengthof the lightwave, respectively, and c is the speed of light in


Fig. 3. Magnitude responses of the trapezoidal family Hm1(z; p = 1; m =1, 2, 3, 4, 5) of the Newton–Cotes optical integrators, where the normalizedfrequency corresponds to ωT/(2π) and Hmp means Hmp(z).

vacuum. From this definition of the normalized frequency, onecan determine the actual values of f and λ from a given T value.

The FSR of the optical integrator is mainly limited by theFSR of the IIR optical filter. The FSR of the IIR optical filter isinversely proportional to the loop length of the IIR filter, whichincreases with the length of the gain element. The FSR of theFIR or optical hybrid filter can be designed to be large. Forexample, a 16-tap FIR or optical hybrid filter with an insertionloss of only 5.6 dB can realize sinc-type and Gaussian-typespectral responses (but not optical integrators) and has an FSRof 517 GHz, which corresponds to ∆L = 400 µm (for silicawaveguide) and T = 1/FSR = 1.93 ps [14]. With advances inthe planar lightwave circuit (PLC) technologies, it would not besurprising that ∆L and, hence, T can be further reduced in thefuture.

B. Application of a Trapezoidal Optical Integrator as anOptical Dark-Soliton Generator

Dark solitons have been found to offer better stability thanbright solitons against fiber loss [15], interactions betweenneighboring solitons [16], and amplified noise-induced timingjitter [17]. Dark-soliton transmission experiments, however, aremuch more difficult to implement than bright-soliton trans-mission experiments because of the difficulty of generatingoptical dark solitons. This is because, unlike the presence ofa sharp peak in a Gaussian-like or soliton (bright soliton) pulse,the presence of a sharp dip in a dark-soliton pulse with acontinuous-wave background is very difficult to generate usinga conventional semiconductor laser or fiber laser [18]. Forthis reason, we demonstrate here that the proposed trapezoidal

optical integrator H11(z) described above can be used as anoptical dark-soliton generator. By definition [19], the integralof a “bright-squared” soliton pulse with amplitude sech2(t) is afundamental dark-soliton pulse with amplitude tanh(t), whichis given by

t∫−∞

sech2(t′)dt′ = tanh(t). (19)

We now show that the trapezoidal optical integrator H11(z)can perform the integral function described by (19). The in-put modulated optical signal (electric-field amplitude) to thetrapezoidal optical integrator H11(z) is assumed to be of anormalized form, which is given by

x(t) =∑

i

(−1)iaiS12

(t

T0− 2iq0

)(20)

where ai ∈ (0, 1) is the binary sequence, q0 is a constant, andT0 is the pulsewidth of the intensity pulse shape (of a “bright-squared” soliton pulse), which is given by

S(t) = sech4(t). (21)

The processed or integrated signal at the output of the trape-zoidal optical integrator is a fundamental dark-soliton signal,which is described by

y(t) = x(t) ∗ h11(t) ∼=∑

i

(−1)iaitanh(t

T0− 2iq0

)(22)

where ∗ denotes the convolution operation and h11(t), which isthe inverse z-transform of H11(z), is the impulse response ofthe trapezoidal optical integrator.

To verify that (22) is correct, we now simulate the perfor-mance of the trapezoidal optical integrator H11(z) to show thatit can function as an optical dark-soliton generator. It is con-sidered that the input amplitude pulse to the trapezoidal opticalintegrator is a “bright-squared” soliton pulse pair (normalized),which is given by

x(t) = −sech2

(t

T0− q0

)+ sech2

(t

T0+ q0

)(23)

which has a full-width at half-maximum (FWHM) of 1.21T0

and a bit time (the inverse of the bit rate) of 2q0T0, in whichq0 = 3.52 is chosen here. Fig. 4 shows the ideal (solid curve),generated (dashed curve), and propagated (dotted–dashedcurve) dark-soliton signals. The solid curve corresponds tothe ideal fundamental dark-soliton signal whose normalizedamplitude pulse is described by [16]

U(t) =

tanh(

tT0

+ q0

), −∞ < t

T0< 0

−tanh(

tT0

− q0

), 0 ≤ t

T0<∞

(24)

where q0 = 3.52, 2q0T0 is the bit period, and 1.76T0 is theFWHM of the dark soliton pulse. The “bright-squared” soliton


