Reverse relative Goos-Hänchen effect

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EUROPHYSICS LETTERS 10 February 1996

Europhys. Lett., 33 (5), pp. 359-364 (1996)

Reverse relative Goos-Hanchen effect

L. Dutriaux, Ph. Balcou, F. Bretenaker and A. Le Floch

Laboratoire d’Electronique Quantique-Physique des Lasers, Unite Associee 1202Centre National de la Recherche Scientifique, Universite de Rennes ICampus de Beaulieu, F-35042 Rennes Cedex, France

(received 2 August 1995; accepted in final form 18 December 1995)

PACS. 42.25Gy – Edge and boundary effects; reflection and refraction.PACS. 42.60Da – Resonators, cavities, amplifiers, arrays, and rings.

Abstract. – A reverse Goos-Hanchen displacement between TE and TM polarizations is pre-dicted and experimentally isolated. The necessary reversal of the evolution of the differencebetween the phase retardances for the TE and TM reflections vs. the angle of incidence isobtained thanks to a Fabry-Perot–like structure. The experimental measurements are in goodagreement with numerical calculations that take fully into account the Gaussian nature of thelight beam.

The Goos-Hanchen effect is a polarization-dependent spatial displacement of a light beamat reflection on an interface [1]. Its small size (a few wavelengths) together with its intriguingproperties have recently motivated new experimental developments in the optical range [2], [3].According to Artmann [4], this displacement originates from the existence of phase retardancesϕ upon reflection. Indeed, the displacement is proportional to the derivative dϕ/di of thephase retardance φ with respect to the angle of incidence i. In the usual case of total internalreflection at angles of incidence i slightly above the critical angle ic, ϕ is given by Fresnel’sformulae [5]. Then, as |dϕTM/di| > |dϕTE/di|, the Goos-Hanchen effect is larger for theTM polarization than for the TE polarization. One may then wonder whether the sign of therelative Goos-Hanchen effect, i.e., the difference between the TM and TE beam displacements,could be reversed. To reach this goal, one must change the sign of (dϕTM/di − dϕTE/di).The aim of this letter is to show that a Fabry-Perot–like structure fulfils this condition forrather important values of the relative displacement. This provides a measurable reverseGoos-Hanchen effect.

Let us first recall the characteristics of the usual Goos-Hanchen displacement for totalinternal reflection on a single prism with index of refraction N (see fig. 1 a)) with e =∞). Forangles of incidence i < ic, where ic = arcsin(1/N) is the critical angle, no phase retardanceoccurs at reflection of a plane wave. On the contrary, for i > ic, a phase retardance appearswhich is larger for the TM than for the TE polarization and increases with i. The differencebetween these phase retardances is plotted in fig. 1 b) (dashed line) for N = 1.409. Except inthe vicinity of the critical angle, the relative Goos-Hanchen shift δGH can be deduced from the

c© Les Editions de Physique

360 EUROPHYSICS LETTERS

TM

TM

TE

TE

relative Goos-Hanchen

shift

glass

glass

air

45.0

20

10

0

– 10

– 20

300

200

100

0

45.1 45.2 45.3 45.4 45.5

45.0 45.1 45.2 45.3 45.4 45.5

incidence angle (degrees)

incidence angle (degrees)

rela

tive

GH

shi

ft

(µm

)

a)

b)

e

c)Φ

TM

– Φ

TE

(deg

rees

)

i

ic

ic

¨

Fig. 1. – a) Double-prism set-up used to reverse the relative Goos-Hanchen shift. b) Reflectionphase difference ΦTM − ΦTE in a standard total-reflection set-up on a single prism (dashed line,e = ∞), and when a second prim is present at a distance e = 30 µm (solid line). c) Correspondingrelative Goos-Hanchen displacements between the TE and TM polarizations obtained with Artmann’sformulae using the phases displayed in b). Notice the peculiar behaviour in the neighbourhood of theresonance (i ≈ 45.12◦).

usual Artmann’s formulae [4]

δGH =λ

2π

(dϕTM

di− dϕTE

di

), (1)

where λ is the wavelength of the incident wave inside the denser medium. δGH is zero fori < ic, and decreases from large positive values with the angle of incidence for i > ic, as shownby the dashed line in fig. 1 c) plotted for Nλ = 3.39 µm. Notice that eq. (1) is valid only forangles of incidence large enough so that all the plane-wave components of the incident beamare totally reflected.

