Albert Einstein during a lecture in Vienna in 1921.
Albert Einstein was one of the foremost scientists in a century dominated by science. His work influenced our understanding of the physics of the universe, modern electronics and the quantum nature of reality. Perhaps it was his boundless imagination that enabled him to have such a broad reach.
Indeed, throughout his life and career, Einstein pursued independent thought. In 1901, he wrote: “German worship for authority … is the greatest enemy of truth.” In Einstein’s Autobiographical Notes, which he wrote at age 67, he noted his “suspicion against every kind of authority…and [that] attitude which has never left me.” This theme is recounted in another Einstein quote from 1925: “The pursuit of knowledge for its own sake, an almost fanatical love of justice, and the desire for personal independence—these are the features of the Jewish tradition which makes me thank the stars that I belong to it.”
While it was Einstein’s relativity theory that put him on the world’s stage, he made several critical contributions to the field of optics. Between 1905 and 1916, he put forth the light quantum hypothesis, demonstrated the wave-particle duality of light, and proposed the concept of stimulated emission—an idea that took 40 years to be proven in the form of the laser.
Einstein’s light quantum and the photoelectric effect
In his 1905 publication, “On a Heuristic Viewpoint Concerning the Production and Transformation of Light,” Einstein used Boltzmann’s statistics to postulate that the entropy of blackbody radiation can be considered as a gas of independent “quanta of energy,” with each quantum proportional to the frequency of the corresponding wave. His concept of discontinuous energy in space contradicted Maxwell’s widely accepted continuous wave theory.
Antecedents to Einstein’s work on light-matter interactions
In 1887, Heinrich Hertz observed that ultraviolet light incident on his spark-gap resonators enhanced their capacity to discharge. His assistant Wilhelm Hallwacks confirmed and expanded on this hypothesis in 1888, when he demonstrated that ultraviolet radiation caused neutral metals to acquire a positive charge.
In 1899, Joseph J. Thomson at Cambridge University investigated the effect of ultraviolet radiation on the production of “corpuscles” [electrons] from a metal plate in a Crookes tube. He measured a current from the plate that increased with the frequency and intensity of the radiation. He was the first to write that the ultraviolet-induced photoeffect results in the emission of electrons.
Meanwhile, at the University of Kiel, Philipp Lenard demonstrated in 1902 that if radiation of sufficiently short wavelength (from a carbon arc lamp) is incident on a metal’s surface, then electrons are emitted. The number of electrons ejected increases with light intensity, and below a specific frequency of radiation no electrons are emitted. The maximum kinetic energy of the emitted electrons is independent of the intensity of the incident radiation over an intensity range of 1,000 and rises with increasing frequency of the radiation.
Einstein set the energy of each light quantum in modern notation as equal to hv, where h is Planck’s constant and v is the frequency of the light. To quote Einstein: “Monochromatic radiation of low density (within the range of validity of Wien’s blackbody radiation formula [valid for hv/kT »1]) behaves, in a thermodynamic sense, as if it consisted of mutually independent radiation quanta of magnitude [hv].”
He wrote: “When a light ray spreads out from a point source, the energy is not distributed continuously over an increasing volume [wave theory of light], but consists of a finite number of energy quanta that are localized at points in space, move without dividing, and can only be absorbed or generated as complete units.” Einstein used the term light quantum (ein Lichtquant), which is an indivisible packet, to describe this.
However, it was the American physical chemist Gilbert N. Lewis who coined the term “photon” in a paper published in the journal Nature in 1926. Only in 1916 did Einstein discuss the momentum, p=hv/c and the zero rest mass of the light quantum.
With Einstein’s light quantum hypothesis, he explained the photoelectric effect in a way that the prevailing wave theory of light could not. In the photoelectric effect, electrons are ejected from the surface of a metal when radiation of a threshold frequency is incident. Einstein then continued: “If monochromatic radiation behaves … as though the radiation were a discontinuous medium consisting of energy quanta of magnitude hv, then it seems reasonable to investigate whether the laws governing the emission and transformation of light are also constructed as if light consisted of such energy quanta. Einstein assumed that light interacts with matter by the emission or absorption of his posited light quantum.
He used his light quantum hypothesis to postulate a new mechanism: Light quanta penetrate the surface layer of matter, and their energy is converted into the kinetic energy of the electrons. In the simplest case, a light quantum transfers its entire energy to a single electron. He assumed an electron in the interior will decrease its kinetic energy when it reaches the material surface. Then it must perform work, φ (a function of each material, called the work function), to overcome the attractive forces holding it in the material in order to leave the surface; the maximum kinetic energy of such electrons is hv – φ.
In modern notation: eV = hv – φ, where e is the charge of the electron and V is the retarding potential necessary to stop the fastest photoelectrons. The work function is the electron binding energy to the material; it varies with different materials. This is the first equation in the quantum theory of radiation-matter interactions. In 1912, Arthur L. Hughes who worked at the Cavendish Laboratory in Cambridge measured the maximum velocity of photoelectrons from various metals and verified Einstein’s photoelectric equation. A modern application of the photoelectric effect is the ubiquitous photomultiplier tube, which is a low-light detector.
