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On asymmetries in optics: fundamental considerations
and some current applications
G Lippi
To cite this version:
G Lippi. On asymmetries in optics: fundamental considerations and some current applications. Jour-nal of Interdisciplinary Methodologies and Issues in Science, JourJour-nal of Interdisciplinary Methodologies and Issues in Science, In press. �hal-01819557v2�
On asymmetries in optics and photonics: fundamental considerations and
some current applications
G.L. Lippi1
1Universit´e Cˆote d’Azur, Institut de Physique de Nice (INPHYNI), CNRS UMR 7010, Nice, France
*Corresponding author: gian-luca.lippi@inphyni.cnrs.fr
DOI: XXXX- 000
Submitted: Month-in-letters Day Year- Published: Month-in-letters Day Year Volume: N - Year: YYYY
Issue: Title of the interdisciplinary issue
Editors: First-name-1 Family-name-1, First-name-2 Family-name-2, First-name-k Family-name-k...
Abstract
The photon, as the basic constituent of light, is presented as an example of a fundamental brick of the physical world which breaks a symmetry. Its properties are illustrated in a heuristic way, to render the discussion accessible to non-specialists, together with the consequences of combining more photons. At the macroscopic scale, when large number of photons are involved – such as in the light coming from the sun – the symmetry is restored through the simultaneous participation of large ensembles of photons. However, we show that macroscopic devices, such as lasers, may reintroduce the break in symmetry even in the presence of very large photon numbers. Finally, we discuss the effects which arise from the consideration of finite numbers of interacting elements in the presence of a transition and draw a parallel with “small” systems from everyday’s experience.
Mots-Cl´es
Photon, Spin, Angular momentum, Symmetry, Asymmetry
I GENERAL REMARKS
Symmetries are one of the deepest properties which characterize the state of most real systems. In general, they are related to the invariance of some quantity. As a concrete example, we can consider the placement of objects in a space, let’s say furniture in a room, relative to its walls. When drawing plans, we can specify the position of each object starting from one chosen corner. However, as long as all positions are measured relative to the same point, the choice of corner is arbitrary, thus there is a symmetry in the problem (the room shape does not need to be symmetric, instead): any corner – or any other point of the room, such as its center – can be taken as reference. This property takes also the name of invariance: the result, i.e., the placement of the furniture in the room, does not change with the choice of reference point (as
long as we don’t make a mistake in the drawings!).
The topic is extremely vast and is the object of deep investigations which involve mathematics, physics, chemistry, biology, etc. In this contribution we are going to focus onto some very specific examples to show how the asymmetry, as opposed to symmetry, is also intrinsic in nature and how it provides us with very interesting examples and applications. Some of them are all around us and part of everyday’s life, even though we might not notice them.
The paper is meant to be accessible to a broad audience, thus it will try to introduce concepts at a simple and heuristic level, without any pretension of rigour. The specialists will know how to fix the imprecisions of the presentation, necessary in order to avoid confusing the general reader with too many details. At the same time, we will try to arrive at some examples and applications of symmetries at a more advanced level. These parts may serve as an overview of some topics, presented in the fashion of a broad review that may be of some use also to more advanced readers, who are therefore invited to skim the better known parts of this manuscript to arrive to those sections which will, hopefully, be of some interest to them.
The paper is organized as follows: we start by presenting the intrinsic properties of photons (sectionII), where we first consider just one photon at the time (section2.1) to then extend the discussion to the superposition of two photons (section 2.3), ending up with large ensembles of photons (section 2.4). A few elementary consequences of the photon’s intrinsic symmetry break in the interaction with matter are simply highlighted in section 2.2. We then turn to the discussion of asymmetries in modern devices (sectionIII), concentrating on the laser as an example (section 3.1). Questions arising from small sample sizes are analyzed in the context of small lasers (section3.2) and their relation to collective behaviour in small human groups is briefly illustrated in section IV, while some highlights of current research in optics related to asymmetries are presented in section3.3.
II PHOTONS
Light is one of the most ubiquitous resources we live with: it allows us to have visual perception, but also enables the growth of organisms, the exchange between carbon dioxide and oxygen, etc. Light consists of very small, elementary and discrete entities called photons, or light quanta (discrete “light pieces”). This discretization does not pertain exclusively to what we call light, i.e. the visible (for us) part of the electromagnetic spectrum. All electromagnetic radiation, extending from radio-waves (long, medium and short waves, radar and microwaves) all the way to x-rays (e.g., used in medical imaging) and γ-rays (e.g., arriving from the Sun), and includ-ing the intermediate frequency ranges correspondinclud-ing to the infrared and ultraviolet part of the spectrum, consists of discrete photons. An interesting chart illustrating the full electromagnetic spectrum and its properties can be found in the internet1.
2.1 One photon at the time
Since we view the photon as an individual piece of light, the first question which comes to mind is: what are its features? Photons possess neither mass nor electrical charge 2 but they 1https://draxe.com/wp-content/uploads/2016/07/electromagnetic-spectrum-article.png: accessed on June 6,
2018.
