Tavrov et al. Vol. 16, No. 4 /April 1999 /J. Opt. Soc. Am. A 919

Method to evaluate the geometricalspin-redirection

phase for a nonplanar ray

Alexander V. Tavrov,* Tsutomu Kawabata, Yoko Miyamoto, and Mitsuo Takeda

Department of Communication and Systems Engineering, The University of Electro-Communications,1-5-1, Chofugaoka, Chofu, Tokyo, 182-8585, Japan

Vladimir V. Andreev

P. N. Lebedev Physical Institute, Leninsky Prospect 53, 117924 Moscow, Russian Federation

Received June 23, 1998; accepted November 24, 1998

An evaluation method is suggested for determining the Berry geometrical spin-redirection phase in systemswith a set of separated optical elements. The evaluation considers the coordinate transformation, taking intoaccount the directions of the beam. The spin-redirection phase contains information about the nonplanarityof beam propagation. This geometrical phase can be determined even if the ray trajectory is traced as anunclosed topological contour on the unit sphere associated with wave-vector directions. © 1999 Optical Soci-ety of America [S0740-3232(99)00204-5]

OCIS codes: 120.3180, 120.2130, 120.0120, 120.5050, 080.2730.

1. INTRODUCTIONTwo types of Berry1 geometric phase in optics have beendescribed and experimentally demonstrated.2 They cor-respond to the rotations of the k wave vector and the Ppolarization (Jones) vector. The best known type of geo-metric phase is the Pancharatnam phase,3–5 which iscaused by the P propagation along the closed contour on aPoincare sphere. This type of propagation always causescyclic wave states. If A is the polarization transforma-tion and the Pancharatnam phase is denoted as wPanch ,this is the case of the eigenvalue, and P is the eigenvector:

AP 5 P exp~iwPanch!. (1)

The second type of geometrical phase is called the spin-redirection phase. It was discovered in 1938–1941 byRytov6 and Vladimirsky,7 but not until after publicationsby Chiao’s group2,8 were investigations of the spin-redirection phase continued. The closed contour in para-metric (Hilbert) space is traced on the unit sphere of di-rections associated with the k-vector directions. Thecontour illustrates the nonplanarity of the ray propaga-tions. According to the Gauss–Bonnet theorem, the geo-metrical phase is proportional to the solid angle that isbased on that closed contour.

The dynamical phases that are due to optical path dif-ferences and the Fresnel shift are usually eliminated toallow visualization of geometrical phases only. The geo-metrical w and dynamical f components satisfy the gen-eral equation for the combined phase Fcomb , which is de-termined from the following interferogram2:

Fcomb 5 w 1 f. (2)

Initial experiments exposed the contribution of the geo-metrical phase to the interferometer image.9,10

0740-3232/99/040919-03$15.00 ©

However, if an optical system consists of several ele-ments and the beam propagates in many planes, it is dif-ficult to evaluate the spin-redirection phase. The defini-tion of the solid angle that is based on many geodesiclines becomes nontrivial and inconvenient. Moreover,this representation does not allow one to operate withvarious polarizations.

In the case of a closed contour on the unit-directionsphere, the spin-redirection phase is well defined. In themore general case of an unclosed contour, the relation ofthe polarization rotation to this spin-redirection phase isconsidered.

The purpose of the present paper is to introduce anevaluation routine and an interpretation of the spin-redirection geometrical phase stipulated by nonplanarray trajectory.

2. POLARIZATION DEFINITION UNDER ASET OF NONPLANAR REFLECTIONSTo operate on the two vectors k and P in three-dimensional space described with the Cartesian coordi-nate set $x, y, z%, the commonly known form of the two-component complex Jones vector has to be denoted inthree-component form:

k 5 S kx

ky

kz

D ⇒ P 5 S Px

Py

Pz

D . (3)

The notation in expression (3) is unusual because the po-larization components P 5 (Px Py) are always mea-sured in the plane normal to k. However if k rotates, Passumes the existence of the third nonzero component

1999 Optical Society of America

920 J. Opt. Soc. Am. A/Vol. 16, No. 4 /April 1999 Tavrov et al.

with respect to the laboratory coordinate set of $x, y, z%.This becomes obvious if a snapshot of the electric field isconsidered.

Let us consider the transformation that describes a re-flection and requires the condition

k1 5 M1k0 ,

P1 5 M1P0 . (4)

The operator M1 in Eqs. (4) means a coordinate transfor-mation from the space of the incident wave state withsubscript 0 to the space of the transformed wave statewith subscript 1. The M1 transforms an arbitrary vectordefined in the 0 basis to the one with corresponding coor-dinates in the 1 basis. The spinor P1 with three nonzerocomponents can be interpreted as a Jones vector bymeans of the P1 projection to the plane normal to k1 .This is the plane x1 y1 of the local coordinate system$x1 y1z1%, where k1 5 (0 0 1). The $x1 y1z1% coordi-nate system is rotated by the angles 2h and j with respectto the laboratory system $x, y, z%.

Let us solve system (4) for a geometry of the nonplanar(skew) reflection illustrated in Fig. 1. Then the uniquesolution for M1 can be factorized by considering the azi-muthal h and the meridional j rotation angles:

M1 5 F cos j 2sin j 0

sin j cos j 0

0 0 1GF cos 2h 0 2sin 2h

0 1 0

sin 2h 0 cos 2hG

3 F cos j sin j 0

2sin j cos j 0

0 0 1G , (5)

M1 5 @j#@h#@2j#. (6)

The form of matrix M1 [Eq. (6)] is similar to the Jonesmatrix calculation routine, in which an optical elementthat is at first determined on its own basis is to be rotatedafterward to a basis of calculations. However, Eq. (6) isgeneralized for the 3D case and is valid for a single in-stance of skew reflection. The only basis here is the co-ordinate system where the units are p and s field compo-nents, the p component remains constant, and the scomponent reverses its sign.

