302 OPTICS LETTERS / Vol. 21, No. 4 / February 15, 1996

All-fiber three-path Mach–Zehnder interferometer

Gregor Weihs, Michael Reck, Harald Weinfurter, and Anton Zeilinger

Institut fur Experimentalphysik, Universitat Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria

Received August 14, 1995

We report the realization of a three-path Mach–Zehnder interferometer using single-mode fibers and twointegrated 3 3 3 fiber couplers. We observed enhanced phase sensitivity, as compared with two-pathinterferometers, with a visibility of the interference pattern of more than 97%. This interferometer has ananalog in two-photon interferometry, and we believe it to be the first nontrivial example of N 3 N multiportinterferometers. 1996 Optical Society of America

Multipath interferometers permit optical realizationsof discrete unitary operators in higher-dimensionalHilbert spaces1 in tests of the foundations of quantummechanics. All fiber quantum cryptography and Bell-inequality experiments have been realized with two-path inteferometers.2 Here we present a multipathinterferometric experiment, using single-mode fibertechnology.

Although conventional two-path interferometers arebeing steadily improved by various techniques, therehave been nearly no efforts to exploit the enhancedsensitivity of many-path interferometers.3,4 One ofthe main reasons is that setting up such a devicewith discrete optical components is a formidable task.However, single-mode optical fiber technology providesan easy way to set up complex interferometers withstandard commercial N 3 N couplers. In this Letterwe demonstrate the basic operational principles ofsuch N 3 N fiber-coupler interferometers for thecase N ­ 3. These f iber interferometers can provideinsights into multipath interferometry that later maybe exploited in integrated optical devices.

Figure 1 shows a schematic of a general N -pathinterferometer consisting of two N 3 N beam splitters(multiports). Each multiport is described by an N 3

N matrix, which is unitary for a lossless device. In ourexperiment the multiports were commercially availablesymmetric 3 3 3 fiber couplers (tritters).

A symmetric device is described by a unimodular ma-trix. A standard form of the tritter matrix T , in whichthe first column and first row are real, is given by5

T ­1

p3

266666641 1 1

1 expµi2p

3

∂exp

µi4p

3

∂1 exp

µi4p

3

∂exp

µi2p

3

∂37777775 . (1)

The operation of the whole setup on the input modes isthen described by the system matrix M , which is theproduct of two tritter matrices T and a phase matrix P :

M ­ TPT ­ T

264 expsif1d 0 00 expsif2d 00 0 expsif3d

375T .

(2)

0146-9592/96/040302-03$6.00/0

In our experiment light is incident on only one input(see Fig. 2) and is linearly polarized. We therefore re-strict our analysis to waves of identical polarization, forwhich a scalar-wave analysis is suff icient. The inputfield Kin ­ sE in, 0, 0d is a vector whose elements arethe amplitudes of the input modes to the interferome-ter. Kin is transformed by the interferometer into anoutput vector sEout

1 , Eout2 , Eout

3 d ­ MKin. We measurethe intensities In sn ­ 1, 2, 3d at the three outputs:

In ­ jEoutn j2 ­ I0y9f3 1 2 cossf1 2 f2 1 und

1 2 cossf2 2 f3 1 und 1 2 cossf3 2 f1 1 undg , (3)

where I0 ­ jEinj2 is the input intensity and the phaseshifts between the outputs are given by su1, u2, u3d ­s0, 22py3, 2py3d. The output intensity depends onlyon phase differences, which we write as w ­ f1 2 f2and x ­ f2 2 f3.

Note that it can be shown that identical equa-tions have been derived for proposed experiments,in which correlated photons from a downconversionsource are fed into two separate tritters and theoutputs are detected in coincidence.6 Such setupsprovide demonstrations of Einstein–Podolsky–Rosencorrelations and tests of Bell’s inequalities in higher-dimensional Hilbert spaces.

Equation (3) describes an ideal lossless interferom-eter. When accounting for losses present in any realexperiment, we need only consider those occurring be-tween the two tritters because all external losses areequivalent to a reduction of detection eff iciency and/orsource strength. The internal losses and the phases

Fig. 1. Schematic of an N -path interferometer.sEin

1 , E in2 , . . . E in

N d are the electric-f ield amplitudes atthe input modes of the f irst N 3 N beam splitter (multi-port), and sEout

1 , Eout2 , . . . Eout

N d are the output amplitudesof the second one. The multiports are described by thematrices T s1d

N and T s2dN . f1, f2, . . . fN are the phases

accumulated by the fields on their paths between themultiports.

1996 Optical Society of America

February 15, 1996 / Vol. 21, No. 4 / OPTICS LETTERS 303

Fig. 2. Schematic of our three-path interferometer setup.The tritters are three-way integrated f iber couplers. Po-larization controllers were used to adjust the polarizationmanually in all three arms for maximal visibility. Piezotubes were used to modulate the interferometer phases fand x in two of the three interferometer arms. The pho-todiode signals and the driving signals were recorded in adigital storage oscilloscope.

w and x can be mathematically represented by thenonunitary phase matrix

P 0 ­

264 t1 0 00 t2 expsiwd 00 0 t3 expsixd

375 , (4)

where t1, t2, and t3 are the amplitude transmission fac-tors s0 # tn # 1d inside each arm of the interferometer.After replacement of P by P 0 in Eq. (2) the intensitiesat the three output ports are given by

In ­ I0y9ft12 1 t2

2 1 t32 1 2t2 cossw 1 und

1 2t3 cossx 1 und 1 2t2t3 cossw 2 x 1 undg . (5)

Gray-scale plots of predicted and measured inten-sities are shown in Fig. 3. The section curves alongthe diagonal w ­ 2x (Fig. 4) show the sidelobes thatare characteristic of a three-path interferometer. Ingeneral, as with the diffraction pattern of an N -slitconfiguration illuminated by a plane wave, there areN 2 2 sidelobes between the main peaks in the inter-ference pattern of an N -path interferometer. Usingthe definition of the sensitivity S as the derivative ofthe output intensity with respect to a phase,7

S ­1I0

ÇdIdw

Ç, (6)

we see that an N -path interferometer provides highersensitivity to phase changes than a two-path inter-ferometer because the slopes of the main peaks aresteeper.