2Current measurements ensure that the mass of the photon, if it exists at all, it is smaller than 10−53Kg, i.e.,
0.00 . . ., where the number of zeroes which follow the dot is 53. For comparison, the mass of an electron, an extremely light particle, is 1022times larger than this upper limit (i.e., 10 followed by 21 zeros, i.e., one thousand billion billions). Similarly, the electrical charge of the photon, if it exists, is smaller than 10−35C, i.e., 1016times
carry energy (by which we distinguish colours) and another quantity which is particular interest for this discussion: spin. The best way of picturing spin is thinking of a toy top: at rest, it lays down on the surface on which it reposes, but when spun, it will turn either clockwise or counterclockwise (since we can choose how to make it rotate). Even though this is mostly an analogy, we can think of the photon as a spinning top which can turn in either direction: for definitness, we will assign the value +1 to a photon spinning counterclockwise and −1 to one spinning clockwise.
In order to get another intuitive picture for the photon’s spin, we are going to think of another analogy: that of a string. The reason is that, as we mentioned earlier, photons are the elementary componentsof electromagnetic waves. Waves exist in many other media (e.g., acoustic waves in air) and in particular on strings. Thus, let us imagine holding a long piece of string attached at one end: one can either let it fall limp or pull it tight, but in neither case this produces a wave. An easy way of producing a nice wave is to rotate the free end of the string (the one in your hand) either clockwise or counterclockwise (cf., e.g., Fig.1). By doing this fast enough, we see our string forming a spiral wave all the way down until the fastened end, just as a corkscrew (or a spiral staircase laying down). If the rotation is counterclockwise, following our previous convention, we assign it the value +1, otherwise −1.
We can further advance our understanding if we now look at the rotating string from its end (imagine having attached it to a transparent, glass, wall). If we just look at the motion of one point of the string, marked for instance by a coloured dot, we see it tracing a circle either with counter- or with clockwise motion (cf. Fig.2). This is another way of picturing the photon’s spin: we assign to it a circular motion with value +1 when rotating in the counterclockwise direction, −1 when clockwise. To this motion we associate the concept of circular polarization, with the same signs used up until now.
Figure 1: Illustration of a left-circularly (spin: -1, left sketch) and right-circularly (spin: +1, right sketch) polarized photon (i.e., wave) propagating in the direction marked by the arrow.
The existence of the two opposite states of polarization represents a spontaneous break of the intrinsic symmetry, since a photon can exist only with either handedness in polarization: left or right.
The question naturally arises whether there is the possibility for the existence of a photon with “0 spin” – this would restore a symmetric state between right and left. Various fundamental
mass and charge are 0 is that measurements are limited by the sensitivity of the apparatus and can never find a true 0, but can give only an upper limit (instrumental sensitivity) to “what we may be missing”. In other words, the instruments reads 0 within the limitation (error bar) of its sensitivity, which becomes the largest value (for mass or charge) that is compatible with the 0 reading.
physical arguments can be given for the reason why such a state cannot exist (although in general any physical system with spin ±1 also has a 0 component – except for the photon!), but this is left for specialists. A good overview of the topic, and some references in the literature, can be obtained from the web 3. Here, we are going to use, again, an analogy to explain the
reason why the “0 spin” state cannot exist for photons.
Going back to our example with the top, we can spin it either clockwise or counterclockwise. However, if we do not impart it any motion, the top is in an unstable state and topples over at the slightest perturbation: we cannot have it in a non-spinning state! We can therefore continue our analogy saying that we cannot have a photon with 0 spin. A different analogy is based again on the string: choosing either counter- or clockwise motion, we get a corkscrew kind of wave, but it is impossible to set up a wave that does not spin – a “counterexample” is going to be discussed in section2.3, where we will clearly see that a non-rotating wave requires something more than a single photon.
Figure 2: Illustration of a left-circularly (spin = -1, left sketch) and right-circularly (spin = +1, right sketch) polarized photon seen from front (head-on). The spiraling outer trajectories illustrate the per-spective (i.e., taking into account the direction of propagation towards the reader – i.e., out of the page), while the thicker circles at the respective figure centers eliminate the perspective, showing that the tra-jectory is entirely circular. The arrows on top of the thicker circles denote the direction of rotation.
2.2 Interaction between one photon and matter
Let us now ask a simple question: what happens when one photon encounters “matter”? Is there any interaction and what are its rules? Since we are focussing on the spin properties, we will particularly concentrate on this point, but first we need to settle the question of energy conser-vation: the interaction between the photon and matter will occur only if the exact energy of the photon can be exchanged back and forth (with matter). In order to keep the discussion as simple as possible, we consider an atom: its energy states are known to be discrete (think of the spec-trum obtainable from various metals: coloured flames in a chemistry lab, coloured fluorescent tubes – the old red neon tubes –, or the emission from a spectral lamp 4, etc.). We schemati-cally represent these discrete states as the two horizontal lines of Fig.3a, where the lower level has less energy than the upper one and the energy jump is equal to the photon’s energy (in mathematical terms: E2− E1 = ¯hω, where ¯h is a numerical constant – Planck’s constant – and
ω = 2πν is proportional to the frequency of oscillation ν of the “photon wave”). The possibility
3https://en.wikipedia.org/wiki/Photon. Consulted on June 12, 2018.
