1272 OPTICS LETTERS / Vol. 24, No. 18 / September 15, 1999

Robust one-beam interferometer with phase-delay control

Jose A. Ferrari, Erna M. Frins, Daniel Perciante, and Alfredo Dubra

Facultad de Ingenierıa, Instituto de Fısica, J. Herrera y Reissig 565, 11300 Montevideo, Uruguay

Received June 14, 1999

A robust one-beam interferometer with external phase-delay control is described. The device resembles aMach–Zehnder interferometer in which the two arms are together in one collimated beam. However, theproposed device is not an amplitude-division interferometer but a wave-front division one. The phase-delaycontrol occurs at the interferometer output with the help of two polarizing beam splitters, a quarter-wave plate,a Faraday rotator, and a polarizer. An additional phase delay is introduced by application of an electricalcurrent to the Faraday rotator or by rotation of the polarizer (the latter is of topological origin), which permitsthe use of techniques of phase-stepping interferometry. 1999 Optical Society of America

OCIS codes: 120.3180, 260.5740, 350.5030, 050.5080.

The purpose of this Letter is to present a novel in-terferometer architecture with external phase-delaycontrol. The proposed device, shown in Fig. 1, resem-bles a Mach–Zehnder interferometer in which the twoarms are together in one collimated beam. However,the proposed device is not an amplitude-division inter-ferometer but a wave-front division one. As shown inFig. 1, the top half of the beam (collimated by lens L1)acts as the reference arm of the interferometer, and thebottom half is the test arm in which the phase object isplaced. Since both arms are parts of the same beamand fewer optical components are involved, the inter-ferometer becomes robust and insensitive to environ-ment vibrations and temperature f luctuations. Theonly alignment required by this interferometer is toplace (or cement) together the two cube beam splitterson a holder to achieve interference. Indeed, the rela-tive positions of the optical elements (e.g., lenses, beamsplitters, polarizer) are not a critical matter as in con-ventional interferometers.

In the proposed device we can induce an arbi-trary phase delay between the reference and the testwaves without varying the physical lengths of theoptical paths. Moreover, the optical elements (l�4-wave plate, Faraday rotator, and polarizer) are crossedby both the reference and the test waves, produc-ing the phase delay; i.e., the elements lie outside theinterferometer.

The light source is a polarized laser with its polar-ization direction at 45± with respect to the polarizationdirection of the polarizing cube beam splitters, whichare used to achieve spatial superposition of the half-beams. Thus, at the interferometer output we havethe addition of two orthogonal linearly polarized wavesof the same amplitude. Then, after the output passesthrough a l�4-wave plate with its fast axis orientedparallel to the polarization direction of the laser, onehas left-handed and right-handed circularly polarizedwaves. Finally, after their passage through the Fara-day rotator and the polarizer, the waves acquire an ex-tra phase delay (in addition to the spatially modulatedphase changes produced in the test object). The addi-

0146-9592/99/181272-03$15.00/0

tional phase delay can be induced by application of anelectrical current to the Faraday rotator or by rotationof the polarizer.

It is easy to demonstrate1,2 that the intensity distri-bution at the output plane (C in Fig. 1) will be given by

I �x, y� � I0�x, y� �1 1 V �x, y�cos�f�x,y� 1 �fR 1 2u��� ,

(1)

where �x, y� are spatial coordinates, I0 is the averageintensity, V is the visibility of the interference, f is theunknown phase profile of the test object, fR is the de-lay between left- and right-circularly polarized wavesinduced in the Faraday rotator, and u is the angle be-tween the transmission direction of the polarizer andthe fast axis of the l�4-wave plate.

The proposed device is suitable to be used in phase-shifting interferometry (PSI),3 since it is possible toacquire a series of interferograms while the phase de-lay is shifted through a series of steps dn�n � 1, 2, . . .�that are controlled by the electrical current that is

Fig. 1. One-beam interferometer: L, polarized He–Nelaser; SF, spatial filter; L1, collimating lens; PO, phaseobject; PBS’s, polarizing cube beam splitters; QW, l�4-wave plate with its fast axis at 45± with respect to thepolarization directions; FR, Faraday rotator; P, polarizerwith its transmission direction at u with respect to the fastaxis of the l�4-wave plate; CS, electrical current source; L2,lens that images PO onto output plane C (where the CCDcamera is placed).

