APPLIED OPTICS. Lecture-1: EVOLUTION of our UNDERSTANDING of LIGHT. Is it a stream of particles?

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1 A. La Rosa Lecture Notes APPLIED OPTICS Lecture-1: EVOLUTION of our UNDERSTANDING of LIGHT What is light? Is it a wave? Is it a stream of particles? A. Light as a particle NEWTON ( ) was the most prominent advocate of this theory Ray of light conceptualized as a stream of very small particles emitted from a source of light and traveling in straight lines. This view was based on the fact that light can cast sharp shadows. But cannot explain well what is now known as Newton s rings B. Light as a WAVE Fig.1 Fig. Newton s rings. 1 CHRISTIAN HUYGENS ( ) contemporary of Newton, championed this view When two beams of light intersect they emerge unmodified (different than the case when particles collide.) 1801 YOUNG s Double-Slit experiment The complex shadows formed by the two slits (in the form of seem to demand an interpretation of light as a wave. Fig.3 Two-slit experiment 1 We know that the contemporary interpretation of light is that it is constituted by photon particles. Can you envision a way to interpret these Newton s rings under this photon particles view?

2 181 FRESNEL Light is a transverse wave Light is polarized Explained the phenomenon of double refraction in calcite. Fig. 4 Independent of this progress in optics, the study of electricity and magnetism was also flourishing JAMES C. MAXWELL ( ) is a genius who condensed the phenomena of electromagnetism into a set of four equations 1 Predicts EM-waves travel at speed = It turns out 1 km 300,000!! S 0 0 Light must be an electromagnetic wave! HEINRICH HERTZ confirms experimentally the existence of electromagnetic waves 1887 A. MICHELSON and E. MORLEY if the speed of light is constant in the aether and the earth presumably moves in relation to the aether (at ~67,000 mi/h) then the speed of light with respect to the earth should be affected by the planet s motion. BUT no motion of the earth with respect to the aether was detected. A. EINSTEIN, special theory of relativity Rejected the aether hypothesis Light always propagates with a definite velocity c (in empty space) which is independent of the motion of the light source Light was then envisaged as a self-sustained electromagnetic wave The 19 th century: served to place the wave theory of light on a firm foundation.

3 C. Interpretation of Light as a WAVE is inconsistent with some experiments However, by the end of 19 th century - beginning of 0 th century: It became evident that the wave theory of light could not explain certain experiments (the blackbody radiation, the photoelectric effect for example.) Indeed, the wave theory of light began to crumble. The ultraviolet catastrophe Predicting the spectral density I() of light inside a cavity at a temperature T. Reflecting walls I() Incident radiation x o Scattered (re-emitted) light q e, m e Atom (modeled as an oscillator) x o is the electron s amplitude of oscillations. m e, q e : electron s mass and charge. : angular freq of oscillations (electron and light). Fig.5 Light intensity existent inside the cavity is absorbed by the atom and re-emitted in all directions. At equilibrium, the rate at which light energy is absorbed and the rate at which the atom re-emit the light must be equal, which requires a particular value of the spectral light intensity I() Classical prediction 6 c Number of modes of frequency : Average energy of each mode : kt ~ f Spectral density ( ) f kt (1) 6 c I However, the experimental results were quite different in the high frequency regime. Only at low frequencies there was an agreement between the classical prediction and the experimental results.

4 I() Classical prediction Experimental results Frequency Fig. The serious discrepancy between the experimental results and the theoretical prediction of the spectral light intensity at high frequencies is called the ultraviolet catastrophe. The photoelectric effect Fig.3 Einstein proposed an explanation based on quantized electromagnetic fields (1905), corroborated by Milikan in The Compton effect ' Scattered light Compton, 193 X-rays incident on a graphite target. Incident x-ray radiation electron Fig. 4 Scattered light contains frequencies different than the incident one.

5 D. Planck s Hypothesis of quantized energy In dealing with the ultraviolet catastrophe problem, it turns out, the difficulty brought by classical physics was that, in general, it assigned an average energy of the oscillator equal to kt, regardless of the natural frequency ( o ) of the oscillator. Planck (1900) realized that he could obtain an agreement with the experimental results if, rather than treating the energy of an oscillator (of natural frequency ) as a continuous variable, the energy states of the oscillator have only discrete values: 0,,, 3, the energy steps would be different for each frequency =( where the specific dependence of in terms of had to be determined.(notice we have dropped the use of the sun-index zero when indicating the natural frequency). Incident radiatio q Planck postulated nergy thao the oscillator (of natural fre ) is quantized Fig.5 An atom (of natural frequency w) takes up energy from the incident radiation in the form of lumps having energy values,, 3, Planck s calculation of average energy of the oscillator: W Planckl ( ω) E n0 E n0 n e e E E n n / k T B B / k T where E n = n(); n= 1, 3,

6 W Planckl ( ω) e 1 kt 1 average energy of an oscillator (atom) of natural frequency. (Notice, it is different than the classical prediction kt) Planck s prediction 6 c Number of modes of frequency : Average energy of each mode : W ~ f Planckl ( ω) e 1 kt 1 ssical diction Spectral density ( ) f W Planck () 6 c I I() Experimental results Particle s and wave s energy quantization Historically. Planck initially (1900) postulated only that the energy of the oscillating particle (electrons in the walls of the blackbody) is quantized. The electromagnetic energy, once radiated, would spread as a continuous. It was not until later that Plank accepted that the oscillating electromagnetic waves were themselves quantized. The latter hypothesis was introduced by Einstein