4The emission spectra of various kinds of atoms (e.g. Sodium, Hydrogen, Calcium and Mercury) obtained from
spectral lamps can be found at: http://www.didamoe.com/mercury-vapor-lamp-spectral-lines/ – consulted on June 12, 2018. Each colour corresponds to a different distance between the energy levels.
for a photon to be absorbed (or emitted) in the interaction with matter requires energy matching, which in the following discussion will be considered as automatically satisfied.
Figure 3: Illustration of atomic states (simplest kind of “matter”) for the interaction with single photons. (a) Two discrete (or “quantized”) atomic states of energy E1 < E2 in the so-called “Kastler
represen-tation”. (b) The upper state of the atom possesses spin with components ±1, 0 (allowed for the atom), identified by the value of m. The double arrows show the possible interactions between circularly polar-ized photons and the atom – the arrows on the circles show the circular polarization of the corresponding photon. No interaction is possible between the state (E2, m = 0) and (E1, m = 0) through the exchange
of a photon. These considerations hold when we keep the reference axis unique for all “elements” in-volved in the discussion (here photons and atoms). When this is not the case, it is possible to induce transitions between states with same value of m, but this issue is too complex for the basic discussion offered here.
Fig. 3b illustrates a somewhat more complex situation, where one atomic level (say the upper one with energy E2) also possesses spin, denoted by the letter m. In the case of the atom, all
three components are allowed (m = ±1, 0) and the interaction with the photon is subject to the restriction that the total spin is conserved: the spin “lost” by the atom in the downward transition has to be taken up by the photon (vice-versa if the photon is absorbed by the atom). Thus, the two possible paths are those indicated by the double arrows, either towards m = +1 or towards m = −1: the interaction between the two m = 0 states is not possible by exchanging a single photon, since the exchange requires a change in spin in the atomic state – due to the nonzero intrinsic photon spin –, while these two states have identical (m = 0) spin values (thus no “spin change” can be involved in this transition). This is an important point which lies at the basis of the questions posed on the chirality (i.e., handedness) of simple organic molecules as found in nature (Sugahara et al.,2018).
In other words, not all interactions are allowed and light selects those possibilities which are compatible with its broken symmetry, thereby imprinting the symmetry break onto matter.
2.3 Two photons
One photon does not carry a large amount of energy (for visible photons Ephot ≈ 2 × 10−19J ,
i.e., ≈ 0.000000000005 times the amount of energy needed to raise one drop of water by 1mm). It therefore makes sense to see what happens if we take more than one photon simultaneously. The first step is to look at two photons. Of course, we are going to discuss the combination of two photons possessing the same energy (otherwise we simply get two “colours”).
If the photons have the same handedness, we simply get twice the energy with the same spin, but if the two photons have opposite handedness we get the result of Fig.4. On the left, we combine two photons starting from a particular “initial position” (i.e., technically “phase” = 0◦): the two dots represent the motion of the two photons on the circle of Fig. 2 (inner circle) rotating in opposite directions (arrows). The composition of the two rotating trajectories gives a trajectory which oscillates in the diagonal plane (tilted to the right), representing the actual motion of the
wave. In the string analogy, there is no longer a helical wave, but an oscillation of the string in the diagonal plane drawn in the figure. This is of course possible, as one can easily test with a piece of rope (it may be difficult to get the direction right, but possible).
Figure 4: Combination of two photons with opposite handedness. The left panel illustrates the rotation of the two photons (in opposite directions) starting from the horizontal axis: the resulting polarization is linear along the diagonal of the first and third quadrant (double-arrow on the bisectrix, here and in the right panel). The right panel shows that a different choice of “starting point” (on the vertical, one point up, the other down) gives rise to a linear polarization along the other diagonal. As explained in the text, all possible linear directions can be obtained with suitable transformations.
Simply starting from a different initial position (phase) it is possible to obtain a different os-cillation plane (right sketch in Fig. 4). When testing the analogy with the rope, one is not simultaneously rotating one’s hand left and right simultaneously, but moving it diagonally up and down to generate the correponding wave. The practical illustration tells us immediately – as can be done with mathematical transformations which we leave for specialized textbooks ( Pe-drotti and PePe-drotti,1993) – that there is an infinite number of planes along which one can make the string oscillate. Thus, there is also an infinite number of planes along which the radiation resulting from the combination of two photons with opposite handedness oscillates!
These considerations highlight several points:
a. The combination of the two photons with opposite handedness results in a state which has lost the apparent symmetry break;
b. The spatial symmetry is, however, lost because there is a preferential plane of oscillation; c. The number of planes is infinite, thus this symmetry break is not a fundamental one,
unlike the one related to the handedness, which rests on only two states.