Fig. 4. Fundamental dark-soliton signals where the relative bit time on the x-axis corresponds to the bit period of 2q0T0. The solid curve shows the idealfundamental dark-soliton signal as described by (24). The dashed curve shows the generated dark-soliton signal at the output of the trapezoidal optical integratorH11(z) at Z = 0. The dotted–dashed curve shows the generated dark-soliton signal by the trapezoidal optical integrator H11(z) after having propagated throughthe single-mode optical fiber over a large distance of Z = 100Z0, where Z is the propagation distance and Z0 is the soliton period.

pulse, as described by (23), is processed by the trapezoidaloptical integrator, and this results in the generated opticaldark-soliton signal (the dashed curve) at Z = 0, where Z isthe propagation distance along the single-mode optical fiber.To show that this generated dark-soliton signal by the trape-zoidal optical integrator is indeed a fundamental dark-solitonsignal, we now test it by propagating it through a large dis-tance of single-mode optical fiber by solving the nonlinearSchrödinger equation using the split-step Fourier method [20].The dispersion-shifted fiber, with a zero-dispersion wavelengthat 1550 nm, and soliton source used in the model are assumedto have the following typical parameter values [18]. The groupvelocity dispersion parameter is β2 = +1.27 ps2/km (for thefiber dispersion parameter of D = −1.0 ps/nm/km), the fibernonlinear coefficient is ξ = 3.2 W−1km−1 (for the nonlinearrefractive index of n2 = 3.2× 10−20m2/W and the effectivecore area of Aeff = 40 µm2), the peak power of the incidentpulse is P0 = 0.494 mW (for a fundamental dark soliton andT0 = 28.4 ps), the soliton period is Z0 = 995 km, and thebit rate is 5 Gb/s, which can easily be increased to a largervalue. The dotted–dashed curve shows the dark-soliton signalafter having propagated through a large distance of Z = 100Z0

(where Z0 is the soliton period) through the optical fiber andthat it resembles the dark-soliton signal (atZ = 0) generated bythe trapezoidal optical integrator. This means that the generateddark-soliton signal (at Z = 0) still preserves the characteristicsof a fundamental dark-soliton signal even after having prop-agated through a large distance of Z = 100Z0 through theoptical fiber. This shows the effectiveness of the trapezoidaloptical integrator in generating high-quality fundamental darksolitons that can propagate stably over a long distance. Tofurther support this claim, Fig. 5 shows the evolution of the

generated dark-soliton signal by the trapezoidal optical integra-tor (the dashed curve in Fig. 4) through the single-mode opticalfiber over 100 soliton periods, where it can be seen that theproperties of the generated fundamental dark-soliton signal arepreserved. It should be noted that, to implement the trapezoidaloptical integrator H11(z), the FIR optical filter structure shownin Fig. 1(a) reduces to a simple two-tap FIR optical filter or anasymmetrical Mach–Zehnder interferometer [11].

V. CONCLUSION

We have presented a theory of the Newton–Cotes opticalintegrators for high-speed optical signal processing. The ar-chitecture of the Newton–Cotes optical integrators consists ofan FIR optical waveguide filter in cascade with an IIR opticalwaveguide filter. By numerical simulations, we have shown thata trapezoidal optical integrator H11(z) can generate fundamen-tal dark solitons that can propagate stably through a very largedistance of single-mode optical fiber. Although our discussionhas been focussed on the optical integrators, the techniques andthe results are applicable to other optical systems based on free-space optics, fiber optics, and fiber gratings. It must be notedthat optical temporal integration is a new concept with manypotential pulse-shaping applications in ultrafast optical signalprocessing for future high-speed optical networks.

APPENDIX

DERIVATION OF A GENERALIZED THEORY OF THE

NEWTON–COTES DIGITAL INTEGRATORS

Here, a classical numerical integration scheme together withthe digital signal processing technique is employed to develop,


Fig. 5. Evolution of the generated dark-soliton signal by the trapezoidal optical integrator H11(z) (the dashed curve in Fig. 4) over 100 soliton periods in asingle-mode optical fiber. Z is the propagation distance, Z0 is the soliton period, and the relative bit time on the axis corresponds to the bit period of 2q0T0.

to our knowledge, for the first time, a generalized theory ofthe Newton–Cotes digital integrators. A definition of numericalintegration is first given and used as a basis in the derivationprocess. The Newton’s interpolating polynomial is then de-scribed, based on which a general form of the Newton–Cotesclosed integration formulas is derived. Finally, a generalizedtheory of the Newton–Cotes digital integrators is derived.

