Impact of random angular jitter on fiber-coupleddifferential phase-shift keying receivers withMach–Zehnder interferometer demodulation

Fang Zhao,* Jing Ma, Siyuan Yu, Liying Tan, and Qiqi HanNational Key Laboratory of Tunable Laser Technology, Harbin Institute of Technology, Harbin 150001, China

*Corresponding author: zhaofang0126@163.com

Received 26 July 2010; accepted 14 September 2010;posted 24 September 2010 (Doc. ID 132233); published 26 October 2010

In an optically preamplified differential phase-shift keying (DPSK) receiver with Mach–Zehnderinterferometer (MZI) demodulation, it is essential to couple free space light into a single-mode fiber,by which the received optical signal can be guided into the amplifier and MZI. Random angular jitterhas a profound impact on fiber coupling efficiency, in that it would significantly deteriorate the receiverperformance. In this paper, the average bit error rate (BER) of fiber-coupled DPSK receivers is obtainedin the presence of random angular jitter. It is shown that the degradation is significant when the randomangular jitter normalized by the mode field radius is larger than 0.3. We further analyze the impact ofother factors, including central obstruction and optical filter transfer functions. Our results show that thesuper-Gaussian filter outperforms other filters and leads to lower average BER at a fixed optical signal-to-noise ratio. This work can be used to improve the design of fiber-coupled DPSK receivers. © 2010Optical Society of AmericaOCIS codes: 060.2605, 060.2430, 260.3160, 010.3310.

1. Introduction

Although the conventional intensity modulation/direct detection (IM/DD) scheme currently is domi-nant in satellite laser communication systems, alter-nate modulation and detection techniques have beenstudied intensively, in an attempt to improve receiv-er sensitivity and expand transport capacity [1–6].Among various modulation formats, differentialphase-shift keying (DPSK) seems to be an attractivechoice because it poses relatively lax requirementson laser linewidth and promises high spectral effi-ciency in dense wavelength division multiplexed(DWDM) systems [7–9]. An optically preamplifiedDPSK receiver with MZI demodulation becomesthe technically most practicable way of achieving(nearly) quantum-limited receiver performance dueto its relatively simple configurations in comparisonwith a coherent receiver.

In an optically preamplified DPSK receiver, it isessential to couple free space light into a single-modefiber, by which a received optical signal can be guidedinto the amplifier. A DPSK receiver using single-mode fiber coupling is referred to as a fiber-coupledDPSK receiver in this paper. In practice, laser com-munication terminals operate in the presence ofsome random angular jitter [10–12], which refersto the uncertainty of the instantaneous direction ofthe received optic axis with respect to a nominal axisof the fiber. Since the random angular jitter has agreat impact on fiber coupling efficiency, which hasbeen discussed in our previous work [13], the jitterwould deteriorate the received optical signal-to-noiseratio (OSNR), and then degrade the BER of the re-ceiver. Therefore, it is important to consider the ef-fect of random angular jitter on the performance ofa fiber-coupled DPSK receiver. In this paper, theaverage BER in the presence of random angular jit-ter is derived.

The paper is organized as follows. Section 2describes the structure of a fiber-coupled DPSK

0003-6935/10/316024-06$15.00/0© 2010 Optical Society of America

6024 APPLIED OPTICS / Vol. 49, No. 31 / 1 November 2010

receiver and presents a model. Section 3 derives theaverage BER in the presence of random angular jit-ter. Section 4 is devoted to numerical analysis anddiscussion. Section 5 presents the most relevant con-clusions of this work.

2. Fiber-Coupled DPSK Receivers

A. Receiver Setup

A fiber-coupled DPSK receiver is represented sche-matically in Fig. 1. The received optical signal is fo-cused into a single-mode fiber (SMF) with mode-fieldradius ω0 by the receiver telescope, which would beequivalent to a thin diffraction-limited lens of focallength f and radius R located in the pupil plane A.The single-mode fiber end is placed in the focal planeO. The received optical signal then passes into a er-bium-doped fiber amplifier with flat gain G, whichadds amplified spontaneous emission (ASE) noise.In order to suppress ASE noise, the amplifier outputis filtered by an optical bandpass filter. The amplifiedoptical signal is demodulated by a MZI, whose differ-ential delay is equal to the bit period. The MZI letstwo adjacent bits interfere with each other in outputports; thus, the preceding bit in a DPSK-encoded bitstream acts as the phase reference for demodulatingthe current bit. Fine-tuning of the differential delayis done to match the transmit laser frequency andachieve perfectly constructive/destructive interfer-ence. The two outputs of the MZI are then separatelydetected by balanced photodetectors. The differenceof the photocurrent is low-pass filtered and sampled.

B. Receiver Model

The low-pass equivalent model for the fiber-coupledDPSK self-homodyne receiver is shown in Fig. 2. xðtÞis the baseband equivalent representation of the re-ceiving signal, which can be expressed as

xðtÞ ¼Xn

Φnpðt − nTbÞ: ð1Þ

The bit sequence is represented by fΦng, whereΦn ∈ fej0; ejπg. pðtÞ represents the elementary pulseshape. In this paper, we only consider the NRZ code,which is given by

pðtÞ ¼� ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Eb=Tb

pt ∈ ½0;TbÞ

0 otherwise; ð2Þ

where Eb denotes the optical energy per transmittedbit and Tb is the bit period.

