Plane wave coupling into single-mode fiber in the presence of random angular jitter

Plane wave coupling into single-mode fiber in thepresence of random angular jitter

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

*Corresponding author: [email protected]

Received 1 July 2009; revised 12 August 2009; accepted 20 August 2009;posted 21 August 2009 (Doc. ID 113519); published 15 September 2009

Efficient coupling of coherent plane waves into single-mode fibers is a key technology for free-space op-tical communication at 1550nm. Here we deal with the influence of random angular jitter on a planewave to single-mode fiber coupling. The expression of mean-coupling efficiency in the presence of randomjitter is handled in the pupil plane. First, an analytical expression of the mean-coupling efficiency isderived for the zero bias error case. Then the bias error was taken into account. By minimizing thecoupling efficiency penalty, the optimum value of design parameter β of fiber-coupled optical systemsin the presence of random jitter is obtained. The results obtained here will be useful in facilitating para-metric estimation and optimization of fiber-coupled optical systems. © 2009 Optical Society of America

OCIS codes: 060.2310, 060.4510, 010.3310.

1. Introduction

With the development of optical fiber communica-tions, especially the maturity of optical fiber am-plifiers and the wavelength division multiplexing(WDM) technology, space optical communication at1550nm becomes a promising solution for futurehigh-speed satellite communication [1–6]. Couplingspace light into single-mode fiber is a key technologyin space optical communication with optical pream-plifiers. It is important to achieve maximum fibercoupling efficiency to obtain the maximum receivedsignal-to-noise ratio (SNR). In practice, laser commu-nication terminals operate in the presence of somerandom angular jitter [6–9], so it is important toknow the influence of random jitter on a plane waveto single-mode fiber coupling and what the optimumdesign parameter of fiber-coupled optical systemsshould be for production of the highest possible aver-age coupling efficiency. Random angular jitter refersto the uncertainty of the instantaneous direction ofthe received optic axis with respect to a nominal axisof the fiber. On the assumption that the bias error is

zero, Toyoshima [10] reported the mean-coupling ef-ficiency expression in the focal plane and discussedoptimum relationships between a focused opticalbeam and the mode field size of optical fiber in thepresence of random angular jitter. In contrast withRef. [10], we evaluated the mean-coupling efficiencyin the presence of random jitter in the pupil plane. Asimpler analytical expression was derived for zerobias error. Moreover, in practice, perfect alignmentof the fiber relative to the lens cannot be achieved;the fiber lateral offset results in static bias error.In a general case, random jitter distribution was aNakagami–Rice distribution [6,7], and the mean-coupling efficiency was numerically simulated. Wediscuss the optimum value of β and how the maxi-mum mean-coupling efficiency varies with the ran-dom jitter.

2. Basic Models for Fiber Coupling and RandomAngular Jitter

To derive a mathematical model for plane wavecoupling into single-mode fiber in the presence ofrandom angular jitter, we first review the conven-tional single-mode fiber coupling models and themodel for random angular jitter.

0003-6935/09/275184-06$15.00/0© 2009 Optical Society of America

5184 APPLIED OPTICS / Vol. 48, No. 27 / 20 September 2009

A. Conventional Coupling Theory

The geometry of the optical system is shown inFig. 1. A uniform optical beam is focused by a thin,diffraction-limited lens of focal length f and radius Rlocated in pupil plane A. The single-mode fiber end isplaced in focal plane O.Coupling efficiency is defined as the ratio of the

power coupled into the fiber to the power available inthe pupil plane. Since the complex field in the pupilplane and focal plane Fourier transform conjugatepairs, according to the Parseval–Plancherel theoremthe coupling efficiency can be evaluated in pupilplane A, which leads to [11–13]

η ¼

��RR E�AðrÞFAðrÞds

��2RR jEAðrÞj2ds ·RR jFAðrÞj2ds

; ð1Þ

where EAðrÞ is the complex incident field in thepupil plane, FAðrÞ is the fiber eigenmode back-propagated to the pupil plane and normalizedas ∬ jFAðrÞj2ds ¼ 1.