A. Definition of Numerical Integration

It is assumed that a continuous-time signal x(t) is given andthat its integral

y(t) =

t∫0

x(t) dt (A.1)

is to be determined from a sequence of samples of thecontinuous-time signal x(t) at the discrete time t = tn where

tn = nT, n = 0, 1, 2, . . . (A.2)

with T > 0 being the period between successive samples. Intu-itively, the integral y(t) cannot be obtained for all t, but onlyfor t = tn. Thus, (A.1) can be written as

yn = y(tn) =

tn∫0

x(t)dt. (A.3)

To simplify the numerical integration algorithm, the integra-tion interval [0, tn] is divided into a number of equal segmentswith each segment having a step size of T . The underlyingprinciple of the numerical integration algorithm is shown inFig. 6.

Fig. 6. Graphical illustration of the numerical integration technique.

From Fig. 6, the integral in (A.3) can be divided into twointegrals as

yn =

tn−p∫0

x(t)dt+

tn∫tn−p

x(t)dt = yn−p + ip (A.4)

where the partial integral ip, which represents the area of thehatched region of Fig. 6, is given by [9]

ip =

tn∫tn−p

x(t) dt. (A.5)

The z-transform of (A.4) is given by [9]

Y (z) =[

11− z−p

]Ip(z) (A.6)

where Y (z) = Z{yn} and Ip(z) = Z{ip} with Z{·} beingthe z-transform of {·}. In (A.6), the z-transform parameter isdefined as z = exp(jωT ) where j =

√−1, ω is the angular


frequency, and T is the sampling period of the integrator [21].The z-transform of the partial integral Ip(z) is to be determinedin Appendix D.

B. Newton’s Interpolating Polynomial

For analytical simplicity, the discrete-time variables in Fig. 6are redefined as

tk = tn−p, k = n− p (A.7a)

tk+p = tn (A.7b)

tk+m = tm. (A.7c)

Using (A.7a) and (A.7b), (A.5) becomes

ip =

tk+p∫tk

x(t)dt. (A.8)

For the time interval [tk, tk+m] as shown in Fig. 6, thecurve x(t) can be approximated by the mth-order Newton’sinterpolating polynomial, which passes through m+ 1 datapoints, as [22]

x(t) =x(tk) +∆x(tk)T

(t− tk)

+∆2x(tk)2!T 2

(t− tk)(t− tk+1) + · · ·

+∆mx(tk)m!Tm

(t− tk)(t− tk+1) · · · (t− tk+m−1)

(A.9)

where the ith discrete-time variable is given by

tk+i = tk + iT, i = 0, 1, . . . ,m− 1 (A.10)

and the qth forward difference equation is given by [19]

∆qx(tk) =q∑

i=0

(−1)i(q

i

)x(tk+q−i), q = 0, 1, . . . ,m.

(A.11)

The binomial coefficient in (A.11) is defined as [19](q

i

)=q(q − 1) · · · (q − (i− 1))

i!=

q!(q − i)!i!

(A.12a)

with (q

0

)=

(q

q

)= 1. (A.12b)

Equation (A.11) can be written in a recursive form as [19]

∆0x(tk) =x(tk) (A.13a)

∆1x(tk) =∆x(tk) = x(tk+1)− x(tk) (A.13b)

∆qx(tk) =∆q−1x(tk+1)−∆q−1x(tk),

q = 2, . . . ,m. (A.13c)

Using (A12) and (A13), (A.9) can be simply expressed as

x(t) = x(tk) +m∑

q=1

∆qx(tk)q!T q

{q−1∏i=0

(t− tk+i)

}(A.14)

which can be further simplified by defining a new quantity

η =t− tkT

. (A.15)

Substituting (A.15) into (A.10), we obtain

t− tk+i = T (η − i), i = 0, 1, . . . ,m− 1. (A.16)

Substituting (A.16) into (A.14) results in

x(t) =x(tk) +m∑

q=1

∆qx(tk)q!T q

{q−1∏i=0

[T (η − i)]

}

=x(tk) +m∑

q=1

∆qx(tk)q!