Let us now consider the Fabry-Perot–like structure schematized in fig. 1a): an air slab oftunable thickness e and refractive index 1 is sandwiched between two glass prisms of identicalrefractive indices N . When a plane wave is incident on the slab at angle of incidence i, thereflectivity of the structure is given by [6], [7]

rk(i) =ρk[1− exp[jζ]]

1− (ρk)2 exp[jζ], (2)

where ρk (k = TE, TM) is the Fresnel reflection coefficient for the glass-air interface, and

ζ = 4πe

Nλ[1−N2 sin2 i]1/2 , (3)

which is either real or imaginary depending on the sign of i− ic. In the domain where i < ic,this Fabry-Perot–like structure is of course expected to exhibit resonances, which must lead to

L. DUTRIAUX et al.: REVERSE RELATIVE GOOS-HANCHEN EFFECT 361

1.2

0.0

90

90

– 90

– 180

15020100– 10– 20– 30– 40

0 10 20 30 40 0 10 20 30 40

0 10 20 30 40

100500

– 50

0.8

0.4

refle

ctiv

ity

refle

ctio

n ph

ase

(d

egre

es)

relative GH shift (µm)

phase difference (degrees)prism separation e (µm)

prism separation e (µm) prism separation e (µm)

a)

b)

c)

Fig. 2. – a), b) Reflectivity and phase retardances undergone by the TE (solid line) and TM (dashedline) upon reflection vs. the prism separation e for i = 45.10◦ = ic − 0.11◦. In b), the dot-dashedline represents the phase difference ΦTM − ΦTE (right-hand side scale). c) Corresponding relativeGoos-Hanchen displacements between the TE and TM polarizations obtained with Artmann’s formulausing the phases displayed in b). Notice the peculiar behaviour in the neighbourhood of the resonance(e ≈ 27 µm).

rapid and important variations of the retardances ΦTE and ΦTM associated with the reflectioncoefficients given by eq. (2). This can be seen in fig. 1 b) (full line), which represents theevolution vs. i of the difference ΦTM − ΦTE between the TM and TE phase retardances fora fixed value of e. Consequently, if we follow Artmann’s derivation, we expect the relativeGoos-Hanchen displacement to follow the derivative of this curve with respect to i, as seen infig. 1 c) (full line). Of course, near resonance, Artmann’s formulae are not rigorously valid.Nevertheless, they provide a physical guideline, as will be seen later. Remarkably, the rapidevolution of ΦTM−ΦTE vs. i leads to negative values of the relative Goos-Hanchen shift betweenthe TM and TE polarizations. Obviously, the negative relative shift predicted here is differentfrom the absolute shift predicted in the case of total reflection near an exciton-polaritonresonance in a semiconductor [8].

In an experiment, it is obviously more convenient to observe the evolution of the relativeGoos-Hanchen shift vs. the air slab thickness e rather than vs. the angle of incidence i. Fora fixed value of i slightly below ic one can again observe the typical Fabry-Perot resonancewhile tuning e. These resonances are shown for the TE and TM polarizations in fig. 2 a)which were computed from eqs. (2) and (3). The system behaves like a Fabry-Perot with afinesse smaller for the TM polarization than for the TE polarization. Here again, the intensityresonances of fig. 2 a) are associated with rapid variations of the phase retardances ΦTE andΦTM at reflection, as can be seen from fig. 2 b). Due to the finesse difference between theTM and TE polarization, the phase difference ΦTM − ΦTE also evolves rapidly vs. e in theneighbourhood of the resonance (dot-dashed line of fig. 2 b)). As can be seen from eq. (3), anincrease in thickness e is quite equivalent to a decrease in angle of incidence i. This meansthat a rapid dispersion-like variation of ΦTM−ΦTE vs. e, such as the one around e = 27 µm infig. 2 b), must lead to important values of the relative Goos-Hanchen displacement, togetherwith a change of sign of this relative shift. This is confirmed by the calculations obtained fromArtmann’s formulae (1) which are reproduced in fig. 2 c). Here again, we expect the relative

362 EUROPHYSICS LETTERS

DΦ

M1

M2PZT2

PZT1active

medium∆p ∆Φ

TMTM

TE

TE

Po

D

Fig. 3. – Experimental setup. Po: polarizer. D: detector.

