Optics Communications 237 (2004) 37–43

www.elsevier.com/locate/optcom

Optical coupling from plane wave to step-indexsingle-mode fiber

Massimo Lazzaroni a,*, Fabio E. Zocchi b

a Dipartimento di Tecnologie dell’Informazione, Universit�a degli Studi di Milano, Via Bramante 65, 26013 Crema (CR), Italyb Media Lario S.r.l, Localit�a Pascolo – 23842 Bosisio Parini (LC), Italy

Received 14 January 2004; accepted 24 March 2004

Abstract

The optical coupling efficiency from a plane wave to a single-mode step-index fiber optics is studied considering the

exact fiber mode field distribution. A general expression relating the coupling efficiency to the fiber parameters is

presented and the optimum coupling efficiency is studied as a function of the fiber normalised frequency and the ratio

between the coupling optics radius and focal length. It is also shown that for small values of this ratio the optimum

coupling efficiency and a corresponding suitably defined optimum design parameter are functions of the fiber nor-

malized frequency only and they do not depend separately on the wavelength, the fiber core radius or the fiber

numerical aperture.

� 2004 Elsevier B.V. All rights reserved.

PACS: 42.30.Kq; 42.81.)i

Keywords: Fourier optics; Fiber optics

1. Introduction

Optical coupling to a single-mode fiber optics is

the subject of several studies concerning both co-

herent and incoherent incident fields [1–9]. An

ideal perfect imaging system that couples a plane

wave to a single-mode fiber is the simplest to be

* Corresponding author. Tel.: +39-02-503-30058; fax: +39-02-

503-30010.

E-mail addresses: lazzaroni@dti.unimi.it (M. Lazzaroni),

fabio.zocchi@media-lario.com (F.E. Zocchi).

0030-4018/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2004.03.092

studied and thus the most suited to a theoretical

analysis. Although in most applications dealing

with fiber optics the field to be coupled is best

described by a Gaussian distribution, a plane wave

input can be taken as representative of real sys-

tems, ranging from optical and infrared astro-

nomical interferometry to high performance FreeSpace Optics communication. In the former ap-

plication fiber optics can be used to guide light

from telescopes to the interferometer [10] whereas

in the latter, currently considered for large band-

width optical inter-satellite links [11], best perfor-

mances are achieved by coherent detection and in

ed.

38 M. Lazzaroni, F.E. Zocchi / Optics Communications 237 (2004) 37–43

this case the coupling of the incident field ap-

proximated by a plane wave to a single-mode fiber

optics seems unavoidable. In both cases the dis-

tance between the source and the receiver is so

large compared to the wavelength and the tele-

scope aperture that the plane wave approximationis fulfilled.

The coupling efficiency for a plane wave by

means of an ideal optical system is well known

when the electric field of the fiber mode is ap-

proximated by a Gaussian function and the par-

axial approximation is valid for the optical

description of the coupling optics. If a plane wave

of free space wavelength k and wave numberk ¼ 2p=k is incident on the coupling optics with

focal length F and aperture radius A, then in the

Gaussian approximation for the fiber mode the

coupling efficiency g, defined [1] as the ratio be-

tween the coupled optical power Pout and the in-

cident one, Pin, is given by [1],

g � PoutPin

¼ 2

v21

�� e�v2

�2

; ð1Þ

where v ¼ Akw=2F , w being the mode radius de-

fined as the distance from the fiber axis at which

the optical intensity drops by a factor 1=e2. It

should be stressed that equation (1) is valid only ifthe coupling system can be described in the par-

axial approximation, restricting the use of (1) to

small values of the ratio A=F .The result provided by (1) gives a simple rule

for optimum design. For v equal to about 1.121

the coupling efficiency g in (1) has a maximum of

81.5%. This value is always achievable by a suit-

able design of the coupling optics, provided v ismade equal to 1.121.