{q−1∏i=0

(η − i)

}

=x(tk) + ∆x(tk)η +∆2x(tk)η(η − 1)

2!+ · · ·

+∆mx(tk)η(η − 1) · · · (η − (m− 1))

m!(A.17)

which can be further simplified to

x(t) =x(tk) +m∑

q=1

(η

q

)∆qx(tk)

=m∑

q=0

(η

q

)∆qx(tk). (A.18)

Thus, for the time interval [tk, tk+m], the mth-orderNewton’s interpolating polynomial of the curve x(t) can besimply described by (A.18).

C. General Form of the Newton–Cotes Closed IntegrationFormulas

Substituting (A.18) into (A.8) results in

ip =

tk+p∫tk

[m∑

q=0

(η

q

)∆qx(tk)

]dt. (A.19)

From (A.15), dt = Tdη and the limits of integration arechanged from t = tk to η = 0 and from t = tk+p to η = p.Substituting these parameters into (A.19) results in

ip =

p∫0

[m∑

q=0

(η

q

)∆qx(tk)

]· Tdη

=Tm∑

q=0

p∫

0

(η

q

)dη ·∆qx(tk)

(A.20)


which can be rearranged to give

ip = T

m∑q=0

Cq(p)∆qx(tk) (A.21)

where the qth coefficient Cq(p) is given by

Cq(p) =

p∫0

(η

q

)dη, q = 0, 1, . . . ,m. (A.22)

The qth forward difference equation, as described by (A.11),can be further simplified by substituting u = q − i or i = q − uinto (A.11) to give

∆qx(tk) = (−1)qq∑

u=0

(−1)−u

(q

q − u

)x(tk+u)

= (−1)qq∑

u=0

(−1)u(q

u

)x(tk+u) (A.23)

where

(−1)−u =(−1)u, u ∈ integer (A.24a)(q

q − u

)=

(q

u

)(A.24b)

have been used. Thus, a general form of the Newton–Cotesclosed integration formulas can be simply described by threeclosed-form formulas, as given in (A.21)–(A.23).

D. Generalized Theory of the Newton–Cotes DigitalIntegrators

Taking the z-transform of (A.21) results in

Ip(z) = Tm∑

q=0

Cq(p)∆qX(z) (A.25)

where ∆qX(z) = Z{∆qx(tk)} is the z-transform of (A.23),which is given by

∆qX(z) = (−1)qq∑

u=0

(−1)u(q

u

)·X(z)z−u (A.26)

where X(z)z−u = Z{x(tk+u)}. Equation (A.26) can be re-arranged to give

∆qX(z)[X(z)(−1)q]

=q∑

u=0

(−1)u(q

u

)z−u

=1− qz−1 +q(q − 1)

2!z−2 + · · ·

+ (−1)rq!

(q − r)!r!z−r + · · ·+ (−1)qz−q.

(A.27)

Note that (A.27) can be recognized as [23]

∆qX(z)[X(z)(−1)q]

= (1− z−1)q (A.28)

or

∆qX(z) = ∆qD(z) ·X(z) (A.29)

where

∆qD(z) = (−1)q(1− z−1)q, q = 0, 1, . . . ,m. (A.30)

Substituting (A.29) into (A.25) gives

Ip(z) = X(z)Tm∑

q=0

Cq(p)∆qD(z). (A.31)

Substituting (A.31) into (A.6b), the pth-order transfer func-tion of the Newton–Cotes digital integrator can be generallydescribed by

Hmp(z)

=Y (z)X(z)

=T

1− z−p

m∑i=0

Ci(p)∆iD(z)

=T

1− z−p

[C0(p) + C1(p)∆D(z) + C2(p)∆2D(z)

+ · · ·+ Cm(p)∆mD(z)] (A.32)

where the ith coefficient is given, from (A.22), as

Ci(p) =

p∫0

(η

i

)dη (A.33)

and the ith difference equation is given, from (A.30), as

∆iD(z) = (−1)i(1− z−1)i (A.34)

for i = 0, 1, . . . ,m and 1 ≤ p ≤ m. In summary, a generalizedtheory of the Newton–Cotes digital integrators has been de-rived, and (A.32)–(A.34) correspond to (3)–(5) in Section II.

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Nam Quoc Ngo, photograph and biography not available at the time ofpublication.

Le Nguyen Binh, photograph and biography not available at the time ofpublication.

Optical realization of Newton-cotes-based integrators for dark soliton generation

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