Optical interferometry is one of the most interest-ing domains of optical measurement techniques.Someone who studies interferometric techniques getsto know their greatest advantage: sensibility and ac-curacy (of wavelength order). Someone who does theinterferometric measurements immediately faces themain disadvantage: the relativity of obtained re-sults. This means that (a) we measure only thedifference between the phases of the measured andreference light waves; (b) we absolutely measurethe value of the light phase limited to the valuesbetween zero and � (wavelength of used light).These indeterminacies could be specified using someadditional sophisticated techniques (white-light in-terferometry is a good example). One of the ideas is touse some phase markers as a specific calibration sys-tem or simply to increase the measurement range. Anoptical vortex (OV)—an isolated point singularity in awavefront phase distribution—is an example of aninteresting and unique type of such a marker.1 Thelight beam containing OVs reveals properties thatmake them interesting tools in modern optics and ininterferometry. In many papers, one can find a num-ber of methods with which the light beam contain-ing OVs can be generated, for example, syntheticholograms,2–4 spiral wave plates,5–7 nonlinear opticalphenomena,8,9 and speckles.10 Also, another method
of OVs net generation based on three homogenousplane-wave interference is presented.11 These meth-ods seem intuitive and simple in analysis but com-plicated in realization due to the use of as many asthree different waves; however, their value has beenconfirmed by small-angle rotation measurement.12
As Masajada and Dubik11 emphasize at the end oftheir work, their proposal was just a starting point forfurther research, and the work presented below is anattempt to elaborate on that further research. Wepresent another realization of the three-beam inter-ferometer using the Wollaston prism (WP) in one ofthe arms of our interferometer, which again makes ita simple two-beam one. It also means using fewerhigh-quality optical elements in the setup and makesthe experiment easier and cheaper.
2. Scheme of the Interferometer
Let us consider the interferometer containing a WP13
as in Fig. 1. Four beam splitters form a classicaltwo-armed Mach–Zender interferometer. In fact, twoof them (marked as 2 and 3 in Fig. 1) could be re-placed with mirrors. The interferometer’s arm be-tween the beam splitters marked as 1 and 2 will becalled as reference (index r in further considerations)while the arm 3-4 will be the object (index o). The WPwith the azimuth angle of 45° of its first eigenvectoris placed in the reference arm, while the transmissionfilter is placed in the object arm to control the relativeintensity of the light beam in both arms.
Let us assume that a fully polarized, monochro-matic plane wave with an azimuth angle of 0° entersthe first beam splitter 1. This wave is split into twowaves: reference and object, and these waves haveamplitudes Ar and Ao, respectively. In fact, we denote,as Ao, the amplitude of the light in the object armafter passing through the filter F. That allows us to
The authors are with the Institute of Physics, Wrocław Univer-sity of Technology, Wybrzeze Wyspianskiego 27, Wrocław 50370,Poland. W. Wozniak’s e-mail address is [email protected].
Received 13 February 2006; revised 20 April 2006; accepted 22April 2006; posted 6 June 2006 (Doc. ID 67898).
We decided to choose the Jones formalism to de-scribe the behavior of the polarized light in our setup.In this formalism, we can describe the Jones vectorErln of the light incident on the WP as
Erln � Ar �10�. (2)
The Jones matrix JWP of the WP with the azimuthangle of the first eigenvector equal to 45° and thephase difference �r between the first and the secondeigenvector is done by
JWP �12 �1 � e�i�r
1 � e�i�r
1 � e�i�r
1 � e�i�r�. (3)
Due to the specific orientation of the WP, thephase difference �r changes linearly (for fixed z coor-dinate) its value along the line y � x, which means aline parallel to the prism axis. The values of thischange depend on the shearing of the used birefrin-gent prism.
The Jones vector ErOut of the light passing throughthis prism is then given by
ErOut �12 �1 � e�i�r
1 � e�i�r
1 � e�i�r
1 � e�i�r�Ar�10��12 Ar�1 � e�i�r
1 � e�i�r�.
(4)
Let us note that we omit the transmission coeffi-cient of the prism. Simply, all of this arm’s transmis-sion coefficients (as well as the reflection coefficients
of the second and third beam splitters) are reduced toone value of Ar.
This form of equation for ErOut will be used in ourfurther consideration, however, to explain some fun-damentals of the proposed setup, we shall rewrite itin another form:
ErOut � Are�i �r�2 � cos
�r
2
sin�r
2 ei��2 �. (42)
It means that the light wave, after passing throughthe WP, changes both its phase and its polarizationstate—the azimuth angles are equal to 0° (or 90°)while the ellipticity angles vary in the same way (asphase) along the line y � x. Then, we can considerthis reference arm in our interferometer as an equiv-alent of two arms in usual three-beam interferometer(as, for example, the one described in Ref. 11). Sincethe WP is a birefringent medium, the outgoing wavecan be treated as a superposition of two orthogonallypolarized waves. From this point of view, one canfinally recognize our interferometer as a type ofthree-beam one. Regardless of the name, we proposethis setup as an alternative build with its advantagesand disadvantages.