The polarization states illustrated in Fig. 4 correspond to what is normally known as linear polarization. The fact that we need two photons with opposite handedness to obtain this state proves that linearly polarized photons do not exist, but that we can obtain linearly polarized light only by superposition of suitably paired photons (even in large number to obtain intense light, as long as each circularly polarized photon has its counterpart). Since the interaction with matter occurs one photon at the time (unless nonlinear, very high-intensity interactions are considered), linearly polarized light plays no role in transferring a symmetry break onto matter.
2.4 Unpolarized photons?
There is no such thing as an unpolarized photon. Unpolarized light, as received for instance from the sun, is the result of the superposition of pairs of photons, each producing a linearly polarized emission but without preference for the orientation of the plane of polarization.
Sticking to the sun as an example, Earth receives approximately 1021photons per square meter per second (over the whole visible spectrum). Given this extremely large number of photons, it
is easy to see how the random orientation of the plane of polarization (i.e., the axis in Fig.4) pro-duces light with no polarization preference. However, there is no such thing as an unpolarized photon, as there is no unpolarized wave – try making a string oscillate without any direction of oscillation, neither circular nor linear!
Thus, it is important to remark upon the fact that the only, truly symmetric state – a photon without polarization – can never exist!
III ASYMMETRIES IN CURRENT OPTICAL APPLICATIONS AND DEVICES The fundamental break in the symmetry originating from the photon properties influences many applications. The symmetry break can be neutral or even beneficial, but there are situations where a large amount of effort is placed into the reestablishment of a symmetric state for par-ticular technologies.
3.1 Lasers
One of the most ubiquitous devices which has entered everyone’s life is the laser. Discovered 58 years ago, and considered “a solution in search for a problem” (Townes, 2003), it has now found an innumerable number of applications, covering telecommunications, medical, sensing, entertainement, construction, etc. Everyone has nowadays manipulated a laser at least as a pointing device or in a computer mouse.
The laser (Light Amplification by Stimulated Emission of Radiation) converts the sponta-neously emitted photons, as those normally emitted by thermal sources, into a collimated beam of monochromatic (i.e., single frequency: one “colour”) radiation possessing a single “phase reference” (for an intuitive picture, think of a long wave which continues without breaks). As such, it has to perform some kind of selection on the kind of photons that it harbors inside and (in part) emits. This is achieved by selectively exciting atoms into one single energy level (the upper one in Fig. 3), so that the emerging photons all possess the same frequency (or wave-length), then recycling the photons – i.e., bouncing them back and forth – with the help of two facing mirrors (forming an optical cavity). The latter strengthens the so-called stimulated emission process, whose existence was first postulated in 1917 by Einstein (Einstein, 1917), responsible for the establishment of laser action.
Stimulated emission is a strongly nonlinear process – i.e., in everyday’s words, its outcome is not simply proportional to the ‘stimulus” but “overreacts” to it – which performs a selection in the photon’s properties, since it amplifies any instantaneous imbalance bewteen the two photon spin states. We have previously seen that when the number of photons is large, there is an equal number of them (statistically) in each polarization state: left or right. The statistical balance, however, exists only when considering averages (over time or over different repetitions), exactly in the same way as we can predict a 50% outcome for head and tails when tossing a coin (alternately, between odd and even numbers). However, at the level of an actual sequence of tosses, we may find a sequence of multiple “heads” (say three or four) before getting a “tail”. Thus, at the “microscopic” scale we always have imbalances.
Even in a weak, barely visible laser the number of photons emitted in a second exceeds 1012 and, on the short timescales on which the laser field builds up, we typically find more than 1000 photons. While the imbalance may be small (think of a “head” and “tails” experiment), there is bound to be one. Because of the “reactiveness” of the amplification by stimulated emission to any excess, it is natural to expect that the photon class (say with “+1” polarization) which has a
larger number of photons will win the competition and the laser will emit exclusively +1 pho-tons. The intrinsic symmetry break is now translated into a macroscopic property, unlike what happens in the emission of large sources (e.g., the sun), thanks to the selectivity of stimulated emission and the overall lasing conditions.
We therefore dispose of a mechanism which transfers the intrinsic symmetry break of photons onto the macroscopic state of light, providing an optical source whose characteristics strongly differ from those of any other thermal source (lamp bulbs, sun or even spectral lamps). The previous considerations would imply that the actual state of polarization of the light would de-pend on the configuration (predominance of +1 or -1 polarized photons) at the moment the laser is turned on. This would indeed be the case if the technological implementation of the system (first of all the mirrors, but also any other element entering into the construction) were perfectly isotropic. This is, in general, not the case and the smallest anisotropy helps breaking the sym-metry and imparting a preferential polarization on the device. In this case, the polarization will always be the same, thus rendering the use of the device more practical. Indeed, having a re-producible control on the characteristics of the emitted photons (in this case, their polarization) helps designing the properties of the optics which follows.
