They have been used to develop some devices basedon existing single-optical-vortex points3 and on thecreation of entire lattices formed by numerous opticalvortex points. The second approach allowed the de-velopment of a new kind of interferometry and anew device—the optical vortex interferometer (OVI)—which uses optical vortices as phase markers. Fordetails concerning OVI the reader is referred to Ref.4. One of the main applications of OVI is measure-ment of wavefront modifications. The tests showedthat the resolution of OVI used for small wave-tilt-angle measurements is competitive when comparedwith that achieved in classical two-beam interferom-etry.5 However, some practical problems occur in op-tical vortex interferometry. The first problem to solveis the development of procedures for localizing thephase markers with sufficient precision, which sig-nificantly affects the accuracy of the measurement.6The second is the generation of a stable regular op-tical vortex lattice. The latter problem seems to becrucial because stability of the generated vortex lat-tice is necessary in most of the measurement ar-rangements. The generation of the optical vortexlattices used in OVI can be achieved by three-plane-wave interference. This approach was presented inRef. 7. The generation of optical vortex lattices byplane-wave interference was analyzed by some other
authors8,9 also, but not in this context. To reducesome disadvantages of the three-wave setup of OVI,such as mutual wave vibration influence, the two-wave interference setup was proposed in Ref. 10. Thequestion is, why not generate the optical vertices in aone-wave setup? This approach would eliminate mostof the problems connected with the stability of theinterferometer and perhaps would also permit newapplications of the interferometer. The generation ofa regular stable optical vortex lattice in a one-wavesetup using birefringent elements is presented in thispaper.
2. Theoretical Analysis
The proposed setup consists of the following elements(Fig. 1): the polarizer P1 with the azimuth angle�P � 0°, the first Wollaston compensator W1 with theazimuth angle �W1 � 45°, the second Wollaston com-pensator W2 with the azimuth angle �W2 � 0°, andthe analyzer A with the azimuth angle �A � 45°. Toobtain the intensity distribution containing forklikepatterns correlated with optical vortices, the refer-ence arm [(II) in Fig. 1] of the interferometer shouldbe constructed. The reference beam arm consists ofthe mirror and the polarizer P2 with the azimuthangle �P � 45°.
Let us assume that the incident plane and themonochromatic wave propagate along the z axis ofthe Cartesian system. Using the Jones vector andmatrix formalism one can show that the Jones vectorof the output light is given by
Eout � �1 11 1��1 0
0 e�i�2��1 � e�i�1 *
1 � e�i�1 *��10�, (2.1)
where the vector
The authors are with the Institute of Physics, Wrocław Univer-sity of Technology, Wrocław, Poland. M. Borwinska’s e-mail ad-dress is [email protected].
Received 29 June 2006; revised 9 October 2006; accepted 25October 2006; posted 26 October 2006 (Doc. ID 72505); published25 January 2007.
�10�represents a polarization state of the light after thepolarizer P1, the matrix
�1 � e�i�1 *
1 � e�i�1 *�is the Jones matrix of the first Wollaston compensa-tor W1, the matrix
�1 00 e�i�2�
is the Jones matrix of the second Wollaston compen-sator W2, the matrix
�1 11 1�
is the Jones matrix of the analyzer A, * denotes un-important elements, and �1,2 are the variable phaseshifts introduced by compensators W1 and W2 alongtheir main axes, respectively. From the azimuth an-gle definition the main axis of the Wollaston compen-sator W2 is in line with the x axis of the Cartesiansystem, whereas the main axis of the Wollaston com-pensator W1 forms an angle of 45° with the x axis (seeFigs. 2 and 3). The Wollaston compensator is a pair ofbirefringent wedges (Fig. 2) that form a cuboid. Inpolarization optics there are two Wollaston prism ap-plications: polarization angular wave splitting andcompensation of the phase shift introduced by bire-fringent media. In the first case the wedge angle � isrelatively large and the wedge material is highly bi-refringent (for example, calcite). In the second case(called here a compensator construction) the angle isrelatively small. In our computer simulations andexperiments the value of the angle is 5° (or 10°) andthe wedges are made of the same material—quartz.Formally, two waves (ordinary and extraordinary)leaving the Wollaston prism are inclined with regardto the last prism surface (its wavevectors are tilted
with regard to the z axis of the Cartesian system), butin the case of the compensator construction consid-ered here this inclining angle (for example, the wedgeangle of 5°) is about �2�40� (for the wavelength of632.8 nm), so we can assume that both ordinary andextraordinary waves propagate without propagationdirection change; i.e., the considered interferencephenomenon results from the polarization propertiesof the Wollaston compensator rather than from itsangular wave splitting properties.
