Observing angular deviations in the specular reflection of a light beam

NATURE PHOTONICS DOI: 10.1038/NPHOTON.2009.75LETTERS〈Δθ p 〉/θ 00.6 Theoryw 0 = 34 μm0.40.2w 0 = 59 μmw 0 = 95 μm0.0−0.2−0.4p s−0.6−15 −10 −5 0 5 10 15(θ − θ B )/θ 0D m − D g (μm)1,2001,00080060040020000.00.1w 0 = 34 μmw 0 = 59 μmw 0 = 59 μm0.2 0.3 0.4 0.5Distance from beam waist (m)Figure 4 | Angular deviation near the Brewster angle u B . The curve showsthe theoretical prediction of equation (5) for the angular deviation kDu p lversus the angle of incidence u. The brackets k...l refer to averaging over theangular beam spread. Abscissa and ordinate have been made dimensionlessby dividing by u 0 , the half-opening angle of the incident TEM 00 beam.Experimental data were obtained with the set-up of Fig. 2c for three valuesof the beam waist w 0 . The insert shows the polarization-analysed reflectedintensityprofilesofabeam(w 0 ¼ 34 mm) incident at u ¼ u B ¼ 56.58.We also measured the angular deviation much closer to theBrewster angle (u B ¼ 56.58), now using now the set-up of Fig. 2cwhere the incident polarization was fixed as p. We effectively calibratedthe split detector by means of a small, controlled rotationof the air–glass interface which was nulled by a lateral (x) shift ofthe detector. In Fig. 4 we plot the theoretical prediction ofequation (5) as kDu p l /u 0 versus (u – u B )/u 0 . In this dimensionlessform the dispersive resonance curve is universal in the sense thatit is independent of the beam divergence u 0 ; the extrema have coordinates(0.5, 0.5) and (–0.5, –0.5). The theory is nicely confirmed bythe experimental data for three values of the beam waistw 0 ¼ 2/(ku 0 ). In absolute terms, the measured angular deviationvaries between 10 25 and 10 22 rad.The inserts in Fig. 4 show the measured polarization-analysedintensity profiles of the reflected beam when the ( p-polarized)input beam is incident at exactly the Brewster angle. In this casethe reflected profile differs greatly from the TEM 00 input profile.The reflected intensity shows a double-peak structure (this hasbeen observed before 21,22 ); according to our theory this is in fact asuperposition of a p-polarized TEM 01 profile and an s-polarizedTEM 10 profile (see Supplementary information). For the ratioof the corresponding beam powers P s and P p at Brewsterincidence, our theory gives (P s /P p ) B ¼ 1 ¼ 0.18 (1 is defined inequation (5)). This value agrees well with our experimental result:(P s /P p ) B ¼ 0.20 + 0.05.According to our theory the angular deviation is centred at theposition of the beam waist and not, as one might naively think,on the glass surface. This can be tested in the set-up of Fig. 2c,where we introduce D m and D g as the beam’s actual x-positionand its geometrical-optics x-position, respectively, both measuredat the split detector. Figure 5 shows the differential x-shift(D m – D g ) measured this way as a function of the distance zbetween the detector and the beam waist, for two values of thewaist. For both values we find proportionality between (D m – D g )and u in , thus confirming the angular nature of the effect.We emphasize that each plane-wave solution in the angular expansionis governed by Maxwell’s equations in the regular manner. Theboundary conditions associated with momentum conservation andSnell’s Law are satisfied as usual when considering all three planewaves: the incident one, the reflected one and the transmitted(¼ refracted) one. The fact that in the beam case the (average)Figure 5 | Verification of the angular nature of the deviation. We plot here(D m 2 D g ) as a function of the distance between the beam waist (atthe asterisk in Fig. 2c) and the split detector. Experimental data arepresented for two values of the beam waist, w 0 ¼ 34 and 59 mm, in thelatter case for two independent experimental runs. D m is the beam positionmeasured on the split detector and D g the corresponding geometrical opticsposition. For measuring the latter we remove the lens from the set-up ofFig. 2c so that we deal with a collimated incident beam with a very largebeam waist, w 0 ¼ 1.64 mm; this represents the geometrical optics limit.Spurious shifts due to removing the lens are prevented by ensuring that thelens, before removing it, is exactly centred on the beam; for this centring weuse a separate quadrant detector.in-plane momentum of a reflected photon is different from that ofan incident photon does not contradict momentum conservation;the difference is balanced by an opposite difference associated withthe transmitted beam (see Supplementary information).The three key elements required for the occurrence of angularnon-specularity can be simply understood in a qualitative sense.The first requirement is a finite-diameter optical beam, becausethis produces a wavevector spread (due to diffraction). The secondis less than 100% reflection, because this leads generally to anangular dependence of the reflectivity (which cannot occur forR(u) ¼ 100%). Within a given beam, the more oblique rays thenhave a different intensity reflectivity than the less oblique ones.This unbalance slightly rotates the centre-of-mass axis of the reflectedbeam away from the geometrical-optics angle (see Fig. 1). The thirdrequirement is oblique incidence, because at normal incidence theangular deviation vanishes for symmetry reasons.These three elements occur quite generally in optical implementationsof angular metrology. The resulting angular deviation, evenif very small, produces, by means of propagation, an unlimitedgrowth in the transverse positional coordinate 5 . This may occur ingeodetic surveying, machine-tool operation, torsion pendulumreadout 23 and cantilever-based surface microscopies (such as atomicforce microscopy, AFM) 24 . Another example is the angular alignmentof gravitational wave detectors such as LIGO (a Michelson interferometerwhere one deals with oblique incidence on a 50%/50%beamsplitter) or LISA (basically a triangular interferometer) 25 .For simplicity we have emphasized in our work the case of anair–glass interface with its associated Brewster resonance.However, angular non-specularity should occur generally for multilayerdielectric stacks and for metal mirrors (which have ,100%reflectivity). Moreover, it should also occur in acoustics 16 andquantum mechanics 12 , where we deal with scalar waves instead ofelectromagnetic vector waves. In the scalar case the Brewster resonancedoes not contribute, so the angular shift is governed byequations (4) and (6) with R s (u) ¼ R p (u). Finally, an argument forthe potential impact of angular non-specularity is that the studyof its positional ‘cousin’, namely the GH effect, has broadlyNATURE PHOTONICS | VOL 3 | JUNE 2009 | www.nature.com/naturephotonics 339© 2009 Macmillan Publishers Limited. All rights reserved.