Goos-Hanchen displacement to reach rather important values and also to change sign in theneighbourhood of the resonance of our Fabry-Perot–like structure.

To test experimentally these predictions, we need i) a light beam with a smooth andperfectly controlled profile, such as the Gaussian TEM00 eigenmode of a laser cavity, andii) a sensitive method able to measure the relative displacement between the TE and TMpolarizations for a single reflection, in order to avoid the spurious beam degradation effectsthat may occur in multiple-reflection experiments [1]. Besides, the high sensitivity of lasereigenstates has recently proved to be the ideal tool to measure the usual Goos-Hanchen effectfor a perfectly controlled Gaussian beam for a single reflection [3]. We consequently introduceour Fabry-Perot–like structure inside a quasi-isotopic cavity, as schematized in fig. 3. Thisstructure is made with two right-angle silica prisms (N = 1.409 at 3.39 µm, ic = 45.21◦) thatface one another, the in between spacing thickness e being accurately controlled by means ofa piezoelectric transducer PZT1. The 1 m long L-shaped resonator consists of a curved mirrorM1 (radius of curvature 1.2 m, 95% reflectivity) and a plane mirror M2 (95% reflectivity).The active medium is a 25 cm long discharge tube closed with quasi-perpendicular silicawindows. It is filled with a 5 : 1 3He-20Ne mixture at a total pressure of 1 Torr. Lasingoccurs on the 3.39 µm Ne line. A diaphragm DΦ selects the fundamental TEM00 Gaussianeigenmode of the cavity. The eigenstates of this laser are a TE-polarized and a TM-polarizedeigenstates which are spatially separated by their relative Goos-Hanchen effect only betweenthe prisms and the plane mirror. By means of a tilted silica plate and a stressed silica plate,we can introduce auxiliary linear TE/TM loss and phase anisotropies to compensate for thoseinduced by the prisms. If the TE/TM loss anisotropy is perfectly compensated for and if asmall overall TE/TM phase anisotropy remains, then the TE and TM eigenstates can oscillatein a vectorial bistability regime, and describe a hysteresis cycle when the cavity length istuned by means of a piezoelectric transducer that carries mirror M1. Then, at the centreof the hysteresis cycle, the intensities of the bistable TE and TM eigenstates are equal onlyif their losses are perfectly equal. Once this situation is achieved, the relative Goos-Hachendisplacement between the two eigenstates can be measured by slightly introducing a knife edge,which is mounted on a calibrated piezoelectric transducer PZT2, into the spatially separatedbeams [9], as shown in fig. 3. This creates a small diffraction loss anisotropy insufficientto switch off any of the eigenstates. Then, the maximum intensities of the two eigenstatesinside the hysteresis cycle become different. One then measures the intensity of the weakereigenstate. The value of the relative Goos-Hanchen shift is the given by the extra displacementof the knife edge necessary to reduce the intensity of the other eigenstate to the same value.The knowledge of this calibrated displacement then permits to obtain quantitative values ofthe relative Goos-Hanchen effect for several values of the air slab thickness e.

L. DUTRIAUX et al.: REVERSE RELATIVE GOOS-HANCHEN EFFECT 363

5 10 15 20 25– 20

0

20

prism separation e (µm)

rela

tive

GH

shi

ft

(µm

)

Fig. 4. – Full squares: experimentally measured relative Goos-Hanchen shifts vs. prism separation efor i = ic−0.11◦. Solid line: result of a numerical calculation taking the Gaussian nature of the beaminto account.

