Jones and Mueller matrices for specular reflection from a chiral medium: determination of the basic chiral parameters using the elements of the Mueller matrix and experimental configurations to measure the basic chiral parameters Elena Georgieva Using Fresnel reflection amplitudes, the Jones and Mueller matrices for reflection from a nonabsorbing gyrotropic medium are presented. Some basic chiral parameters are defined by using the elements of the Mueller matrix; experimental configurations are described. Key words: Rotatory power, gyrotropy, polarization, reflection, Mueller matrices. Introduction By definition an object is chiral if it cannot be brought into congruence with its mirror image by translation or rotation. Chiral objects occur in two versions, so that one is a mirror image of the other, and they are called enantiomorphous. All such objects have the property of right handedness or left handedness. The gyrotropic (chiral) medium may be natural or induced (electrically or magnetically). The origin of the asym- metry in the structure of such a medium is a quan- tum interference effect between electric and magnetic dipole transitions connecting parity noninvariant states. The usual constitutive equations D = EE and B = , H. used for nongyrotropic media, are modified in the case of chirality. The theoretical description of the gyrotropy is necessarily provided by the symmetric set 3 of constitutive or material relations that con- nect the electric field and magnetic induction E, B to the electric displacement and magnetic field D, H: D = e(E + 3V x E), B = (H + V x H), where e is the dielectric constant, ,uis the permeabil- The author is with the Department of Physics of Condensed Matter, Faculty of Physics, University of Sofia, 5, Boulevard A. Ivanov, Sofia BG-1126, Bulgaria. Received 22 February 1990. 0003-6935/91/345081-05$05.00/0. c 1991 Optical Society of America. ity, and 1 is a measure of the chirality of the medium. This set involves light-induced electric polarization and magnetization terms. In an isotropic gyrotropic medium the incident linear polarization splits into two circular polariza- tions that propagate with different velocities, result- ing in rotation of the plane of polarization. In a birefringent gyrotropic medium tihe incident linear polarization splits into two elliptical polarizations whose major axes are normal and the direction of the elliptical polarizations is opposite. The measurement of gyrotropy so far has been limited to well-established light-transmission meth- ods: the optical rotatory dispersion and circular dichro- ism. Scattering can provide additional information for the gyrotropic properties of the medium that is not obtained from transmission experiments. The problem is of considerable importance for strongly absorbing (opaque) substances, where traditional methods are of no help. Because of the experimental difficulties reliable experiments have not been carried out so far, to the best of our knowledge. A number of papers 7 have appeared recently on the subject of gyrotropy. Of particular importance are the theoretical results of Silverman, which we believe to be the major breakthrough on the subject. The objectives of this paper are to obtain the Jones and Mueller matrices for reflection from a gyrotropic medium, the definition of the basic chiral parameters by using the elements of the Mueller matrix, and experimental configurations for their determination by using the powerful Stokes-Mueller method. 1 December 1991 / Vol. 30, No. 34 / APPLIED OPTICS 5081
Transcript
Jones and Mueller matrices for specular reflectionfrom a chiral medium: determination of the basicchiral parameters using the elements of theMueller matrix and experimental configurations tomeasure the basic chiral parameters
Elena Georgieva
Using Fresnel reflection amplitudes, the Jones and Mueller matrices for reflection from a nonabsorbinggyrotropic medium are presented. Some basic chiral parameters are defined by using the elements of theMueller matrix; experimental configurations are described.
By definition an object is chiral if it cannot be broughtinto congruence with its mirror image by translationor rotation. Chiral objects occur in two versions, sothat one is a mirror image of the other, and they arecalled enantiomorphous. All such objects have theproperty of right handedness or left handedness. Thegyrotropic (chiral) medium may be natural or induced(electrically or magnetically). The origin of the asym-metry in the structure of such a medium is a quan-tum interference effect between electric and magneticdipole transitions connecting parity noninvariantstates.
The usual constitutive equations D = EE and B =, H. used for nongyrotropic media, are modified in the
case of chirality. The theoretical description of thegyrotropy is necessarily provided by the symmetricset 3 of constitutive or material relations that con-nect the electric field and magnetic induction E, B tothe electric displacement and magnetic field D, H:
D = e(E + 3V x E),
B = (H + V x H),
where e is the dielectric constant, ,u is the permeabil-
The author is with the Department of Physics of CondensedMatter, Faculty of Physics, University of Sofia, 5, Boulevard A.Ivanov, Sofia BG-1126, Bulgaria.
