1290 J. Opt. Soc. Am. B/Vol. 22, No. 6 /June 2005 Pillon et al.
Goos–Hänchen and Imbert–Fedorov shiftsfor leaky guided modes
Frank Pillon, Hervé Gilles, Sylvain Girard, and Mathieu Laroche
Equipe Lasers, Instrumentation Optique et Applications, Centre Interdisciplinaire de Recherche Ions Laser, CentreNational de la Recherche Scientifique-Commissariat à l’Energie Atomique-Ecole Nationale Supérieure
d’Ingénieurs de Caen Unité Mixte de Recherche (UMR 6637), 6 boulevard du Maréchal Juin,14050 Caen Cedex, France
Robin Kaiser
Institut Non Linéaire de Nice, Centre National de la Recherche Scientifique/Unité Mixte de Recherche (UMR 6618),1361 route des Lucioles, F-06560 Valbonne, France
Azra Gazibegovic
Prirodno Matematicki Fakultet, Zmaja od Bosne 53, 71000 Sarajevo, Bosnia-Herzegovina
Received October 27, 2004; revised manuscript received December 17, 2004; accepted December 29, 2004
. INTRODUCTIONhe Goos–Hänchen (GH) shift1 is the small displacementhat a light beam undergoes when it is totally reflected onhe interface separating two infinite half-spaces, one with
higher refractive index than the other. Such spatialhift is attributed to the evanescent wave that travelslong the interface. It appears as if the incident light pen-trates first into the lower-refractive-index medium as anvanescent wave before being totally reflected back intohe high-index space. If we consider such physical inter-retation, the GH effect appears as clear experimentalvidence that the ray model is simply a first approxima-ion, and that only the wave approach could allow a com-lete description of the total reflection. The order of mag-itude of the GH shift during a total internal reflection inhe optical domain is relatively small (typically of the or-er of the wavelength, i.e., a few micrometers in the vis-ble or near infrared) and is therefore difficult to observexperimentally after a single total reflection. However,espite the difficulties, several original techniques haveeen developed for an experimental observation of thisundamental optical effect. The most common methodonsists of using multiple reflections in a slab2 or a prism3
tructure to increase the shift by a factor corresponding tohe number of successive reflections. But this allows rela-ively poor control of the spatial beam geometry along theifferent reflections, which is a disadvantage when the
xperimental results are compared to theoretical models.ore recently, enhancement of the effect has been ob-
ained as a result of total internal reflection inside a laseravity.4 Despite this technique’s being a very original ap-roach to the problem of measuring the GH shift after aingle total reflection, it appears complicated to modifyhe incidence angle, as it is necessary to realign the entiresotropic cavity between measurements.
Our group has developed a totally different approachimed at the improvement of the optical detection sensi-ivity to very small spatial shifts of a light beam. It isased on the use of a special shaped photodiode called aosition-sensitive detector (PSD) and on a periodicwitching between two orthogonal states by use of anlectro-optic modulator (Pockels cell or liquid-crystalalve (LCV)). This technique was first used to measurerecisely the longitudinal GH shift5 versus the angularncidence when the polarization states switch between TEnd TM. More recently, the Imbert–Fedorov6 (IF) shift,hich corresponds to a transverse displacement perpen-icular to the incident plane, was also precisely measuredhen the polarization states were switched periodicallyetween left and right circular states.7 Until now, theechnique has been used only to measure the geometricalhifts after a total reflection on a single dielectric inter-ace.
It has also been pointed out in earlier works on the GH
005 Optical Society of America
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Pillon et al. Vol. 22, No. 6 /June 2005/J. Opt. Soc. Am. B 1291
hift that peculiar behavior could also be observed whenotal reflection occurs on structures containing multipleielectric layers.8,9 One of the interesting cases involves aimple guiding structure formed by a substrate with aigh refractive index covered by a single dielectric layerith a low refractive index. If the incident angle inside
he substrate is correctly chosen to obtain refraction onhe substrate–layer interface and total reflection on theayer–air interface, some resonance could appear on theH shift for a discrete series of specific angles.10 This be-avior has already been investigated because it repre-ents interesting opportunities to make atomic mirrors11
r optical sensors.The present paper is devoted to a full reinvestigation of
he interesting properties of such guiding structure withur original experimental setup. It is divided into threeain parts. The first part contains a theoretical descrip-
ion of the GH shift on a single dielectric layer. Differentodels initially developed to describe the total reflection
n a single interface—such as Artmann’s model12 basedn a phase argument or Renard’s model13 based on an en-rgetic approach—have been modified to take into ac-ount the effect of the dielectric layer. This major revisionill be complemented by some comments on importantarameters that could influence the resulting displace-ent, such as the distributed losses into the layer.The second part of the paper is dedicated to a descrip-
ion of the experimental results obtained on a single di-lectric layer with our setup based on both periodic polar-zation switching and PSD detection. The mainxperimental results are obtained in the case of polariza-ion switching between TE and TM states, because thesewo states correspond naturally to the polarization eigen-tates of a planar waveguide. The experimental GH shiftsre then compared with the theory, and some commentsre offered on the observed resonance. Complementaryeasurements are also reported on the experimental ob-
ervation of the transverse IF shift in such structure. Theifferent parameters (thickness, losses, refractive index ofhe substrate or the thin-film, etc.) that could have a sig-ificant influence on the effect of resonance observed inhe layer used here are discussed.
