Spectrum analysis in cross-talk of series Fabry±Perot sensors inpath-matching di�erential interferometry

Yu-Lung Lo *, Ming-Hong Tsai, Chih-Chiang Tsao

Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan

Received 26 August 1998; received in revised form 22 October 1998; accepted 26 October 1998

Abstract

The signals between cascaded Fabry±Perot sensors have serious cross-talk problems due to complex phase interactions

between sensors. However, cross-talks in cascaded Fabry±Perot sensors for demodulation in a path-matching di�erentialinterferometry (PMDI) can be limited by corresponding path-matching read-out systems. This paper presents a method of usingspectrum analysis to design the series Fabry±Perot sensors without cross-talk. The spectrum transfer function in a light source

and Fabry±Perot cavities are simulated by a Gaussian distribution spectrum and a low-®nesse re¯ectivity, respectively. It showsthat two series Fabry±Perot sensors without cross-talk can be designed precisely and following the same procedure, even moreseries Fabry±Perot sensors without cross-talks in PMDI can be achieved. In-line ®ber etalon (ILFE) as a Fabry±Perot cavity is

used for multiplexing in the experiment. Experimental data shows that the cross-talk between two series Fabry±Perot sensors islimited. This design in two series Fabry±Perot sensors in PMDI can be applied to a two-parameter optical ®ber sensors. # 1999Elsevier Science Ltd. All rights reserved.

Keywords: Cross-talk; Fabry±Perot sensors; PMDI; Spectrum analysis

1. Introduction

The demodulation schemes for common-path inter-

ferometry akin to that used in Fabry±Perot sensors are

relatively limited and perhaps the simplest and most

robust demodulator is based on the concept of white

light interferometry and PMDI [1]. The primary ad-

vantage of using PMDI is that it uses a broad band

source which enhances the system robustness and obvi-

ates the use of long gage length required for most

demodulation schemes and, in addition allows coher-

ence based multiplexing [2]. Several Fabry±Perot sen-

sors in multiplexing have a serious problem in cross-

talks. To solve this problem in wavelength demodula-

tion, Humblet and Hamdy [3] analyzed crosstalk in

the optimization of single- and double-cavity Fabry±

Perot ®lters for wavelength-division multiple-access

(WDMA) networks. Di�erent from the design of

Fabry±Perot ®lters in networks, Chang and Sirkis [4]present several multiplexing systems to decoupleFabry±Perot sensors in parallel for phase demodula-tion. The capability of using PMDI with a singleFabry±Perot read-out interferometer to multiplex sev-eral sensors has been investigated by designing a singlehigh-®nesse Fabry±Perot resonator that produces mul-tiple path-match conditions. Also, coherence divisionmultiplexing using a single read-out resonator andcoherence/frequency division multiplexing using asingle phase modulator were designed for extractingsignals from the parallel Fabry±Perot sensors.However, the parallel Fabry±Perot sensors have lesscross-talk problem compared to the series ones. Later,Singh and Sirkis [5] designed two Fabry±Perot sensors:an in-line ®ber etalon (ILFE) and an intrinsic Fabry±Perot (IFP) ®ber in series for a two-parameter sensorto measure the axial strain and temperature simul-taneously. However, without a quantitative analysis intwo cavity lengths of Fabry±Perot sensors for a PMDImodulation, an estimated pair was designed to preventcross-talk.

Optics & Laser Technology 30 (1998) 395±401

0030-3992/98/$19.00 # 1999 Elsevier Science Ltd. All rights reserved.

PII: S0030-3992(98 )00068 -1

* Corresponding author. Tel.: +886-6-2757575 ext. 62123; fax:

+886-6-2352973; e-mail: loyl@mail.ncku.edu.tw

In this paper, quantitative design in lengths of cav-ities for series Fabry±Perot sensors in PMDI usingspectrum analysis is presented. Experimental data alsodemonstrates that it is feasible to use spectrum analysisto design series Fabry±Perots without cross-talk.

2. Analysis of cross-talk in multiple Fabry±Perotsensors

Multiple-beam fringes with two parallel plates hadbeen analyzed by Born and Wolf [6] in the transmittedmode. It was described that the two plane parallelplates, with surface of high re¯ectivity, were placedone behind the other and illuminated with plane wavesof monochromatic light. The optical system illumi-nated with a white-light source in Fig. 1 can be ana-lyzed in a similar manner by using the spectrumanalysis. The analysis of the sensing system is accom-plished by treating the sensor and read-out interferom-eters as spectral transfer functions. Following thisapproach, the time-dependent intensity function at thedetector is found from [7]

I � Z�H R

r �k�H Sr �k�i�k� dk; �1�

where I is the intensity function at the detector; Z isthe system transmission loss; k is the wave number;Hr

