Circular and Near-Circular Polarization States of Evanescent Monochromatic Light Fields in Otalt Internal Reflection

Home , Circular polarization, Elliptical polarization, Evanescent field

University of New Orleans ScholarWorks@UNO

Electrical Engineering Faculty Publications Department of Electrical Engineering

11-20-2011

Circular and near-circular polarization states of evanescent monochromatic light fields in otalt internal reflection

R. M. A Azzam University of New Orleans, [email protected]

Follow this and additional works at: https://scholarworks.uno.edu/ee_facpubs

Part of the Electrical and Electronics Commons, and the Optics Commons

Recommended Citation R. M. A. Azzam, "Circular and near-circular polarization states of evanescent monochromatic light fields in total internal reflection," Appl. Opt. 50, 6272-6276 (2011).

This Article is brought to you for free and open access by the Department of Electrical Engineering at ScholarWorks@UNO. It has been accepted for inclusion in Electrical Engineering Faculty Publications by an authorized administrator of ScholarWorks@UNO. For more information, please contact [email protected]. Circular and near-circular polarization states of evanescent monochromatic light fields in total internal reflection

R. M. A. Azzam Department of Electrical Engineering, University of New Orleans, New Orleans, Louisiana 70148, USA ([email protected])

Received 12 August 2011; accepted 19 September 2011; posted 4 October 2011 (Doc. ID 152799); published 18 November 2011

Conditions for the production of near-circular polarization states of the evanescent field present in the rarer medium in total internal reflection of incident monochromatic p-polarized light at a dielectric– dielectric planar interface are determined. Such conditions are satisfied if high-index (>3:2) transparent prism materials (e.g., GaP and Ge) are used at angles of incidence well above the critical angle but sufficiently below grazing incidence. Furthermore, elliptical polarization of incident light with nonzero p and s components can be tailored to cause circular polarization of the resultant tangential electric field in the plane of the interface or circular polarization of the transverse electric field in a plane normal to the direction of propagation of the evanescent wave. Such polarization control of the evanescent field is sig- nificant, e.g., in the fluorescent excitation of molecules adsorbed at solid–liquid and solid–gas interfaces by total internal reflection. © 2011 Optical Society of America OCIS codes: 240.0240, 240.6690, 260.0260, 260.5430, 260.6970.

t 1. Introduction Ei ¼½Eix Eiy Eiz Total internal reflection (TIR) of an incident mono- t ¼½ðsin ϕÞEip Eis ð− cos ϕÞEip ; ð1Þ chromatic plane wave of arbitrary polarization state (with nonzero p and s components present) at the planar interface between two transparent media generates an evanescent field in the rarer medium t Ee ¼½Eex Eey Eez ; ð2Þ whose state of polarization is in general elliptical and three-dimensional [1–4]. In this paper, we con- t E E sider conditions that lead to circular or near-circular where indicates the transpose. In Eq. (1), ip and is p s polarization of the evanescent refracted wave. The are phasors that represent the time-harmonic and ϕ transparent media of incidence and refraction of re- components of the incident electric field and is the fractive indices n0 and n1 are assumed to be linear, angle of incidence that exceeds the critical angle of ϕ > ϕ ¼ −1ðn =n Þ homogeneous, optically isotropic, and nonmagnetic. TIR, c sin 1 0 . Application of boundary jωt x y z The e time dependence, Nebraska–Muller conven- conditions to the normal ( ) and tangential ( and ) – tions [5,6], and the xyz coordinate system shown in components of the electric field at the dielectric Fig. 1 are adopted. dielectric interface leads to the following relations The 3 × 1 generalized Jones vectors of the incident between the Cartesian components of the evanescent (i) and evanescent (e) refracted waves are written as and incident fields:

