Overlap relation between free-space LaguerreGaussian modes and step-index fiber modesROBERT BRÜNING,1 YINGWEN ZHANG,2 MELANIE MCLAREN,3 MICHAEL DUPARRÉ,1 AND

ANDREW FORBES3,*1Institute of Applied Optics, Abbe Center of Photonics, Friedrich Schiller University Jena, Fröbelstieg 1, D-07743 Jena, Germany2Council for Scientific and Industrial Research, National Laser Center, P.O. Box 395, Pretoria 0001, South Africa3School of Physics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa*Corresponding author: andrew.forbes@wits.ac.za

Received 19 June 2015; revised 29 July 2015; accepted 29 July 2015; posted 30 July 2015 (Doc. ID 243373); published 20 August 2015

We investigated the overlap relation of the free-space Laguerre–Gaussian modes to the corresponding linearlypolarized modes of a step-index fiber. To maximize the overlap for an efficient coupling of the free-space modesinto a fiber, the scale-dependent overlap was theoretically and experimentally determined. The presented studiespave the way for further improvement of free-space to fiber optical connections. © 2015 Optical Society of America

OCIS codes: (060.2310) Fiber optics; (090.1760) Computer holography; (140.3295) Laser beam characterization.

http://dx.doi.org/10.1364/JOSAA.32.001678

1. INTRODUCTION

Multi-mode fibers (MMF) are widely used in a plethora ofapplications, such as optical sensors [1], fiber lasers [2], andoptical communication [3]. Besides the possibility of all-fiberdevices, the connection between the fiber and its supportedmodes with the free-space modes is important to control theoptical properties of the emerging beam or the excited fielddistributions at the fiber input. The simplest connection isgiven by fibers with a parabolic refractive index profile, typicalof graded-index fibers, whose modes can be described byLaguerre–Gaussian (LG) functions and are solutions of thefree-space wave equation [4]. However, in all other cases,the fiber modes do not match the free-space solutions, resultingin no stable propagation of the emerging beams in free space. Inaddition, in general, free-space modes are not suitable to exciteselective pure fiber modes, and additional beam shaping tech-niques are required, such as the use of computer generatedholograms [5]. The main disadvantages of such beam shapingtechniques are low efficiency and the fact that they result inhigh transformation losses. For the common type of step-indexfibers, which show the same cylindrical symmetry as in free-space, it is possible to approximate the fiber modes by suitablefree-space modes [6].

In this paper, we investigate the approximation of step-indexfiber modes by free-space modes theoretically as well as exper-imentally, to evaluate the quality and the limitation of thisapproach. We determine the overlap relation between bothmode sets as a function of the scale parameter for the free space,as well as the fiber parameter, V . For the theoretical investiga-tions, we develop an analytical solution for this overlap

problem, which allows us to study a wide parameter range.Additionally, we investigate experimentally the overlap relationby applying the correlation filter method [7] and verify the ana-lytical solution. Our results will be of interest to studies inwhich fiber to free-space links are necessary.

2. FUNDAMENTALS

At the transition from free space to fiber, and vice versa, onealways has mode coupling between the free-space modes on theone side and the fiber modes on the other. To achieve a maxi-mized coupling efficiency and a low crosstalk between modes ofdifferent orders, the scaling of the free-space beam has to beadapted to the fiber.

In the case of a weakly guiding step-index fiber, the fibermodes are given by the linearly polarized (LP) mode set.For that, the field distribution, Flp�r;φ�, is given by thesolution of the scalar Helmholtz equation. Considering a cylin-drically symmetric fiber with a core radius a, the solution isgiven by [4]

LPlp�r;φ� � N lp

8<:

Jl�νlpra

�∕Jl�νlp� for r < a

K l

�μlpra

�∕K l�μlp� for r ≥ a

× eilφ;

(1)

where N lp is a normalization constant, Jl denotes the lth-order Bessel function of the first kind, and K l denotes thelth-order modified Bessel function of the second kind, withνlp and μlp being the normalized propagation constants of

1678 Vol. 32, No. 9 / September 2015 / Journal of the Optical Society of America A Research Article

1084-7529/15/091678-05$15/0$15.00 © 2015 Optical Society of America

the core and cladding, respectively. The exact expression for thenormalization constant is derived in Appendix A. For the de-scription of the LP modes, the additional fiber parameter V isrequired, which is defined as

V 2 � ν2 � μ2 ��2π

λa�

2

�n2core − n2cladding�; (2)

where ncore and ncladding are the refractive indices of the core andcladding material, respectively. This fiber parameter defines theamount of modes and their propagation constants.