The first confirmation of the quantum hypothesis occurred in 1907 when Einstein showed that energy quantization can be applied to materials (condensed matter). In addition, he explained the anomalous temperature dependence of the specific heats of solids (i.e., that they decreased with reducing temperature) by assuming that they consist of a lattice of quantized oscillators. Einstein’s formula was in good agreement with the experimental results provided in 1910 by Walther Nernst and his assistant Frederick A. Lindemann. This was the first confirmation of the quantum hypothesis in an area of physics other than radiation.
Einstein’s 1922 Nobel Prize in Physics cited his 1905 paper on the photoelectric effect “for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect.” But in 1905, many physicists—including Max Planck, Hendrik A. Lorentz, Max von Laue, Wilhelm Wien and Arnold Sommerfeld—doubted Einstein’s hypothesis. They based their objections on the interference of light, which required a wave interpretation.
Both Planck and Lorentz accepted that radiation interacts with matter in a quantized manner, but they could not conceive of individual light quanta that propagated as a wave. Interestingly, in 1909, Johannes Stark proposed localized energy quanta in X-rays, and he also supported Einstein’s light quantum hypothesis. Einstein at the 1911 Solvay congress stated: “I insist on the provisional character of this concept [light-quanta].”
In 1916, Millikan’s experiments validated Einstein’s equation for the photoelectric effect; nevertheless, he did not accept the hypothesis of light quanta. The physics community did not completely embrace Einstein’s theory until Compton published his paper on the Compton effect in 1923.
Einstein’s theory of wave-particle duality
The origins of the wave-particle duality of light are found in Einstein’s 1909 paper on energy fluctuations. Einstein built on his theoretical methods to calculate fluctuations of energy and momentum with respect to his analysis of Brownian motion (1905), and he applied these methods to blackbody radiation. He boldly generalized his 1905 fluctuation theory of mechanical systems (Brownian particles) to nonmechanical blackbody radiation. These insights led in 1909 to Einstein’s conception of light with both particle and wave attributes.
He investigated the energy fluctuations of blackbody radiation that was contained in partial volume V of an isothermal cavity at temperature T. Starting with Planck’s blackbody distribution law, Einstein wrote the variance of the energy fluctuations as:
where 〈 〉 represents the statistical average, E is the radiation energy of frequency between v and v + dv, and c is the vacuum velocity of light. This is known as Einstein’s fluctuation formula for blackbody radiation. He reasoned from statistical mechanical analysis that the first term after the equals sign referred to the quantum properties of the radiation. This term, linear in average energy, is found in the high frequency limit in which Wien’s law is valid. Einstein concluded that radiation, in particular its energy fluctuations, is consistent with a gas of independent particles—i.e., light quanta.
He reasoned from dimensional analysis that the second term, quadratic in average energy, is from the interference of waves. This term is obtained in the limit of low frequency radiation. In 1916, H.A. Lorentz rigorously derived the second term. Einstein’s equation showed that both quanta and waves occur in blackbody radiation; this later became the modern concept of “wave-particle duality.” In 1909, Einstein wrote “…the next phase in theoretical physics will bring us to a theory that light can be interpreted as a kind of fusion of the wave and emission theory…”
Arthur Compton (left) with graduate student Luis Alvarez in 1933.
Experiments validate the quantum nature of radiation
In 1916, Robert A. Millikan experimentally verified Einstein’s photoelectric theory with high precision, extending the previous experiments of Lenard. He showed that the maximum kinetic energy of the emitted electrons is proportional to frequency. His plots of the stopping voltage for photoemission vs. the frequency of the incident radiation followed Einstein’s predicted linear dependence, and for different metals, the values of “h” were in agreement and equal to the numerical value that Planck calculated in his 1901 paper. He also showed that the number of photoelectrons is proportional to the intensity of the radiation.
Arthur Holly Compton investigated the scattering of X-rays and γ-rays by light elements. His theory of 1923, which used Einstein’s light quantum hypothesis, showed that the energy of the scattered quantum is less than that of the incident quantum, and the difference is the increased kinetic energy of the recoiled scattering electron. Compton derived an equation that related the increased wavelength of the scattered beam to the angle between the incident and scattered beam.
He assumed that each electron scatters a complete quantum of radiation. An X-ray quantum of frequency v0 is scattered by an electron of mass m. The scattering electron is assumed to be initially at rest; after the collision with the quantum of radiation, the electron recoils. The energy of each ray is hv0, and the momentum of the incident ray is hv0/c, where c is the velocity of light, and h is Planck’s constant, and that of the scattered ray is hvθ/c as an angle θ with the initial momentum.