In Fig. 2, η denotes the coupling efficiency, which isdefined as the ratio of the power coupled into the fi-ber to the power available in the pupil plane; thus theenergy coupled into fiber is E0

b ¼ η · Eb, and we getsðtÞ ¼ ffiffiffiηp

xðtÞ. wðtÞ represents the ASE noise as com-plex AWGN with two-sided power spectral density,

N0 ¼ nspðG − 1Þ

Ghν ≈ nsphν for G ≫ 1: ð3Þ

In this paper, the optical filters are assumed to beFabry–Perot (FP), Gaussian, and super-Gaussian fil-ters. The transfer function of the FP filter is [14]

Hoðf Þ ¼1�

1þ j 2fBo

� ; ð4Þ

where Bo is the 3dB bandwidth (full width at halfmaximum) of the bandpass optical filter. For theGaussian filter, we have

Hoðf Þ ¼ exp�−

�f

0:85B0

�2�; ð5Þ

For the super-Gaussian filter, we have

Hoðf Þ ¼ exp�−

�f

0:65B0

�4�; ð6Þ

The transfer functions for MZI from its input portto its sum (constructive) and difference (destructive)ports are

Hcðf Þ ¼exp½−j2πf Tb� þ 1

2; ð7Þ

Hdðf Þ ¼exp½−j2πf Tb� − 1

2: ð8Þ

Fig. 1. Schematic of fiber-coupled DPSK receiver.

1 November 2010 / Vol. 49, No. 31 / APPLIED OPTICS 6025

Heðf Þ is the (low-pass equivalent) transfer functionof postdetection filters, which is assumed to be afifth-order Bessel type [14]

Heðf Þ ¼945

j~f 5 þ 15~f 4 − 105j~f 3 − 420~f 2 þ 945j~f þ 945;

ð9Þwhere ~f ¼ 2:43f =Be and Be is the 3dB cutofffrequency.

3. Average BER in the Presence of Random AngularJitter

In an earlier paper [13], we pointed out that the in-cident beam tilt by angle θ can be equivalent to thelateral shift Δr of the fiber axis from the optical axisof the lens in the coupling plane. The relationship be-tween Δr and θ is

Δr ¼ θf : ð10ÞWhen there is a lateral shift Δr in the fiber axis,

the coupling efficiency is given by

ηΔr ¼

����R REAðrÞFAðr;ΔrÞrdrdφ

����2πR2ð1 − ε2Þ ; ð11Þ

where FAðr;ΔrÞ is the backpropagated fiber mode inthe presence of the fiber lateral shift Δr, which isgiven by

FAðr;ΔrÞ ¼ffiffiffiffiffiffiffiffiffi2

πω2a

sexp

�−r2

ω2a

�

× exp�j2πλf cosðφ − ΩÞrΔr

�: ð12Þ

Here φ is the angle between~r and x axis, Ω is the an-gle between Δ~r and the x-axis, and ωa is the mode-field radius of the backpropagated fiber mode, givenby ωa ¼ λf =πω0.

Substituting FAðr;ΔrÞ from Eq. (12) into Eq. (11)and recalling the integral formulae,

J0ðxÞ ¼12π

Z2π0

expðjx cos θÞdθ; ð13Þ

the fiber coupling efficiency in the presence of fiberlateral shift Δr is given by

ηΔr ¼

����RRεR

ffiffiffiffi8πω2a

qexp

�−

r2

ω2a

�J0

�2πλf rΔr

�rdr

����2πR2ð1 − ε2Þ ; ð14Þ

where ε is a linear central obstruction, ε is equal tozero for an unobstructed pupil.

Introducing the normalized radial positionρ ¼ r=R, Eq. (14) yields

ηΔr ¼

����R 1ε

ffiffiffiffiffiffi8π

pβ expð−β2ρ2ÞJ0

�2β Δr

ω0ρ�ρdρ

����2πð1 − ε2Þ ; ð15Þ

where β ¼ R=ωa ¼ πRω0=λf is the design parameter,which is defined as the ratio of the aperture radius tothe radius of the backpropagated fiber mode. Whencentral obstruction ε ¼ 0, the optimum design para-meter βopt for maximum fiber coupling efficiencyin the presence of jitter is derived in our previouswork [13]:

βopt ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:25642σr2ω20þ 1

vuut ε ¼ 0: ð16Þ

Similarly, the optimum design parameter βopt forε ¼ 0:15 can also be determined as

βopt ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:19582σr2ω20þ 1

vuut ε ¼ 0:15: ð17Þ

When the instantaneous direction of the receivedoptic axis with respect to a nominal axis of the fibervaries randomly, the lateral shift probability distri-bution can be given by the Rayleigh distribution withσr rms,

f ðΔrÞ ¼ Δr

σ2rexp

�−Δr2

2σ2r

�ð18Þ

Therefore, the unconditional BER in the presence ofjitter should be averaged with respect to the prob-ability density function of the lateral shift Δr, whichcan be expressed as

hBERi ¼Z

∞

0BERðηΔrÞ · f ðΔrÞdΔr: ð19Þ

Fig. 2. Low-pass equivalent receiver model.