Finally the conclusion deals with future possible appli-ations for such components based on the observation ofodification of the resonance. Other original structures
resenting a potentially enhanced Goos–Hänchen shiftill be briefly reviewed and the opportunity to study such
ystems with our experimental technique will be dis-ussed.
. THEORETICAL MODELS OF THEOOS–HÄNCHEN SHIFT ON A SINGLEIELECTRIC LAYER
n the past two main models have been investigated to de-cribe accurately the GH or IF shifts observed after a to-al internal reflection on a dielectric interface. The firstpproach, usually called Artmann’s model,12 is based onn argument considering the phase shift of the reflectedeam with respect to the incident beam. To summarizehe main results obtained with Artmann’s model, we canimit our discussion to the particular case of an incident
eam well collimated with a divergence of only a few mil-iradians. In this case the GH effect corresponds simply to
displacement LGH along the interface of the incidentlane (see Fig. 1) that is given by
LGH =l
2pn1 cos i1SdF
di1D , s1d
here l is the wavelength in vacuum, n1 is the refractivendex of the substrate, F is the phase shift during the to-al reflection for an incident plane wave, and i1 is the in-idence angle.
The second approach, usually called Renard’s model,13
s based on an argument with respect to the energy flow ofhe evanescent wave along the interface. As already re-orted elsewhere (see for example the explanation de-ailed in Ref. 7) the GH displacement LGH on the interfaceay be expressed as
LGH =1
kSrxlE−`
0
kStzldx, s2d
here Srx is the Poynting’s vector component for the re-ected beam along the axis normal to the interface andtz is the Poynting’s vector component for the evanescentave along the axis aligned on the interface and in the
ncident plane. The Poynting’s vectors are time averagedsymbol k l).
. Adaptation of Artmann’s Model for a Singleielectric Layerhe layer structure investigated during the present workonsists of a thin, homogeneous dielectric film of thick-ess h and refractive index n2 deposited on a substrateith a refractive index n1 (see Fig. 2). The index of the
ubstrate n1 is higher than the index of the thin film n2.here is no cladding covering the thin film and the me-ium above the layer is simply air, which could be consid-red a dielectric medium with a refractive index n3 of 1.he incident light is injected through the substrate intohe thin film thanks to a specifically designed prismatictructure for the substrate as shown in Fig. 3(a). If we fol-ow an approach similar to Artmann’s model already men-ioned for a single interface, we can calculate the spatialhift that results from the phase shift F between the in-ident and the reflected waves. The amplitude of the total
ig. 1. Schematic representation of the Goos–Hänchen effect aspure spatial shift for the reflected light beam. This simplified
epresentation is correct when the incident beam has lowivergence.
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1292 J. Opt. Soc. Am. B/Vol. 22, No. 6 /June 2005 Pillon et al.
eflection coefficient for the entire system (the two inter-aces and the homogeneous film) is classically calculatedy an approach similar to that used in the case of aabry–Perot interferometer [see Fig. 3(b)]. The amplitudef the complex reflection coefficient is described as
rtot = r12 +t12t21r23 expsjwd
1 − r21r23 expsjwd= urtotuexpsjFd, s3d
here rij represents the reflection coefficient for the inter-ace separating media i and j, tij is the correspondingransmission coefficient, and w is the phase shift betweenwo successively reflected waves due to the optical pathifference for a round trip into the layer. The reflection rijnd transmission tij coefficients are related to the refrac-ive index ni and nj and to the incidence angle i1 accord-ng to the classical Fresnel formulas.14 The phase shift wepends on the thickness h of the layer, its refractive in-ex n2, and the refracted angle i2 into the layer, as well ashe wavelength l according to
w =2pn2
l2h cos i2. s4d
The complex nature of the total reflection coefficient isue simultaneously to the total reflection on the secondnterface described by the complex coefficient r23 and tohe interference term that is directly related to the walue. If we start with Eq. (3), it is straightforward to cal-ulate numerically the argument of the complex coeffi-ient rtot. This argument F corresponds to the total phasehift that could be inserted into Eq. (1) to simulate theH shift for the thin film by use of Artmann’s model. Of
ourse the second part of Eq. (3) contains the interferenceerm and therefore represents possible resonance effectshat depend on the incidence angle.