When the exact distribution of the electric field

of the fiber optics mode is taken into account,

numerical evaluation [3] shows that the optimum

plane wave to single-mode fiber coupling effi-

ciency is a function of the fiber normalised fre-

quency V , in contrast with the Gaussian

approximation result for which an optimum effi-ciency of 81.5% is always achievable, irrespective

to the value of V . However, in [3] the calculation

was performed in paraxial approximation for

small values of the ratio A=F and for a particular

fiber optics, with given core radius and numerical

aperture, and the conditions for optimum cou-

pling were not analysed.

When high performance optical systems are

considered, a deeper theoretical investigation of

the coupling efficiency and of the conditions for

optimum coupling is desirable. To this aim thecoupling efficiency of an ideal system coupling a

monochromatic plane wave to single-mode step-

index fiber optics is here calculated taking into

account the exact electric field distribution of the

fiber mode. An expression relating the coupling

efficiency to the fiber parameters is presented and

plotted and the optimum coupling efficiency is

studied as a function of both V and A=F , up tolarge values of the latter ratio. It is also shown that

for small values of A=F the optimum coupling ef-

ficiency and a corresponding suitably defined op-

timum design parameter are functions of the fiber

normalized frequency only and they do not depend

separately on the wavelength, the fiber core radius

or the fiber numerical aperture.

2. Optical coupling

2.1. Coupling field

The geometry of the optical coupling system is

shown in Fig. 1. The optical axis z has its origin in

the focal plane of an ideal coupling optics withfocal length F and aperture radius A. The z axis

coincides with the axis of a single-mode fiber op-

tics, the entrance facet of which is on the focal

plane at z ¼ 0. Polar co-ordinates (q0;u0) and

(q;u) are used on the exit aperture r0 of the cou-

pling optics and on the entrance facet r of the fi-

ber, respectively.

The effect of the ideal coupling optics is tomultiply the incident wave by a pure phase factor

e�ikr for q06A;

0 for q0 > A;

�

where r ¼ ðF 2 þ q02Þ1=2 is the distance from the

origin. If the incident field is a monochromaticplane wave travelling along the z axis, Ei ¼ E0e

ikz,

then the field at the exit aperture r0 of the imaging

system is Ee ¼ E0eikz�ikr for q0

6A.

Fig. 1. Geometry of the ideal optical system coupling a plane wave to a single-mode fiber optics (not to scale); longitudinal and

transversal cross-sections (a) and three-dimensional view (b).

M. Lazzaroni, F.E. Zocchi / Optics Communications 237 (2004) 37–43 39

The electric field Ef in air at the entrance facet rof the fiber optics is obtained from the Rayleigh–

Sommerfeld integral [13]

EfðQÞ ¼1

ik

Zr0EeðPÞ

eikR

Rz � Rdr0; ð2Þ

where R is the distance between the point of inte-

gration P and the point Q at which the field is

evaluated. R and z are the unit vectors along R and

z, respectively. Since the field is different from zero

only in a very small area on the entrance facet of thefiber with a linear extension of the order of few radii

a of the fiber core, the scalar product between z andR is almost equal to the product between z and r,the unit vector along r, that is z � R ffi z � r ¼ cos#.In addition it can be assumed that R ffi r in the

denominator of the integrand of (2) and that

R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ q2 � 2r � q

pffi r � r � q

¼ r � q sin# cosðu0 � uÞ

40 M. Lazzaroni, F.E. Zocchi / Optics Communications 237 (2004) 37–43

in the exponent. The above approximations are

standard in optics when the angle # is small.

However in the assumption that F is much greater

than the fiber core radius a these approximations

are also valid for large values of # or, equivalently,of the ratio A=F .

Inserting the above approximations into (2)

along with dq0 ¼ ðF = cos2 #Þd# and q0=r ¼ sin#,the field at the entrance facet r of the fiber becomes

Ef q;uð Þ ¼ FE0e�ikF

ik

Z #0

0

sin#

cos#d#

�Z 2p

0

e�ikq sin# cosðu0�uÞ du0

¼ �ikFE0e�ikF

Z #0

0

J0ðkq sin#Þsin#

cos#d#;

ð3Þ

where J0 is the Bessel function of the first kind oforder zero and #0 ¼ atanðA=F Þ.

