Spectral-domain intermodal interference and the effect of a low-resolution spectrometer

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Optik

Optics

Optik 116 (2005) 469–474 www.elsevier.de/ijleo

Spectral-domain intermodal interference and the effect of a low-resolution spectrometer Petr Hlubina Department of Physics, Technical University Ostrava, 17. listopadu 15, 708 33 Ostrava-Poruba, Czech Republic Received 21 December 2004; accepted 15 February 2005

Abstract Spectral-domain intermodal interference is analysed theoretically at the output of a few-mode optical fibre alone and at the output of the optical fibre in a tandem configuration with a Michelson interferometer. The theoretical analysis is performed under general measurement conditions when a broadband source and a low-resolution spectrometer of a Gaussian response function are considered and when the first- and second-order intermodal dispersion effects in the optical fibre are taken into account. The theoretical analysis is performed for two different examples of dispersion curves of a two-mode optical fibre and the effect of the limiting factors is specified. r 2005 Elsevier GmbH. All rights reserved. Keywords: Broadband source; Spectral interference; Optical fibre; Dispersion curve; Michelson interferometer; Spectrometer; Response function; Equalization wavelength

1. Introduction Spectral-domain characteristics and parameters of optical fibres such as losses, cut-off wavelengths for different fibre modes, wavelength dependences of both the beat lengths and intermodal group optical path differences (OPDs), etc., are important from the point of view of the development of new types of optical fibres and fibre-optic sensors. Spectral-domain interferometry [1] with channelled spectrum detection plays a crucial role in measuring either dispersion characteristics of standard optical and photonic crystal fibres [2–8] or intermodal interference [9,10] and using it for sensing applications [6]. Spectral-domain interferometry has been used to measure polarization-mode dispersion in single-mode optical fibres [2], birefringence dispersion in Tel.: +420 59 732 3134; fax: +420 59 732 3139.

E-mail address: [email protected] (P. Hlubina). 0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2005.03.004

polarization-maintaining fibres [3–5] and intermodal dispersion in two-mode optical fibres [6–8]. Recently, a new measurement technique employing a low-resolution spectrometer at the output of a tandem configuration of a Michelson interferometer and a twomode optical fibre, has been used to measure the intermodal dispersion in circular-core [11], ellipticalcore [12,13] and bow-tie [14] optical fibres. In comparison with the standard time-domain tandem interferometry [15], the technique of spectral-domain tandem interferometry [16,17] uses a series of the recorded spectral interferograms to resolve the so-called equalization wavelengths [11–14]. At these wavelengths, the overall group OPD in the tandem configuration is zero so that the wavelength dependence of the intermodal group OPD in a two-mode optical fibre can be obtained directly. More recently, the technique was used to measure the dispersion of birefringence in optical fibres [13,18].

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P. Hlubina / Optik 116 (2005) 469–474

In this paper, spectral-domain intermodal interference is analysed theoretically at the output of a few-mode optical fibre alone and at the output of the optical fibre in a tandem configuration with a Michelson interferometer. The theoretical analysis is performed under general measurement conditions when a broadband source and a low-resolution spectrometer of a Gaussian response function are considered and when the first- and secondorder intermodal dispersion effects in the optical fibre are taken into account [16,17]. The theoretical analysis is performed for two different examples of dispersion curves of a two-mode optical fibre and the effect of the limiting factors is specified.

2. Theory

Consider now that the optical field at the output of the optical fibre is analysed by a spectrometer characterized by a response function Rðo o0 Þ of a given spectral bandpass. The spectral intensity IðR; z; oÞ as a function of the angular frequency o adjusted by the spectrometer can be expressed by the convolution relation Z 1 IðR; z; oÞ ¼ SðR; z; o0 ÞRðo o0 Þ do0 . (4) 0

Substituting Eq. (2) into Eq. (4), and assuming that the modal spectral density contributions are within the spectrometer bandpass constant, i.e. S k ðR; o0 Þ ¼ S k ðR; oÞ, for the spectral intensity at the output of the optical fibre alone we obtain X IðR; z; oÞ ¼ Sm ðR; oÞGmm ð0; oÞ m