6026 APPLIED OPTICS / Vol. 49, No. 31 / 1 November 2010

4. Numerical Results and Analysis

Because of the intersymbol interference (ISI) and thenonstationarity of the signal-ASE beat noise term,the BER is different in each bit. The unconditionalBER can be written as

BERðηΔrÞ ¼1N

XNk¼1

PkðηΔrÞ; ð20Þ

where PkðηΔrÞ is the BER associated with the kth bit,and N is the length of the sequence.

The Karhunen–Loéve series expansion (KLSE)method is employed here to calculate PkðηΔrÞ. Theessence of the KLSE method is to evaluate themoment-generating function (MGF) of the outputsample, which is obtained through a Karhunen–Loéve series expansion of noise. From the MGF,the BER can be computed using the method of stee-pest descents, also referred to as saddle point approx-imation. In the interest of brevity, the mathematicaldescription of the KLSE method, explained in detailin Refs. [14–16], is not provided here. In the calcula-tion, the bit sequence is specified as a 25 bit de Bruijnsequence to accurately account for ISI. The band-width of optical/electrical filters, the sampling in-stant, and the decision threshold are all kept attheir optimum values to minimize the BER.

Assuming that a FP filter is used, Fig. 3 shows theaverage BER as a function of Eb=N0 with a differentdesign parameter, β. The solid and dotted curves cor-respond to the optimum design parameter βopt in thepresence of random angular jitter and β ¼ 1:12 opti-mized for zero random jitter, respectively. It is clearthat the average BER can be improved when the

design parameter is optimum, and the improvementis significant when random jitter is relatively large.The optimum design parameter βopt leads to maxi-mum coupling efficiency, so that the energy coupledinto fiber increases when the design parameter isoptimized, in which case the average BER can beimproved.

Figure 4 shows the average BER as a function ofEb=N0 and normalized random jitter σr=ω0. Thecurve labeled η ¼ 1 corresponds to the situation withno fiber coupling loss. As shown in Fig. 4, the averageBER for a given Eb=N0 increases with the rise in ran-dom jitter, and it degrades significantly when therandom jitter is large. The deviations among thevarious σr=ω0 curves broaden as Eb=N0 increases,which means that random jitter can further deterio-rate the performance of the system for a largerEb=N0. The power penalty at a fixed BER is the in-crease in energy per bit required to maintain theBER value when random jitter is present. The powerpenalty as a function of normalized random jitterσr=ω0 is given in Fig. 5. It is shown that powerpenalties drastically increase when random jitterσr=ω0 ≥ 0:3 and that as the BER increases, so dothe power penalties.

Figure 6 gives the average BER as a function of thenormalized random jitter σr=ω0 at various central ob-struction ε, when Eb=N0 ¼ 12dB. As shown in Fig. 5,the average BER decreases with enlargment of thecentral obstruction. Because the pupil central ob-struction contributes to an energy leak from the cen-tral area to the sidelobes of the spot, the fundamentalfiber mode will receive less energy when there is acentral obstruction. Thus, the average BER will de-crease when a central obstruction is present. We canalso see that the influence of the central obstructionis relatively small when the random jitter is large,

Fig. 3. Average BER as a function of Eb=N0; solid and dottedcurves correspond to optimum design parameter βopt in thepresence of random angular jitter and β ¼ 1:12 optimized for zerorandom jitter, respectively.

Fig. 4. Average BER as a function of Eb=N0 and normalizedrandom jitter σr=ω0.

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because it is outweighed by the effect of the randomjitter.

The average BER as a function of the normalizedrandom jitter σr=ω0 for different optical filters isshown in Fig. 7. It can be seen from Fig. 5 thatthe super-Gaussian filter outperforms the other fil-ters and leads to a lower average BER at a fixedEb=N0, while the FP filter has the worst perfor-mance. The behavior observed is due to the fact thatthe super-Gaussian filter has a sharp dropoff of itstransmission, which matches the power spectrumof NRZ signals, while the FP filter has a fairly mod-erate dropoff of its transmission.

5. Conclusion

In this article, the influence of random angular jitteron fiber-coupled DPSK receivers withMach–Zehnderinterferometer demodulation is analyzed. The aver-age BER in the presence of random angular jitter isderived. The numerical results show that the aver-age BER degrades in the presence of random jitter.The power penalties increase significantly when therandom angular jitter normalized by the mode fieldradius is larger than 0.3, and as the BER increases,the power penalties increase. If the residual trackingerror is determined in the design of the optical track-ing system, the design parameters β can be optimizedto improve the average BER. Central obstruction canfurther deteriorate the average BER, but it is not cri-tical when the random jitter is large. Comparing dif-ferent optical bandpass filters, our results show thatthe super-Gaussian filter outperforms the other fil-ters and leads to a lower average BER at a fixedEb=N0. The results obtained here will be helpful inthe design of fiber-coupled DPSK receivers.

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