6272 APPLIED OPTICS / Vol. 50, No. 33 / 20 November 2011 2 From Eqs. (1)–(4) and (7), T11 ¼ N ð1 þ rpÞ;

T22 ¼ð1 þ rsÞ; 2 χe ¼ðcot ϕ=N Þ½ðrp − 1Þ=ðrp þ 1Þ ð8Þ T33 ¼ð1 − rpÞ: ð4Þ is obtained. With rp ¼ expðjδpÞ, Eq. (8) simplifies to In Eqs. (4), N ¼ n0=n1 is the high-to-low refractive r r 2 index ratio of the two dielectrics, and p, s are the χe ¼ðcot ϕ=N Þ½j tanðδp=2Þ: ð9Þ TIR interface Fresnel coefficients that are pure phase factors: Substitution of tanðδp=2Þ from Eqs. (6) in Eq. (9) leads to the simple result rp ¼ expðjδpÞ; rs ¼ expðjδsÞ: ð5Þ 2 1=2 χe ¼ jb=a ¼ je ¼ j½1 − ðcsc ϕ=NÞ : ð10Þ The TIR phase shifts δp, δs are functions of N, ϕ given by [7] χe of Eq. (10) describes an ellipse of polarization a b 2 2 1=2 with semimajor and semiminor axes and aligned tanðδp=2Þ¼N sec ϕðN sin ϕ − 1Þ ; x z ð6Þ with the and axes, respectively, and with ellipti- −1 2 2 1=2 e ¼ b=a tanðδs=2Þ¼N sec ϕðN sin ϕ − 1Þ : city . In Appendix A we provide an alternate derivation of Eq. (10), which is based on Gauss’s ~ The rest of the paper proceeds as follows: In law ∇ · Ee ¼ 0: Section 2, incident p-polarized light (Eis ¼ 0) is as- Near-circular polarization is achieved if e ¼ b=a ≥ sumed; such light produces elliptical polarization 0:95: For illustration, Fig. 2 shows a circular state of the evanescent refracted field in the xz plane of in- (CS) and an elliptical near-circular state (ENCS) that cidence (plane of the page in Fig. 1)[1,8]. We show correspond to e ¼ 1 and e ¼ 0:95, respectively. that near-circular polarization is achieved at suffi- Figure 3 shows e ¼ b=a of Eq. (10) plotted as a ciently large but realistic values of N, ϕ. In Section 3, function of ϕ from the critical angle ϕ ¼ ϕc ¼ the incident polarization state χi ¼ Eis=Eip that arcsinð1=NÞ to grazing incidence ϕ ¼ 90° at constant causes circular polarization of the resultant of the values of the index ratio N from 2 to 6 in steps of 1. y and z tangential components of the electric field Note that e ¼ 0, which represents linear polarization in the plane boundary is determined. In Section 4, in the x direction, occurs at the critical angle ϕc and the incident polarization state χi ¼ Eis=Eip that that e asymptotically reaches maximum saturation x causes circular polarization of the combined and value y transverse components of the electric field of the evanescent wave propagating in the z direction is 2 1=2 e ¼½1 − ð1=NÞ ¼ cos ϕc ð11Þ also obtained. Polarization control of the evanescent max field is important in studies of fluorescent excitation ϕ ¼ 90 N – – at °. The value of required to achieve a of molecules adsorbed at solid liquid and solid gas e interfaces using TIR [8–10]. Finally, Section 5 gives certain max is readily obtained from Eq. (11)as a brief summary of this work. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 N ¼ 1= 1 − emax: ð12Þ 2. Near-Circular Polarization of the Evanescent Field Generated by Incident p-Polarized Light The state of polarization of the evanescent refracted 1 p E ¼ 0 E field generated by incident -polarized ( is ) light 0.8 x is defined by the complex ratio [6] 0.6

χe ¼ Eez=Eex: ð7Þ 0.4

0.2 CS ENCS E 0 z

-0.2

-0.4

-0.6

-0.8 Fig. 1. Total internal reflection of p- and s-polarized light at a -1 dielectric–dielectric interface at angle of incidence ϕ and the -1 -0.5 0 0.5 1 reference xyz coordinate system used to describe the incident Fig. 2. (Color online) Circular polarization state (CS, e ¼ 1) and and evanescent refracted wave fields. an elliptical near-circular state (ENCS) with e ¼ b=a ¼ 0:95.