Consider a random scalar field, represented by a statistical ensemble fEðR; z; oÞg. We take EðR; z; oÞ to be the complex representation of a linearly polarized optical field at the angular frequency o, propagating in the positive z direction along the axis (R is the position vector in the transverse x2y plane) of a lossless optical fibre (see Fig. 1), in which no birefringence effects are present. The measurable quantity, the spectral density SðR; z; oÞ, which is defined as [1] SðR; z; oÞ ¼ hjEðR; z; oÞj2 i,

(1)

where the angular brackets denote the ensemble average and the brackets j j represent the modulus, can be expressed under both the conditions of optical fibre excitation by a spatially coherent source field and the mode guiding without mode conversion in the form [16] SðR; z; oÞ X XX ¼ Sm ðR; oÞ þ 2 ½Sm ðR; oÞS n ðR; oÞ1=2 m

m4

ð2Þ

where Dbmn ðoÞ is the difference in the propagation constants between mth and nth mode, and Sk ðR; oÞ are modal contributions to the spectral density given by the expression S k ðR; oÞ ¼ c2k ðoÞE 2k ðR; oÞGðoÞ ¼ I k ðR; oÞGðoÞ,

XX m4

½S m ðR; oÞS n ðR; oÞ1=2

n

RefGmn ðz; oÞg,

ð5Þ

where Z

1

Rðo o0 Þ expf i½Dbmn ðo0 Þzg do0 .

Gmn ðz; oÞ ¼

(6)

0

The knowledge of the spectral density SðR; z; oÞ at the output of the optical fibre alone enables us to express the spectral density SM ðR; z; tM ; oÞ and the spectral intensity I M ðR; z; tM ; oÞ of the optical field at the output of the optical fibre in a tandem configuration with a Michelson interferometer (see Fig. 2). The interferometer is characterized by the time delay between beams tM ¼ DM =c expressed via the OPD DM adjusted between beams and the velocity c of light in vacuum. From our previous theoretical considerations [16] we have S M ðR; z; tM ; oÞ

n

Ref i½Dbmn ðoÞzg,

þ2

(3)

¼ SðR; z; oÞ þ RefSðR; z; oÞ expð iotM Þg,

ð7Þ

where SðR; z; oÞ is given by Eq. (2). The spectral intensity I M ðR; z; tM ; oÞ as a function of both the optical delay tM and the angular frequency o adjusted by the spectrometer can be expressed as Z 1 S M ðR; z; tM ; o0 ÞRðo o0 Þ do0 . I M ðR; z; tM ; oÞ ¼ 0

where ck ðoÞ and E k ðR; oÞ are the real excitation coefficients and the spatial field configurations, respectively, of the orthonormal guided eigenmode k, and GðoÞ is the source field spectral density.

(8)

M1 BS Spectral fringes

WLS

Spectral fringes

WLS L1

P

L2

FUT

A

Fig. 1. Experimental set-up for measuring intermodal interference in a fibre under test (FUT).

L1

MI M2

P L2

FUT

A

Fig. 2. Experimental set-up with a Michelson interferometer (MI) for measuring intermodal interference in a fibre under test (FUT).

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Substituting Eqs. (2) and (7) into Eq. (8) we obtain I M ðR; z; tM ; oÞ ¼ IðR; z; oÞ þ

X

3. Wavelength-domain limiting values Let us consider a two-mode optical fibre characterized by the wavelength-dependent difference between propagation constants Db10 ðlÞ. The spectral intensity at the output of the two-mode optical fibre alone is

S m ðR; oÞRefGmm ð0; tM ; oÞg

m

XX þ ½S m ðR; oÞS n ðR; oÞ1=2 m4

n

RefGmn ðz; tM ; oÞ þ Gmn ð z; tM ; oÞg,

ð9Þ

where IðR; z; oÞ is given by Eq. (5) and Gmn ðz; tM ; oÞ is defined as Z 1 Rðo o0 Þ expf i½Dbmn ðo0 Þzg Gmn ðz; tM ; oÞ ¼

IðR; z; lÞ ¼ I ð0Þ ðR; lÞf1 þ V ðz; lÞV ðR; lÞ cos ½b10 ðlÞz þ f10 ðz; 0; lÞg,

where I ðR; lÞ is the reference spectral intensity and V ðz; lÞ ¼

ð10Þ

Let us assume now that the spectrometer response function is a Gaussian function with half-width GR   R0 ðo o0 Þ2 0 Rðo o Þ ¼ 1=2 exp . (11) p GR G2R Next, let us assume that the difference in the propagation constants between mth and nth mode can be expanded in a Taylor series around the mean frequency o 0

