Revisiting Reflection

Alternative title: My place in the optics’ sun or me standing on the shoulders of giants…

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I’m leaving the Quantum Optics group of the Leiden University by October 2011.  Here, I’ll write things that kept me busy for the last 2 years. No, it’s not about my travels. :)
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When I was about to leave for a postdoctoral fellowship almost two years ago, a perplexed former professor asked me, “What is still there in reflection that needs to be studied?”   This is after I told him the topic which I will be working on.

I don’t blame him.

Around 300 BC, Catoptrics by Euclid already had a mathematical description of reflection [1]. Since then it has been a central dogma of geometric optics. With respect to an imaginary line that is perpendicular to the surface of the reflecting object, light will be reflected at the same angle and at the same plane as it was incident. This is the law of reflection.

I. Newton, Opticks 1704.

In the 17th century, Isaac Newton revisited reflection in his book Opticks [2]. Although not contradicting Euclid, he conjectured that light can begin to be reflected before hitting the surface (Query #4). By this, light would appear to have been reflected behind the actual point of reflection predicted by geometric optics.  This means a shift in the point of reflection of light. There was no experiment which tested the inference of Newton.  Experiments on the shift only happened after the last world war.

The Goos-Hänchen shift

Goos and Hänchen in 1947 reported the first quantitative measurement of such a shift of the centroid of the beam [3].  Now aptly called the Goos-Hänchen shift, they observe a physical light (a beam of light with finite extent) reflected multiple times inside a strip of glass and they measured the acquired displacement of the beam relative to the predictions of geometric optics.

What they observed was that the light that is reflected by the interface between a higher and a lower index of refraction (such as when light is inside an optical fiber) behaves as if the light is reflected after the interface.

Goos-Hänchen Shift

The measurements gave a different position with respect to the guess of Newton.  Newton says that the light is bent before the interface but Goos and Hänchen’s measurements say otherwise, bending happens after the interface.  This is at least in their measurements for a glass-air interface.

“The measurements prove that light after reflection is displaced.”

The displacement when the number of reflection was factored in, is in the order of a wavelength (~10-9 – 10-7).  This displacement was attributed to the tunneling of photon in the evanescent wave before being reflected.  Think of this as light “walking” a little bit on the interface before commencing reflection. Artmann soon after explained this shift by relating it to the phase of the amplitude of the reflected beam [4].
If light “walked” parallel to the plane of incidence, could it also “walk” perpendicular to it? 

Right hand circularly polarized light. Courtesy: Wiki

The Imbert-Fedorov shift

Yes, but the light must be in a particular spin state (i.e. elliptically polarized).  Newton did not guess that this is possible.  He was not particularly a fan of the polarization of light since it is a phenomenon associated with the wave nature of light (attributed to Huygens).  It was opposite to what he was advocating, the particle theory.

The perpendicular excursion of the beam with respect to the plane of incidence after reflection only started to be mentioned in literature in 1912.  Several conflicting theoretical calculations were put forth and it took almost 60 years before experiments were done to finally reveal which interpretation is correct.

Imbert-Fedorov Shift

In 1972, Imbert with a clever procedure presented the first measurements [5].  These agreed with predictions from the initial calculations of Fedorov done in 1955 [6].

This experiment is much harder because the displacement they measured was less than half the magnitude of the measurement by Goos and Hänchen. They also had to prepare the spin state of their incident light.  This beam shift is now called the Imbert-Fedorov shift.

By the 1980’s, these beam shifts are already textbook materials. This is where my professor is coming from.  Only a few scientists continue doing experiments and theoretical work.  Most think that nothing can be added in the general understanding of reflection anymore.  Although from a theoretical point of view, there is still a question on the apparent contradiction between the law of reflection (from Maxwell’s equations) and the conservation of angular momentum.  From the practical aspect, the effects are too small to matter anyway unless of course one works in the nanometer scale.

A new look at an old problem?

In the mid-2000’s, several literatures on the Imbert-Fedorov beam shift started to appear again but this time not in the context of evanescent wave or our “walking” light analogy. It is treated as the effect of the geometric phase (Pancharatnam-Berry phase) on the trajectory of a ray of light upon reflection (transmission) [7].  The geometric phase of light in general is the phase acquired by light when it is slowly returned to its original polarization state using two intermediate states.

O. Costa de Beauregard, Physics Review 139 (1965)

The Spin Hall effect of Light

This particular phenomenon results to a splitting of a linearly polarized light into two circularly polarized light (left and right circularly polarized light) after reflection, as verified by Osten and Kwiat [8].