20 November 2011 / Vol. 50, No. 33 / APPLIED OPTICS 6273 1 Circular polarization of the tangential field is 0.9 N = 6 achieved if

0.8 χet ¼j: ð16Þ 0.7 N = 3 χ ¼þj 0.6 We take et and use Eqs. (5), (13), and (15)to obtain e = b/a 0.5

0.4 N = 2 χi ¼ Eis=Eip

Ellipticity 0.3 ¼ cos ϕ½sinðδp=2Þ= cosðδs=2Þ exp½jðδp − δsÞ=2: 0.2 ð17Þ

0.1 Use of χet ¼ −j instead of χet ¼þj simply changes 0 π 0 102030405060708090 the phase angle of the last term of Eq. (17)by . Angle of Incidence (degrees) φ Equation (17) specifies the incident elliptical polar- Fig. 3. (Color online) Ellipticity e ¼ b=a of the evanescent electric ization state that causes the tangential electric field field [Eq. (10)] is plotted as a function of ϕ from the critical angle to of the evanescent wave in the yz plane to be circularly grazing incidence at constant values of N from 2 to 6 in steps of 1. polarized. From Eqs. (6)and(17) the required ratio of amplitudes of the s and p components of incident light is given by Equation (12) indicates that emax ≥ 0:95 is achieved N ≥ 3:2026: if jχ j¼jE j=jE j As a specific example consider TIR at the GaP–air i is ip interface. GaP is transparent in the 620–760 nm red ¼ðN2 sin2 ϕ − 1Þ1=2=½ðN2 þ 1Þ sin2 ϕ − 11=2; ð18Þ visible spectrum and also in the near- and middle-IR and has a refractive index of 3.3 at the He–Ne laser and the associated phase difference is wavelength λ ¼ 633 nm [11]. With N ¼ 3:3, Eq. (11) e ¼ 0:953 e 2 2 1=2 gives max . And according to Eq. (10), in- δis − δip ¼ arctan½ðN sin ϕ − 1Þ =ðN sin ϕ tan ϕÞ: creases slightly from 0.950 at ϕ ¼ 76:04°toe ¼ max ð19Þ 0:953 at ϕ ¼ 90°. This confirms that ENCSs can be achieved in TIR at the GaP–air interface at incidence From Eq. (18) it is apparent that jχij < 1. Figure 4 angles sufficiently below 90°. shows jχij plotted as a function of ϕ from the critical As another example, consider TIR of IR radiation angle ϕ ¼ ϕc ¼ arcsinð1=NÞ to grazing incidence – N ≈ 4 at the Ge air interface for which over a broad ϕ ¼ 90° for constant values of N from 2 to 6 in steps bandwidth [12]. Substitution of N ¼ 4 in Eq. (11) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi of 1. Notice that jχij initially rises steeply just above e ¼ 15=16 ¼ 0:968 gives max . Equation (10) indi- the critical angle and saturates at larger angles as cates that e increases from 0.950 at ϕ ¼ 53:19°to emax ¼ 0:968 at ϕ ¼ 90°. Once again this proves that ENCSs are realizable in TIR of IR radiation at the 1 Ge–air interface at incidence angles well below 90°. 0.9 N = 6

0.8 ) i χ 3. Circular Polarization of the Tangential Electric Field 0.7 of the Evanescent Wave in the Plane of the Interface 0.6 N = 3 For TIR of incident monochromatic light of arbitrary polarization, 0.5 0.4 χi ¼ Eis=Eip; ð13Þ 0.3 N = 2 the polarization state of the tangential (t) electric Abs( Ratio Amplitude 0.2 field of the evanescent wave in the plane of the inter- 0.1 face is given by 0 0 102030405060708090 φ χet ¼ Eez=Eey: ð14Þ Angle of Incidence (degrees) Fig. 4. (Color online) Amplitude ratio jχij¼Eis=Eip [Eq. (18)] re- From Eqs. (1)–(4), (13), and (14) we obtain quired to produce circular polarization of the tangential electric field in the plane of the interface is plotted as a function of ϕ from N χ ¼ð ϕ=χ Þ½ðr − 1Þ=ðr þ 1Þ: ð Þ the critical angle to grazing incidence for constant values of from et cos i p s 15 2 to 6 in steps of 1.