As an example, think of polarizing sunglasses: their principle of operation is to remove one of the two polarized components out of the sunlight. Given the “unpolarized” state of sunlight (equivalent number of photons emitted in all planes of polarization) this would remove half of the light which arrives to the eye. However, reflection from surfaces tends to privilege one polarization component over the other (Pedrotti and Pedrotti, 1993). Hence, aligning the axis of the polarization filter (i.e., the “sunglass”) to remove the reflected components attenuates the light even further, lending comfort to the eye. Those who have used polarizing sunglasses may, however, have noticed that it is impossible to read at certain angles digital, liquid-crystal-based displays, such as those of wristwatches. These displays are also polarized, because of construction needs, and if your wrist happens to be turned at the wrong angle, the light arriving from your wristwatch will be entirely filtered: in such a case, the diplay appears to be completely dark.
While the control of the broken polarization state is useful for many practical purposes, there are interesting applications for the balanced situation, where equal number of photons in each polarization state are emitted by a laser. In this case, the macroscopic equilibrium is reestab-lished, not through “unpolarized photons”, but through the balance in the number of photons of each kind. This is an extremely difficult condition to satisfy because of its intrinsic lack of stability, which finds its origin in the sensitivity of the nonlinear amplification process to any imbalance, as described above. One can picture the difficulty in achieving this condition with the following analogy: you can hold a long rod on the tip of your finger, as long as it stays perfectly vertical. Since external influences will perturb it away from that position, you need to continuously compensate by moving your finger to keep balance. The same is needed in an “unpolarized” laser and requires serious efforts. Such efforts are, however, repaid by the sen-sitivity of the resulting device to external perturbations. One example for possible uses of the “unpolarized” laser (or, more properly, “polarization isotropic laser”) is its sensitivity in mon-itoring tiny magnetic fields (Cotteverte et al., 1990). More recently, applications of isotropic lasers for detection in biophotonics have been envisaged (Cerd´an et al.,2016).
3.2 Small lasers
In the last couple of decades there has been a keen technological interest in considerably re-ducing the volume of the laser cavity. The biggest push behind this engagement comes from a considerable reduction in energy required to supply a tiny laser, as compared to a macroscopic one, accompanied by lower thermal dissipation and a diminished footprint (i.e., the portion of space occupied by the device). The main scope of these investigations is to reach devices so small as to fit into a chip the size of an electronic one, with the ultimate aim of replacing elec-tronics with light (i.e., photonics). Indeed, the speed (of light) at which the information could flow in such a chip is about two orders of magnitude larger than the electron speed in a circuit, thereby offering large savings in the time needed for performing the different operations. Small volumes are necessary to pack a large number of components – to be competitive with electron-ics – while low consumption and low thermal dissipation are required for compatibility with dense packing. Such chips are now already being built (Smit et al.,2012), but the light sources (lasers) are still external and feed the optical circuit through fibers.
The reduction in cavity volume, however, carries with it an additional problem related to the small number of “elements” 5 participating in the laser emission. This deviation from large number laws, which characterize the preceding discussions, introduces an additional element of randomness which induces more complex transitions between the states of the system. We are going to touch on a couple of problems and will conclude, in the next section, with an analogy in other systems which can be interpreted with the help of physics and optics.
We first analyze an example within the framework of the polarization states of the photon and look at the transition between random light (like a Light Emitting Device – LED) and lasing in a semiconductor laser (specifically a VCSEL: Vertical Cavity Surface Emitting Laser). When the cavity volume is very small, due to the reduced total number of photons, the suppression of the weaker population is less effective. Thus, at low emission levels it is possible to observe the simultaneous emission of photons of both polarizations, albeit in different numbers. Ramping up the power, i.e. increasing the photon number, the population which contains a large number of photons enjoys larger gain and can eventually completely suppress the weaker one. Manufac-turing imperfections, common in these kinds of devices, can also induce a switch in polarization (predominance of one or the other of the symmetry–broken photons populations) with simple bistabilility (i.e., the stable emission of either polarization) or with dynamics (spontaneous tem-poral alternation between the two states) (Sondermann et al.,2003). In our group, we are in the process of investigating these questions in microlasers in order to understand, on the one hand the physics of the interactions and, on the other hand, the issues at stake related to finite size effects (cf. below).
As the cavity size is reduced, another form of competition apparears, illustrating an aspect not related to the photon polarization states. Since the collective interaction which induces the transition towards lasing relies on the existence of a large number of elements participating in the interaction, the numerical reduction coming from the small cavity size opens questions connected to the resulting small numbers (finite size effects). It has been known for nearly 30 years that the transition between the two kinds of behaviour (LED-like light vs. laser light) is not so clearly defined for small lasers (Yokoyama and Brorson,1989;Bj¨ork and Yamamoto,1991) and fundamental questions on the nature of the transition between the two regimes (another form of symmetry breaking) still remain open (Ning,2013;Ma et al.,2013).
5By “elements” we mean, here and in the following discussion, emitters (e.g., atoms or equivalent elementary
We have been able to show, in the past couple of years (Wang et al., 2016, 2018), that the transition between the two states goes through a sequence of intermediate steps (Barland et al.,
2018) which appear and can be identified also in other “physical” and “non-physical” systems, as exemplified in the discussion of sectionIV.