Since the step-index fiber has cylindrical symmetry, theadapted free-space mode set is given by the LG modes, whichare the solutions of the paraxial Helmholtz equation in a cylin-drically symmetric coordinate system. For that, the solution atthe waist position is given by

LGpl�r;φ� � Mpl

�2r2

w20

�jlj2

Ljljp

�2r2

w20

�e− r2

w20eilφ; (3)

where Ljljp are the associated Laguerre polynomials, w0 is thefundamental Gaussian radius, andMpl � 1

w0� 2p!π�l�p�!�

12 is a nor-

malization factor.Comparing the modes described by Eq. (1) and Eq. (3), we

notice that they show the same azimuthal dependence and acharacterization of the radial order by the amount of root pointsin the intensity distribution. Hence, corresponding modescan be found by choosing the same azimuthal order and thefield functions with the same amount of roots in radialdirection.

Since both mode sets differ in the actual shape of the radialfunction, an adaption of the scale parameters is needed for thebest possible matching of both mode sets. The matching of thescale can be evaluated by the overlap relation:

ηn �ZZ

LGn�r;φ�LP�n�r;φ�dA; (4)

where ηn defines the amount of power that is coupled from onemode into the other at the transition between both mode sets.It can reach values between one when the fields are perfectlymatched, and zero for orthogonal fields. The amount of power1 − ηn which is not coupled into the desired mode goes in othernonorthogonal, reflecting, or radiating modes to satisfy energyconservation.

3. ANALYTICAL ANALYSIS

To optimize the mode overlap, we derive an analyticalexpression for the overlap relation as a function of the modeparameters. First, we separate the problem into a core and acladding part, corresponding to the solution of the fiber modesEq. (1). The eilφ term ensures that the LP and LG modesmust have the same azimuthal index l. Hence, to solve theoverlap relation, the radial part of Eq. (4) becomes the impor-tant one.

For deriving the solution for the core region, we note thatthe Bessel function of the first kind can be written as an infinitesum,

Jl�x� �X∞m�0

�−1�mm!Γ�m� l� 1�

�x2

�2m�l

; (5)

with Γ the gamma function, whereas the Laguerre polynomialcan be written as its generating function,

Ljljp �x� � 1

p!

�ddξ

�p 1

�1 − ξ�jlj�1exp

�−xξ1 − ξ

�����ξ�0

: �6�

Using Eq. (5) and Eq. (6) for the description of the mode func-tion, Eq. (1) and Eq. (3), respectively, the overlap of both modesets within the core is given by

ηcorelp � N lp

Jl�ν�Mpl

p!�

ffiffiffi2

p�l�ddξ

�p 1

�1 − ξ�jlj�1

×X∞m�0

� �−1�mm!Γ�m� l� 1�

�νw0

2a

�2m�l

�1 − ξ

1� ξ

�m�l�1

×Γ�m� l� 1� − Γ

�m� l� 1;

a2

w20

1� ξ

1 − ξ

������ξ�0

;

(7)

with Γ�a; z� being the incomplete gamma function; N lp andMpl being the normalization constant of the LP and LGmodes, respectively; a being the fiber core radius; and w0 beingthe Gaussian radius.

For the cladding region, an analytical solution for the over-lap integral can be found by a similar approach. For that, wemake use of the following relation between modified Besselfunctions of the second kind K l and the modified Bessel func-tions of the first kind Il:

K l�x� � limn→l

π

2 sin�nπ� �I −n�x� − In�x��; (8)

where the In can be written as an infinite sum,

In�x� �X∞m�0

1

m!Γ�m� n� 1�

�x2

�2m�n

: (9)

Using Eq. (8) and Eq. (9) to describe the field distribution ofthe fiber modes in the cladding region, together with the pre-viously used representation of the LG modes given by Eq. (3)and Eq. (6), the overlap of the cladding part becomes

ηcllp � limn→l

π

2 sin�nπ�N lp

K n�μ�Mpl

p!�

ffiffiffi2

p�n�ddξ

�p

×1

�1 − ξ�jlj�1

X∞m�0

8<:−1

m!