B. Random Angular Jitter Model

Random angular jitter refers to the uncertainty ofthe instantaneous direction of the received optic axiswith respect to a nominal axis of the fiber. The PDFofthe random angular jitter of σ rms is the Nakagami–Rice distribution and is given by [7–9]

f ðθ;ϕÞ ¼ θσ2 exp

�−θ2 þ ϕ2

2σ2�I0

�θϕσ2

�; ð2Þ

where θ is the random jitter angle, ϕ is the biaserror angle from the nominal axis of the fiber, andI0ð Þ is the modified Bessel function of order zero.The bias error angle can even be regarded as the zeromean random value, which then causes the probabil-ity density function (PDF) to reduce to the Rayleighdistribution:

f ðθÞjϕ¼0 ¼ θσ2 exp

�−

θ22σ2

�: ð3Þ

3. Coupling Efficiency in the Presence of Jitterwithout Bias Error

A. Mean-Coupling Efficiency

The incident beam tilt by angles θ results in a lateralshift of the focused mode EoðrÞ by Δr ¼ θf in the cou-pling plane, as shown in Fig. 2(a), which is equivalentto lateral shift Δr of the fiber axis from the opticalaxis of the lens in the coupling plane, as shown inFig. 2(b). Here we assume that the bias error canbe regarded as zero, and then the lateral shift dis-tribution that is due to random jitter becomes aRayleigh distribution [6,7] with σ rms and can be ex-pressed as

f ðΔrÞ ¼ Δr

σ2 exp�−Δr2

2σ2�: ð4Þ

Under some conditions [12,14], the fiber eigen-mode can be approximated as a Gaussian field distri-bution with mode-field radius ω0. A lateral shift ofΔrleads to an additional factor of expfj2π½ðxΔx=λf Þ þðyΔy=λf Þ�g in the expression of the backpropagatedfiber mode FAðrÞ [12]. Defined as x ¼ r cos θ, y ¼r sin θ, Δx ¼ Δr cosΩ, Δy ¼ Δr sinΩ, the backpropa-gated fiber mode can easily be found and normalizedas ∬ jFAðrÞj2ds ¼ 1:

FAðrÞ ¼ffiffiffiffiffiffiffiffiffi2

πω2a

sexp

�−r2

ω2a

�exp

�j2πλf cosðθ − ΩÞrΔr

�;

ð5Þwhere λ is the wavelength, θ is the angle between~rand the x axis, Ω is the angle between Δ~r and the xaxis, and ωa is the mode-field radius of the back-propagated fiber mode given by ωa ¼ λf =πω0. Thus,the expected value of the coupling efficiency shouldbe averaged with respect to the PDF of lateral shiftand is given by

Fig. 1. (Color online) Coupling geometry.

Fig. 2. Incident beam tilt by angles θ results in a lateral shift ofthe focused mode EoðrÞ by Δr ¼ θf in the coupling plane, which isequivalent to lateral offsetΔr of the fiber axis from the optical axisof the lens.

20 September 2009 / Vol. 48, No. 27 / APPLIED OPTICS 5185

hηi ¼

��RR EAðrÞFAðrÞds��2�

�RR jEAðrÞj2ds��RR jFAðrÞj2ds

�

¼

��RR EAðrÞFAðrÞf ðΔrÞrdrdθdΔr

��2�RR jEAðrÞj2ds� : ð6Þ

Substituting f ðΔrÞ and FAðrÞ from Eqs. (4) and (5)into Eq. (6), we find the fiber coupling efficiency,which is given by

hηi ¼ 1

πR2

��ffiffiffiffiffiffiffiffiffi2

πω2a

s ZR0

Z∞0

Z2π0

Δr

σ2 exp�−r2

ω2aþ j

2πλf rΔr

× cosðθ − ΩÞ −Δr2

2σ2�rdθdΔrdr

��2: ð7Þ

Recalling the integral formulas [15],

J0ðxÞ ¼12π

Z2π0

expðix cos θÞdθ: ð8Þ

Equation (7) can easily be simplified. Integrating theterm with respect to θ and Δr, the equation can berewritten as