1999 Optical Society of America

September 15, 1999 / Vol. 24, No. 18 / OPTICS LETTERS 1273

Fig. 2. Experimental results: interferograms of butanegas f lowing through a tube acquired with the polarizer at(a) u � 0±, (b) u � 45±, and (c) u � 90±. Clearly, the brightfringes of interferograms (a) and (c) are p (rad) shiftedin phase.

applied to the Faraday rotator. In standard PSI sys-tems, phase shifting is usually accomplished by move-ment of one of the mirrors of the interferometerwith a piezoelectric transducer, but even good piezo-electric transducers exhibit nonlinearity and oftenhysteresis.4 The nonlinearity and hysteresis gener-ate phase-shift miscalibration, which is an importantsource of error in PSI. The main advantages of theproposed system are that it does not involve mov-ing parts, since the current that passes through theFaraday rotator can precisely control the phase de-

lays, and the phase-shift mechanism indeed lies out-side the interferometer, which contributes to the me-chanical stability of the system. Also, the opticalelements that are necessary for phase control (e.g.,l�4-wave plate, polarizer) do not need to have highoptical quality.

A less efficient (but cheaper) way of phase shifting isto mechanically rotate the polarizer. In this case theinduced phase shift can be interpreted as a manifes-tation of the so-called Pancharatnam phase.1,2,5,6 Thecombination of a l�4-wave plate and a rotating polar-izer was also used by Jin et al.7 for speckle-patterninterferometry and three-dimensional profilometry byuse of fringe projection and photoelasticity but not forPSI; this was later proposed elsewhere.2

To demonstrate the feasibility of the proposed one-beam interferometer we conducted experiments witha device similar to that shown in Fig. 1 but withoutthe expensive Faraday rotator. The phase object wasa stationary f low of butane gas coming through atube. The polarizer was mounted on a rotatory stagewith a resolution of �1 arcmin. We recorded threeinterferograms with a CCD camera by rotating thepolarizer in equal angular steps of 45± betweeninterferograms. The three frames are shown inFigs. 2(a)–2(c). To show that the phase shift betweeninterferograms is actually p�2 (rad) we intentionallyproduced straight fringes by cementing together thecube beam splitters with a small inclination. Clearly,the bright fringes of the interferograms in Figs. 2(a)and 2(c) are p (rad) shifted in phase.

It can easily be demonstrated3,4 that the phaseprofile of the f lowing gas is given by

f�x, y� � tan21∑I2�x,y� 2 I3�x, y�I1�x, y� 2 I2�x, y�

∏2 p�4 , (2)

where I1, I2, and I3 are the interferograms obtainedfor u � 0±, u � 45±, and u � 90±, respectively. A

Fig. 3. Three-dimensional plot of phase profile of the gasf low, calculated with a three-frame algorithm.

1274 OPTICS LETTERS / Vol. 24, No. 18 / September 15, 1999

three-dimensional plot of the calculated phase profileis shown in Fig. 3.

In conclusion, we have described a robust new wave-front division interferometer with phase-delay controlwithout the need for physically changing the opti-cal path lengths. Our system is more stable against(thermical and mechanical) f luctuations of the opticalpath lengths than other interferometers that have beenproposed in the literature. The proposed system hasoptical elements for phase control outside the interfero-meter, and thus great optical quality is not required,as is the case in other PSI systems that use a combi-nation of several birefringent plates. The problem ofphase-shift calibration is reduced to the measurementof the current through the Faraday rotator or the rota-tion angle of a polarizer, which can be done with greatprecision. Effects such as nonlinearity and hysteresisin the plane-shifting operation are absent, which al-lows reliable phase reconstruction.

J. A. Ferrari’s e-mail address is jferrari@fing.edu.uy.

References

1. H. Schmitzer, S. Klein, and W. Dultz, Phys. Rev. Lett.71, 1530 (1993).

2. E. M. Frins, W. Dultz, and J. A. Ferrari, Pure Appl. Opt.7, 53 (1998).

3. K. Creath, in Progress in Optics, E. Wolf, ed. (Elsevier,Amsterdam, 1988), Vol. XXVI, pp. 349–393.

4. E. Greivenkamp and J. H. Bruning, in Optical ShopTesting, D. Malacara, ed. (Wiley, New York, 1992),pp. 501–598.

5. G. N. Ramachandran and S. Ramaseshan, in Handbuchder Physik, S. Ramaseshan, ed. (Springer-Verlag, Berlin,1961), Vol. XXV/1, pp. 7–11.

6. M. V. Berry, Proc. Soc. London Ser. A 329, 45 (1984);J. Mod. Opt. 34, 1401 (1987).

7. G. Jin, N. Bao, and P. S. Chung, Opt. Eng. 33, 2733(1994).