Dbmn ðo Þ  Dbmn ðoÞ þ þ

Db0mn ðoÞðo0 2 0

00 1 2 Dbmn ðoÞðo

oÞ

oÞ ,

ð12Þ

where Dbmn ðoÞ, Db0mn ðoÞ and Db00mn ðoÞ are the difference, the first and second derivatives of the difference the propagation constants between mth and nth mode, respectively, at the mean frequency o. Substituting Eq. (12) into Eq. (10) we obtain [17]   1 pðtM þ Dtmn Þ2 Gmn ðz; tM ; oÞ ¼ exp

2t2c ð1 þ r2mn Þ ð1 þ r2mn Þ1=4 expf i½Fmn ðz; tM ; oÞg,

ð13Þ

where Dtmn ¼ Dtmn ðz; oÞ ¼ Db0mn ðoÞz

(14)

is the intermodal group delay between mth and nth mode, rmn ¼ rmn ðz; oÞ ¼ p

Db00mn ðoÞz t2c

(15)

is the second-order dispersion parameter with tc ¼ ð2pÞ1=2 =GR as the modified coherence time [1], and Fmn ðz; tM ; oÞ is the phase function given by Fmn ðz; tM ; oÞ ¼ otM þ Dbmn ðoÞz þ fmn ðz; tM ; oÞ,

(16)

where fmn ðz; tM ; oÞ ¼

1 pðtM þ Dtmn Þ2 rmn arctanðrmn Þ 2 2t2c ð1 þ r2mn Þ

(17)

is an additional phase term due to the first- and secondorder intermodal dispersion.

ð18Þ

ð0Þ

0

expð io0 tM Þ do0 .

471

1 ½1 þ r10 ðz; lÞ2 1=4 ( ) p½Dg10 ðz; lÞ2 exp 2ðtc cÞ2 ½1 þ r210 ðz; lÞ

ð19Þ

is a visibility term, which is responsible for the visibility reduction due to the first- and second-order intermodal dispersion and V ðR; lÞ ¼ ¼

2½S 0 ðR; lÞS 1 ðR; lÞ1=2 ½S 0 ðR; lÞ þ S1 ðR; lÞ 2½I 0 ðR; lÞI 1 ðR; lÞ1=2 ½I 0 ðR; lÞ þ I 1 ðR; lÞ

ð20Þ

is a photometric visibility term, which is responsible for the visibility reduction due to the modal overlap. Using Eq. (14) we can express the wavelength-dependent intermodal group OPD Dg10 ðz; lÞ l2 d½Db10 ðlÞz ¼ DN 10 ðlÞz (21) dl 2p or we can introduce the so-called fibre equalization wavelength l0 [17] at which the intermodal group OPD is zero, i.e. Dg10 ðz; l0 Þ ¼ 0. Similarly, using Eq. (15) we can express the wavelength-dependent second-order dispersion parameter r10 ðz; lÞ   p d½Dg10 ðz; lÞ DlR 2 r10 ðz; lÞ ¼ 2 dl l   p DlR 2 ¼ DN 010 ðlÞz , ð22Þ 2 l Dg10 ðz; lÞ ¼

where DN 010 ðlÞ is the derivative of the group refractive indices difference DN 10 ðlÞ. We can also express the length zr¼1 of the optical fibre for which r10 ðz; lÞ ¼ 1 zr¼1 ¼

2 l2 1 , p jDN 010 ðlÞj Dl2R

(23)

or the corresponding maximum length zr¼1;max of the optical fibre. Next, if fibre length z5zr¼1;max , the wavelengthdependent visibility term V ðz; lÞ is (   ) p2 DN 10 ðlÞzDlR 2 V ðz; lÞ ¼ exp (24) 2 l2

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and we can express the length zV ¼0:1 of the optical fibre for which V ðz; lÞ ¼ 0:1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 1 2 ln 10 zV ¼0:1 ¼ , (25) p jDN 10 ðlÞj DlR

equalization wavelength corresponding to the change dðDM Þ of the OPD

or the corresponding maximum length zV ¼0:1;max of the optical fibre. We can also express the wavelengthdependent period of the spectral modulation LðlÞ

or the corresponding maximum length zmax of the optical fibre for the known minimum resolved shift dl1 min of the equalization wavelength l1 . Similar relations hold for the equalization wavelength l2 .