Our interferometer (Fig. 2) was built from twosingle-mode fiber tritters (3 3 3 couplers, manu-factured by SIFAM Ltd., England) with symmetricpower division ratios at the operational wavelength of633 nm. We used piezo cylinders, each wound withtwo loops of f iber, as phase shifters in two of the threearms. The manual polarization controller in eachpath consisted of three tiltable disks, each bearing twoloops of fiber. Induced birefringence8 in the coiledfiber permits any state of polarization to be reached bya proper setting of the three disks.9 To provide easy

handling of the setup, we equipped the outputs of thefirst tritter and the inputs of the second with f iberconnectors.

Our light source was a 10-mW He–Ne laser coupledto the fiber by use of a standard M20 microscopeobjective. The phase shifters were driven with RCgenerators, which had a common trigger to ensurephase stability, followed by high-voltage amplifiers.To make the collected data easy to interpret, we drovethe two phases at different frequencies; these werelimited on the lower end (1 Hz) by the stability ofour RC generators and on the higher end (100 Hz)by the cutoff frequency of the piezo tubes and thefiber response. The generators were therefore set tofrequencies of approximately 2 and 80 Hz. The useof triangular waveforms resulted in a linear phasesweep over the whole parameter space of f0, 2pg 3f0, 2pg. At the outputs of the second tritter the lightwas detected by standard photodiodes. Their signalsas well as the piezo voltages were displayed andrecorded on a digital storage oscilloscope.

By tilting the disks of the three polarization con-trollers we adjusted the state of polarization in eacharm to attain maximum visibility of the interferencepattern. First, one of the two arms containing a phaseshifter was disconnected, leaving only the interfer-

Fig. 3. Experimental data as gray-scale plots (minimum,white; maximum, black) of the measured intensities I1 –I3(see Fig. 2) as functions of the voltages applied to the phaseshifters (left column). The corresponding theoretical func-tions given by Eq. (3) are shown in the right column. Thephase axes were calibrated by separate measurements withone arm of the interferometer disconnected. The experi-mental data show only slightly reduced visibility, resultingin smaller white areas in the plot compared with the theo-retical calculation.

304 OPTICS LETTERS / Vol. 21, No. 4 / February 15, 1996

Fig. 4. Diagonal cut through the three-dimensional plotsreveals the typical structure of a three-path interferometerwith two peaks in each 2p phase interval. The higherphase sensitivity stems from the fact that the resultingpeaks are thus steeper than the sinusoidal signal of a two-path interferometer. The diagonal phase-space cuts showa lower visibility than the total interference patterns sincethe absolute minima and maxima are not exactly on thediagonal line (see Fig. 3). This is due to slight differencesin the transmissions of the three interferometer arms.Again, the experimental data compare well with theoreticalpredictions.

ence pattern of a conventional two-path interferome-ter. Adjusting the polarizations in the two remainingarms yielded interference visibilities of greater than99%. The second arm containing a phase shifter wasthen disconnected and the formerly blocked path reac-tivated. After adjusting only the polarization in thelatter arm we were able to achieve equally high visi-bilities. Finally, after reconnecting all interferometerarms and fine tuning the polarization, we attained amaximum visibility of 97% for the three-beam Mach–Zehnder interferometer.

For each measurement, we collected 5000 data pointsper channel. Because of the hysteresis of the piezos,we divided the data into parts corresponding to therising and falling slopes of the triangular waveforms.All data collected at the peaks of the triangular wave-forms, and hence showing nonlinear phase shifts, hadto be discarded. The remaining data (,900 points perchannel) were spread homogeneously over the wholephase space. Contour plots of the experimental dataare shown in Fig. 3.

By using a nonlinear least-squares f it of Eq. (5) toour raw data we were able to extract the corresponding

parameters I0, t1, t2, and t3 of the model. From theseparameters we calculated the three section curvesshown in the right column of Fig. 4. They showthe characteristic sidelobes predicted by the theo-retical analysis. Despite the losses in our system therelative phase sensitivity S had a maximum value of0.60 6 0.01 rad21, which is less than the theoreticalvalue of 0.78 rad21 but significantly more than thetheoretical value of 0.50 rad21 for a lossless two-pathinterferometer.

In summary we have realized what to our knowl-edge is the f irst all-f iber three-path Mach–Zehnderinterferometer. We have shown that the outputintensities satisfy the same equations as the correla-tions of two photons in a quantum optical experimentwith two separate tritters.6 We have further shownthat a three-path interferometer displays enhancedsensitivity, which is clearly important in manyapplications of interferometric measurement. Fur-ther experiments in the foundations of quantumphysics and the realization of higher-dimensionalsystems with optical fiber multiports are under way.

This study was supported by the Fonds zurForderung der Wissenschaftlichen Forschung (Aus-trian Science Foundation), Schwerpunkt Quantenop-tik, project S6502.

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