6274 APPLIED OPTICS / Vol. 50, No. 33 / 20 November 2011 80 ϕ → 90°. From Eq. (18) the saturation value of jχij at N = 6 ϕ ¼ 90° is given by 70 o )

2 1=2 ip 60 jχij90 ¼ðN − 1Þ =N: ð20Þ δ N = 4 - is jχ j δ 50 Also of interestpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is i at the angle of incidence 2 ϕm ¼ arcsin 2=ðN þ 1Þ at which the TIR differen- 40 N = 3 tial phase shift (δp − δs) is maximum [7]. At ϕ ¼ ϕm, 30 Eq. (18) simplifies to N = 2 20 2 1=2 2 1=2 jχij¼ðN − 1Þ =ðN þ 1Þ : ð21Þ Phase Difference ( 10 Figure 5 shows the amplitude ratios given by 0 Eqs. (20) and (21) plotted versus N for 2 ≤ N ≤ 6. 0 102030405060708090 In Fig. 6, the phase difference between the incident Angle of Incidence (degrees) φ s and p components at which the tangential electric Fig. 6. (Color online) Phase difference δis − δip [Eq. (19)] between field becomes circularly polarized [Eq. (19)] is plotted the incident s and p components of incident light, which is required ϕ to produce circular polarization of the tangential electric field, is as a function of from the critical angle to grazing ϕ N plotted as a function of from the critical angle to grazing inci- incidence for constant values of from 2 to 6 in steps dence for constant N values from 2 to 6 in steps of 1. of 1. Figures 4 and 6 indicate that to produce circular – polarization (χet ¼þj) of the tangential electric field From Eqs. (1) (4), (13), and (22) we obtain in the plane of the boundary the incident polarization χ ¼ χ ðN2 ϕÞ½ðr þ 1Þ=ðr þ 1Þ: ð Þ χi is limited to the first quadrant of the unit circle in i eT sin p s 23 the complex plane. A polarization state generator, e.g., a combination of a linear polarizer and a quar- Circular polarization of the transverse field is χ ¼j: χ ¼þj ter-wave retarder, is used to polarize the incident achieved when eT We take eT and follow light in any desired state [6]. steps similar to those used in Section 3 to obtain

2 2 1=2 jχij¼jEisj=jEipj¼ðN sin ϕÞ=½ðN þ 1Þ sin ϕ − 1 ; 4. Circular Polarization of the Transverse Electric ð24Þ Field of the Evanescent Wave The polarization of the transverse (T) electric field of the z-traveling evanescent wave in the xy plane 2 2 1=2 δis − δip ¼ arctan½ðN sin ϕ − 1Þ =ðN sin ϕ tan ϕÞ is given by þ π=2: ð25Þ χeT ¼ Eey=Eex: ð22Þ If χeT ¼ −j is selected instead of χeT ¼þj the right- most term of Eq. (25) changes from þπ=2 to −π=2. 1 Equation (25) differs from Eq. (19) only in the þπ=2 added term. Figure 7 shows jχij of Eq. (24) plotted as a function 0.95 φ = 90 ) i ϕ χ of from the critical angle to grazing incidence at constant values of N from 2 to 6 in steps of 1. Note jχ j > 1 N ϕ 0.9 that i for all values of and and that the limiting values of jχij at ϕc and 90° are N and 1, respectively. For χeT ¼þj, χi occupies the region of φ 0.85 m the second quadrant of the complex plane outside the unit circle.

Amplitude Ratio Abs( 0.8 5. Summary Near-circular polarization of the evanescent field present in the rarer medium under conditions of 0.75 22.533.544.555.56 TIR of incident p-polarized light at a dielectric– Refractive Index N dielectric planar interface is achieved with high- Fig. 5. (Color online) Amplitude ratio jχij¼Eis=Eip for circular index (>3:2) transparent prism materials (e.g., ϕ ¼ 90 ϕ ¼ polarizationpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the tangential electric field at ° and m GaP and Ge) and at angles of incidence well above arcsin 2=ðN2 þ 1Þ [Eqs. (20) and (21)] is plotted versus N in the critical angle but sufficiently below grazing inci- the range 2 ≤ N ≤ 6. dence. Elliptical polarization of incident light with

20 November 2011 / Vol. 50, No. 33 / APPLIED OPTICS 6275 7 For incident p-polarized light the evanescent wave field is given by 6 N = 6 ) i ~ t χ Eeðx; y; zÞ¼½Eex 0 Eez expð−kexxÞ expð−jkezzÞ: N = 5 5 ðA5Þ

4 N = 4 In the absence of free charge in the medium of refraction, the wave field of Eq. (A5) must satisfy Gauss’s law 3 N = 3 ∇ E~ ¼ 0: ð Þ