¼

l2 , jDN 10 ðlÞzj

dðDM Þ DN 010 ðl1 Þz

(30)

(26)

or the corresponding minimum length zmin of the optical fibre when L  lmax lmin . Finally, the spectral intensity I M ðR; z; DM ; lÞ at the output of a tandem configuration of a Michelson interferometer and a two-mode optical fibre of length z can be expressed in the wavelength domain as I M ðR; z; DM ; lÞ (

 ¼ I ð0Þ ðR; lÞ exp ðp2 =2Þ½DM DlR =l2 2 cos½ð2p=lÞDM 

 þV ðR; lÞ exp ðp2 =2Þ½Dg10 ðz; lÞDlR =l2 2 cos½Db10 ðlÞz

n

2 o þ 0:5 exp ðp2 =2Þ ðDM Dg10 ðz; lÞÞDlR =l2 cos½ð2p=lÞDM Db10 ðlÞz n

2 o þ 0:5 exp ðp2 =2Þ ðDM þ Dg10 ðz; lÞÞDlR =l2 ) cos½ð2p=lÞDM þ Db10 ðlÞz . ð27Þ The spectral intensity I M ðR; z; DM ; oÞ represents the superposition of the four spectral interference fringes which have different visibilities at different measured wavelengths l and different OPDs DM adjusted in the Michelson interferometer. If a suitable OPD DM , which fulfils the conditions DM  jDg10 ðz; lÞj and jDg10 ðz; lÞjb l2 =DlR , is adjusted in the interferometer, the equalization wavelengths l1 and l2 [17], which differ from the fibre equalization wavelength l0 , can be introduced at which the intermodal group OPDs match the OPD DM , i.e. DM ¼ Dg10 ðz; l1 Þ and DM ¼ Dg10 ðz; l2 Þ. We can express the spectral width Dl0 in which the interference fringes are resolved around l0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ln 10 l20 1 Dl0 ¼ . (28) 0 p jDN 10 ðl0 Þzj DlR We can also express the spectral width Dl1 in which the interference fringes are resolved around wavelength l1 pffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ln 5 l21 1 Dl1 ¼ (29) 0 p jDN 10 ðl1 Þzj DlR or the corresponding maximum length zmax of the optical fibre for the known minimum width Dl1 min . Moreover, we can express the change dl1 of the

4. Modelling: Two-mode optical fibres First, let us consider a two-mode optical fibre characterized by the wavelength dependence of the difference between propagation constants Db10 ðlÞ as shown in Fig. 3. The corresponding wavelength dependence of the intermodal group OPD per unit fibre length Dg10 ðz; lÞ=z is shown in Fig. 3 by the bold curve. Fig. 4 shows the theoretical spectral intensity IðR; z; lÞ given by Eq. (18) when V ðR; lÞ ¼ 0.8 and z ¼ 0:1 m, and a low-resolution spectrometer with DlR ¼ 3 nm [19] (bold curve) or a high-resolution spectrometer with DlR  0 nm is used. We clearly see the effect of the resolving power of the spectrometer on the visibility of the interference fringes. Using Eqs. (23) and (25) in the specified wavelength range, we obtain for the maximum lengths zr¼1;max and zV ¼0:1;max of the optical fibre values 10 and 0.15 m, respectively. Next we choose the case of a low-resolution spectrometer and the fibre length z ¼ 1 m so that no spectral fringes are resolved at the output of the fibre alone. The corresponding spectral intensity IðR; z; lÞ is shown in Fig. 5 by the dashed curve. The same figure shows the spectral intensity I M ðR; z; DM ; lÞ at the fibre output when the OPD adjusted in the interferometer is DM ¼ 500 mm. The interference fringes are resolved around the equalization wavelength l1 . 500 −100

550

600

650

700

750

800 2000

−300 1600 −500

1200

−700

800

−900 −1100 500

∆β10(λ) (m-1)

l2 jDg10 ðz; lÞj

∆g10(z;λ)/z (µm m-1)

LðlÞ ¼

dl1 ¼

400

550

600

650

700

750

0 800

λ (nm)

Fig. 3. Wavelength dependence of the difference between propagation constants Db10 ðlÞ and the intermodal group OPD per unit fibre length Dg10 ðz; lÞ=z (bold curve).