Getting back to our calculations—the Jones vectorEoOut of the light in the interferometer’s object armafter passing through the filter F is described as
EoOut � Ao�e�i�o
0 �, (5)
where �o denotes the phase-shift distribution of themeasured object wave.
At the interferometer’s output, these two wavesErOut and EoOut interfere and the result of this inter-
Fig. 1. Scheme of the interferometer setup and orientation of coordinate axes: P, polarizer; WP, Wollaston prism; A, analyzer; F,transmission filter; BS1, BS2, BS3, BS4, four beam splitters.
ference should be observed using a linear analyzerwith an azimuth angle equal to 0°. The Jones matrixof this analyzer is done by
JA � �10 00�. (6)
Note that we omit the analyzer’s transmission co-efficient because both of the interfered waves are at-tenuated at the same value. This fact does not affectthe contrast of the interference pattern. Then, theJones vector of outgoing light is done by
EOut � JA�ErOut � EoOut�, (7)
which finally leads to the following equation:
EOut �12 Ar�1 � e�i�r � 2ae�i�o
0 �. (8)
Now our task is to find the formulas for phase-shifts �r and �o, which allow us to obtain the opticalvortices in the phase front distribution of EOut. Suchsingularities could be formed only in those locationsin which the isolated points of zero intensity of thelight appear. It means that
IOut � EOutEOut* � 0, (9)
where the asterisk denotes a conjugation of complexquantity of EOut. Equation (9) is fulfilled if only bothreal and imaginary part of EOut equal zero:
Re�EOut� � 0,
Im�EOut� � 0, (10)
which immediately leads to the following equations:
1 � cos �r � 2a cos �o � 0,
sin �r � 2a sin �o � 0. (11)
Equations (11) have a pair of solutions as follows:
cos �o � �a, cos�r
2 � �a,
cos �o � �a, cos�r
2 � �a.(12)
depending only on the relative amplitude a of the twointerfering waves ErOut and EoOut. We can change thisparameter using the transmission filter F. As men-tioned above, this beam could be treated as two wavesleaving the WP and that would be the explanation ofdifferent behavior of phase-shifts �r and �o in Eq. (12).The same notation could be made for factor 1�2 inthis equation.
We have obtained a set of points in the interferenceimage at the output of our setup in which the inten-sity of the light is equal to zero. We claim these pointsas optical vortices; however, we have decided to provethis in an experimental way (see Section 4). In thistheoretical section, another problem arises worthmentioning. As follows from Eqs. (12), the parametera—the relative amplitude of the reference and objectbeams—is very important for the process of obtaininggood quality minima of intensity in the interferenceimage. One can, for example, calculate the sensitivityof the phase shift �o according to the changes of theparameter a as
��o
�a �1
1 � a2. (13)
Because the parameter a represents the relativeamplitude of both beams in the interferometer whilethe relative intensity is usually controlled and mea-sured, another quantity could be calculated:
��o
�I ���o
�a�a�I �
1
1 � I
12
1
I�
1
2I�1 � I�, (14)
where I � a2 denotes the relative intensity coefficientof both interfering waves. It means that if we couldcontrol the relative intensity with accuracy equal to�I, the accuracy of the (measured) object wave ��o
could be equal to
��o ���o
�I �I �1
2I�1 � I��I, (15)
and could be minimized for I � 0.5:
��o,min � 0.25�I. (16)
The same formulas could be obtained for referencewave �r:
��r ���r
�I �I �1
I�1 � I��I, (15�)
��r,min � 0.5�I. (16�)
Both accuracies are minimal for the same value ofI � 0.5. Because we are able to control the relativeintensity I with accuracy �I equal to 1%, we canobtain the accuracies of both waves as
Note that these accuracies could not be treated asan accuracy of determination of the measured phase-shift �o but as calculation errors due to the inaccuracyof filter F in the setup.
3. Numerical Analysis
As shown in Section 2, a set of isolated intensity zeroscould be obtained in the interference image at theoutput of the proposed setup. Equations (12) give thesolution for the phase shifts in both arms of our in-terferometer, for which the minima of intensity couldbe observed. Note that Eqs. (12) could be solved onlyfor the value of the relative amplitude parameter abetween 0 and 1. If this parameter exceeds the valueof 1, no OVs could be observed. However, we shouldprove that these points indeed represent the OVs—singularities in a wavefront phase distribution. A nu-merical calculation of the phase and intensity in theinterference image has been done for the simple caseof the object wave. Due to the specific orientation ofthe WP, the phase shift of reference beam �r is equalto the shearing of this prism and changes linearlyparallel to the line y � x. As an object wave �o, weassumed the plane wave inclined to the y axis, whichsimply means that the phase difference �o changeslinearly along this axis. We calculate the phase in the
interference plane using the formula for the (only) xcomponent of the output electric field [see Eq. (8)]:
Ex,Out � 1 � e�i�r � 2ae�i�o. (18)
The field Ex,Out is a complex number and could berepresented in trigonometric form as a product of areal complex modulus (here, the intensity) and a com-plex argument (here, the phase). Some results of ourcomputer calculations of intensity and phase distri-butions in the interference plane for the differentrelative intensity coefficient I are presented in Fig. 2.The gray background of these figures represents theintensity of the light, while the gray curves denoteconstant phases. The white circles and triangles inFigs. 2(a)–2(c) denote the OVs of the opposite signs. Asshown in the following figures, these points move alongthe straight line while changing the relative intensitycoefficient I and finally they annihilate14 [whitesquares in Fig. 2(d)]. To clearly illustrate the appear-ance of the OVs in these locations, we show Fig. 3, inwhich the phase constant lines around the single OVare presented in detail (and labeled). Observed phasediscontinuities in marked locations are good evidencethat the singularities (OVs) indeed exist.