2.2. Coupling efficiency

If the incoming wave is propagating in air with

impedance Z0, the optical power incident on the

coupling system is given by Pin ¼ pA2E20=2Z0. The

power coupled to the fiber is calculated asPout ¼ jCj2, C being the coupling integral [1]

C ¼ s2

ZrEf �H� � zdr

¼ pnsZ0

Z 1

0

EfðqÞE�ðqÞqdq; ð4Þ

where � denotes complex conjugation and s is the

electric field transmission coefficient from air to

the fiber. In the previous equation E and H are the

electric and magnetic field of the fiber mode and

EðqÞ is the transversal component of E. Finally n isthe index of refraction, assumed to be approxi-

mately the same in the core and in the cladding

of the fiber optics; for example in the Corning�

SMF-28TM single-mode fiber [12], considered the

standard fiber in optical networks, the relative re-

fraction index ratio between the core and the

cladding is about 0.35%.

The fiber fields E and H in (4) are normalizedsuch that

1

2

ZrE �H� � zdr ¼ pn

Z0

Z 1

0

EðqÞj j2qdq ¼ 1: ð5Þ

The second equalities in (4) and (5) are valid

provided the incident field is linearly polarized

along an axis orthogonal to the optical axis z. In-serting (3) into (4) gives

jCj ¼ pnkFE0

Z0

s

�����Z #0

0

sin#

cos#d#

Z 1

0

J0 kq sin#ð ÞE� qð Þqdq����:ð6Þ

The electric field EðqÞ of a step-index single-

mode fiber optics, normalized as in (5), is given by[14]

EðqÞ ¼vaV

ffiffiffiffiffi2Z0pn

q1

J1ðuÞJ0 uq=að Þ for 06 q6 a

vaV

ffiffiffiffiffi2Z0pn

qJ0ðuÞ

J1ðuÞK0ðvÞK0 vq=að Þ for q > a;

8<:

ð7Þwhere u and v are determined by the relations [14]

u2 þ v2 ¼ V 2; ð8aÞuJ1ðuÞK0ðvÞ � vK1ðvÞJ0ðuÞ ¼ 0: ð8bÞIn (7), (8a) and (8b) V is the normalized frequencyof the fiber, V ¼ kaðn2co � n2clÞ

1=2, nco and ncl being

the refractive index in the core and in the cladding,

J1 is the Bessel function of the first kind of order

one and K0 and K1 are the modified Bessel func-

tions of the second kind of order zero and one,

respectively.

Using (7) and (8b) the integral with respect to qin (6) can be evaluated to be [15]

2ffiffiffiffiffiffipn

p

affiffiffiffiffiZ0

pZ 1

0

J0 kq sin#ð ÞE�ðqÞqdq

¼ 2Vv

v2 þ n2� �

u2 � n2� � � uJ0ðnÞ

�� nJ1ðnÞ

J0ðuÞJ1ðuÞ

�

� I n; Vð Þ;

where n ¼ ka sin#. Since u and v are completely

defined through (8a) and (8b) once V is given, the

function Iðn; V Þ depends only on the normalized

frequency V . Finally by inserting the expression of

the above integral into (6), the coupling efficiency

is found to be

M. Lazzaroni, F.E. Zocchi / Optics Communications 237 (2004) 37–43 41

g � PoutPin

¼ Tk2a2

tan2 #0

�Z ka sin #0

0

I n; Vð Þ n

k2a2 � n2dn

� �2

; ð9Þ

where T ¼ njsj2 is the intensity transmission coef-

ficient [16]. In the following T will be assumedequal to unity. Equation (9) can be evaluated as a

function of A=F ¼ tan#0 and the results plotted,

giving similar curves as obtained for small value of

A=F by Barrel and Pask in [2]. However, in Fig. 2

the coupling efficiency g, as calculated from (9), is

compared with the result (1) following from the

Fig. 2. Coupling efficiency g as a function of the ratio A=F for

k ¼ 1550 nm and a ¼ 5 lm. The normalized frequency is

V ¼ 2:4 in the upper plot and V ¼ 1 in the lower one. The solid

line refers to the solution presented in the paper whereas the

dotted line is based on the Gaussian approximation (1).