ARTICLE IN PRESS P. Hlubina / Optik 116 (2005) 469–474 500 160

2

473 550

600

650

700

750

800 −3400

80

−160

−3600

g

0.8

−80

∆β10(λ) (m−1)

10

1.2

−3500

0

∆

I(R,z;λ) (a. u.)

−1

(z;λ)/z (µm m )

1.6

−240

0.4

−320

500

0 500

550

600

650

700

750

−3700 800

λ (nm)

550

600

650

700

750

800

(nm)

Fig. 4. Theoretical spectral intensity IðR; z; lÞ for two different resolving powers of the spectrometer.

Fig. 6. Wavelength dependence of the difference between propagation constants Db10 ðlÞ and the intermodal group OPD per unit fibre length Dg10 ðz; lÞ=z (bold curve).

2 500 2

550

600

650

700

750

800 2

1.2

1.2

0.8

0.8

0.4

0.4

0 500

550

600

650

700

750

I(R,z;λ) (a. u.)

1.6 I(R,z;λ) (a. u.)

IM(R,z,∆M;λ) (a. u.)

1.6 1.6

1.2

0.8

0.4

0 800

0 500

550

λ (nm)

Fig. 5. Theoretical spectral intensities IðR; z; lÞ I M ðR; z; DM ; lÞ for z ¼ 1 m and DM ¼ 500 mm.

600

650

700

750

800

λ (nm)

and

Fig. 7. Theoretical spectral intensity IðR; z; lÞ for two different resolving powers of the spectrometer.

Second, let us consider a two-mode optical fibre characterized by the wavelength dependence of the difference between propagation constants Db10 ðlÞ as that shown in Fig. 6. The corresponding wavelength dependence of the intermodal group OPD per unit fibre length Dg10 ðz; lÞ=z is shown in Fig. 6 by the bold curve. Fig. 7 shows the theoretical spectral intensity IðR; z; lÞ given by Eq. (18) when V ðR; lÞ ¼ 0.8 and z ¼ 0:5 m, and a lowresolution spectrometer with DlR ¼ 3 nm [19] (bold curve) or a high-resolution spectrometer with DlR  0 nm is used. We clearly see the effect of the resolving power of the spectrometer on the visibility of the interference fringes. Using Eqs. (23) and (25) in the specified wavelength range, we obtain for the maximum lengths zr¼1;max and zV ¼0:1;max of the optical fibre values 16 and 0.41 m, respectively. Finally, we choose the case of a low-resolution spectrometer and the fibre length z ¼ 2:5 m so that the

spectral fringes are resolved at the output of the fibre alone only around the equalization wavelength l0 . The corresponding spectral intensity IðR; z; lÞ is shown in Fig. 8 by the dashed curve. The same figure shows the spectral intensity I M ðR; z; DM ; lÞ at the fibre output when the OPD adjusted in the interferometer is DM ¼ 250 mm. The interference fringes are resolved around three different equalization wavelengths l1 , l0 and l2 .

5. Conclusions In this paper, spectral-domain intermodal interference was analysed theoretically at the output of a few-mode optical fibre alone and at the output of the optical fibre

ARTICLE IN PRESS P. Hlubina / Optik 116 (2005) 469–474

IM(R,z,∆M;λ) (a. u.)

500 2

550

600

650

700

750

800 2

1.6

1.6

1.2

1.2

0.8

0.8

0.4

0.4

0 500

550

600

650

700

750

I(R,z;λ) (a. u.)

474

0 800

λ (nm)

Fig. 8. Theoretical spectral intensities IðR; z; lÞ I M ðR; z; DM ; lÞ for z ¼ 2:5 m and DM ¼ 250 mm.

and

in a tandem configuration with a Michelson interferometer. The theoretical analysis was performed under general measurement conditions when a broadband source and a low-resolution spectrometer of a Gaussian response function were considered and when the firstand second-order intermodal dispersion effects in the optical fibre were taken into account. The theoretical analysis was performed for two different examples of dispersions curves of a two-mode optical fibre and the effect of the limiting factors was specified. The results obtained are important from the point of view of both measuring dispersions of specialty fibres, such as microstructured ones, and proposing new schemes of fibre-optic sensors employing a low-resolution spectrometer.

6. Acknowledgement The research was partially supported by the Grant Agency of the Czech Republic, Project No. 202/03/0776.

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