and (from the definition) the output intensity Iout ishere given by
Fig. 1. Interferometer setup: P1, P2, polarizers; M1, M2, mirrors;W1, W2, Wollaston compensators; BS1, BS2, beam splitters; A,analyzer; CCD, camera. (I) Object arm of the interferometer; (II)reference arm.
Fig. 2. Wollaston compensator setup: the z axis is a wave prop-agation axis; the � axis is a main Wollaston axis; arrows indicatethe optical axis orientation in the wedges. Lines with constantphase difference introduced by the Wollaston compensator havebeen marked. For the first Wollaston compensator the angle be-tween the � axis and the x axis of the Cartesian system is equal to45°, whereas the second angle is 0°.
Fig. 3. Main axis orientations of the Wollaston compensators W1and W2 with regard to the observation Cartesian coordinate (x, y)system.
Iout��1, �2� � �ExEx* � EyEy*� � 1 � sin �1 sin �2,(2.3)
and the phase distribution � of the output light is theargument of the complex amplitude in Eq. (2.2). FromEq. (2.3) it follows that the intensity Iout is zero if
sin �1 sin �2 � �1, (2.4)
which means that zero-intensity points create a rect-angular lattice in ��1, �2� coordinates or a rhomboidallattice in �x, y� coordinates. Moreover, these zero-intensity points are optical vortices because the wavephase � in these points is undetermined (aroundthese points it changes from 0 to 2�).
3. Numerical Simulations
The considerations presented above were used to gen-erate numerical interferograms. Exemplary intensityand phase distributions carried out in the objectbeam arm [(I) in Fig. 1] of the interferometer arepresented in Figs. 4(a)–4(d). In Figs. 4(a) and 4(b) aquotient of the wedge angles �1 and �1 of the Wollastoncompensators W1 and W2 is equal to 1 ��1��2 � 1�,whereas in Figs. 4(c) and 4(d) this quotient equals2 ��1��2 � 2�.
It is easy to notice the regular lattice of zero-intensity points [Figs. 4(a) and 4(c)]. Each of thesezero-intensity points corresponds to the singularitypoint in the calculated phase distribution; i.e., thephase � around the considered point changes from 0to 2�, but it remains undetermined exactly in thispoint [Figs. 4(b) and 4(d)]. The points with such a
phase singularity and intensity equal to zero are op-tical vortices. The localization of optical vortices wascarried out automatically by means of a minima pro-cedure (see Ref. 6 for details).
4. Experimental Results
The experimental interferograms were recorded inthe setup as presented in Fig. 1(I). It is impossible toreceive phase distributions in experiments as op-posed to numerical simulations. Therefore to ensurethe zero-intensity points [see Fig. 5(a)] are the opticalvortices, the reference beam was added [arm (II) inFig. 1]. As can be seen in Fig. 5(b) the obtained in-tensity distribution contains the forklike patterns.2The optical vortex localization was also carried out bymeans of the minima method.
It is worth noting that the obtained optical vortexlattice is regular. The lattice geometry modification isstrictly connected with the modification of the objectbeam. Thus by tracing the shift of the vortex lattice orits deformation, the reconstruction of the object beamdeformation is possible.