Gaussian approximation of the fiber mode and the

paraxial description of the coupling optics. The

curves in Fig. 2 are evaluated for two different

values of the normalized frequency, V ¼ 2.4 and

V ¼ 1; in both cases k ¼ 1550 nm and a ¼ 5 lm.The field radius w in the Gaussian approximation

is chosen as the 1=e radius of the actual fiber

mode. When V ¼ 2:4, the positions of the two

maxima are almost at the same value of the A=Fratio, 0.101 for the solution given by (9) and 0.103

for the Gaussian approximation, whereas the

corresponding optimum coupling efficiencies are

78.4% and 81.5%, respectively. For V ¼ 1, themaximum efficiency is 66.6% at A=F ¼ 0:0194 for

the solution given by (9) and 81.5% at

A=F ¼ 0:0397 for the Gaussian approximation.

The optimum coupling efficiency can be studied

numerically from (9). For a given normalized fre-

quency V and a given ratio A=F ¼ tan#0, the

maximum value gopt of the coupling efficiency gcan be found from (9) by a suitable choice of theproduct ka. Since, the derivation of (9) is valid for

both small and large value of A=F , equation (9)

can be used to trace the optimum coupling effi-

ciency gopt up to large value of A=F . Fig. 3 shows

the plots of gopt as a function of A=F up to

A=F ¼ 10 for three different values of the

Fig. 3. Optimum coupling efficiency gopt as a function of the

ratio A=F for three values of the normalized frequency V ¼ 1,

1.7 and 2.4. The dashed line is the optimum efficiency, inde-

pendent from V , when the fiber mode is approximated by a

Gaussian function and the incident filed is given by (3).

Fig. 4. Optimum coupling parameter nopt and optimum cou-

pling efficiency gopt as a function of the normalized frequency Vfor small value of A=F . The dashed line is the optimum effi-

ciency of 81.5% in the Gaussian approximation, independent

from V .

42 M. Lazzaroni, F.E. Zocchi / Optics Communications 237 (2004) 37–43

normalized frequency, V ¼ 1, 1.7 and 2.4. The

optimum coupling efficiency is a decreasing func-

tion of A=F . It reaches the highest values when the

ratio A=F is small whereas for very large values of

A=F it approaches zero asymptotically. Even if the

use of A=F values greater than about 1 is notrecommended, the study of the coupling efficiency

when the A=F ratio approaches 1 could be of in-

terest in inter-satellite free space optical commu-

nications where both the requirement of large

collecting area and the dimensional constraints

limiting the focal length should be taken into ac-

count. In addition increasing the value of A=Fimproves the field of view of the optical systemallowing the implementation of a fine fast tracking

system by just moving the fiber optics on the focal

plane. At design stage a trade-off should be per-

formed between these benefits and the drawback

represented by the decreasing of the coupling effi-

ciency at increasing values of A=F .The dashed line in Fig. 3 is the optimum effi-

ciency when the fiber mode is approximated by aGaussian function whereas the incident field is

given by (3). In this case the optimum efficiency is

independent from the normalized frequency V that

determines the width of the Gaussian approxima-

tion of the fiber mode. Indeed for a given value of

V and of A=F , it is possible to select the wave

number k in (3) to match the Gaussian shape of

the fiber mode. At low values of the ratio A=F(paraxial approximation) the incident field (3)

approaches an Airy pattern and the coupling effi-

ciency is approximated by (1). Consequently, as

indicated in Fig. 3, for small values of A=F the

dashed line approaches the efficiency of 81.5% gi-

ven by the maximum of Eq. (1).