Furthermore the complex reflection coefficient is polar-zation dependent with two polarization eigenstates (TE,ith an electric field perpendicular to the incident plane,nd TM, with a magnetic field perpendicular to the inci-ent plane). This is because the dephasing factor duringhe total reflection on the film–air interface depends onhe polarization states. Therefore the resulting GH dis-lacement will also be slightly dependent on the polariza-ion states. The difference DLGH between the displace-ents for TE and TM states is equal to the amplitude of
ig. 2. Size of the incident beam d adjusted exactly such thathe spatial shift corresponds to an incident beam that completelyenetrates into the lower-refractive-index medium as an evanes-ent wave (Renard’s model).
eriodic spatial movement along the substrate–film inter-ace when the polarization state of the incident beam iseriodically switched between TE and TM. It can be ex-ressed as
DLGH =l
2pn1 cos i1FSdFTE
di1D − SdFTM
di1DG , s5d
here FTE and FTM are respectively the arguments of theotal reflection coefficients for TE and TM modes, and thether parameters correspond to those used in Eq. (1). Ourxperimental setup allows measuring DdGH. The differ-nce DLGH is directly deduced following DLGHDdGH/cos i1 (see Fig. 2).
. Adaptation of Renard’s Model for a Single Dielectricayeror Renard’s model the electromagnetic waves propagat-
ng in the film and in the air are considered a single, in-omogeneous electromagnetic field distribution. Renard’s
ormula can be presented as
LGH =1
kSrxlSE−h
0
kS2zldx +E−`
−h
kS3zldxD , s6d
here LGH represents the absolute longitudinal shiftlong the substrate–film interface compared with the po-ition of the incident beam, Srx corresponds to the compo-ent along the x axis of the Poynting’s vector for the re-ected beam inside the substrate, S2z is the componentlong the z axis of the Poynting’s vector for the guidedave inside the film, and S3z is also the component along
he z axis of the Poynting’s vector, but for the evanescent
ig. 3. Schematic representation of the thin layer deposited onsubstrate: (a) the prismatic structure of the substrate, (b) the
eflected wave that can be calculated with an approach similar tohe one usually used in Fabry-Perot interferometer with the mul-iple reflections into the thin film.
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Pillon et al. Vol. 22, No. 6 /June 2005/J. Opt. Soc. Am. B 1293
ave in the air. Therefore because Renard’s model is en-irely based on the expression of the Poynting’s vectors, itecomes necessary to determine completely the expres-ion of the electromagnetic field in the substrate, the film,nd the air. In the film the electromagnetic wave resultsrom the superposition of two counterpropagating waves.ne wave propagates in the direction of the incidenteam while the second corresponds to a wave backre-ected by the film–air interface. As in the case of any op-ical waveguide, each eigenmode can be considered atanding wave along the transverse x axis and a propa-ating wave along the longitudinal z axis. Moreover, thencidence angle i1 should satisfy specific conditions to pro-uce a well-controlled standing wave along the x axis.owever, in the present system there is no total internal
eflection on the interface between the film and the sub-trate; consequently the eigenmodes are progressively ra-iated back into the substrate. Therefore these eigen-odes are progressively attenuated along the z axis evenithout any losses into the waveguide. For this reason
hey can be considered leaky guided modes.Of course, as soon as a guiding effect into the film ap-
ears, the GH displacement will be enhanced. As alreadyescribed in the previous paragraph, the effect is polar-zation dependent. Therefore it is necessary to distinguishhe effect that depends on the incident polarization state.ifferent expressions must be established for a completeescription of the electric and magnetic fields reflectednto the substrate, or injected into the thin film and intohe air (See Appendix A for the expression of the reflectednd evanescent waves and Appendix B for the guided fieldnto the thin layer). The expressions for the electric fieldistribution inside the dielectric layer and in the air areependent on the incidence angle. As soon as this inci-ence angle corresponds to a resonance, the amplitude ofhe electric field inside the thin dielectric layer becomesnhanced. Starting with an incidence angle correspondingo the critical angle for the layer–air interface, the differ-nt resonant modes appear as the angle i1 is increased.he first resonance corresponds to TE0, the second to TE1,nd so on for a TE incident polarized beam.As examples, two electric field distributions were calcu-
ated for TE0 and TE1 respectively and are illustrated inig. 4. The amplitude of the electric field is significantlynhanced compared with the amplitude into the substratehen the light is guided into the structure.Appendixes A and B also contain the detailed expres-
ions for Poynting’s vectors. From these expressions, it
ig. 4. Calculated electric field distribution inside the guidingtructure for TE0 and TE1 modes.
