1382 OPTICS LETTERS / Vol. 17, No. 19 / October 1, 1992

Fiber-stress-insensitive interferometricsetup for displacement measurements

Umberto Minoni

Dipartimento di Elettronica per 1'Automazione, Facolta di Ingegneria, Universitb degli Studi di Brescia, Via Branze, 25133 Brescia, Italy

Received June 10, 1992

A novel optical setup suitable for the implementation of interferometric sensors with the use of a remote lasersource and photodetectors is presented. A high-birefringence fiber is used as the linking medium between themeasuring head (the interferometer) and the controlling head (laser and detectors). The proposed setup makesthe system insensitive to fiber bending and fiber temperature variations.

Miniaturization, geometrical flexibility, high accu-racy, and extended measuring range are key require-ments for optical metrology in the application ofdimensional gauging of both workpieces and end prod-ucts in modern industrial automatic production sys-tems. Ordinary interferometric systems meet bothextended-range and high-accuracy requirements butdo not provide enough geometrical flexibility owing tothe rigid paths of the optical beams. In this context,the use of optical monomode fibers is appealing, inthat it makes it possible to keep the laser sourceapart from the measuring head. This holds truealso in the case of semiconductor lasers, in whichmode stabilization makes the laser source cumber-some. Monomode fibers, however, suffer from a lackof polarization state control. Passive stabilizationschemes proposed to overcome this drawback' 2 resultin quite complex and bulky systems, which are notsuitable for implementing miniaturized measuringheads. The use of high-birefringence (HiBi) fibersmakes it possible to implement polarimetric schemeshaving good rejection of environmental factors.3 4

Recently, HiBi fibers have been proposed in aninterferometric setup as the link medium betweenthe laser source and the measuring head, which inthis case can include the photodetectors.5 A modi-fied optical scheme permits the use of the samefiber for delivering the interference signals back tothe laser heads, while keeping the photodetectorstogether with the detection electronics inside thelaser head.6

HiBi fibers provide adequate control of the polar-ization state of the beams, provided that the inputpolarization state is correct with respect to orienta-tion of the fiber optical axes. In this Letter a novelscheme is proposed that uses only one axis of thefiber for feeding the interferometer and both axesfor delivering back the interference signals. Thisscheme has been proven to be insensitive to fiberbending and to fiber temperature changes, whichmakes the fiber alignment a quite easy task.

Figure 1 shows the optical layout of the system.The laser head provides a linearly polarized beamwith its polarization axis parallel to the drawing

plane. This beam is amplitude divided by beamsplitter BS1, and one of the resulting beams is usedto feed the sensor head. Gradient-index lens Li isused to launch the laser beam into the optical fiber.This is a HiBi fiber mounted with its fast axis parallelto the beam polarization at the input end. In thisconfiguration the fiber is used to deliver the polarizedlaser beam to the sensor head and, at the sametime, to deliver the interference signals back to thephotodetectors.

The measuring head is based on a Michelson inter-ferometer. The output fiber end is oriented with itsfast axis parallel to the drawing plane. The beamexiting the fiber is collimated by gradient-index lensL2 and then amplitude divided by beam splitter BS2.One of the resulting beams is directed to the signalarm, while the other is directed to the reference arm.A A/8 retarding plate is inserted into the signal armto provide a returning beam that is circularly polar-ized. In contrast, a A/4 retarding plate is insertedinto the reference arm to provide a returning beamthat is linearly polarized with the polarization planeat 45° with respect to the drawing plane.

PD2

P 1 B CL HiBirfiberPD1

PBS |Ll I

Cu~ ~ ~ ~ ~ ~~Lzu,_ ~~~RM 1b 7 : S2

SM i11

Fig. 1. Optical layout of the system. SM, signal armmirror; RM, reference arm mirror; BS1, BS2, nonpolar-izing beam splitters; PBS, polarizing beam splitter; PD1,PD2, photodetectors; L1, L2, gradient-index lenses.

0146-9592/92/191382-03$5.00/0 © 1992 Optical Society of America

October 1, 1992 / Vol. 17, No. 19 / OPTICS LETTERS 1383

After traveling through the interferometer arms,the two beams recombine at beam splitter BS2. Heretwo interference signals (one from the parallel andone from the perpendicular polarization plane) out ofphase by 90° are produced. The fiber is now usedto deliver the intensity signals to the photodetectorswhile keeping them separated. Finally, the polariz-ing beam splitter PBS separates the two signals anddirects them to the photodetectors.