R(k) is the re¯ective transfer function of Fabry±Perotread-out interferometer; Hr

S(k) is the re¯ective transferfunction of Fabry±Perot sensor and i(k) is the powerspectrum of the source. Assuming a Gaussian distri-bution spectrum (a reasonably good approximation formany broad-band sources), the spectral power densityof the light source is given by [8]

i�k� � 1������psp exp

�ÿ�kÿ k0

s

�2�; �2�

where s is the spectral width that is the halfwidth

between points at which the spectral density is equalto the peak value divided by e. The source is assumedto have unit intensity and a Gaussian spectral distri-bution symmetric about wave number k0. Hr

R(k) canbe written as a Fourier expansion of an Airy transferfunction for a Fabry±Perot read-out system shownas [6]

H Rr �k� � 2

R

1� Rÿ 2

1ÿ R

1� R

X1m�1

Rmcos�2mkLR�; �3�

where R is the re¯ection coe�cient of the Fabry±Perotcavity and LR is the cavity length of a read-out system.Also, a re¯ective transfer function of low ®nesseFabry±Perot sensor, Hr

S(k), can be written as

H Sr �

FS

2�1ÿ cos�2kLS��; �4�

where FS is the ®nesse of the Fabry±Perot sensor andLS is the cavity length of a sensor. It can be seen thatthe low ®nesse of the Fabry±Perot cavity can betreated as a two-beam interferometry. SubstitutingEqs. (2)±(4) into Eq. (1) and considering the two-beaminterferometry in the read-out system, the integral isevaluated by using the forms from Gradshteyn andRyzhik [9], and then the intensity, I, is given by

I � ZFS

2

�2R

1� R

ÿ 2R�1ÿ R�1� R

cos�2k0LR�exp�ÿs2L2R�

ÿ 2R

1� Rcos�2k0LS�exp�ÿs2L2

S�

� R�1ÿ R�1� R

cos�2k0�LS � LR��exp�ÿs2�LS

� LR�2� � R�1ÿ R�1� R

cos�2k0�LS

ÿ LR��exp�ÿs2�LS ÿ LR�2�

�5�

where Z=1/16 is the fractional of intensity power inthe system and it can be explained that the intensitypower is divided by 2 when the light is passing througha 3 dB 2�2 coupler. In PMDI, the coherent length ofthe optical source is speci®cally chosen to be less thanLS and LR; and LSÿLR is adjusted so that it is lessthan the coherent length. Under these conditions,Eq. (5) reduces to

I � ZFS

2

�2R

1� R� R�1ÿ R�

1� Rcos�2k0�LS

ÿ LR��exp�ÿs2�LS ÿ LR�2��; �6�

Fig. 1. Con®guration of PMDI.

Y.L. Lo et al. / Optics & Laser Technology 30 (1998) 395±401396

Following the same approach, the two time-dependentintensity functions as illustrated in Fig. 2 at the twodetectors are found from

I1 � Z��H S1

t �H S2r �H S1

t �H S1r � �H R1

r � i�k� dk �7�

and

I2 � Z��H S1

t �H S2r �H S1

t �H S1r � �H R2

r � i�k� dk �8�

where I1 and I2 are the intensity functions at detectors1 and 2, respectively; H S1

r and H S1t are the re¯ective

and transmitted spectrum transfer functions in theFabry±Perot sensor S1; H

S2r is the re¯ective spectrum

transfer functions in the Fabry±Perot sensor S2; HR1r

and H R2r are the re¯ective spectrum transfer functions

in the read-out systems with respect to sensors S1 andS2. Accordingly, to obtain the real signals withoutcross-talk, the terms with cross-talk e�ects in Eqs. (7)and (8) are forced to be zero. For the sake of sim-plicity, the two series Fabry±Perot sensors and tworead-out systems can be assumed as low-®nesseFabry±Perots and the transfer functions in re¯ectionscan be expressed as

H Sir �

FS

2�1ÿ cos�2kLSi��; �9�

and

H Rir �

FR

2�1ÿ cos�2kLRi

��; �10�

where LSi and LRi(i=1, 2) are the cavity lengths in

sensors and read-out systems. Assuming that theenergy is conserved and no loss occurs in Fabry±Perots, the condition Hr+Ht=1 is required.Substituting Eqs. (9) and (10) into Eq. (7), the integralis given by