ecomes straightforward to calculate the GH shift for theE mode and for the TM mode by Eq. (6).The difference between the GH shifts for TE and TModes can therefore be deduced as
DLGH = LGHTE − LGH
TM, s7d
ith
LGHTE = tan i1
E−h
0
uU' cos ax + V' sin axu2dx + uteva' u2
d
2
urtot' u2
,
LGHTM = tan i1
E−h
0
uUi cos ax + Vi sin axu2dx + utevai u2
d
2
urtoti u2
.
s8d
he different parameters are explained in Appendix B.he difference between the two polarization states is im-ortant since it allows a direct comparison with the ex-erimental results, as detailed in the following. Finally, itan be pointed out that only negligible differences appearn the calculated GH shift deduced for the total internaleflection on a thin dielectric film between the two meth-ds and that the two curves corresponding to the twoodels are completely overlapped in Fig. 5. Following Re-ard’s model and formulas (6)–(8), the GH shift results
rom two main contributions:1. the Poynting’s vector into the film, which corre-
ponds to a guided mode near a resonance,2. the Poynting’s vector corresponding to the evanes-
ent wave.It is clear that, as soon as a resonance appears in the
lm, the resulting guiding effect significantly increaseshe displacement. In this case, the main contribution isue to the first part of Eq. (6). Of course, the second termue to the evanescent wave also shows a resonance but itsagnitude is very small compared with that due to therst term. On the other hand, far away from any reso-ance, the second part corresponds to the classical GHhift already mentioned. Moreover, it is obvious starting
ig. 5. Calculated GH shift by Artmann’s or Renard’s modelsersus the incidence angle for two polarization eigenstates: TEode, TM mode.
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1294 J. Opt. Soc. Am. B/Vol. 22, No. 6 /June 2005 Pillon et al.
rom Fig. 4 that, as soon as the amplitude of the electriceld for TE0 is higher than it is for TE1, the GH shifthould be higher for TE0 compared to TE1 (see Fig. 5).hese different specific behaviors will be further eluci-ated and compared to theory below.For any other polarization states (elliptical or circular),
t is also possible to calculate the GH shift based on Re-ard’s model by the following technique. The incidentlectric field is decomposed as a linear combination on theasis of the two orthogonal polarized states TE and TM.hen the electric and magnetic fields into the thin filmnd into the air can be deduced by the approach alreadyescribed in Appendixes A and B. Finally the Poynting’sectors into the film and in the air may be expressed. Byse of Eq. (6), it becomes possible to calculate the GHhift for any intermediate state. Moreover, it is possible toalculate the transverse IF shift that also appears for anlliptical (or circular) polarization state by
LIF =1
kSrxlSE−h
0
kS2yldx +E−`
−h
kS3yldxD . s9d
quation (9) must be related to Eq. (9) in Ref. 7 and wille useful to calculate the transverse IF shift when, in-tead of linearly polarized light, the thin film is periodi-ally illuminated with a left- or right-handed circularlyolarized beam.
. EXPERIMENTAL RESULTSo measure precisely the polarization dependence of suchnhanced GH shift, we used the experimental setup al-eady described elsewhere.5,7 This setup is represented inig. 6 and will be only briefly described here. The colli-ated beam is emitted by a Spectra Diode Laboratories
aser diode (model 6702 H1; l=1.083 mm) driven by apectra Diode Laboratories power supply (model SDL00). With a birefringent Glan-Thomson prism, the linear
ig. 6. Experimental setup used to measure the GH and IFhifts on a thin dielectric layer.
olarization state emitted by the laser diode is further im-roved to obtain a high polarization extinction ratio.40 dBd. The light beam passes through a LCV fromisplaytech, Inc., (model LV1300) to periodically switch
ts polarization state. The valve allows periodic polariza-ion switching between TE and TM modes. By locating auarter-wave plate (QWP) just after the LCV, it also be-omes possible to obtain circularly or elliptically polarizedtates before entering the prism. If the angle of incidencen the prism is close to 0°, the polarization state remainsnchanged after transmission through the entrance inter-
ace; therefore, it also becomes the incident polarizationtates on the thin film.