A formal description of the interferometer is givenin the following, with the use of the Jones matrixformalism: a right-oriented arrow is used to identifya forward transformation matrix, and a left-orientedarrow is used for the reverse transformation. Thefiber can be described as a variable linear retarderinserted between two polarization rotators. The firstrotator, placed at the input end, takes into accountorientation a of the input fiber end with respect to thex axis. The other rotator, placed at the output end,takes into account orientation /8 of the output fiberend with respect to the x axis. The fiber is thereforedescribed by the matrix

describes the mirror of both the signal and the ref-erence arm. Evaluation of Eqs. (5)-(7) gives thefollowing equation for the reference arm matrix:

T [1 1 ] (8)

The signal arm of the interferometer is described bythe matrix

Tsig = RAI8RaMRaRAI8,

where matrices

- _- i (1 + 2)RA/8 = 1

-i(1 +± )ftA/8- =

(9)

1

-i(l + 12-)

1 ]i(l + d2)

(10)

describe the A/8 retarder at 450 to the x axis, andmatrix

F- Tilf RfoutX,

where

Fcos a sin aTin = .-sin a cos J

F cos/3 - sinS 1T It I ,

sinm, cos,8 J

1 0Rf = O exp(i8S) .

(1)RRa = =Ra = exp(icF)I (11)

describes the optical path difference between the two(2) interferometer arms; the phase difference is indicated

with (D. With the use of Eqs. (9)-(11), the signalarm matrix becomes

(3)

(4)

Tsig = exp(i2)[ I i 1 (12)

The electric-field vector, before the polarizing beamsplitter PBS of Fig. 1, is given by

Ef oc F(Tsig + Tref)"Ein = F[ 11 + exp(i2CD)

+ exp[i(2CD - 7r12)

1 + exp[i(2CF + ir/2)] 1- (1 - expO(i2) Fj., (13)

In Eq. (4), a = 8fast - 3slow is the phase difference be-tween the E0, and Eo, components of the beam exitingthe fiber. Obviously, this phase difference is a func-tion of fiber stress induced either by fiber bending ortemperature changes. The interferometer referencearm is described by the following matrix:

Tref = RA/4MRA/4 , (5)

where matrices

RA/4 =1 -iV

[- 1

1]

-1 1-(1 + iV/)] (6)

describe the A/4 retarder at 22.50 to the x axis, andmatrix

M =[ 1 (7)

where

under the initial condition that the beam enters thefiber with its polarization plane parallel to the fiberfast axis.

If the input and output ends of the fiber are wellaligned, angles a and ,8 of Eqs. (3) and (4) equal zero,and therefore, from Eq. (1) we have

(14)

If Eq. (14) is taken into account, Eq. (13) becomes

E = F 1~~ + exp(i2(D) Ef _ ._ exp(i8){1 + exp[i(2CD - vr/2)]} [Ef,

(15)

]'

P = i7= 1 00 exp(i8)

1384 OPTICS LETTERS / Vol. 17, No. 19 / October 1, 1992

PD1 PD2

time (1 ms/div)

(a)

amplitute (20 mV/div)

(b)Fig. 2. Output signals from photodiodes PD1 and PD2 ofFig. 1: (a) plot of amplitude versus time, (b) plot of PD1signal versus PD2 signal.

Beam splitter PBS in Fig. 1 separates the two com-ponents of Ef and sends them to the two photodiodesPD1 and PD2. The signals at photodiodes obtainedby polarization separation are as follows:

VPD1 cX Ef.Ef* = [1 + cos(2CF)],

VPD2 a EfyEfy *[1 + cos(2CD + ?T/2)].

(16)

(17)

It is clear from relations (16) and (17) that if thefiber ends are correctly aligned, the signals at thephotodetectors are insensitive to any fiber opticalpath variation; therefore, the displacement obtainedfrom these signals is not affected by perturbationsinduced into the fiber such as those due to mechanicalor thermal stress.

A laboratory prototype of the system, which usesa He-Ne laser, has been developed and tested.Figure 2 shows the plots, obtained from a digitalscope, of the interference signals measured when alinear displacement is applied to the signal mirror.As expected from relations (16) and (17), the twosignals are out of phase with each other, and asshown in Fig. 2(b) the phase difference is -r/4.

In conclusion, a new optical setup has been pro-posed that permits an efficient and flexible remotecontrol of an interferometric sensing head. This is ofparticular interest, among other metrological applica-tions, when hostile targets have to be dimensionallymonitored with interferometric accuracy.

The author thanks F. Docchio for the fruitful dis-cussions during preparation of this manuscript andA. Taroni for careful reading of the manuscript.

References1. N. C. Pistoni and M. Martinelli, Opt. Lett. 16, 711

(1991).2. A. D. Kersey, M. J. Marrone, and M. A. Davis, Elec-

tron. Lett. 20, 518 (1991).3. M. Corke, A. D. Kersey, K. Liu, and D. A. Jackson,

Electron. Lett. 20, 69 (1984).4. A. D. Kersey, M. Corke, and D. A. Jackson, Electron.

Lett. 20, 211 (1984).5. H. Kitajima, J. Takagi, and T. Yamashita, Sensors Ac-

tuators A22, 442 (1990).6. F. Farahi and D. A. Jackson, Rev. Sci. Instrum. 61,

753 (1990).

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