I1 ��FR

2

�FS

2

�2� 1

2cos�S1 ÿ R1� � 1

2cos�S2 ÿ R1�

�ÿ F 2

S

2

�1� 1

2cos�S1 ÿ R1� ÿ 1

4cos�S1 ÿ S2 � R1�

�� F 3

S

8

�1� 1

2ÿ 1

4cos�2S1 ÿ S2� � cos�S1 ÿ R1�

ÿ 1

2cos�S1 ÿ S2 � R1� ÿ 1

8cos�2S1 ÿ S2 � R1�

� 1

8cos�2S1 ÿ S2 ÿ R1�

��i�k� dk; �11�

where S1=2k0LS1 ; S2=2k0LS2 and R1=2k0LR1. For

a white-light interferometry, the coherence length of ashort coherence light source, Lc, is much smaller thanthe length of cavities LS1 , LS2 and LR1

in Eq. (11).Therefore, Eq. (11) is simpli®ed by the terms incos(2k0LS1 ), cos(2k0LS2 ) or cos(2k0LR1

) integrated withi(k) approaching zero in the condition of the white-light interferometry. The terms in cos(S1ÿR1)integrated with i(k) in Eq. (11) are the real signals. Itcan be seen that those terms cos(S2ÿR1),cos(S1ÿS2+R1), cos(2S1ÿS2), cos(2S1ÿS2+R1) andcos(2S1ÿS2ÿR1) are the cross-talk e�ects; hence, todiminish the e�ects of cross-talk, the following con-ditions are listed under path-matching LS11LR1

andLS21LR2

as

j2LS1 ÿ LS2 j � Lc; �12a�

j3LS1 ÿ LS2 j � Lc; �12b�

jLS1 ÿ LS2 j � Lc: �12c�Following the above equations, Eq. (11) can bereduced to

I1 � I0FRFS

4

�2ÿ FS � 3F 2

S

8

�

� I0FRFS

4��1

2� F 2

S

4ÿ FS

2

�cos 2k0�LS1 ÿ LR1

��13�

It can be seen that under the proper designs in elimi-nating the cross-talk e�ects, the real signals withoutcross-talk can be obtained. Similarly, the intensity I2 inFig. 2. Con®guration of two series ILFEs in PMDI.

Y.L. Lo et al. / Optics & Laser Technology 30 (1998) 395±401 397

Eq. (8) can be expressed as

I2 ��FR

2

�FS

2

�2� 1

2cos�S2 ÿ R2� � 1

2cos�S1 ÿ R2�

�

ÿ F 2S

2

�1� 1

2cos�S2 ÿ R2� ÿ 1

4cos�S1 ÿ S2 ÿ R2�

�

� F 3S

8

�1� 1

2ÿ 1

4cos�2S1 ÿ S2�

� 1

2cos�S2 ÿ R2� ÿ 1

2cos�S1 ÿ S2 ÿ R2�

ÿ 1

4cos�2S1 ÿ R2� � 1

2cos�S2 ÿ R2�

� 1

8cos�2S1 ÿ S2 ÿ R2�

��i�k� dk;

�14�where R2=2k0LR2

. The terms in cos(S2ÿR2) inte-grated with i(k) in Eq. (14) are the real signals. Also, itcan be seen that those terms in cos(S1ÿR2),cos(S1ÿS2ÿR2), cos(2S1ÿS2), cos(2S1ÿR2), andcos(2S1ÿS2ÿR2) are the cross-talk e�ects; similarly,to diminish the e�ects in cross-talk, the following con-ditions are required,

j2LS1 ÿ LS2 j � Lc; �15a�

j2LS2 ÿ LS1 j � Lc �15b�

jLS1 ÿ LS2 j � Lc: �15c�Therefore, the intensity I2 in Eq. (14) can be reducedto

I2 � I0FRFS

4

�2ÿ FS � 3F 2

S

8

�

� I0FRFS

4

�1ÿ FS � F 2

S

4

�cos�2k0�LS2 ÿ LR2

�� �16�

Combining Eqs. (12a)±(c) with Eqs. (15a)±(c), the ®nalconditions for eliminating the cross-talk e�ects arelisted as

jLS1 ÿ 2LS2 j � Lc; �17a�

j2LS1 ÿ LS2 j � Lc; �17b�

j3LS1 ÿ LS2 j � Lc; �17c�

jLS1 ÿ LS2 j � Lc: �17d�

It is concluded that to design two series Fabry±Perot sensors in PMDI, the cavity lengths in LS1 andLS2 must be much longer than the coherence length of

light source, Lc. In addition, the condition shown inEqs. (17a)±(d) is necessary.