Received 22 February 1990.0003-6935/91/345081-05$05.00/0.c 1991 Optical Society of America.
ity, and 1 is a measure of the chirality of the medium.This set involves light-induced electric polarizationand magnetization terms.
In an isotropic gyrotropic medium the incidentlinear polarization splits into two circular polariza-tions that propagate with different velocities, result-ing in rotation of the plane of polarization. In abirefringent gyrotropic medium tihe incident linearpolarization splits into two elliptical polarizationswhose major axes are normal and the direction of theelliptical polarizations is opposite.
The measurement of gyrotropy so far has beenlimited to well-established light-transmission meth-ods: the optical rotatory dispersion and circular dichro-ism. Scattering can provide additional informationfor the gyrotropic properties of the medium that isnot obtained from transmission experiments. Theproblem is of considerable importance for stronglyabsorbing (opaque) substances, where traditionalmethods are of no help. Because of the experimentaldifficulties reliable experiments have not been carriedout so far, to the best of our knowledge.
A number of papers 7 have appeared recently onthe subject of gyrotropy. Of particular importance arethe theoretical results of Silverman, which we believeto be the major breakthrough on the subject.
The objectives of this paper are to obtain the Jonesand Mueller matrices for reflection from a gyrotropicmedium, the definition of the basic chiral parametersby using the elements of the Mueller matrix, andexperimental configurations for their determinationby using the powerful Stokes-Mueller method.
Jones and Mueller Matrices for Reflection from aGyrotropic Medium
Several authors8 9 have pointed out that incident s- orp-polarized light is reflected from a nonabsorbingchiral medium as an elliptical polarization, i.e., theincident s- or p-polarized light generates ellipticalpolarization. We exclude the generation of ellipticalpolarization caused by absorption, surface roughness,and defect scattering. An ideal nongyrotropic surfacehas a diagonal Jones matrix in the p, s frame ofreference. However the Jones matrix of a gyrotropicsurface has (although small) nonzero off-diagonalelements. It was shown in Ref. 4 that it is impossibleto satisfy the conditions of continuity of tangentialcomponents of E and H across the boundary unlessthe reflected wave contains in general s- and p-polar-ized components.
In Fig. 1(a) the amplitude E of the incident light isresolved into Es (perpendicular to the plane of inci-dence) and E (parallel to the plane of incidence)components. n our frame of reference the directionof Es coincides with the X direction and E, coincideswith the Y direction. The light propagates in the Zdirection. The incident Es light [Fig. 1(b)] gives rise tothe reflected E "s2 = rsE, and E"p = rE light, whichwhen combined constitutes the reflected ellipticallight caused by the incident Es light. Similarly [Fig.1(c)] the incident E light generates the reflectedE" = rEp and E"p = r E light. Here r r , rp andrpp are the Fresnel reflection amplitudes, where r.. andrpp are the big terms and rs and r are the small terms(the small terms are due to the gyrotropy of the
Elt
p(a)
EB
E'0
(b)EPs
En(c) -1
Fig. 1. (a) Amplitude E of the incident light is resolved into E.(perpendicular to the plane of incidence) and E, (parallel to theplane of incidence) components. (b) Incident light E. gives rise tothe reflected light E",, = rE 8 and E`,s = rE. (c) Incident light E,generates the reflected light E" = rEP and E"PP = rPPEP.
medium). Therefore the reflected s- and p-polarizedlight is
E" = Es + E,p = rsE + rEp,E ̀ = E + E . = rpsE, + rEP. (1)
With the Jones matrix and the Maxwell column wewrite Eqs. (1) in the form
[E'= [, Fs [E . [rssr + irssi rspr + irsp[ E, 1E"pL p Jpp EP rps, + irp.. rppr + irppi EP
where the complex Jones elements are written withtheir real and imaginary parts.
The correspondence between our designation andthat of Silverman is
rssra,, rpr a2 , rps, -bl, rppr b2,
r -si a11, rpi - 21 rp - I, rppi P2-
The Fresnel amplitudes of a gyrotropic nonabsorbingmedium were first derived in Ref. 5.