Amplitude Ratio Abs( Ratio Amplitude · e A6 2 N = 2 Setting the divergence of the right-hand side of 1 Eq. (A5) equal to zero leads to 0 102030405060708090 Angle of Incidence (degrees) φ χe ¼ Eez=Eex ¼ jkex=kez: ðA7Þ Fig. 7. (Color online) Amplitude ratio jχij¼Eis=Eip [Eq. (24)], which is required to produce circular polarization of the transverse If kez and kex of Eqs. (A2) and (A4) are substituted electric field of the evanescent wave, is plotted as a function of ϕ in Eq. (A7) we obtain from the critical angle to grazing incidence for N values from 2 to 6 in steps of 1. 2 1=2 χe ¼ j½1 − ðcsc ϕ=NÞ ; ðA8Þ nonzero p and s components can be selected to pro- in agreement with Eq. (10). duce circular polarization of the resultant tangential electric field in the plane of the interface or circular polarization of the transverse electric field in a plane normal to the direction of propagation of the evanes- References cent wave. Control of the polarization of the evanes- 1. F. de Fornel, Evanescent Waves (Springer, 2001). cent light field in TIR influences its interaction – 2. L. Novotny and B. Hecht, Principles of Nano-Optics with oriented molecules adsorbed at solid liquid and (Cambridge, 2006). solid–gas interfaces via the dot and cross products of 3. L. Józefowski, J. Fiutowski, T. Kawalec, and H.-G. Rubahn, the optical electric field with the electric dipole mo- “Direct measurement of the evanescent-wave polarization ments of such molecules [13]. state,” J. Opt. Soc. Am. B 24, 624–628 (2007). 4. A. Norrman, T. Setälä, and A. T. Friberg, “Partial coherence Appendix A and partial polarization in random evanescent fields on loss- The complex wave vector of the evanescent refracted less interfaces,” J. Opt. Soc. Am. A 28, 391–400 (2011). plane wave in the Cartesian coordinate system of 5. R. H. Muller, “Definitions and conventions in ellipsometry,” Fig. 1 is given by Surf. Sci. 16,14–33 (1969). 6. R. M. A. Azzam and N. M. Bashara, Ellipsometry and ~ Polarized Light (North-Holland, 1987). ke ¼ −jkexx^ þ kez^z; ðA1Þ 7. R. M. A. Azzam, “Phase shifts that accompany total internal reflection at a dielectric-dielectric interface,” J. Opt. Soc. Am. and x^, ^z are unit vectors along the positive x and z A 21, 1559–1563 (2004). axes. Phase matching across the boundary requires 8. E. H. Hellen, R. M. Fulbright, and D. Axelrod, “Total internal that the z components of the incident and refracted reflection fluorescence theory and applications at biosurfaces,” wave vectors be equal, i.e., in Spectroscopic Membrane Probes, L. M. Loew, ed. (CRC Press, 1988), Vol. II, pp. 47–79. k ¼ k ¼ð2πn =λÞ ϕ; ð Þ 9. K. H. Drexhage, “Interaction of light with monomolecular dye ez iz 0 sin A2 layers,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1974), Vol. – λ 12, pp. 163 232. where is the vacuum wavelength of light. We also 10. S. E. Sund, J. A. Swanson, and D. Axelrod, “Cell membrane have orientation visualized by polarized total internal reflection fluorescence,” Biophys. J. 77, 2266–2283 (1999). ~ ~ 2 2 2 11. A. Borghesi and G. Guizzetti, “Gallium phosphide,” in ke · ke ¼ð2πn1=λÞ ¼ kez − kex: ðA3Þ Handbook of Optical Constants of Solids, E. D. Palik, ed. – From Eqs. (A2) and (A3) (Academic, 1985), pp. 445 464. 12. W. J. Tropf, M. E. Thomas, and T. J. Harris, “Properties of ” 2 2 1=2 crystals and glasses, in Handbook of Optics, M. Bass, ed. kex ¼ð2πn1=λÞðN sin ϕ − 1Þ ðA4Þ (McGraw-Hill, 1995), Vol. I, Chap. 33. 13. J. Michl and E. W. Thulstrup, Spectroscopy with Polarized is obtained, where N ¼ n0=n1: Light (VCH, 1995).

6276 APPLIED OPTICS / Vol. 50, No. 33 / 20 November 2011