Duality between spatial and angular shift in optical reflection
Aiello, A.; Merano, M.; Woerdman, J.P.
Citation
Aiello, A., Merano, M., & Woerdman, J. P. (2009). Duality between spatial and angular shift in optical reflection. Physical Review A, 80, 061801. doi:10.1103/PhysRevA.80.061801
Version: Not Applicable (or Unknown)
License: Leiden University Non-exclusive license Downloaded from: https://hdl.handle.net/1887/60048
Note: To cite this publication please use the final published version (if applicable).
Duality between spatial and angular shift in optical reflection
A. Aiello,1,2,
*
M. Merano,2and J. P. Woerdman21Max Planck Institute for the Science of Light, Günter-Scharowsky-Straße 1/Bau 24, 91058 Erlangen, Germany
2Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands 共Received 14 September 2009; published 7 December 2009兲
We report a unified representation of the spatial and angular Goos-Hänchen and Imbert-Fedorov shifts that occur when a light beam reflects from a plane interface. We thus reveal the dual nature of spatial and angular shifts in optical beam reflection. In the Goos-Hänchen case we show theoretically and experimentally that this unification naturally arises in the context of reflection from a lossy surface共e.g., a metal兲.
DOI:10.1103/PhysRevA.80.061801 PACS number共s兲: 42.79.⫺e, 41.20.Jb, 42.25.Gy, 78.20.⫺e
I. INTRODUCTION
In the 17th century Newton was the first to surmise that the center of a reflected beam should present a small spatial shift ⌬ in the plane of incidence relative to its geometrical optics position 关1兴. More than two centuries afterward, in 1947 Goos and Hänchen共GH兲 关2兴 were able to quantitatively measure such a shift共see 关3兴 for a literature survey since that time兲. The GH shift is typically in the subwavelength do- main; it has become technologically important in recent years since it directly affects the modes of optical waveguides and microcavities 关4,5兴 and has great potential for 共bio兲sensor applications 关6兴. Theoretically, the GH shift has been explained at various levels and several generaliza- tions have been discovered 关7–9兴. Among the latter it was predicted that the axis of the reflected beam should display a small angular deviation⌰ from the law of specular reflection
inc=ref关7,10兴. Interestingly, it took about 50 years since the original experiment performed by Goos and Hänchen to ac- tually observe such angular shift in the microwave关11兴 and the optical关3兴 regimes. Presently, it is common wisdom that spatial and angular GH shifts are two different phenomena observables in two mutually exclusive regimes: the spatial GH shift occurs in total reflection 共reflected intensity
= incident intensity兲 关12兴, while the angular GH shift appears in partial reflection 共reflected intensity⬍incident intensity兲 关10兴.
In this Rapid Communication we show that this separa- tion is artificial. We present a unified description for the spa- tial and angular GH shifts that will appear as two aspects of a unique beam-propagation phenomenon. We show that such duality between spatial and angular shift is rather general and also applies to the Imbert-Fedorov 共IF兲 effect, which is a shift normal to the plane of incidence 关13兴 that has drawn considerable interest lately关14–16兴. Finally, for the GH shift we show that unification naturally arises in the context of reflection from lossy surfaces. For this case we also furnish an experimental demonstration that the spatial and angular shifts occur simultaneously.
Our Rapid Communication is structured as follows: we give first a qualitative picture of the envisaged unification.
Then, we furnish a rigorous theoretical analysis of the beam-
propagation problem and show that the unified description actually holds for both the GH and the IF shifts. Finally, we demonstrate, both theoretically and experimentally, that for the GH case lossy reflecting surfaces naturally induce unifi- cation.
II. QUALITATIVE PICTURE
Consider a system consisting of two homogeneous isotro- pic media of dielectric constants 1 and2 filling the half- spaces z⬍0 and zⱖ0, respectively, as shown in Fig. 1. A monochromatic beam of light of wavelength0and waist w0 propagates along the central wave vector k0 in the region z
⬍0 before impinging upon the plane interface of equation z = 0 that separates medium 1 from medium 2. Detailed deri- vations of the angular and the spatial displacements⌰ and ⌬ for this system have already been reported elsewhere 关7–9,12,17–19兴 and will not be repeated here. We merely quote the basic results in the form
⌬=0Im关D兴, ⌰= −共02/2兲Re关D兴, 共1兲 where0=0/共2兲, and0= 20/w0is the angular spread of the incident beam 关20兴. The expressions above are valid for both the GH and the IF shifts where the coefficient D is equal to
*andrea.aiello@mpl.mpg.de
q q
x
z y
k0 k0
1 1
ε = Q
D
l tan Q ~ l Q l
2 r i i
ε = ε + ε
k0
yr
xr zr
.