3.3 Highlights of some other current research topics
This section highlights a few other current research topics which deal with symmetries in optics (and more generally, electromagnetism). I am grateful to an anonymous reviewer who, propos-ing a couple of very interestpropos-ing references as possible extensions to the brief overview offered in sectionIII, stressed the benefits of offering a somewhat broader summary of today’s research in this field. The selection proposed here is in any case arbitrary and I apologize in advance for the omissions and for the necessarily very scant bibliography, which cannot do justice to any research field. I hope that this extremely cursory presentation will at least convey an idea of the importance of symmetries in current optical research and of the vigour with which they are being investigated not only for fundamental research, but also in the perspective of extremely powerful applications.
The behaviour of a photon interacting with a beamsplitter (a plate which separates an optical beam into two parts, permitting to one fraction to continue straightahead while redirecting the complementary fraction) is a preliminary step at the base of numerous current applications. While the interpretation of the operation of a beamsplitter on a “macroscopic” optical beam – i.e., containing a very large number of photons – is straightforward, its conterpart with one or two photons is not trivial. The symmetries which are present in this problem would have qualified this topic for dedicated discussion in sectionII, but a pedagogically sound discussion would have required too much space to justify its presence in a paper for a general audience. I am therefore going to attempt a brief simplified description which may not be entirely accessible to a larger readership.
A photon impinging onto the beamsplitter can be described as being simultaneously in two states: one transmitted, the other one reflected, and described by suitable complex coeffi-cients (Weihs and Zeilinger, 2001). The measurement after the beamsplitter, obtained with two detectors – one placed on the transmitted, the other on the reflected path –, provides the so-called collapse of the wavefunction (i.e., of the superposition of states): one detector clicks in response to the arrival of the photon on its surface, while the other does not receive anything. This way, one definite path is chosen between the two possibilities (transmitted or reflected), as indicated by the detectors’ response. Repeating the experiment N times provides N2 counts on each path, if we have taken the reflectivity to be 50% (i.e., same probability of transmission or reflection; we will keep this particular choice in all discussions that follow). In other words, the symmetry of the superposition state is broken by the action of checking, with the help of the detector, which path the photon has taken. The same symmetry break takes place when send-ing a ssend-ingle photon onto two slits: lettsend-ing the photon arrive onto a screen through the two slits simultaneously, one sees the expected interference pattern (symmetric state), while checking the path chosen (by inserting a transparent detector in one slit, thus getting one or zero clicks) breaks the state’s symmetry and the interference disappears.
If the experiment is repeated by sending two photons, possibly coming even from different directions, the result is that interference will force the two photons into the same “channel” (or path), i.e. they will be both reflected or transmitted (Hong et al., 1987), as long as the two considered photons are identical and simultaneous. It is therefore easy to see why when large
number of photons are involved, half of the total number would follow one path and the other half another. Indeed, in the large ensemble involved in this case, only the photon subgroups which are arrive simultaneously collectively interact at the beamsplitter, but different subgroups will behave independently. Since there is a priori no preferential direction in a symmetric beamsplitter, then there will be equal probability for each subgroup to be reflected or transmitted and, in average, we collect half the photons in one channel and half in the other.
The reason why these considerations are important in current research is that the detection of single, or very few, photons is steadily gaining importance: its applications range from the de-tection of very weak optical beams, preferred to minimize energy consumption, to the encoding of secure (quantum) information which can be done only with single photons. With these very weak signals, information is collected through the statistical features of the photon field ob-tained through the simultaneity (coincidence) in the arrival of photons traversing beamplitters (one or more) onto two (or more) detectors. Sophisticated techniques, striving to reduce the number of detectors, are being developed to reduce costs and complexity of the setups (Laiho et al.,2012) and are all based on the breaking of the intrinsic symmetry inherent in the photon wavefunction.