Let us consider the next instance of reflection, which isdescribed by the operator M2 . If the second reflection di-rects the beam out of the first reflection plane, it is impos-sible to factorize the product M2 M1 to the form of just

Fig. 1. Geometry of the skew reflection.

two rotations as given by Eq. (6). The M2 M1 decompo-sition requires the third rotation, j8, which is associatedwith the geometrical phase:

M 5 M2 M1 5 @j218 #@h21#@2j21#. (7)

In Eq. (7) the angles h21 and j21 are no longer connectedwith the optical-element planes. Instead, these anglescorrespond to the rotation of k2 with respect to k0 (h21)and to the orientation of the p and s components with re-spect to the laboratory system $x, y, z% (j21).

Therefore, under the nonplanar set of skew transfor-mations P2 5 MP0 , the polarization vector is rotated byDj21 5 j218 2 j21 .

In the case of an interferometer, to define the geometri-cal phase jab8 for the two arms a and b, we multiply thematrices for each: M 5 Ma Mb

21.

3. COORDINATE TRANSFORMATIONASSOCIATED WITH BEAM PROPAGATIONThe solution of the Eqs. (4) for a set of separated elementsis an operator M 5 Mi11Mi...M1 that describes the coor-dinate transformation associated with beam propaga-tions. A convenient method for calculating M is based ona particular proof of the Gauss–Bonnet theorem and canbe denoted as

M 5 Fi11EiTFi ...E1

TF1E0T . (8)

In the notation of Eq. (8), Ei , Fi11 are two coordinate setsrotated about the vector

n 5ki 3 ki11

iki 3 ki11i:

Ei 5 @ki , ~n 3 ki!, n#,

Fi11 5 @ki11 , ~n 3 ki11!, n#,

where the wave vectors ki and ki11 correspond to the onesbefore and after the ith beam transformation, iki3 ki11i is norm of the vector, and T transposes the ma-trix.

E0 determines the orientation of the laboratory coordi-nate system with respect to the initial vector k0 , whichhas there the coordinates

k0T 5 ~0 0 1 !. (9)

4. INTERPRETATION OF THEPOLARIZATION ROTATION IN TERMS OFAMPLITUDE AND PHASELet us consider that the initial k0 and the final kM areparallel: kM 5 Mk0 5 k0 and M 5 Mi11Mi...M1 :

M 5 F cos 2j8 2sin 2j8 0

sin 2j8 cos 2j8 0

0 0 1G . (10)

Nevertheless, the components PxM, PyM

are rotated toform an angle 2j8 with respect to Px0

, Py0.

Tavrov et al. Vol. 16, No. 4 /April 1999 /J. Opt. Soc. Am. A 921

Let us first consider the initial circular polarization,P0 5 (i 1 0). The polarization PM transformed bymatrix (10) also remains circular; however, PM Þ P0 :

PM 5 P0 exp~62jj8!. (11)

To illustrate the physical meaning of Eq. (11), let usconsider the schematic diagram in Fig. 2, where the z axisis normal to the plane of the sketch. Let us denote thefollowing: The Px0

, Py0units coincide with x, y, the PxM

,PyM

components are rotated to form angle 2j8, and thephase of A is w 5 0 at time t 5 0.

The local coordinate system that follows the beam is as-sociated with the directions $PxM

PyMz%. The direc-

tions are rotated according to matrix (10). Therefore att 5 0, the position of electric field vector in the trans-formed beam is the point B. As a result, the geometricalspin-redirection phase shift in the PM vector with respectto the P0 appears.

Let us stress that the above shift of the phase compo-nent under linear polarization is, unfortunately, invisible,though the linear polarization is used primarily in practi-cal interferometry. The linear polarization is always su-perposed in two orthogonal circular polarizations. Asboth the left- and the right-hand circular polarizationsproduce a geometrical phase with opposite signs, thephase component is compensated and only amplitudecomponents will include an j8 rotation:

MP linear ⇒ MS 10 D 5 S cos 2j8

2sin 2j8 D .

Therefore direct visualization of the geometrical phase re-quires an operation with the circular polarizations.

5. CONCLUSIONSA definition of the spin-redirection phase is proposed foran arbitrary nonplanar ray trajectory in a set of discreteoptical elements. If the pairs of wave vectors ki and ki11before and after the ith element are known, the proposedmethod is valid both for reflected and refracted beams.The method also allows evaluation for different polariza-

Fig. 2. Delay of the circular polarized beam resulting from anonplanar propagation.

tions. A limitation of the routine is that polarizationchanges caused by skew reflections are not considered.However, the authors hope to extend this approach totake into account Fresnel shifts and anisotropic cases.The calculation of the spin-redirection phase has beenperformed with the geometrical-optics ray approach. Aconsideration of this situation with the imaging approachrequires further investigation.

ACKNOWLEDGMENTSFinancial support from the Japanese Society for Promo-tion of Science under grant P97332 is gratefully acknowl-edged. Substantial financial support from NanoTech Co.Ltd., Russia, allowed us to initiate these investigations in1994. The authors are grateful to the Institute of Struc-tural Macrokinetics, Russian Academy of Science, Cher-nogolovka, Russian Federation to Vladimir Maltzev, andto all the specialists who have previously worked on thesubject.

*On leave from Institute of Structural Macrokinetics,Russina Academy of Sciences.

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