7 (1905) in the context of explaining the photoelectric effect, which was corroborated later by Millikan (1914). E. Quantum Mechanics Schrodinger Equation i t V ( x,t) m x This equation marked a historic moment constituting the birth of the quantum mechanical description of matter. The great historical moment marking the birth of the quantum mechanical description of matter occurred when Schrodinger first wrote down his equation in 196. For many years the internal atomic structure of the matter had been a great mystery. No one had been able to understand what held matter together, why there was chemical binding, and especially how it could be that atoms could be stable. (Although Bohr had been able to give a description of the internal motion of an electron in a hydrogen atom which seemed to explain the observed spectrum of light emitted by this atom, the reason that electrons moved this way remained a mystery.) Schrodinger s discovery of the proper equations of motion for electrons on an atomic scale provided a theory from which atomic phenomena could be calculated quantitatively, accurately and in detail. Feynman s Lectures, Vol III, page F. Quantum Electrodynamics (QED) Reference: The following description is taken from Feynman, QED, The strange theory of light and matter, Princeton University Press (1985). Quantum mechanics was a tremendous success (could explain chemistry). However, the description of light-matter interaction still faced difficulties. [Maxwell theory of electricity and magnetism had to be changed to be in accord with the new properties of quantum mechanics. The theory of light-matter interaction, called quantum electrodynamics was finally developed in 199]. But the theory was troubled.

8 Right after Schrodinger, Dirac developed a relativistic theory of the electron that did not take into account the effects of the electron s interaction with light. But it was expected to provide a good starting point. Quantum electrodynamics was straightened out by Julian Schwinger, Sin-Itiro Tomonaga and Feynman in This is the theory that Feynman describes in his QED, The strange theory of light and matter, Princeton University Press (1985). Such theory has been tested over a wide range on conditions. Aside from gravitation and nuclear physics, QED can explain every phenomenon accurately. QED is also the prototype for new theories that attempt to explain things going on inside the nuclei of atoms. It turns out, quarks, gluons,, etc. all behave in a certain style, the quantum style. F1. QED analysis of light phenomena Photons: Particles of Light QED considers light to be made of particles (as Newton originally thought), but the price of this great advancement of science is a retreat by physics to the position of being able to calculate only the probabilities that a photon will hit a detector. Event: Light travels from the source S, reflects from the surface at X, and arrives to the detector at D We assign an amplitude probability (a complex number) that such an event will occur. n=1 D S 1 m x X Fig. Light reflected from a mirror QED RULE-1 (Assignment of amplitude probability) How to do such an amplitude probability assignment? i) Pictorial view We assign to the amplitude probability an arrow. To obtain the arrow we do the following: Let s imagine that we have a stopwatch that can time a photon as it

9 moves. It has a handle that moves rapidly. When the photon leaves the source, we start the watch. As long as the photon moves, the handle turns. When the photon arrives at the detector we stop the watch. The handle ends up pointing in a certain direction. We then draw a corresponding arrow of magnitude 1 (blue arrow in the figure). ii) More formal view Amplitude probability (photon starts at S and arrives at D via the path S X P) A(SXP) = e i tx) where is the angular frequency of the light and t(x) is the time the light takes to travel via SXP. (1) That is, a phasor e i tx) probability A. (a complex number) is assigned to the amplitude QED RULE- (For events that have the same initial and final states) Since the photon has many optional paths available to go P from X, the total amplitude probability is given by, Total amplitude probability A (photon starts at S and arrives at P via any path joining S and P) all X A(SXP) = X e i tx) () where is the angular frequency of the light and t(x) is the time light takes to travel from S to P passing through X (the latter located at the interface). QED RULE-3 The probability for an event to occur is given by the square of the final amplitude: P = A (3) Consider the reflection of light coming from a surce S and reaching a detector via reflection from a mirror. We want to calculate the chance that the detector will make (4) a click after a photon has been emitted by the source

10 Classical view: The mirror will reflect light where the angle of incidence is equal to the angle of reflection Classical way: In the graph above, it would appear that the ends of the mirror contribute to nothing to the reflection phenomena. QED view: Every possible path contributes to the amplitude probability. Should the reflections path involving the center of the mirror have more weight than the once reflecting from the edges? Answer: No. All the path have equal chance. QED view: QED view assigns an equal amplitude probability to each possible path

11 For the analysis of a mirror, an infinite number of path would have to be considered. To simplify the problem, let s divide the mirror into a number of smaller discrete strips. S P Each amplitude will be represented by an arrow of a standard (arbitrary) size While the size of the arrow will be essentially the same, its orientation will be different for the different reflection point selected. This is because it takes a different time for as photon to go through a differente path that have different length.

12 C D A B E F G H I L J M K Top: Figure shows each possible path the photon could take to go from the source to that point in the mirror and then to the detector. Middle: A plot of the corresponding time for each possible path. Below the graph is the direction of each amplitude probability (arrow). Bottom: The result of adding all the arrows. Notice the major contribution to the total arrow comes from arrows E through I, whose directions are nearly the same because the timing of their path is nearly the same. This also happens to be where the total time is least. It is therefore approximately right to say that light goes where the time is least.

13 Why do the edges appear to make no contribution? We zoom in to see in more detail the contribution to the total amplitude probability from the edges of the mirror. Notice when the arrows are added, they go in a circle, hence adding up nearly to nothing. The above gives us a clue to engineer a way to get contribution from th e edges. As we move from left to right, we notice the arrows have a bias orientation to the right then to the left, and so on. If only the section with arrows biased to the right are kept (etching away the sections with arrows to the left), then a substantial amount of light will be reflected. Such a mirror with preferentially etched regions is called a diffraction grating.

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