Because of this observation, it was called the Spin Hall effect of light, analogous to the electronic Spin Hall effect.

The spin of the photon (polarization) plays  the role of the  spin of the charge. The potential gradient in the latter is the refractive index gradient in the former.

This was not actually new from the point of view of theorists.  In 1965, Costa De Beauregard proposed this splitting and termed it as a translational inertial spin effect of photons [9].

With weak measurement

The Weak Measurement

The measurement of Hosten and Kwiat uses the weak measurement.  It is a quantum mechanical technique that uses pre- and post selection of states.

In their setup, they crossed the polarization axis of polarizers placed at the incident and at the transmitted light, respectively. (There is by the way a one to one mapping between reflected and transmitted light.) And because the beam has split into two circular polarized light with opposite helicity, there will be a portion of the beam that can pass through the second polarizer.  What comes out is a two-crescent patch of intensity. The intersection of the split light will not pass through because it will have the same polarization as the incident light (right circular + left circular polarized light with equal phases equals linearly polarized light).

The distance between the center of the crescents is the shift of the beam.

Hosten and Kwiat employed a setup that enhances the SHEL. This enhancement can actually be described classically.

A classical description and a real deviation from the law of reflection

In 2008, my current group embarked on a purely classical analysis of the splitting reported by Hosten and Kwiat [10].  A classical description is more accessible to the metrology community.  The shifts are displacements of the centroid of the beam.

Angular Goos-Hänchen Shift

The most significant result of this theoretical exercise is the articulation that the GH and the IF shifts increase with the divergence (opening angle) of the beam and with the beam’s propagation.

The more focused the beam, the larger the angular deviation.  Where the focus is at does not matter.

An increasing shift or displacement from geometric optics prediction upon propagation can only happen when the beam does not follow the law of reflection.  Instead of equal angles of incidence and reflection, the beam is reflected with a small angular deviation. The angle of incidence of a focused incident beam is defined as the angle of incidence of the center of that beam.

This angular deviation is actually a diffraction correction of the beam.

A year after, our group presented an experimental verification on external reflection on a smooth dielectric  surface [11].  The angular deviation was measured as a difference in the shift between s- and p-polarized light.  The p-polarized light’s shift is enhanced near the Brewster’s angle.

Now, it is well known that there are four shifts that can happen when light is reflected.  These are the two spatial shifts (spatial GH and IF shifts) and the two angular shifts (angular GH and IF shifts).

These shifts might happen together or separately.  This depends up to this point on: 1) the polarization of the incident beam; 2) the index gradient seen by the beam; and 3) the degree of the beams’ divergence.  That ‘up to this point’ comment will be revisited a few paragraphs down (or by October when we publish our data).

About this time also, the group managed to measure the GH shift for metallic reflection using gold [12].

Reflection from metallic interface

What is the difference between metallic and dielectric materials (example, glass)?  In the language of optics, it has to do with the difference in the nature of the index of refraction.  Metals have complex indices of refraction compared to the purely real refractive indices of dielectric materials.

From a practical point of view, most reflectors are metals or make use of thick metallic films.

Goos-Hänchen Shift in metallic reflection. The spatial and angular shifts are exaggerated.

What was observed was that for a particular polarization state, light travels backwards or seemed to be reflected on top of the surface.  The energy flow is backward. This shift is strangely similar to the Query of Newton.

Also, the GH shifts that were measured for different metallic surfaces show signatures of plasmonics effect when rough gold films were used [13].

When the beam is focused, both the angular and the spatial shifts can simultaneously occur.  This is a consequence of the index of refraction of the metal.

The experimental evidence provided by the measurement of the GH shift in gold with focused beams shows that the angular shifts and the spatial shifts are actually not that different. They can be described by a common formalism [14].

Spin Hall effect of light in metallic reflection

In our recent publication, we have measured the Imbert-Fedorov shift/Spin Hall effect of light when a beam is incident on gold.  This is the first measurement of SHEL for metallic reflection [15].

A. Nugrowati made this sketch of the setup. (Hermosa, et al., Optics Letters 2011)

We did not use the weak measurement. We cleverly look at the shift of a circularly polarized light instead of looking at the splitting of a linearly polarized light.  The linearly polarized light splits into helically opposite polarization anyway (see above discussion). So why not use a circularly polarized light at the start?