�μw0

2a

�2m−l

�1 − ξ

1� ξ

�m�1

×

24Γ

�m� n� 1; a

2

w20

1�ξ1−ξ

�Γ�m� n� 1�

�μw0

2a

�2n�1 − ξ

1� ξ

�n

×Γ�m� 1; a

2

w20

1�ξ1−ξ

�Γ�m − n� 1�

359=;������ξ�0

: (10)

The complete overlap relation of core and cladding is thengiven by the sum of Eqs. (7) and (10).

Research Article Vol. 32, No. 9 / September 2015 / Journal of the Optical Society of America A 1679

4. CORRELATION FILTER ANALYSIS

For a direct measurement of the overlap relation betweenthe LP and the corresponding LG modes, the correlationfilter method (CFM) was used. This optically performsthe integral relation given by Eq. (4) and allows theexperimental investigation of the overlap between LP andLG modes [7].

The experimental setup used for measuring the scale-dependent overlap relation is shown in Fig. 1. A plane wavehas illuminated a phase-only spatial light modulator (SLM).Applying the phase-only coding technique for complex valuedfield distributions proposed by Arrizòn et al. [8] enables theshaping of arbitrarily scaled LG modes. The generated LGmodes were then imaged by a telescopic 4f setup onto the cor-relation filter (CF), which had the field distribution of the LPmodes implemented as the transmission function. The CF wasrealized by a static binary computer generated hologram, wherethe encoding technique by Lee [9] was used, since, for the CF,no dynamic change of the transmission function was required.After an optical Fourier transformation with a lens in 2fconfiguration, the correlation signal was accessible at theCCD sensor as the intensity value on the optical axis,which was proportional to the overlap relation betweenboth modes.

Since the encoding method of the hologram required all thegenerated LG modes to be normalized to unit amplitude,energy conservation is, therefore, violated. To ensure the com-parability of the measured results by an appropriated powerscaling, a correction parameter for each encoded field was in-troduced, which followed by the renormalization of the modefields. The encoded mode fields, ψ̃n�r�, differ from the basismode fields, ψn�r�, by a constant factor, ψ̃n�r� � ψn�r�∕αn,where αn � maxfjψn�r�jg. An appropriate power scalingand correction of the measured intensities Imeas were achievedby multiplying it by the square of the correction parameter αnto produce the normalized intensity Inorm � Imeasα

2n [5].

Finally, the measured and corrected overlap values have tobe normalized to one, which is given by the overlap betweenthe LP mode with itself. For that, the LP mode encoded in theCF was generated with the SLM, and the measured intensitywas used to normalize the overlap relations obtained for thedifferent scaled LG modes.

This procedure for the generation of dynamically scaledLG modes and the evaluation of the overlap with a set ofLP modes with a fixed scale enable the measurement of thescale-dependent relation between both mode sets.

5. RESULTS

We applied the correlation filter method onto an LP mode setwith an underlying V parameter of 4.72, resulting in theappearance of six guided modes. In Fig. 2, the experimentallymeasured scale-dependent overlap is shown, together with thetheoretical curve calculated by our analytical solution. It can beseen that, for all LP modes of the investigated fiber, the corre-sponding LG modes are good approximations at a beam-to-core radius ratio of 0.75. In this fiber, the overlap is about0.99 for the LP01, LP11 and, LP21 modes, and about 0.98for the LP02 mode. A comparison of the scale-dependent over-lap of the four fiber modes shows explicit differencesbetween them. Noticeable are the differences in the rate ofdecrease of the overlap when the LG mode sizes are not ideal,whereas for the fundamental mode, LP01, the overlap falls rel-atively slowly, while the higher-order modes show an obviousfaster decline. Additionally, the optimal ratio between beamwidth and core radius for maximal overlap differs slightly forthe investigated modes from 0.70 for the LP21 [see Fig. 2(d)],up to 0.81 for the fundamental mode LP01 [see Fig. 2(a)].Further, the maximal reachable overlap for the higher-orderradial, LP02, mode [see Fig. 2(b)] with about 0.98 slightly lowercompared to the other modes. Since the comparison betweenthe experimental and theoretical results showed a goodagreement that demonstrates the reliability of our method,we used the theoretical expressions to study further theinfluences of the mode order or the V parameter on the overlaprelation.