The beam is totally reflected at the base of a speciallyesigned prism [see Fig. 3(a)]. The substrate is made of aigh-refractive-index glass sn1=1.9d while the base of therism is covered with a thin, porous silicate film with aefractive index n2=1.45. The thickness h is 1.68 mm. Thencidence angle is precisely controlled with a rotationtage. After the total reflection at the prism, the averageeam position is photodetected by a two-dimensional PSDAdvanced MicroElectronics/UDT Sensors, Inc., modelIL-C4DG) that allows the simultaneous observation andeasurement of both longitudinal and transverse shifts.Because the resonant angles are slightly different be-
ween TE and TM modes (corresponding to the effect ofolarization dispersion of the optical guide), the reso-ance peaks are not exactly coincident. Moreover, theirmplitudes are different. The present technique allowseasuring the difference between the GH shifts for TE
nd TM polarization states. The measured relative GHisplacement is represented in Fig. 7. The abscissa showshe angular orientation representing the internal inci-ence angle compared with the axis normal to the base onhich the thin film is deposited. The measured points
how clearly identifiable peaks corresponding to the an-ular condition for a leaky guided mode into the planarielectric film. Compared with the theoretical curve ob-ained by use of formulas (5)–(8), the experimental resultsre in good agreement with theory and correctly repro-uce the observed structures as well as the amplitude ofhe resonance.
As can be deduced from a comparison between Fig. 5nd Fig. 7, the difference between the GH shift for TE and
ig. 7. Difference between the GH shifts for TE and TM polar-zation states versus the incidence angle on the thin film: com-arison between theory (solid curve) and experiment (filledquares).
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dttitststteto
Ft=
Fts
Pillon et al. Vol. 22, No. 6 /June 2005/J. Opt. Soc. Am. B 1295
M modes is smaller than the absolute displacement. Ofourse, this result is not very surprising because, as al-eady mentioned, the difference between the resonancengles is very small. The difference can be observed onlyhen the electromagnetic field into the thin film is stud-
ed versus the incidence angle. As the amplitude of theH shift is smaller for TM compared with TE polarization
tates, the experimental spatial shift observed corre-ponds mainly to the GH shift for the TE mode. For ex-mple, the maximum absolute GH shift for the TE0 modes about 55 mm, whereas it is only about 30 mm for theM0 mode. The relative GH shift between TE0 and TM0 isnly 25 mm but it matches relatively well the observedisplacement. Even though the difference is smaller thanhe absolute displacement, it is still perfectly measurableith our technique, and the resonance peaks correspond-
ng to the modes (TE0–TM0 and TE1–TM1) can be easilydentified in Fig. 7. Moreover, it appears that when the or-er n of the leaky guided mode TE–TMn increases, thebsolute (as well as the relative) GH shifts decrease. Thiss expected because when the order increases, the ampli-ude of the electromagnetic field decreases inside the thinlm as is shown in Fig. 4 for TE0 and TE1. Therefore, themplitude of the associated Poynting’s vector S2z also de-reases and it is straightforward, with Eq. (6), to see thathe GH shift will become smaller. On the other hand, theidth of the resonant peak becomes larger when the orderincreases. For example, on the present thin film, the an-
ular tolerance, defined as the FWHM of the resonanceeak is Di1=0.8° for sTE0–TM0d and Di1=2° forTE1–TM1d.
To check that the displacement measured is undoubt-dly attributed to the GH effect and not to a spurious ef-ect, the QWP just after the LCV was turned around itsxis and the corresponding longitudinal shift was mea-ured versus the angular orientation of the QWP, andherefore versus the incident polarization states. The re-ults are shown in Fig. 8. In particular, the periodicity ob-erved corresponds to the expected value. As soon as theeutral axes of the QWP are oriented along the linear po-
arization states emerging from the LCV, a maximum lon-itudinal shift is observed. This is not surprising since inhis case the polarization states correspond to TE–TM,he two polarization eigenstates of the planar waveguide.herefore, a guiding effect becomes possible (as long as
ig. 8. Longitudinal shift measured versus the angular orienta-ion of the QWP located in front of the LCV; measured for i148.5° corresponding to the first resonance peak.
he incidence angle also corresponds to a resonant one).n the other hand the effect is nullified when the incidentolarization states are circular, that is to say when theeutral axes of the QWP are oriented at 45° comparedith those of the LCV. This behavior has previously been
bserved for the GH shift on a single interface and wasecognized as a good way to check the viability of the mea-ured effect.