The measurement of correlations, based on the previous considerations, opens up the field to in-numerable kinds of applications. One that is particularly fascinating, for its concepts as well as for its potential impact, is the so-called ghost imaging6. Initially predicted as a test of quantum mechanics (Klyshko,1988;Pittman et al.,1995), it has evolved over the years to cover various techniques which are based on the use of light beams either with classical properties (macro-scopic beams) or with quantum properties (e.g., single photons) (Gatti et al., 2004;D’Angelo and Shih,2005;Erkmen and Shapiro, 2008). A good overview of this field is given byPadgett and Boyd(2017). A very schematic illustration of the principle is the following: a laser beam is divided into two parts by a beamsplitter and on each path a “detector” is inserted. The first one is a camera capable of forming an image of desired resolution (even very high), the other one is a single (i.e., “one-pixel”) detector which simply clicks when it receives light – the two “detectors” are isolated from each other and may even be located very far apart. In front of the single detector we place an object, while the camera receives the unimpeded light coming from the laser beam (thus does not see the object). Scanning the laser beam, to point at each camera pixel one at a time, before the beamsplitter – thus simultaneously performing the same scan-ning operation on the other path – and measuring the coincidences (detection of light in both paths) one can make the high-resolution camera see the shadow of the object placed in front of the single detector. Indeed, the absence of light on the single detector, due to shadowing, will tell the high-resolution camera that the corresponding pixel is “dark”, while the arrival of light onto the “one-pixel” detector signals a bright pixel on the camera. The breaking of the symmetry in the detection is used here to signal the presence of an obstacle, while the symmet-ric state tells the camera that in the corresponding position there is no shadow, thus providing information on the profile of the object. If the detector is capable of measuring light levels, then grey shades will be possible. Extensions of the technique include the removal of the camera – thus reducing the whole apparatus to a single detector (Bromberg et al.,2009) –, or the use of quantum-correlated photons (Walborn et al.,2010), which can be used to “announce” the arrival of the photon whose detection (or absence thereof) is going to tell about the shadowing by the obstacle (or not) (Tasca et al.,2013). The potential for applications of this particular implemen-tation of symmetry break is enormous, and range from extremely low intensity imaging, where
the camera receives a sufficiently intense beam to ensure a sharp contrast while the object is illuminated by as little as a single photon (Morris et al.,2015), to microscopy at very low light level (Aspden et al.,2015) and three-dimensional image reconstruction (Gong et al.,2016;Sun et al.,2016). The frequency conversion opportunities offered by ghost imaging, where the sens-ing path (i.e., the light beam impinging on the object) and the detecting path (where the light is detected by the “camera”) may operate in widely different parts of the electromagnetic spec-trum (e.g., terahertz, infrared or even x-rays on the sensing path and visible on the detection path), promise amazing imaging opportunities (Aspden et al., 2015) where the image is col-lected with adequate illumination levels in the visible range, while extremely reduced photon fluxes, at potentially dangerous frequencies, are used on the sensing path enabling for instance very low-level medical x-rays (Pelliccia et al.,2016).
Symmetries are at the base of another huge field of development to which the European Union has devoted an entire flagship: quantum technologies 7. One of the main components of this research is quantum information and data security. The latter is based on the preparation of superposed states, called entangled, conceptually similar to those of the photon impinging on a beamsplitter: the system (again photon or other quantum system) will end up in one of states at the base of the superposition once it is measured. And, again, the measurement breaks the intrinsic symmetry of the entangled state in an irreversible way, thereby ensuring the detectabil-ity of any attempt at espionage: the action of reading the private information is irreversibly and unavoidably discovered by the disappearence of the entanglement. One of the numerous pos-sibilities for preparing entangled states rests on mixing photon polarization states (Ikuta et al.,
2018;Huber et al.,2018), thereby reconnecting this line of investigation directly to our present discussion.
Beams possessing orbital angular momentum (the same physical quantity characterizing the centrifugal force, for instance) have been the object of research for more than a couple of decades (and are related to the 2018 Physics Nobel Prize awarded to Arthur Ashkin). The angular momentum of an optical beam – one constituted of a large number of photons – is the analogon of the photon’s spin and corresponds to the breaking of the rotational symmetry. These beams currently find applications in the non-invasive (i.e., devoid of mechanical contact) ma-nipulation of small objects and are therefore particularly interesting for micromechanics (Ogura et al.,2001) in special environments, and for biological manipulations (He et al., 1995;Friese et al., 1998; Grier, 2003; Ashkin, 2006; Barnett et al., 2016). They also hold promise for in-creasing the bandwidth in data transmission (Jensen and Wallace, 2004;Yan et al.,2014) – in free space or in fibers, at radio or optical frequencies – since the conservation of angular mo-mentum ensures that information encoded in one state will not mix with that of another one. This way, it is possible to multiply the number of transmission channels without creating new frequency channels – the standard way used up until now. Introducing for instance three angu-lar momentum states, say (0, ±1), would immediatly multiply by 3 the quantity of information transmissible in the same frequency bands. Following the optical angular momentum research line, the generation of optical vortices in the extreme ultraviolet, though fraught with serious technical difficulties, presents a strong interest since it paves the way for numerous applications, such as stimulated emission depletion microscopy, optical manipulation, bypassing the diffrac-tion limits, to name only a few. Hence, a large amount of effort has been recently devoted to
7For more information, the reader can consult:
https://ec.europa.eu/digital-single-market/en/news/european-commission-will-launch-eu1-billion-quantum-technologies-flagship, together with a summary offered in: https://en.wikipedia.org/wiki/Quantum technology. Both sites have been consulted on November 8, 2018.
this field which is starting to yield very encouraging and original results (G´eneaux et al.,2016;
Ribi˘c et al.,2017;Gauthier et al.,2017).
Recent developments in nanotechnology now offer the possibility of creating new structures capable of generating light, or modifying its properties, in novel ways. One particularly fruitful line of investigation resides in the use of the interaction between light and structures made of dielectrics (i.e., insulating materials) and conductors: a new kind of “wave”, the plasmon, is generated at the interface between these materials. The resulting field of plasmonics allows for the generation of polarization states of light coupled to general directional properties (e.g., lenses made of flat materials!) out of suitably shaped nanoantennas (Yu and Capasso, 2014). Nanostructured holograms are also used to create light with symmetry-broken properties (Lin et al.,2013) and plasmonic polarizers can be integrated on top of semiconductor lasers, thereby manipulating the symmetry properties of the emitted light directly at the source (Yu et al.,2009). Optical chirality is attracting a large amount of interest, not only for the questions which pertain to protolife (Sugahara et al., 2018), but also for the possibilities that it opens in detection of molecular species (Tang and Cohen, 2011) or in imaging applications (Khorasaninejad et al.,
2016). The technical aspects of the generation and detection of light with chiral properties are often coupled to the use of relatively complex plasmonic techniques (Khorasaninejad et al.,
2016), but relatively simple setups have been devised (Tang and Cohen,2011), including the use of DNA (3D DNA-origami) as a model for self-assembling metal nanoparticles with nanometric accuracy in nanoscale helices (Kuzyk et al.,2012).