As seen in Fig. 2, the overlap relation changes in shape,maximum position, and maximum value for different modes.

Fig. 1. Experimental setup for the measurement of the overlaprelation between the LP and LG modes, where the LG modes are gen-erated with the SLM and decomposed by the CFM. PW, plane wave;SLM, spatial light modulator; L1−3, lenses; CCD, camera.

1

0.8

0.6

0.4

0.2

0

1

0.8

0.6

0.4

0.2

0 0.5 1 21.5 0 0.5 1 21.5

1

0.8

0.6

0.4

0.2

0.5 1 21.5 0 0.5 1 21.5

1

0.8

0.6

0.4

0.2

over

lap

η01

over

lap

η02

over

lap

η 11

over

lap

η 21

beam-to-core radius ratio beam-to-core radius ratio

beam-to-core radius ratio beam-to-core radius ratio

meastheory

meastheory

meastheory

meastheory

01/LG00 02/LG10

11/LG01

(a) LP (b) LP

(c) LP (d) LP21/LG02

Fig. 2. Comparison between theoretical and experimental deter-mined overlap relation.

1680 Vol. 32, No. 9 / September 2015 / Journal of the Optical Society of America A Research Article

This means that there is always an optimal match, based on thefundamental Gaussian beam size, for the corresponding LGmode set. Applying our analytical expression, we have deter-mined the scale-dependent overlap for different higher-ordermodes for a fiber with V � 50 to demonstrate some effects.As shown in Fig. 3 for higher-order radial modes with azimu-thal index l � 0 and higher-order azimuthal mode with radialindex p � 1, the position of the optimal beam size becomessmaller with increasing azimuthal and radial indices. Thisbehavior is as expected since the beam width of LG modesof higher order follows w � w0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p� l� 1

pwith w0 the

Gaussian width, while, on the other hand, the LP modesare well confined within the core resulting in a decrease ofthe needed Gaussian width to match the scale of the corre-sponding modes. A second effect which can be seen is thatthe region with high overlap shrinks with increasing mode or-der, and a good adaptation of the scale parameters becomesmore crucial. Especially for modes of higher radial order, themaximal achievable overlap decreases with increasing modeorder, indicating the limits of the applied approximation.Hence, the approximation of the LP modes of the step-indexfiber by LG modes cannot be extended for arbitrary higher-order modes. Therefore, an acceptable coupling loss has tobe considered in possible applications where higher-ordermodes are used. Nevertheless, the results demonstrate thatthe low-order LP modes can be accurately approximated byLG modes.

Additionally, we investigated the influence of the underlyingV -parameter on the quality of our suggested approximation.We have calculated the optimal beam-to-core radius ratioand the maximal achievable overlap for different modes as afunction of the V parameter, as shown in Fig. 4. In Fig. 4(a),the change of the optimal beam-to-core radius ratio is depicted.It can be seen that, for low V values, in the case of LP01 or Vvalues close to the cutoff of a mode, the optimal beam-to-coreratio becomes larger than one indicating the bad confinementof the mode inside the core. For increasing V values, optimalbeam-to-core ratio rapidly decreases. In Fig. 4(b), it can be seenthat, when V increases and moves away from the cutoff of amode, the maximum overlap of that mode rapidly increases to apeak, then slowly starts to decrease. This drop in the maximumoverlap is faster for higher-order modes.

Finally, we have also compared the intensity profiles of LGand LP modes with maximum overlap. Four possible modes for

a fiber with a V parameter of 4.72 can be seen in Fig. 5, wherethe different mode profiles are normalized to unit intensity foreasier comparison. It can be seen clearly that the conformitydecreases with increasing mode order. Figure 5(a) illustratesthat the profiles of the fundamental modes, LP01 and LG00,are well matched, whereas the distinctions become more signifi-cant for the higher-order modes. For modes of higher-azimuthal order, such as the LP11 and LG01 modes shownin Fig. 5(b), the maxima of the ring-like intensity patternare slightly displaced. In addition for higher-order radial modes,structural differences occur, such as the displacement of theminima and the side lobe maxima, as seen in Fig. 5(c). Thedistinction becomes more prevalent with increasing modeorder, as seen in Fig. 5(d) for the higher-order radial andazimuthal modes, LP12 and LG21, where the side lobes shownotable deviations resulting in a lower achievable overlaprelation.