As already mentioned, the IF shift can also be observedhen a circularly polarized beam is totally reflected on aielectric interface. One may wonder whether some pecu-iar behavior such as guiding effects could be observed on
thin film deposited on a substrate. While the longitudi-al GH shift can be described using either Artmann’s orenard’s models, the theoretical description of the IF shiftn thin film is possible only by use of Renard’s model.ithout going too much into details, the technique used is
asically similar to the method already described in Ref. 7or a single interface. And because the PSD used duringur experiment has a two-dimensional structure, it allowshe simultaneous measurement of both the longitudinalnd the transversal shifts. Therefore, rotating the QWPt 45° allows periodic switching of the polarization stateetween s+ and s−. In the event the circular polarizationtates are not strictly speaking polarization eigenstatesor a dielectric optical waveguide, it is obvious that noeal guiding effect could be expected. However, when theateral displacement between s+ and s− is calculated ver-us the incidence angle, some peculiar structures are pre-icted as are represented on the theoretical curve shownn Fig. 9.
The experimental points measured in the transverseirection are in relatively good agreement with theheory, especially if we consider the order of magnitude ofhe observed effect. However, it becomes more difficult todentify clearly the predicted structure when consideringhe experimental points. To check further that the ob-erved experimental shift can actually be attributed tohe IF effect, the transverse displacement was also mea-ured versus the angular orientation of the QWP. First,he amplitude of the transverse shift appears in quadra-ure with the longitudinal shift, which is exactly what isxpected theoretically. Moreover, the periodicity is halfhat observed for the longitudinal shift because the signf the relative lateral displacement reverses between
ig. 9. Difference between the IF shift for s+ and s− polariza-ion states versus the incidence angle on the thin film: compari-on between theory (solid curve) and experiment (filled squares).
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fsThftshlhlmmmitsscbmaF=sHtTta
bsotinritris
Fttatm
1296 J. Opt. Soc. Am. B/Vol. 22, No. 6 /June 2005 Pillon et al.
+–s− and s−–s+. If the amplitude of the transverse shiftn a thin dielectric film is compared with the same effectbserved on a single interface, the orders of magnitudere almost the same with no specific enhancement attrib-ted to the film structure.To understand better the theoretical results obtained
or the IF effect on a thin dielectric film, we have decom-osed the calculated transverse shift into two terms. Therst represents the flux of the Poynting’s vector into thehin film. This term could be considered as the contribu-ion of the geometrical part of the shift due to multiple re-ections inside the thin film. The second part is due to thevanescent wave in the air surrounding the film. It corre-ponds to the effect also observed on a single interface, ex-ept that the electric field into the thin film replaces thencident field and could be significantly enhanced as a re-ult of the guiding effects into the structure. Figure 10epresents the two contributions as well as the resultingotal transverse shift, which was already shown in Fig. 9.uch decomposition allows us to see clearly that the twoontributions have opposite sign, which means that theyartly compensate for each other. On the other hand, ithows that the two contributions present similar struc-ures (with the exception of the opposite sign) and revealshat some form of guiding must appear even when usingircularly polarized light.
To confirm this guiding effect, the absolute longitudinalhift has been calculated for circularly polarized light andeveals a strong resonance as soon as the incidence angleorresponds to the guided modes’s TE or TM. Of course,his longitudinal shift can not be directly observed withur experimental setup because the longitudinal shiftsre equal for the two orthogonal polarized s+–s− states.or circularly polarized light, the transverse shift is notnhanced in the waveguide structure, even when the lights physically guided inside the film. The reason for thiseculiar behavior is that the polarization state will beodified along the guide each time the light is totally re-
ected at the film–air interface. At the beginning if theransverse shift is positive, it becomes negative after only
ig. 10. The two contributions for the Imbert-Fedorov shift at-ributed, respectively, to (a) the multiple reflections inside thehin film (dashed curve), (b) the evanescent wave along the film–ir interface (dotted curve). The solid curve represents the sum ofhe two contributions and should be compared with the experi-ental results.
few reflections. Therefore, the transverse shift is peri-dically counterbalanced and no enhancement could bexpected.