FIG. 1. 共Color online兲 Scheme of the beam reflection at the plane interface. Here k˜
0and k0are the central wave vectors of the reflected beam as predicted by geometrical and wave optics, respec- tively. The waist of the incident beam is located at the origin of the laboratory Cartesian frame xyz.
1050-2947/2009/80共6兲/061801共4兲 061801-1 ©2009 The American Physical Society
D=ln r
= 1 R
R
+ i
⬅ DR+ iDI, 共2兲 in the GH case, and to D= 2i关共rP+ rS兲/r兴cotin the IF case 关21兴. Here DR⬅Re关D兴, DI⬅Im关D兴, and the index is a label for the two linear polarizations parallel 共= P or TM兲 and perpendicular 共=S or TE兲 to the central plane of inci- dence x-z. Moreover, r⬅r共兲=Rexp共i兲 is the Fresnel reflection coefficient 关22兴 evaluated at the central angle of incidence , where R=兩r兩 and= arg r.
From Fig.1it follows, using elementary geometrical con- siderations, that the total beam displacement observed at dis- tance l from the origin is expressible as a linear combination of ⌬and⌰共supposedly ⌰Ⰶ1兲 of the form
␦共l兲 = ⌬+ l⌰. 共3兲 From now on, in order to avoid trivial repetitions, we will consider explicitly the GH case only. However, all our con- clusions for the GH case can be straightforwardly extended to the IF case with minimum effort.
Equation 共3兲 supports the hypothesis that ⌬ and⌰are two different manifestations of a unique phenomenon, as they can be connected by such an elementary geometric re- lation. From Eq.共2兲 it follows that in the case of total reflec- tion 共R= 1兲, the reflection coefficients reduce to a pure phase factor r= exp共i兲, thus, DR= 0 and the reflected beam undergoes a purely spatial shift ⌬. Vice versa, if the reflection coefficients are strictly real, as in the case of air- to-glass partial reflection, then 苸兵0, ⫾其⇒DI= 0, and the beam undergoes a purely angular deflection ⌰. How- ever, when the reflection coefficients are complex, then both
⌬⫽0 and ⌰⫽0, and a unified description of spatial and angular GH shifts becomes mandatory. Since for a plane in- terface between two lossless media the Fresnel coefficients are either purely real or pure phase factors关22兴, then we have either␦= l⌰or␦=⌬, respectively, and spatial and angu- lar GH shifts are mutually exclusive. On the other hand, when either one or both of the media are lossy, the separation between spatial and angular GH shifts becomes artificial, and unification naturally occurs.
III. THEORETICAL DESCRIPTION
As a first step toward unification, in this section we aim to deduce Eq. 共3兲 from a perfectly general ab initio calculation, without restrictions on the media forming the interface or on the transverse shape of the light beam. It is well known that under minimal hypotheses 关9,20兴 it is possible to obtain a valid angular spectrum representation for the electric field of the reflected beam of the form A共X,Z兲=共2兲−1兰兰A共, Z兲ei·Xd2, where A共, Z兲 is uniquely determined by the angular spectrum of the incident beam and the Fresnel reflection coefficients关19兴. A Cartesian reference frame attached to the reflected beam with a scaled coordinate system is utilized in which X = k0xr, Y = k0yr, Z
= k0zr, and k0=兩k0兩, as shown in Fig. 1. Moreover, we have defined X⬅Xxˆr+ Yyˆr, and = k − zˆr共zˆr· k兲⬅1xˆr+2yˆr is the transverse part of the unit wave vector kˆ⬅k/k0=+3zˆr with respect to zˆr⬅k˜0/k0, with
3=共1−·兲1/2. Here k˜
0= k0− 2zˆ共zˆ·k0兲 is the central wave vector of the reflected beam as ruled by geometrical optics.