With the relation between the Jones and Muellermatrices of a definite optical component, 0 the deriva-tion of the elements of the Mueller matrix for reflec-tion from a nonabsorbing gyrotropic surface is amatter of a straightforward calculation:
Definition of the Chiral Parameters in Terms of theMueller Matrix Coefficients
We use the fact introduced in Ref. 6 that the differentchiral parameters for reflection from a gyrotropicsurface can be defined by Mueller matrix elements.
The differential linear reflection, which is definedas the differential reflection of the incident s- andp-polarized light (I, is the intensity of the reflectedlight from incident s-polarized light and I is theintensity of the reflected light from incident p-polar-ized light):
Experimental Configurations to Measure the BasicChiral Parameters
The definition of the three basic chiral parameters(differential reflection of right and left circularlypolarized light, skewness, and differential linear reflec-tion) using the elements of the Mueller matrix of achiral medium allowed us to propose experimentalconfigurations to determine these parameters. Fourdifferent configurations are needed because with twoconfigurations we find relations between the mea-sured signals and the three parameters and withanother two we determine the Bessel functions J, (A)and J2 (A) entering these relations.
Configuration I
Configuration I is shown in Fig. 2. Light polarized at450 to the X axis is incident upon a phase modulator'2
with its fast axis at 90° with respect to X Here X isperpendicular to the plane of incidence, Y is parallelto the plane of incidence, and Z is along the incidentlight. The Mueller matrix of the specimen, the modu-
lig-htFig. 2. Configuration I: X is perpendicular to the plane ofincidence, Y is parallel to the plane of incidence, the light is alongthe Z axis, P is at 450 with respect to X, and the modulator fast axisis at 90° with respect to X.
lator, and the Stokes column of the light incidentupon the modulator are
M12 M13 M14 1
M22 M 23 M2 4 0
M 32 M33 M34 0
M42 M43 M44 o
0 0 0 1
1 0 0 0
0 cos S -sin 8 1
0 sin 8 cos 8 0 ,
where 8(t) = A sin wt is the phase shift of thephotoelastic modulator, the driving frequency of whichis f = w /2 r. The first element of the Stokes column ofthe light incident on the photomultiplier is
M1l + M13 cos 8 + Ml4 sin 5. (5)
Using the relations
sin 5(t) = sin(A sin wt)
= 2J1(A) sin wt + 2J3(A) sin 3wt
+ 2J, (A) sin 5wt +....
cos 5(t) = cos(A sin wt)
= Jo(A) + 2J2 (A) cos 2wt + 2J4 (A) cos 4wt + . . .,
expression (5) becomes
Ml, + 2M,,J 2 (A) cos 2wt + 2J,(A)M 4 sin wt, (6)
where only the Bessel functions J (A) of orders oneand two are considered. Here our choice is A = 137,8°and this makes Jo(A) = 0.
Therefore the intensity of the light incident on thephotomultiplier and the voltage across its load resis-tance may be written as
I, = Ml, + 2M1 2 J2 (A) cos 2wt + 2M1 4 J1 (A) sin wt,
U = p,[M1, + 2M,,J 2 (A) cos 2wt + 2M1 4J1 (A) sin wtI, (7)
where p, includes the cathode efficiency and gain ofthe detector. We write Eqs. (7) in the form (constantdirect-voltage mode)
where CO = plMj = (Ud,), is the direct voltage. FromEq. (8),
UMw| Ud . =2J,(A)(M, 4 /M 1 ) = 2J1 (A)d0,
lu(2w)lUd |, 2J2 (A)(MM3/Mll) = 2J2 (A)s..
(9)
(10)
Configuration II
Configuration II is shown in Fig. 3. It is the same asconfiguration I except that it has a second polarizerafter the modulator, oriented at an angle Ar/4 withrespect to X The matrices and the Stokes column forthis case are
M11 M12 M13 M14 1
M21 M 22 M2 3 M2 4 0
M31 M32 M3 3 M34 1
M41 M42 M4 3 M4 4 0
0 1 0 1
0 0 0 00 1 0 0
0 0 0 0
0
1
0
0
0
0
cos 8
sin 8
0 '1*
0 0
-sin 8 1
cos 8 0 .
The intensity of the light incident on the detectormay be written as
I2 = (M11 + MI3)(1 + cos ) = (M11 + M13)[1 + 2J2 (A) cos 2wt].