To compare our experimental results with theory, we must fully take into account the Gaus-sian nature of the laser beam. The spatial dependence of the electric field of the reflected beamcan be written in an angular-spectrum representation (for one given TE or TM polarization):

Er(x) =∫ +∞

−∞duAr(u) exp[−jkux] , (4)

where k = 2π/λ, x is directed along the interface in the plane of incidence, u is the sinusof the reflection angle, and the Fourier components Ar(u) of the reflected beam can bewritten as Ar(u) = αr(u) exp[jψr(u)] using real amplitudes αr(u) and phases ψr(u). Then,the Goos-Hanchen shift for one polarization is given by the mean abscissa of the reflectedbeam with respect to the centre of the geometrically reflected beam [10]

xr =

∫ +∞−∞ dxEr(x)xE∗r (x)∫ +∞−∞ dxEr(x)E∗r (x)

=1k

∫ +∞−∞ dudψr

du α2r (u)∫ +∞

−∞ duα2r (u)

. (5)

It is worthwhile to notice that eq. (5) means that the actual rigorous displacement just resultsfrom an averaging of Artmann’s formulae (1) over the angular energy density spectrum ofthe reflected Gaussian beam. This means that our qualitative discussion based on Artmann’sformulae and shown in fig. 2 must lead to physically correct predictions for slightly divergentGaussian beams.

This fact is confirmed experimentally by the measurements performed for i = 45.10◦ =ic− 0.11◦ vs. e which are reproduced in fig. 4 (squares). These results are well consistent withwhat was expected from fig. 2c): the relative Goos-Hanchen shift is first small and positive,increases gradually to a maximum with the sign of the usual displacement (δGH > 0), andthen suddenly decreases down to negative values: the relative Goos-Hanchen effect has beenreversed. We stopped our measurements at values of e of the order of 20 µm because, forlarger values of e, our Fabry-Perot–like structure becomes really resonant (see fig. 2a)), andthe losses of the laser are too important. These measurements are in good agreement withthe theoretical curve (full line of fig. 4) which was obtained from eq. (5). These calculationswere performed with values of the parameters taken from the experiment, except for the exactvalue of the position of the interface relative to the beam waist which was adjusted to −70 cm.

In conclusion, we have shown that the fact that the Goos-Hanchen displacement at reflectionis due to a variation of the phase retardance with the angle of incidence can lead to surprisingconsequences when the phase is produced by a structure different from usual totally reflectinginterfaces. Indeed, in the case where reflection occurs on a Fabry-Perot–like structure, the

364 EUROPHYSICS LETTERS

behaviour of the phase near a resonance leads to the existence of a reverse relative Goos-Hanchen effect between the TE and TM polarizations. This reverse Goos-Hanchen effect hasbeen experimentally measured for a well-controlled Gaussian beam and a good agreement hasbeen obtained with theory. As the usual Goos-Hanchen effect is known to occur in many otherfields of physics [11], reversal effects analogous to the one demonstrated here may therefore beof interest also in acoustics, quantum mechanics, and seismology.

***

This work is partially supported by the Direction de la Recherche et de la Technologie andby the Conseil Regional de Bretagne.

REFERENCES

[1] Goos F. and Hanchen H., Ann. Phys. (Leipzig), 1 (1974) 333; Goos F. and Lindberg-

Hanchen H., Ann. Phys. (Leipzig), 2 (1949) 87.

[2] Bretenaker F., Le Floch A. and Dutriaux L., Phys. Rev. Lett., 68 (1992) 931.

[3] Pfleghaar E., Marseille E. and Weiss A., Phys. Rev. Lett., 70 (1993) 2281.

[4] Artmann K., Ann. Phys. (Leipzig), 6 (1948) 87.

[5] Born M. and Wolf E., Principles of Optics, 3rd edition (Pergamon, Oxford) 1965.

[6] Zhu S., Yu A. W., Hawley D. and Roy R., Am. J. Phys., 54 (1986) 601.

[7] Pomer F. and Navasquillo J., Am. J. Phys., 58 (1990) 763.

[8] Birman J. L., Pattanayak D. N. and Puri A., Phys. Rev. Lett., 50 (1983) 1664; Puri A.,

Pattanayak D. N. and Birman J. L., Phys. Rev. B, 28 (1983) 5877. See also the special issueon Propagation and Scattering of Beam Fields, J. Opt. Soc. Am. B, 3 (1986).

[9] Dutriaux L., Le Floch A. and Bretenaker F., Europhys. Lett., 24 (1993) 345.

[10] Hugonin J. P. and Petit R., J. Opt. (Paris), 8 (1977) 73.

[11] See, for example, Lotsch H. K. V., Optik, 32 (1971) 299.