The angular spectrum in the plane Z is determined by its value at Z = 0 via the relation 关20兴
A共,Z兲 = A共,0兲exp共− iHZ兲, 共4兲 whereH=−3is the so-called optical Hamiltonian that gov- erns the well-known Hamilton equations of motion for light rays in vacuum关23兴:
d
dZ= −H
X = 0, dX dZ =H
=
3
. 共5兲
The analogy between Eq. 共4兲 and the expression for the time evolution of the wave function of a quantum system in the Schrödinger picture suggests the use of an enlightening quantumlike notation 关24–26兴 by writing A共, Z兲=具兩A共Z兲典 and A共X,Z兲=具X兩A共Z兲典, where 兩典=兩1,2典 and 兩X典=兩X,Y典 are the basis vectors in the transverse momentum and position space, respectively. In Eq.共4兲 the longitudinal coordinate Z has the role of a dimen- sionless time, then we can write 兩A共Z兲典=exp共−iHˆZ兲兩A共0兲典, where Hˆ is the Hamiltonian operator defined via 具兩Hˆ兩⬘典=−共1−·兲1/2␦共−⬘兲. In quantum mechanics unitary evolution implies that probabilities are conserved along with propagation, namely,具A共Z兲兩A共Z兲典=具A共0兲兩A共0兲典.
In our case, this means that the flux of the electric-field en- ergy density across any plane Z = const is independent from Z, namely,兰兰兩A共X,Z兲兩2dXdY =兰兰兩A共, 0兲兩2d2= 1, where we have renormalized the field amplitude of the reflected beam to ensure 具A共0兲兩A共0兲典=1. At any given coordinate Z the electric-field energy density 兩A共X,Z兲兩2 gives the spatial beam profile in the observation plane X-Y. The Z-dependent centroid of such energy distribution 具X典共Z兲=兰兰X兩A共X,Z兲兩2dXdY measures the deviation of the beam axis with respect to the central axis zˆr关7,17兴 defined by geometrical optics. If we define the transverse position and momentum operators such that Xˆ 兩X⬘典=X⬘兩X⬘典 and Kˆ 兩⬘典=⬘兩⬘典, respectively, then the centroid of the beam can be evaluated as 具X典共Z兲=具A共Z兲兩Xˆ兩A共Z兲典
=具A共0兲兩eiHˆZXˆ e−iHˆZ兩A共0兲典=具A共0兲兩XˆH共Z兲兩A共0兲典, where XˆH共Z兲 is the position operator in the Heisenberg picture:
具兩XˆH共Z兲兩⬘典 =
冉
−1i + Z
3
冊
␦共−⬘兲. 共6兲The first term on the right side of Eq.共6兲 coincides with the momentum representation of the Z-independent position op- erator Xˆ in the Schrödinger picture 关27兴. Therefore, from the very definition of 具X典共Z兲, it follows that the expectation value 具A共0兲兩Xˆ兩A共0兲典=具X典共0兲 gives both the GH and IF spatial shifts具XˆH共0兲典 and 具YˆH共0兲典, respectively. The second term, which is linear in Z, is proportional to the right side of the second equation in Eq. 共5兲 which determines the direc- tion of propagation of classical rays of light, since
AIELLO, MERANO, AND WOERDMAN PHYSICAL REVIEW A 80, 061801共R兲 共2009兲
061801-2
/3= tan1xˆr+ tan2yˆr. As we expect small angular devia- tions from geometrical optics predictions, we can write tani⬇i共i=1,2兲, and the exact relation between wave and geometrical optics established by Eqs.共4兲 and 共5兲 allows us to identify具XˆH共Z兲典/Z⬇1 with the angular GH shift and
具YˆH共Z兲典/Z⬇2 with the angular IF shift. Such identifica- tion becomes even more clear by noting that from 关Kˆ,Hˆ兴=0 and Eq. 共6兲 it follows that
具X典共Z兲 = 具Xˆ典 + Z具Kˆ共1 − Kˆ · Kˆ兲−1/2典, 共7兲
which reduces to具X典共Z兲⬇具Kˆ典+Z具Kˆ典 foriⰆ1 关28兴, and the angular brackets indicate expectation values with respect to the state 兩A共0兲典. Equation 共7兲 has the same form 具X典共Z兲=⌬+Z⌰ as Eq. 共3兲, where ⌬ depends on the position operator Xˆ , and ⌰ on the momentum operator Kˆ solely, thus, defining unambiguously both a spatial and an angular vector shift of the beam equal to ⌬=具X典共0兲 and ⌰=具X典共Z兲/Z, respectively. Note that the dependence of Eq.共7兲 on the char- acteristics of the reflecting surface and on the polarization of the incident beam is contained in the form of the state兩A共0兲典.