Fig. 2. Results of numerical analysis: intensity (gray levels) and phase distributions (curves) in the interference image for different valuesof intensity coefficient I: (a) I � 0.25; (b) I � 0.5; (c) I � 0.75; (d) I � 1 (see detailed description in text).
Numerical analysis is always used to check the cor-rectness of the proposed new experimental setups aswell as to predict their behavior accordingly to pos-sible changes and modifications. However, a goodpractice in all fields of physics is to make experi-ments. During these experiments some remarksshould be made about the conditions that should besatisfied by the elements of the setup.
Owing to the use of the polarized light (light source:the He–Ne laser), whose polarization parameters
should be precisely controlled, a set of good qualitypolarization beam splitters have to be applied in theproposed setup. We assume that all the beam split-ters do not change the polarization state of transmit-ted and reflected light. Instead of using a good qualityfilter with a changeable transmission coefficient, wehave used a set of two pairs of polarizers—accordingto the Malus law, we have been able to obtain allpossible values of intensities of both beams by simplyrotating the first polarizers in both arms. In themeantime, the second polarizers allowed us to obtainthe proper polarization states on the third and fourthbeam splitters. A CCD camera (pixel size: 6.45 m 6.45 m) connected to the PC was used to registerand store obtained interferograms. An additionalmirror was placed next to the beam splitter 3 to leadto the interference of our final EOut wave (with sin-gularities) with an additional reference plane wave.This interference allows us to obtain (or not) charac-teristic forks in the interference patterns that con-vinced us about the existence (or not) of OVs. Thefinal configuration of our experimental setup is pre-sented in Fig. 4. Some results of our experiments arepresented in the following figures. Intensity distribu-tions in the interference image for different intensi-ties of reference and object waves are shown in theleft part of Fig. 5. The circles mark the points inwhich the minimum intensities have been detectedusing well-known methods of image analysis.15 Theright part of Fig. 5 shows the interferograms obtainedby the interference with additional plane waves. Fig-
Fig. 3. Lines of phase distribution in the interference image forthe intensity coefficient I � 0.5 around the optical vortex (en-larged); numbers denote phases in degrees.
Fig. 4. Scheme of the experimental setup: P, polarizer; WP, Wollaston prism; A, analyzer; BS1, BS2, BS3, BS4, four beam splitters; Pr1,Po1, polarizers used to change the intensities of the light in reference and object arms, respectively; Pr2, Po2, polarizers used to obtain thes- or p-polarization states on beam splitters; C, CCD camera connected to the computer; M, additional mirror to obtain the forks (anglesand distances between the elements are not preserved).
ure 5(a) illustrates the situation in which the relativecoefficient I between the intensities of both interfer-ing waves in our interferometer is nearly equal tothe ideal value of 0.5 and separated points of mini-mum intensity could be easily found. In the right partof Fig. 5(a), one can see the characteristic forks—discontinuities in the interference fringes that couldbe good evidence for the existence of OVs. Figure 5(b)illustrates another situation in which the relativecoefficient I differs much more from the ideal value of0.5 and only the usual minima of intensity appear (noOVs in the left part of Fig. 5(b) and no forks in theright part of Fig. 5(b)—except maybe the top leftcorner of the figure in which one fork probably stillexists).
5. Conclusions
In this paper, we have proposed a new setup of theinterferometer in which the set of specific opticalmarkers called the optical vortices could be gener-ated. The classical Mach–Zender two-beam setup hasbeen modernized with the use of the polarization el-ements. The main element of this setup is the WP,
which allows us to obtain a specific phase distributionin one of the arms of our interferometer. As shown inthe theoretical analysis, the optical vortices couldeasily be obtained for a wide range of both beamparameters such as phase distributions and relativeintensities. The numerical analysis has confirmedour theoretical predictions. Finally, we carried outsome experiments that convinced us about the prac-tical value of the proposed setup, and we obtainedstable sets of OVs for different values of relative in-tensity coefficients I.
This work was supported by the Polish Ministry ofScientific Research and Information Technology un-der grant 3T10C04829.
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Fig. 5. (left) Intensity distributions in the interference image fortwo different relative intensities I of the reference and objectwaves; the circles mark the points in which the minimum inten-sities have been founded; (right) same after adding an additionalplane wave: (a) I nearly equals the ideal value of 0.5; characteristicforklike fringes in the right figure; (b) I too far from the ideal valueof 0.5; no forks.