More importantly, the weak measurement technique is impractical for metallic reflection. Remember that a circularly polarized light has s and p components that are π/2 shifted from each other.  Upon reflection from a lossy material such as gold, the difference in the phase acquired by the s and p component of the waves varies gradually with the angle of incidence between 0 and π.  The reflected light will be generally elliptically polarized and at a certain angle, is linearly polarized. The shift may not be detected after the post selection in the weak measurement.

The spatial IF shifts that we measured are below 100 nanometers. Spatial shifts can be observed for circularly polarized light and light with -450/450 polarizations. It is a different story for the angular IF shift.  While light with -450/450 polarizations exhibit angular deviations, the circularly polarized light does not. The angular deviations are in the sub-microradian range.

When -450/450 polarized focused light is used, spatial and angular IF shifts are observed simultaneously.

Contrary to the case of the GH shift for gold, we did not observe a backward energy flow in our measurements.  The reason is that the shifts are perpendicular to the incoming wave vector. Hence, the sign of the flow is independent of the permittivity of the material.

 Click here to go to part 2.

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References:

[1] Hecht, Eugene. Optics. 4th Edition.  Addison Wesley, 2002.

[2] Newton, Isaac. Opticks: A Treatise of the Reflexions, Refractions, Inflexions and Colours of Light. London: Royal Society, 1704.

[3] Goos, Hermann Fritz Gustav, and H Hänchen. “Ein neuer und fundamentaler Versuch zur Totalreflexion.” Annalen der Physik  436 (1947): 333-346. DOI: 10.1002/andp.19474360704.

[4] Artmann, K..“Berechnung der Seitenversetzung des totalreflektierten Strahles.” Ann. Phys. 437 (1948) :87–102.

[5] Imbert, Christian. “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam.” Phys. Rev. D 5 (1972): 787–796. DOI: 10.1103/PhysRevD.5.787.

[6] Fedorov, Fedor Ivanovič. “K teorii polnogo otrazheniya.” Dokl. Akad. Nauk SSR 105 (1955): 465.

[7] Onoda, M, S Murakami, and N Nagaosa.“Hall effect of light.” Phys. Rev. Lett. 93 (2004): 083901. DOI: 10.1103/PhysRevLett.93.083901 .

[8] Hosten, O. and  P Kwiat. “Observation of the spin hall effect of light via weak measurements.” Science 319 (2008):787–790. DOI: 10.1126/science.1152697.

[9] Costa de Beauregard, O.”Translational Inertial Spin Effect with Photons.” Phys. Rev. 139 (1965):B1443–B1446. DOI:10.1103/PhysRev.139.B1443.

[10] Aiello, A, and J P Woerdman. “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts.” Optics Letters 33  (2008) : 1437-1439.  DOI: 10.1364/OL.33.001437

[11] Merano, M,  A Aiello, M P van Exter and J P Woerdman.”Observing angular deviations in the specular reflection of a light beam.” Nature Photonics 3 (2009): 337 – 340. DOI: 10.1038/nphoton.2009.75.

[12] Merano, M,  A Aiello, G W ’t Hooft, M P van Exter, E R Eliel, and J P Woerdman.”Observation of Goos-Hänchen shifts in metallic reflection,” Opt. Express 15  (2007): 15928-15934.  http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15928.

[13] Merano, M, J B Götte, A Aiello, M P van Exter, and J P Woerdman. “Goos-Hänchen shift for a rough metallic mirror,” Opt. Express 17 (2009)10864-10870.  http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-13-10864.

[14] Aiello, A, M Merano, and J P Woerdman. “Duality between spatial and angular shift in optical reflection.” Phys. Rev. A 80 (2009):061801. DOI: 10.1103/PhysRevA.80.061801.

[15] Hermosa, N, AM Nugrowati, A Aiello, and J P Woerdman, “Spin Hall effect of light in metallic reflection,” Optics Letters 36 (2011):3200-3202.  DOI: 10.1364/OL.36.003200

4 thoughts on “Revisiting Reflection”

  1. pardon my short attention span, but what caught my attention was the intro to this post. really, you’re coming home soon? otherwise, i’m keen to read what you’ve written about this topic. see you nath. :)

  2. johnrob said:

    Oi Nath!! Welcome home. Back to NIP?

    Reflection in the level of atoms is obviously not as geometric as it seems. There must really be a shift from the geometric approximation in the real nano-scopic level.

    Cheers and safe voyage!

    • Thanks Johnrob.

      The plasmonic signature of the Goos-Hänchen shift (as observed by our group) will be briefly mentioned when I get to around 2005-2009.

      For now, enjoy the brief history of these shifts.

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