3. Experimental setup and results

This section describes the optical arrangement andimplementation of a demodulation scheme speci®callydesigned for in-line ®ber etalon (ILFE) sensors inmultiplexing. ILFE sensors, which are a variation ofthe EFPI sensor [10], are composed of a hollow core®ber fused in between two standard single mode opti-cal ®bers (see Fig. 3) [5]. The small air gap (typicallybetween 30 and 400 mm in length) acts as a low ®nesseFabry±Perot cavity. The broad-band light(FWHM050 nm; coherence length is around 30 mm)from a pigtailed super luminescent diode source with anominal wavelength 1.3 mm, 150 mW optical powerwas passed through a ®ber-optic coupler to two in-line-®ber-etalon (ILFE) strain sensors as illustrated inFig. 2. The re¯ected light from the ILFEs are againpassed through a ®ber-optic coupler that formed twolow ®nesse Fabry±Perot cavities with mirrors for read-out systems. The mirror in the Fabry±Perot cavity isbonded to a PZT stack that is placed on a linear trans-lation stage. This stack is used to provide a phase gen-erated carrier for ILFE demodulation and thetranslation stage is used to adjust the optical path ofthe read-out interferometer to match that of the ILFE.This con®guration is used to test the strain measure-ment capability of the ILFE sensor by driving the PZTstack with a 6 kHz ramp function and using a singlechannel phase tracker [11]. Subsequently, the demodu-lation system was tested by mounting sensors on acantilever beam. It should be noticed that the Fabry±Perot cavities are all beyond the coherence length, 30mm, for a white-light interferometry. A 90 mm cavitylength ILFE sensor and a 2-mm gauge length resist-ance strain gauge are bonded side-by-side 4 cm fromthe root of a 15�1.3�0.17 cm aluminum cantileverbeam. The dynamic strain response of the ILFE sensoris tested by exciting the beam to experience it's ®rstmode of vibration. Fig. 4 shows that the resultingstrain responses of the ILFE sensor and resistancestrain gauge are in good agreement.

Fig. 3. ILFE sensor.

Y.L. Lo et al. / Optics & Laser Technology 30 (1998) 395±401398

Fig. 4. Experimental data from cantilever beam tests.

Fig. 5. Experimental data for cross-talk tests in exciting a 100 mm ILFE sensor.

Y.L. Lo et al. / Optics & Laser Technology 30 (1998) 395±401 399

To verify the cross-talk in di�erent pairs of twoseries Fabry±Perot sensors, the experimental setup isillustrated in Fig. 2. Two cantilever beams are designedto test the two series ILFE sensors. It should benoticed that the lead-in/out ®bers connecting twoILFE sensors are much longer than the ILFE sensorsso that the e�ect of the spectrum transfer in the spec-trum analysis can be neglected. The cavity lengths oftwo ILFE sensors are designed, LS1 as 100 mm and LS2

as 75 mm, and they are bonded to two cantileverbeams, respectively, for cross-talk measurements. Thecross-talk between respective sensors in this system areevaluated in the following manner. First, a cantileverbeam with a 100 mm ILFE sensor is excited using ashaker with a 4 Hz sinusoidal tip displacement. Then,

the strain responses from the 100 and 75 mm read-outstages are measured as shown in Fig. 5(a) and (b).Fig. 5(b) shows a negligible response in the 75 mmILFE, with an approximate cross-talk level of ÿ33 dB.Similarly, the cantilever beam with the 75 mm ILFEsensor is loaded with a sinusoidal tip displacement andthe strains recorded by the 75 and 100 mm read-outstages are measured and shown in Fig. 6(a) and (b),respectively. The cross-talk level in this case is alsoapproximately ÿ31 dB. The other pair of two ILFEsensors are designed LS1 as 80 mm and LS2 as 100 mmthat also meets the condition in Eqs. (17a)±(d). The ex-periment is conducted in the same manner as describedabove, and the results show that the cross-talk is alsonegligible.

Fig. 6. Experimental data for cross-talk tests in exciting a 75 mm ILFE sensor.

Y.L. Lo et al. / Optics & Laser Technology 30 (1998) 395±401400

4. Conclusions and discussions

Cross-talks in series Fabry±Perot sensors fordemodulation in path-matching di�erential interfero-metry (PMDI) can be limited by corresponding path-matching read-out systems. In this paper, spectrumanalysis is used to design the series Fabry±Perot sen-sors without cross-talk. More series Fabry±Perot sen-sors in PMDI can be achieved by following the sameprocedure to eliminate the cross-talk e�ects.Experimental data shows that the design complyingwith the special condition is in a limited cross-talk.This design in two series Fabry±Perot sensors inPMDI without cross-talk can be applied to a two-parameter optical ®ber sensors.

Acknowledgements

The author would like to acknowledge Dr. J.S.Sirkis in Smart Materials and Structures ResearchCenter (SMSRC) at the University of Maryland,College Park, for the support of hollow core ®bers.This work has been partially supported by NationalScience Council Grant No. NSC 86-2621-E-006-037 tothe National Cheng Kung University, Tainan, Taiwan.

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