2.3. Optimum coupling at small angles

Since the best coupling conditions are reached

at small values of A=F ¼ tan#0, it is worth

studying (9) when the angle #0 is small. Under

this assumption a very general relation for the

optimum coupling can be found. When #0 is

small the denominator of (9) can be simplified as

follows,

k2a2 � n2 ¼ k2a2 1�

� sin2 #�ffi k2a2;

since n ¼ ka sin# and 06#6#0. Thus (9) can be

written as

g ffi 1

n20

Z n0

0

I n; Vð Þndn� �2

; ð10Þ

with n0 ¼ ka tan#0 ffi ka sin#0. The condition that

g is at maximum gives the following relation be-tween the optimum value nopt of n0 and the nor-

malized frequency V ,Z nopt

0

I n; Vð Þndn ¼ I nopt; V� �

: ð11Þ

Solving (11) numerically gives the relation betweennopt and V that is shown in Fig. 4 where the cor-

responding optimum value gopt of g is also plotted.

The dashed line in the figure represents the opti-

mum efficiency of 81.5% in the Gaussian approx-

imation, independent from V . It can be noted that

for a weakly guided mode, at low value of V , theoptimum coupling efficiency calculated above is

well below the approximate Gaussian prediction of81.5%.

The plot of gopt in Fig. 4 has been reported in

[3]. In that case however, the calculation was done

for a particular fiber core radius a ¼ 2 lm and

numerical aperture NA ¼ 0:11. Here the result

follows from (10) and (11) and it is thus shown to

be of general validity, dependent on V but not

M. Lazzaroni, F.E. Zocchi / Optics Communications 237 (2004) 37–43 43

separately on a, k or NA. In addition in Fig. 4 the

value nopt of the design parameter n0 ¼ ka sin#0

that gives the optimum coupling efficiency is also

plotted.

The two curves in Fig. 4 can be very well fitted

by a second order polynomial in V . By applyingthe Least Square Method, the following two ap-

proximations are found:

nopt ¼ a1 þ a2 � V þ a3 � V 2; ð12Þgopt ¼ b1 þ b2 � V þ b3 � V 2; ð13Þ

where: a1 ¼ �1:928, a2 ¼ 2:784, a3 ¼ �0:472,b1 ¼ 48:398, b2 ¼ 21:895, b3 ¼ �3:889. The maxi-

mum deviation from the exact curves is 0.027 for

nopt and 0.345% for gopt. The two above relations

(12) and (13) can be used to easily determine the

optimum coupling parameter and the corre-sponding coupling efficiency for a given value of

the normalized frequency V .

3. Conclusion

A general expression for the coupling efficiency

of plane wave light into a single-mode step-indexfiber optics has been presented allowing the study

of the conditions of optimum coupling for this

specific scenario that finds application in astron-

omy and in inter-satellite free space optical com-

munications. It is shown that the highest efficiency

is obtained, as expected, for small value of the ratio

between the coupling optics radius and focal

length. In this condition the optimum couplingefficiency is function of the normalised frequency

only and it can be achieved by a proper choice of a

design parameter, which again depends only on the

normalised frequency. Consequently it has been

possible to write two simple quadratic approxi-

mations for these two quantities allowing an easy

determination of the optimum coupling conditions.

The Gaussian approximation of the fiber modeis one of the sources of error in the evaluation of

the coupling efficiency of light into a single-mode

step-index fiber optics. Since the field mode in a

parabolic graded-index fiber is itself Gaussian,

further investigation in this direction can possibly

show that a higher theoretical value for the opti-

mum coupling efficiency can be obtained in thistype of fibers. In particular it is possible that the

coupling efficiency represented by the dashed line

in Fig. 3 can be achieved in graded-index fiber.

This would be a consistent improvement with re-

spect to the solid lines corresponding to low values

of V in step-index fibers.

Acknowledgements

The authors thank Fabio Marioni for helpful

discussion.

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