This modification of the polarization states during eachotal internal reflection is especially pronounced in ourase because we are far away from the critical angle. Onhe other hand the longitudinal displacement is alwaysositive and therefore can not be cancelled as a result ofhe modification of the polarization state during propaga-ion. This will lead to a strong enhancement of the longi-udinal displacement as soon as the light is guided, be-ause it will benefit from the cumulative effect in the casef multiple reflections. In fact, only the structured depen-ence of the transverse displacement versus the angularrientation should be considered as the peculiar behavioror the IF effect on thin film. Therefore, further develop-ent would be necessary to identify fully the predicted
tructure.One way to improve the identification of the peculiar ef-
ects attributed to the thin film is to choose a specifictructure to enhance further the guiding properties.herefore, let us consider the main parameters that canave an influence on the GH shift and quantify their ef-
ects on the order of magnitude of the displacement. Thehickness h of the layer, the refractive index n1 of the sub-trate, and the distributed losses into the thin film couldave significant influence on the guiding properties of the
ayer and therefore on the resulting GH displacement. Weave decided to keep constant the refractive index of the
ayer because the resulting effect is very similar to aodification of the refractive index of the substrate. Theost sensitive parameter is the thickness h of the layer,ainly because it can be modified in a wide range. When
t is increased the number of allowed eigenmodes into thehin film becomes higher and the resonance angles corre-ponding to the successive orders become closer. This re-ult is expected because, as soon as the thickness in-reases, the multimode character of the waveguideecomes more pronounced. Moreover, a thicker waveguideeans correspondingly more energy stored in the layer
nd a larger amplitude for the resulting displacement.or example, when the thickness increases from h1.7 mm to 3.4 mm, the absolute amplitude of the GHhift increases from 55 mm to 550 mm for the TE0 mode.owever, the resonance condition becomes more sensitive
o the incidence angle and the peaks become narrower.herefore, the present thin film is a good compromise be-ween a well-pronounced resonance and a relatively widengular acceptance.The other important parameter is the difference
etween the refractive indices of the film and theubstrate. When the contrast becomes higher, the finessef the Fabry–Perot interferometer corresponding tohe thin film increases. Therefore, when the refractivendex n1 of the substrate significantly increases above2, the GH shift becomes higher, and simultaneously eachesonant peak becomes narrower. Moreover, with anyncrease of n1, the range between the total reflections onhe film–air and substrate–film interfaces becomes nar-ower, but not so much as to cause difficulties in observ-ng the effect. Once again the refractive index n1=1.9eems a relatively good compromise to get enough con-
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Pillon et al. Vol. 22, No. 6 /June 2005/J. Opt. Soc. Am. B 1297
rast between the substrate and the film to increase theH shift.Finally, the distributed losses could also have a signifi-
ant influence on the amplitude of the resonance but onlyf the losses become high (typically a few percent peround trip into the layer). Because the thickness of thelm is limited to a few micrometers the losses should alsoave limited impact on the amplitude of the resonanceeak.However, this investigation is mainly theoretical and it
s still necessary to check experimentally the effect of pos-ible adjustments of the main parameters such as thick-ess h or index n1.
. CONCLUSIONn the present paper, the geometrical displacement thatccompanies a totally reflected light beam onto a thin di-lectric film is studied. In the particular case of a thin filmith a refractive index lower than that of the substrate onhich it is deposited, it is demonstrated both theoreticallynd experimentally that under specific incidence angles,he longitudinal GH shift presents some resonance peakshat strongly depend on the internal structure of the di-lectric layer. These peaks correspond to the leaky guidedigenmodes of the planer waveguide. The experimentaletup, previously used to measure GH and IF effects on aingle interface, is based on a periodic polarizationwitching technique. In fact, it permits measuring onlyhe difference between the TE and TM shifts, but it is per-ectly adapted to characterize fully the resonance due tohe differences between the two polarization eigenstates.
The transverse IF shift was also observed and mea-ured versus the incidence angle. The experimental trans-erse shift presents no resonance, thus confirming theheoretical calculations made by use of Renard’s model.owever, some peculiar structures versus the incidencengle—not observed in the case of a single interface—areredicted in such dielectric thin film. These dependencesend to appear on the measured curves, but improve-ents in methods are needed to identify them better.The evanescent wave near a thin dielectric film has
ong been recognized as an interesting tool to measure ahemical or biological reaction near the interface. Usuallyhe detection is based on interferometric techniques. Ourpproach, developed initially to observe and measure theH or IF shifts, could also be useful for optical sensing. It
onsists of detecting a spatial shift instead of a phasehift and could be very sensitive, especially when coupledo a thin-film structure observed near a resonance. More-ver, other structures presenting giant or unusual GHhifts could also be investigated with the present tech-ique, such as the effects predicted at an interface withight-handed and left-handed materials (or negative re-ractive index).15,16
PPENDIX A: EXPRESSIONS FOR THELECTROMAGNETIC FIELD ANDESULTING POYNTING’S VECTOR
. For the Reflected Wave in the Substratehe complex amplitude Er of the reflected wave on the in-erface between substrate and film sx=0d can be ex-
ressed by use of Eq. (3). Therefore, the Poynting’s vectorssociated with the reflected beam can be directly de-uced from
Sr =n1E0
2
2m0curtotu21cos i1
0
sin i12 , sA1d
here E0 is the incident electric field, n1 is the refractivendex in the substrate, m0 is the magnetic permeability ofacuum, c is the velocity of light, and rtot is the total re-ection coefficient defined by use of Eq. (3).