Equation 共7兲 establishes the first part of the announced unification by reproducing Eq. 共3兲 that was naively deduced on the grounds of simple geometric reasoning. We emphasize that, because of its vector form, Eq. 共7兲 describes both GH and IF shifts. The next step is to demonstrate that reflection from lossy surfaces induces simultaneously both spatial and angular GH shifts.
IV. LOSS-INDUCED UNIFICATION
Consider again the system shown in Fig.1. Assuming air as medium 1, we can write the dielectric constant of medium 2 as2=r+ ii. Then, we can distinguish that medium 2 is a dielectric, namely, r⬎1, or a metal with r⬍0. For both cases, from Eq.共2兲 and the well-known expressions for the Fresnel reflection coefficients关22兴, it follows that
DP= 2 sin
冑
共r− sin2兲 + ii⫻ i
2+r共1 − r兲 + ii共1 − 2r兲 共r− sin2兲 + 共i
2−r
2兲cos2+ ii共1 − 2rcos2兲,
DS= 2 sin
冑
共r− sin2兲 + ii. 共8兲
To elucidate the role of the losses, we expand Eq. 共8兲 in a Taylor series around i= 0 and keep terms up to the first order ini. For the sake of simplicity, from now on we will consider only the metal case for which the Taylor expansion furnishes
DPR=i
sin共1 − r兲2 共sin2−r兲3/2
2 sin4−r共sin2+rcos2兲 共sin2−r兲 + r
2cos2 , 共9兲
DPI = − 2rsin
冑
sin2−r共sin2−rcos2兲, 共10兲 for P polarization, and DSR=i2 sin/共sin2−r兲3/2, DSI= −2 sin/冑
sin2−r, for S polarization. From the ex- pressions above we can see that for an ideal lossless metal 共i= 0兲 we have that D is purely imaginary and only the spatial shift occurs 关29兴. However, when losses are “turned on” by letting 0⬍iⰆ1, a first-order real part must be added to DI causing the simultaneous existence of both spatial and angular shifts. Such loss-induced coexistence between⌬ and⌰ is clearly illustrated in Fig. 2 where Eq. 共3兲 is displayed for reflection at an air-gold interface. The central共red兲 curve represents the total beam shift at distance l from the origin and it is given, according to Eq.共3兲, by the sum of the upper 共green兲 curve 共angular shift兲 and the lower 共blue兲 one 共spatial shift兲. This completes the second part of our unification pro- gram.
V. EXPERIMENT
The experimental setup is sketched in Fig.3. The 820 nm output of a superluminescent light-emitting diode共SLED兲 is spatially filtered by a single-mode optical fiber 共SMF兲 to
0 20 40 60 80
1500
1000
500 0 500 1000 1500
Θ degrees
∆Pl∆Slnm
FIG. 2. 共Color online兲␦P共l兲−␦S共l兲 at l=11.9 cm for a funda- mental Gaussian beam with wavelength 0= 820 nm, waist w0= 32 m, incident on a gold surface with 2= −29.02+ i2.03.
Lower 共blue兲 line, ⌬P−⌬S; upper共green兲 line, l共⌰P−⌰S兲; central 共red兲 line,␦P−␦S; black triangles, measured␦P−␦S.
FIG. 3.共Color online兲 Layout of the experimental setup.
select a TEM00 mode. This is then collimated by a micro- scope objective and sent through a Glan polarizing prism 共PBS兲 to fix its polarization to P. A polarization modulator 共PM兲 switches the beam polarization between P and S to exploit the fact that both spatial and angular GH shifts are polarization dependent. Then, a lens is used to focus the beam to the desired spot size in front of the mirror. While the spatial GH shift is unaffected by such beam focusing, the angular GH shift depends on the beam angular aperture. The reason for this behavior is evident from the02factor in Eq.
共1兲, and its physical origin is explained in 关3兴. Finally, the difference signal from a quadrant detector共QD兲 is fed into a lock-in amplifier in order to detect the beam displacement in the plane of incidence when polarization is switched between S and P. See关3,30兴 for further details.
The results of the measurements are shown in Fig.2along with theoretical prediction. The good agreement confirms the simultaneous occurrence of the spatial and the angular GH shifts in the lossy regime.