5. Potential Applications for Determining theProperties of Birefringence Media
Since the presented setup is of the polariscope type,one should expect that it can also be used to deter-mine the properties of birefringence media (pairs ofthe element P1–W1 and W2–A form, generally speak-ing, a so-called elliptical polariscope, but here withmodulated ellipticity). Let us assume then that alinearly birefringent medium is placed between the
Fig. 4. (a), (c) Intensity I and (b), (d) phase � distributions for theexemplary numerically, generated interferograms. (a) and (b) referto �1��2 � 1, whereas (c) and (d) refer to �1��2 � 2. Points that areoptical vortices are marked.
Fig. 5. Experimental interferograms recorded on a CCD camera(a), (c) without the reference beam and (b), (d) with the referencebeam added. (a) and (b) refer to �1��2 � 1, whereas (c) and (d) referto �1��2 � 2. Spatial data for the presented cases have been de-noted.
Wollaston compensators W1 and W2. Then the out-put light intensity Iout is given by
Iout��1, �2� � 1 � sin��1 � 1�sin��2 � 2�, (4.1)
where 1 and 2 are components of a phase markerdisplacement vector � � �1, 2� in the Wollastoncompensator axis basis (�1, �2).
One can easily show that these components arecorrelated with the medium parameters as follows:
tan 1 � tan sin 2�, (4.2a)
tan 2 � tan cos 2�, (4.2b)
where � is an azimuth of the first medium eigenvec-tor and � is a phase shift introduced by the examinedmedium. Knowing the displacement vector � coordi-nates [Eq. (4.2)] one can immediately calculate themedium parameters using the equations
tan2 � tan2 1 � tan2 2, (4.3a)
tan 2� �tan 1
tan 2. (4.3b)
For some cases, especially when the azimuth angleof the medium does not change, one can place theexamined medium with the presented polariscopesetup with a known azimuth angle that correspondsto the specific Wollaston compensator main axisangles, � � �45°, for the first compensator W1 or� � 0°, 90° for W2:
� � �45° I � 1 � sin �2 sin��1 � �, (4.4a)
� � 0° I � 1 � sin �1 sin��2 � �, (4.4b)
� � 90° I � 1 � sin �1 sin��2 � �. (4.4c)
Thus if the main axis of the examined medium is inline with or differs [Eqs. (4.4a)–(4.4c)] from one of thecompensator axes (�1 or �2) by 90°, the marker latticemoves parallel to one of these axes and this transla-tion is simply proportional to the phase shift � intro-duced by this medium. Moreover, this translationdirection allows us to recognize the orientation of themedium’s first eigenvector. For example, if the medi-um’s first eigenvector azimuth angle is � � �45° andthe marker lattice moves in the opposite direction tothe �1 axis, the azimuth angle � is �45°, and when itmoves in the same direction, the angle � equals �45°.
6. Conclusions
The new OVI arrangement presented in this paperhas some advantages with regard to previously pro-posed solutions applied in optical vortex interferom-etry. First of all, in the one-wave setup, the influenceof vibration is remarkably lower than in the two- orthree-wave setups. Moreover, the possibility of put-ting all the main optical elements (the polarizers andthe Wollaston compensators) together could addition-ally reduce the interference setup’s instability. Thepresented setup extends the possibility of OVI appli-cations. It enables measurements of some differentwave parameters (such as wavefront deformation).What is more, it also allows measuring the proper-ties of the optical media (such as birefringence). Bychanging the shearing of the Wollaston compensa-tors, one can do measurements on a global scale aswell as on a quasi-local scale (in the sense of twomarker distance resolution). Because of the inabil-ity to separate the interfering waves, it is not pos-sible to measure the phase distribution as it is donein two- or three-wave OVI arrangements (for exam-ple, Refs. 7 and 10).
This work was supported by the Polish Ministry ofScientific Research and Information Technology un-der grant 3T10C04829.
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