This brief overview has highlighted some of the current research topics related to symmetries in optics. However new and exciting these new topics, some questions remain unsolved. One of particular interest in the context of this paper is the transfer of the collective symmetry (or broken-symmetry) properties of macroscopic beams back to the single photon, following a re-verse path compared to our previous discussions. Indeeed, in sectionIIwe have examined the intrinsically broken symmetry in the photon angular momentum, restored only by the combi-nation of photons with opposite spin. The optical angular momentum carried by macroscopic beams is a feature which does not find an obvious correspondence at the level of the single photon and the way this global symmetry break transfers onto the individual photon is still a point open to debate.
IV AN INTERESTING ANALOGY
The transition between states in the presence of a small number of “elements” (in a generic sense here) is something which is part of everyday’s experience, even though it is not necessarily recognized as such. For lack of space, I am going to limit this discussion to one single example which is known to everyone: audience clapping at a performance.
It is well-known and recognized that depending on the degree of enthousiasm, a group of people will clap in a more or less energetic way. What is less immediately recognized, however, is the fact that different clapping regimes exist, ranging from desultory and weak clapping to the so-called standing ovation and rythmic clapping (N´eda, Ravasz, Brechet, Vicsek, and Barab´asi,
2000;N´eda, Ravasz, Vicsek, Brechet, and Barab´asi,2000;Mann, Faria, Sumpter, and Krause,
2013). The cited investigations have studied the transition from disorderly clapping to rhythmic applause in small test groups, but it is known from experience that only large enough audiences
will spontaneously make the transition to a regular applause8, thus exemplifying the influence
of the system size on the nature of the transition, similarly to what discussed for small lasers in section3.2.
The analogy with the transition between LED-like and laser light can serve as good guidance, as there appears to be interesting similarities in the behaviour of the different systems (Barland et al.,2018), which we cannot detail here, with the added benefit that the laser is not subject to external influences such as psychology, group pressure, external signals, and the like.
V CONCLUSIONS AND REFERENCES
We have presented in a qualitative way the fundamental symmetry break intrinsic to the nature of light and shown its implications into the various forms under which we either receive and perceive radiation, as well as its influence on its interaction with matter. The intrinsic symmetry break often disappears when we look at large ensembles, such as the light which we receive from the sun. However, current technological devices (such as lasers) through their peculiar interactions transfer the microscopic photon properties to the macroscopic scale. Finally, we have shown that when considering systems of reduced size, the transitions become less well-defined and have given an example from everyday’s life showing how it is possible to draw interesting parallels between optical/physical systems and the dynamics of small groups. These analogies potentially offer great advantages in the interpretation of observations concerning social behaviour, thanks to the simpler analysis offered by phenomena which are not influenced by human nature.
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A ACKNOWLEDGMENT
I am grateful to T. Wang, H. Vergnet and G.P. Puccioni for the collaborative work on the physics of small lasers and to an anonymous reviewer for suggesting the idea of writing section3.3. I am further indebted to G.P. Puccioni for LaTeX help.
B BIOGRAPHY
Gian Luca Lippi (Laurea in Fisica, University of Florence, Italy, 1984; Ph.D. Bryn Mawr College, USA, 1990; Habil. Dir. Rech. Universit´e de Nice-Sophia Antipolis, France, 1998) is currently Distinguished Professor at the Physics Department of the Universit´e Cˆote d’Azur and member of the Institut de Physique de Nice (formerly Institut Non Lin´eaire de Nice) since 1994. Between 1990 and 1993 he was active as Post-Doctoral Fellow at the Institut f¨ur Angewandte Physik (Westf¨alische Wilhelms-Universit¨at M¨unster, Germany) first as Alexander-von-Humboldt Fellow, then with DFG support. His research covers laser dynamics, nonlinear dynamics in optical systems, laser-matter interactions, particle trapping, physical properties of nanolasers, and biophotonics mainly from an experimental point of view, but including modelling aspects. He is co-author and author of over 60 papers in refereed journals, 24 conference proceedings and nearly 120 contributions to conferences (22 invited presentations). Adviser and co-Adviser of eight Ph.D. theses, he has been director of a French Doctoral School in Sciences and subsequently Director of the European Doctorate EDEMOM. Former International Consultant for the project “Laser Trapped Mirror Proposal” (MSMT, NASA), past referee for INTAS (EU) and EPSRC (UK) programmes, referee for all the main international physics and optics research journals, he is currently member of the Strategic Council of REA.