hηi ¼ 1

πR2

��2ffiffiffiffiffiffi2π

p

ωa

ZR0

exp��

−2π2σ2λ2f 2 −

1

ω2a

�r2�rdr

��2:ð9Þ

Introducing the design parameter β ¼ R=ωa ¼πRω0=λf , which is defined as the ratio of the apertureradius to the radius of the backpropagated fibermode, we derived an analytical expression of themean-coupling efficiency:

hηi ¼ 2�2

�σω0

�2þ 1

�2β2

×�1 − exp

−�� σ

ω0

�2þ 1

�β2�

2: ð10Þ

In the absence of jitter, that is, σ=ω0 ¼ 0, Eq. (10)yields

hηi ¼ 2

β2 ½1 − expð−β2Þ�2: ð11Þ

An optimum coupling efficiency of η0 ¼ 0:81 canbe obtained for a design parameter of β0 ¼ 1:121[11–13]. Figure 3 shows the degradation of coupling

efficiency as a function of normalized random jitterσ=ω0 for different design parameter β. It is shownthat a suboptimum design parameter (β < β0) leadsto improved coupling efficiency if the normalized ran-dom jitter is relatively large. Considering the curveof β ¼ β0, the efficiency reduces to approximately halfof ηmax with σ=ω0 ¼ 0:6, however, half of ηmax isobtained with σ=ω0 ¼ 0:9 for the curve of β ¼ 0:7β0.The reason being that the mode-field radius of thebackpropagated fiber mode ωa increases as the valueof β decreases, which relaxes the jitter tolerances.Figure 4 shows the coupling efficiency as a functionof design parameter β with normalized random jitterσ=ω0 ¼ 0, 0.2, 0.5, 1 as an example. As expected, thecoupling efficiency decreases as the random jitter in-creases. Obviously, an optimized pair ðηmax; βoptÞ ex-ists for each σ=ω0.

Fig. 3. (Color online) Fiber coupling efficiency as a function ofrandom jitter σ normalized by mode field radius ω0 for differentdesign parameters β.

Fig. 4. (Color online) Fiber coupling efficiency as a function of de-sign parameter β and random jitter σ normalized by mode fieldradius ω0.


B. Optimum β and Maximum Mean-Coupling Efficiency

As shown in Fig. 3, there is a maximum value of thecoupling efficiency as a function of β, and the opti-mum value of β varies with the normalized randomjitter σ=ω0. Differentiating hηi by β, we find

dhηidβ ¼ ð2xþ 1Þe−x − 1; ð12Þ

where x ¼ ½ðσ=ω0Þ2 þ 1�β2, according to dhηi=dβ ¼ 0,the maximum coupling efficiency is achieved whenx ¼ 1:2564. Consequently, the relationship betweenoptimum β, at which the mean-coupling efficiency ismaximized, and normalized random jitter is given by

β ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:25642σ2ω20þ 1

vuut : ð13Þ

Given the mode-field radius ω0 and jitter rms σ,the optimum β can be directly derived from Eq. (13).Figure 5 shows the optimum design parameter β as afunction of the normalized random jitter basedon Eq. (13). As the random jitter increases, the valueof optimum β decreases. Substituting Eq. (13) intoEq. (10), we obtained the maximum coupling effi-ciency as a function of normalized random jitterσ=ω0: �

η� σω0

��max

¼ 0:81451

2σ2ω20þ 1

: ð14Þ

Figure 6 shows the maximum coupling-efficiencyas a function of the normalized random jitter whendesign parameter β is optimum. As expected, themaximum value of the coupling efficiency decreasesas the random jitter increases. Without random jit-ter, the theoretical maximum fiber coupling effi-ciency of η ¼ 0:8145 is achieved. A normalizedrandom jitter equal to one already reduces the effi-ciency to η ¼ 0:2715, which is one third of the theo-retical maximum value.