LP01 LP11 LP21 LP31LP02 LP12

V paramter V paramter

beam

-to-

core

rad

ius

ratio

max

imal

ove

rlap

3 4 5 6 7

1

.90

.92

.94

.96

.98

1.3

1.1

0.9

0.7

0.6

0.8

1.0

1.2

3 4 5 6 78 9 8 9

(b)(a)

Fig. 4. V parameter dependence of the overlap relation between theLP and corresponding LG modes. (a) Optimal ratio between theGaussian beam width and the core radius for a maximized overlap.(b) Maximal achievable overlap in dependence of V .

beam-to-core radius ratio-1 10

0

1

inte

nsity

[a.

u.]

.5

0

1

inte

nsity

[a.

u.]

.5

0

1

inte

nsity

[a.

u.]

.5

0

1

inte

nsity

[a.

u.]

.5

beam-to-core radius ratio-1 10

beam-to-core radius ratio-1 10

beam-to-core radius ratio-1 10

(b)(a)

(d)(c)LP02LG10

LP01LG00

LP11LG01

LP12LG11

Fig. 5. Comparison of the intensity profiles between theLP modes and the corresponding LG modes for the maximizedoverlap.

5 10 15

0.2

0.4

0.6

0.8

10 20 30 40

0.1

0.3

0.5

0.7

0.9be

am-t

o-co

re r

adiu

s ra

tio

radial order p azimuthal order

Fig. 3. Size-dependent overlap relation for modes with increasingradial and azimuthal order. The trend of the optimal beams size ishighlighted by the green line.

Research Article Vol. 32, No. 9 / September 2015 / Journal of the Optical Society of America A 1681

6. CONCLUSION

We have shown that the LP modes of a step-index fiber can beapproximated by a scale-adapted set of LG free-space modes.The validity of this approximation is shown experimentallyfor a fiber with a V parameter of 4.72. Further, we used ananalytical solution of the overlap problem to investigate thelimits of the approach for a wide spectrum of the LP modesand V parameters. We have found that the approximationis good but with certain limitations, which are set by the re-quirements on acceptable coupling losses and mode purityof the specific application. To provide high overlap, the LPmodes have to be far from their cutoff condition and of lowradial order. Additionally, the optimal beam-to-core radius ratiochanges for each corresponding mode pair. For a low V fiberwith few allowed modes (such as the example fiber), thisdependence is weak, and a good approximation of all modescan be found for an appropriately adapted LG mode set. Inthe case of highly multimode fibers, the LG mode set has tobe optimized individually for each corresponding mode pairor at least for different groups of them.

APPENDIX A

To evaluate the overlap relation between the LG and LP modes,we also need an analytical expression for the normalizationconstant, N lp, in Eq. (1). For that, we use the normalizationcondition,

1 �ZZ

LPlp�r;φ�LP�lp�r;φ�dA; (A1)

which gives

N 2lp �

1

2π

8<:Ra0

hJl�νlpra

�i2rdr

jJl�νlp�j2�

R∞a

hK l

�μlpra

�i2rdr

jK l�μlp�j2

9=;

−1

:

(A2)

The integrals are evaluated to be

2π

Za

0

Jl

�νra

�2

rdr � πa2

νlp�νJl�ν�2 − 2l Jl�ν�Jl�1�ν�

� νJl�1�ν�2� (A3)

and

2π

Z∞

a

K l

�μra

�2

rdr � limn→l

π2a2

4μ2 sin �nπ�2 fπμ2I 1−n�μ2�

�2πμI 1−n�μ��−μIn−1�μ� � nIn�μ��− πμI−n�μ�2 � π�2nμIn−1�μ�In�μ�− �4n2 � μ2�In�μ�2 � μ2I 1�n�μ�2��2μI−n�μ��2nK 1−n�μ� sin�nπ�� πμIn�μ��g; (A4)

with In�x� being the modified Bessel function of the first kind.Inserting Eqs. (A3) and (A4) into Eq. (A2) yields the analyticalexpression for the normalization constant for the solution of theoverlap relation for the core and cladding region, Eqs. (7) and(10), respectively.

Acknowledgment. The authors thank Darryl Naidoo fortechnical assistance and Sigmund Schröter for fabrication of thecorrelation filter.

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1682 Vol. 32, No. 9 / September 2015 / Journal of the Optical Society of America A Research Article