VI. CONCLUSIONS
In this Rapid Communication we have presented a unified description for spatial and angular Goos-Hänchen shifts oc- curring in light beam reflection from lossy surfaces. Such description applies to the Imbert-Fedorov effect as well. The unification theory has been worked out for a weakly absorb- ing metal and the corresponding GH shift has been observed experimentally in reflection from an air-gold plane interface.
关1兴 I. Newton, Opticks 共Dover, New York, 1952兲; see query 4 on p.
339.
关2兴 F. Goos and H. Hänchen, Ann. Phys. 436, 333 共1947兲.
关3兴 M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, Nat. Photonics 3, 337共2009兲.
关4兴 D. H. Foster, A. K. Cook, and J. U. Nöckel, Opt. Lett. 32, 1764共2007兲.
关5兴 J. Unterhinninghofen, J. Wiersig, and M. Hentschel, Phys. Rev.
E 78, 016201共2008兲.
关6兴 X. Yin and L. Hesselink, Appl. Phys. Lett. 89, 261108 共2006兲.
关7兴 M. A. Porras, Opt. Commun. 131, 13 共1996兲.
关8兴 W. Nasalski, T. Tamir, and L. Lin, J. Opt. Soc. Am. A 5, 132 共1988兲.
关9兴 M. McGuirk and C. K. Carniglia, J. Opt. Soc. Am. 67, 103 共1977兲.
关10兴 J. W. Ra, H. L. Bertoni, and L. B. Felsen, SIAM J. Appl. Math.
24, 396共1973兲.
关11兴 D. Müller, D. Tharanga, A. Stahlhofen, and G. Nimtz, Euro- phys. Lett. 73, 526共2006兲.
关12兴 K. Artmann, Ann. Phys. 437, 87 共1948兲.
关13兴 O. C. de Beauregard and C. Imbert, Phys. Rev. Lett. 28, 1211 共1972兲.
关14兴 M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901共2004兲.
关15兴 K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. Lett. 96, 073903 共2006兲.
关16兴 O. Hosten and P. Kwiat, Science 319, 787 共2008兲.
关17兴 A. Aiello and J. P. Woerdman, Opt. Lett. 33, 1437 共2008兲.
关18兴 A. Aiello, M. Merano, and J. P. Woerdman, Opt. Lett. 34, 1207 共2009兲.
关19兴 A. Aiello and J. P. Woerdman, e-print arXiv:0903.3730.
关20兴 L. Mandel and E. Wolf, Optical Coherence and Quantum Op- tics, 1st ed. 共Cambridge University Press, Cambridge, UK, 1995兲.
关21兴 In this case we denote with ⌬and⌰, respectively, the spatial and angular separation between the two right-circularly and left-circularly polarized components of the reflected beam gen- erated by the reflection-induced splitting of the -polarized 共苸兵P,S其兲 incident beam 关13兴.
关22兴 M. Born and E. Wolf, Principles of Optics, 7th ed. 共Cambridge University Press, Cambridge, UK, 2003兲.
关23兴 T. Sekiguchi and K. B. Wolf, Am. J. Phys. 55, 830 共1987兲.
关24兴 D. Gloge and D. Marcuse, J. Opt. Soc. Am. 59, 1629 共1969兲.
关25兴 D. Stoler, J. Opt. Soc. Am. 71, 334 共1981兲.
关26兴 M. V. Berry, J. Mod. Opt. 50, 63 共2003兲.
关27兴 All the operators without the subscript “H” must be intended in the Schrödinger picture.
关28兴 The reader knowledgeable about paraxial optics will easily recognize from Eq.共7兲 in the limitiⰆ1, and from the rela- tion Kˆ
H共Z兲=Kˆ, the free-space propagation ABCD-matrix law:
共KXˆˆH
H兲共Z兲=共10 1Z兲共XKˆˆ兲.
关29兴 M. Merano, A. Aiello, G. W. ’t Hooft, M. P. van Exter, E. R.
Eliel, and J. P. Woerdman, Opt. Express 15, 15928共2007兲.
关30兴 M. Merano, J. B. Götte, A. Aiello, M. P. van Exter, and J. P.
Woerdman, Opt. Express 17, 10864共2009兲.
AIELLO, MERANO, AND WOERDMAN PHYSICAL REVIEW A 80, 061801共R兲 共2009兲
061801-4