4. Coupling Efficiency when Bias Error Is Considered

Lateral offset r0 is the static radial offset of the fiberfrom the nominal axis of a lens; the fiber lateral offsetresults in static bias error. The random jitter dis-tribution with respect to Δr and r0 is given by aNakagami–Rice distribution [6,7]:

f ðΔr; r0Þ ¼Δr

σ2 exp�−Δr2 þ r20

2σ2�I0

�Δr; r0σ2

�: ð15Þ

Substituting f ðΔr; r0Þ and FAðrÞ from Eqs. (5) and(15) into Eq. (6), with the use of Eq. (8) we deter-mined that the fiber coupling efficiency is given by

hηi ¼ 1

πR2

��ffiffiffiffiffiffiffiffiffi2

πω2a

sexp

�−

r202σ2

�ZR0

Z∞0

Δr

σ2 exp�−r2

ω2a−Δr2

2σ2�

×2πJ0

�2πλf rΔr

�I0

�Δr;r0σ2

�rdΔrdr

��2: ð16Þ

Recalling the integral formulas [15]:

Z∞0

xe−α0x2IνðβxÞJνðγxÞdx ¼ 1

2α0 exp�β2 − γ2

4α0�Jν

� γβ2α0

�;

½Reα0 > 0;Reν > −1�: ð17Þ

We can simplify Eq. (16) as

hηi ¼ 1

πR2

��2ffiffiffiffiffiffi2π

p

ωa

ZR0

exp��

−2π2σ2λ2f 2 −

1

ω2a

�r2�

× J0

�2πλf r0r

�rdr

��2: ð18Þ

Introduction of the normalized radial positionρ ¼ r=R, Eq. (18) yields

Fig. 5. Relationship between optimum design parameter β andrandom jitter σ normalized by mode field radius ω0.

Fig. 6. Maximum coupling efficiency as a function of normalizedrandom jitter σ=ω0 when design parameter β is optimum.


hηi ¼ 8β2��Z10

exp−β2

�2

� σω0

�2þ 1

�ρ2

× J0

�2r0ω0

βρ�ρdρ

��2: ð19Þ

Although Eq. (19) cannot be solved analyticallyto obtain the optimum β and the maximum mean-coupling efficiency by use of the condition dhηi=dβ ¼ 0, numerical results were readily obtainedand are plotted in Figs. 7 and 8. As shown in Fig. 7,with the increase of normalized bias error, the valueof optimum β decreases sharply for small σ=ω0,whereas it changes slowly for large σ=ω0.

As shown in Fig. 8, the maximum value of the cou-pling efficiency decreases as the bias error increases.Small bias error is particularly critical for efficientcoupling, especially for small σ=ω0. When σ=ω0 ¼ 0:2,a bias error equal to one mode-field radius alreadyreduces the efficiency to approximately half of themaximum value.

5. Conclusion

The mean-coupling efficiency in the presence ofrandom jitter was evaluated in the pupil plane. Ananalytical expression of the mean-coupling efficiencywas derived for zero bias error. Optimum relation-ships between design parameter β and normalizedrandom jitter σ=ω0 were obtained. A maximum aver-age coupling efficiency with optimum β as a functionof σ=ω0 was analytically derived. One can calculatethe fiber coupling efficiency and evaluate the opti-mum parameter in a random jitter environmentwithout complicated numerical calculation. In a gen-eral case, the bias error should be taken into account.The optimum value of β and the maximum mean-coupling efficiency as a function of normalized biasoffset r0=ω0 have been analyzed by numerical si-mulation. Our results will be useful for parametricestimation and optimization of fiber-coupled opticalsystems.

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Fig. 8. Maximum coupling efficiency as a function of normalizedbias error r0=ω0 and random jitter σ=ω0 when design parameter βis optimum.

Fig. 7. Optimum value of design parameter β as a function of nor-malized bias error r0=ω0 and random jitter σ=ω0.


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Plane wave coupling into single-mode fiber in the presence of random angular jitter