. For the Evanescent Wave in the Airith an approach similar to that used to calculate the re-
ected electric field, the thin dielectric layer could be con-idered a Fabry–Perot interferometer, and a transmissionoefficient teva for the evanescent wave can be calculatedo obtain the amplitude of the electric field at the inter-ace between the layer and the air. The evanescent waveeing inhomogeneous, its amplitude decreases exponen-ially when the coordinate x decreases from −h to −`. Theorresponding Poynting’s vector for the evanescent waveecomes simply
S3 =n3E0
2
2m0cutevau2 expF2sx + hd
dG1
0
0
n1
n3sin i12 , sA2d
here d is the penetration depth defined exactly as for alassical total reflection for a single interface,7 and teva de-ends on the incident polarization states (TE or TM).
PPENDIX B: EXPRESSIONS FOR THELECTROMAGNETIC FIELD ANDHE RESULTING POYNTING’S VECTOR
NTO THE THIN DIELECTRIC LAYERhe electric and magnetic fields into the thin dielectric
ayer are respectively called E2 and B2, and they can bealculated by assuming the superposition of a descendantave and an ascendant wave into the structure. The waveectors associated with these two waves are, respectively,
k2d = 1− a
0
b2 ,
k2a = 1a
0
b2 , sB1d
ith a= s2p /ldn2 cos i2 and b= s2p /ldn2 sin i2. Two polar-zation eigenstates for a planar optical waveguide muste distinguished: the TE mode with the electric field ori-nted along the y axis and the TM mode with the mag-etic field along the y axis.. Case of TE ModeStarting with the hypothesis that the electric field can
e described as
Iu
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Tls
2
Io
Tl
a
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A
ttt
s
R
1298 J. Opt. Soc. Am. B/Vol. 22, No. 6 /June 2005 Pillon et al.
E2 = E0sU' cos ax + V' sin axd 3 expf− jsvt − bzdg 3 10
1
02 .
sB2d
t is straightforward to express the magnetic field B2 byse of the Maxwell’s equation CurlsEd=−]B /]t. We have
B2 =5−
E0
cn2 sin i2sU' cos ax + V' sin axd
3expf− jsvt − bzdg
0
− jE0
cn2 cos i2s− U' cos ax + V' sin axd
3expf− jsvt − bzdg
6 .
sB3d
he continuity of components at the interface betweenayer and air imposes the following boundary conditions:
E2ys− hd = E3ys− hd,
B2zs− hd = B3zs− hd, sB4d
nd the constants U' and V' can be deduced as
U' = teva' Hcos ah +
n3
n2 cos i2FSn1
n3sin i1D2
− 1G1/2
3 sin ahJ ,
V' = teva' H− sin ah +
n3
n2 cos i2FSn1
n3sin i1D2
− 1G1/2
3 cos ahJ . sB5d
he resulting associated Poynting’s vector can be calcu-ated by S2=1/2m0 ResE23B2
*d and can be expressed ver-us the parameters U' and V' as
S2 =E0
2
2m0cn1 sin i1sU' cos az + V' sin azd2 3 10
0
12 .
sB6d
. Case of TM ModeSimilarly, the magnetic field can be described as
B2 =n2E0
csUi cos ax + Vi sin axd 3 expf− jsvt − bzdg 3 10
1
02 .
sB7d
t is straightforward to express the electric field E2 by usef the Maxwell’s equation CurlsBd=n 2 /c2]E /]t. We have
2
E2 =5E0sUi cos ax + Vi sin axd
n1
n2sin i1
3expf− jsvt − bzdg
0
jE0 cos i2s− Ui cos ax + Vi sin axd
3expf− jsvt − bzdg6 . sB8d
he continuity of components at the interface betweenayer and air imposes the following boundary conditions:
E2zs− hd = E3zs− hd,
B2ys− hd = B3ys− hd, sB9d
nd the constants Ui and Vi can be deduced as
Ui = tevai Hn3
n2cos ah +
1
cos i2FSn1
n3sin i1D2
− 1G1/2
3 sin ahJ ,
Vi = tevai H−
n3
n2sin ah +
1
cos i2FSn1
n3sin i1D − 1G1/2
3 cos ahJ . sB10d
he resulting associated Poynting’s vector can be calcu-ated by use of S2=1/2m0 ResE23B2
*d and can be ex-ressed in terms of Ui and Vi as
S2 =E0
2
2m0cn1 sin i1sUi cos az + Vi sin azd2 3 10
0
12 .
sB11d
CKNOWLEDGMENTSR. Kaiser thanks V. S. Bagnato, S. Muniz, and the Op-
ics Center at Sao Carlos, University of Sao Paulo, forheir help in the preparation of the coated prisms andheir assistance during his stay in Sao Carlos.
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