The voltage across the load resistance may be writtenas
U2 = 2(M11 + M13 )[1 + 2J 2 (A) cos 2wt] = C + 2J 2 (A) cos 2wt,
where CO = p2 (Mll + M13 ) = p1 Mj, is the direct voltageof the second configuration, made equal to the direct
Y
PiIYT fast axis
lig]Fig. 4. Configuration III: X is perpendicular to the plane ofincidence, Y is parallel to the plane of incidence, the light is alongthe Z axis, P is at 45° with respect to X, the modulator fast axis is at90° with respect to X, the fast axis of the X/4 plate is at 90° withrespect to X, and A is at 450 with respect to X.
voltage of the first configuration by changing thevoltage supply of the photomultiplier or the incidentlight intensity (constant direct-voltage mode). There-fore for configuration II we have
U(2w)lUd. =2J 2 (A). (11)
Configuration III
Configuration III is shown in Fig. 4. The differencebetween configuration II and configuration III is theX/4 plate (at azimuth 90° with respect to X) insertedbetween the modulator and the second polarizer. Thematrix array and the first element of the Stokescolumn of the light incident on the detector are
M, M M13 M14 1 0 1 0 1 0
M21 M22 M M2 4 0 0 0 0 0 1
M3 M32 M M,4 1 0 1 0 0 0
M41 2 M43 M4 4 0 0 0 0 0 0
z
PMT
m~odulator fast axisa
.1 light
Fig. 3. Configuration II: X is perpendicular to the plane ofincidence, Y is parallel to the plane of incidence, the light is alongthe Z axis, P is at 45° with respect to X, the modulator fast axis is at90° with respect to X, andA is at 450 with respect to X.
13 = (M11 + M13)(1 - sin 5),
U3 = p 3(M1 + M13)(1 - sin 8) = p 3(M11 + M13)
- p3(M11 + M13)2J,(A) sin wt = C0 - 2CQJ1 (A) sin wt
(constant direct-voltage mode).For configuration III we write the relation
UdW = 2J1 (A). (12)
Configuration IV
Configuration IV is shown in Fig. 5. The polarizer isat azimuth zero and the modulator is at azimuth r/4
Fig. 5. Configuration IV: X is perpendicular to the plane ofincidence, Y is parallel to the plane of incidence, the light is alongthe Z axis, P is at 00 with respect to X, the modulator fast axis is at450 with respect to X.
with the matrix with respect to X. The matrix array is
Ml
M21
M31
M41
M 12 M 13 M 14 1
M2 2 M2 3 M24 0
M 32 M 33 M 34 0
M42 M 43 M 44 0
0
cos 8
0
sin 8
0
0
1
0
0 '1
-sin S 1
0 0
CosS8 0.J
From this array,
I4 = (M11 + M12 cos S + M14 sin 8),
U4 = p 4 (M11 + 2J 2 (A)M12 cos 2wt + 2J(A)M 14 sin wt)
- C0 + 2CoJ 2(A)(T 2/M11) cos 2wt
+ 2C0J(A)(M 14/M 11) sin wt.
Therefore from configuration IV we have
U(w) = 2J(A)(M 4 /M11) = 2JJ(A)d, (13)
U(2w) 2J 2 (A)(M12/M11) = 2J 2 (A)d,. (14)
From Eqs. (9) and (12) or (12) and (13) we obtainedthe differential circular reflection d. From Eqs. (10)and (11) we obtained the skewness s, and from Eqs.(11) and (14) we obtained the differential linearreflection d,.
Conclusion
The traditional experiments of studying gyrotropicmedia are the well-established transmission methodsof measuring rotatory dispersion or circular dichro-ism. Additional information for the chirally asymmet-ric responses of a gyrotropic medium can be obtained
by specular light reflection, which is of special inter-est for absorbing substances.
The basic chiral parameters (the differential circu-lar reflection, the skewness, and the differential lin-ear reflection) are expressed by the elements of theMueller matrix of the medium, and experimentalconfigurations are proposed for their practical deter-mination.
The differential reflection of right and left circu-larly polarized light is a useful characteristic of bothtransparent and opaque chiral media. We need config-urations I and III or III and IV to measure differentialreflection. On the other hand skewness S, character-izes the absorptive chiral substances that can bemeasured by configurations I and II or II and IV. Theasymmetric light reflection of the incident s- andp-polarized light is described by the linear differentialreflection d1, for which configurations II and IV areneeded. The specimen is transparent in a definitewavelength region and is opaque in another wave-length region. This dictates which chiral parametershould be measured.
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