He then invoked the conservation of energy and momentum for the scattering process, and derived his scattering equation in terms of wavelength:
Compton validated his theory with a series of precise measurements and wrote: “The beautiful agreement between the theoretical and experimental values of the scattering is more striking … There is not a single adjustable constant connecting the two sets of values.” He found that the increase of wavelength is independent of the wavelength. Compton then concluded: “The scattering of X-rays is a quantum phenomenon.” Furthermore,
“… the theory indicates very convincingly that a radiation quantum carries with it directed momentum as well as energy.” The quantity h/mc is called the Compton wavelength and has a value ≈ 0.0241 Å.
For his discovery, which soon after came to be known as the Compton effect, he shared the 1927 Nobel Prize in Physics with Charles Thompson Rees Wilson, who invented the cloud chamber method of visualizing ions and the tracks of ionizing particles.
Nine years after Einstein conceived of the light quantum, he returned to the problem of light-matter interaction; specifically, the transitions between energy states of molecules and the role of light quanta in these processes. In 1916, he postulated stimulated or induced emission—which is, of course, the basis of laser function.
In induced emission, a molecule in an excited state interacts with an electromagnetic field, causing the molecule to transition to the lower energy state; its energy is transferred to the field. The photon created in the process is identical in frequency, phase, polarization and direction as the photons in the incident field. Typically, at thermal equilibrium, the rate of light absorption is greater than that of stimulated emission, since lower energy states of the atom are more populated than higher energy states.
But for the special case of a laser, a population inversion is achieved, in which the number of higher energy states is greater than that of lower energy states; in that instance, the rate of induced emission is greater than that of absorption. This condition permits optical amplification, a prerequisite for laser function.
How did Einstein derive induced emission? Einstein’s 1916 publication, “Emission and absorption of radiation in quantum theory,” contained his so-called “Einstein coefficients,” and his prediction of the process of induced emission. This paper introduced a probabilistic approach to quantum physics. At that time, the concept of energy transitions in atoms mediated by the absorption and the emission of light quanta was not commonly accepted by the physics community. Bohr’s theory of the hydrogen atom did not utilize the idea of the photon; in fact, Bohr rejected that concept until the early 1920s.
Einstein assumed a collection of molecules in thermal equilibrium; they can absorb radiation of frequency vnm and go from state Zn to state Zm, and they can emit radiation of frequency vmn and go from state Zm to state Zn. He assumed a process with upper and lower energy states, and transitions between the two. The number of molecules in the upper and the lower states are constant. At thermal equilibrium, the same number of molecules per unit time will absorb radiation as the number that will emit radiation.
He then distinguished between two types of transitions. The first is when the emission of radiation occurs in the absence of external influences. He made the analogy to Rutherford’s law of radioactive decay. In this instance, the number of transitions per unit time is: AmnNm, where Amn is a constant characteristic of the combination of states Zm and Zn, and Nm is the number of molecules in the state Zm. In modern terms, this is the process of spontaneous emission, which occurs when no external radiation is present, and Amn is called the “Einstein A coefficient.” The resulting photon can be emitted in any direction.
The second transition is due to the interaction between the molecules and incident radiation; in modern terms, it refers to the processes of absorbing radiation and the stimulation of radiation. Einstein assumed that the effect of the incident radiation is proportional to the radiation density ρ (the spectral density or the radiation energy per unit volume in the frequency range between v and v+dv) or the effective frequency to drive the transition between states.
The radiation field can cause a loss or gain of energy in the molecules. While the gain of energy of a molecule (absorption) is intuitive, the process by which radiation interacts with a molecule in the excited state and induces a transition to the ground state is not. The rate of stimulated emission is proportional to the radiation energy density.
Derivation of the Einstein coefficients
Einstein began by asking: How can the principle of microscopic reversibility—which states that the rate of energy transfer from lower to upper states must equal that from upper to lower states for every transfer—be applied to radiation in equilibrium?
In stimulated emission, the emitted photon has the same frequency, direction and polarization as the incident one. Einstein required that the process of stimulated emission occur in order for the energy levels of a molecule in equilibrium with the radiation field to be given by the Boltzmann distribution and to be consistent with the Planck blackbody distribution law.
If the Einstein coefficient for stimulated emission was zero, there would not be a Boltzmann distribution of states at thermal equilibrium. The number of transitions Zn → Zm per unit time is Bmn Nnρ, and the number from Zm → Zn is Bmn Nmρ, where Bnm (stimulated emission) and Bmn (stimulated absorption) are constants related to the combinations of states Zn and Zm. In modern terms, these are called the Einstein B coefficients. At thermal equilibrium, the number of molecules gaining energy must equal the number losing energy; the following equation describes all of these processes: Amn Nm + Bmn Nmρ = Bnm Nnρ.
Einstein remarked on the simplicity of the hypotheses and the generality of the analysis. He invoked the “Boltzmann principle:” The probability Wn of state Zn is given by: Wn = pne–En/kT, where En is the energy of state n, k is the Boltzmann constant, pn is the statistical weight of the state Zn, and T is the absolute temperature. This gives the numbers of molecules in the two states in terms of their energy difference:
He deduced that the probabilities of induced absorption and emission are equal: pnBnm= pmBmn. The Einstein coefficients are independent of radiation density. Einstein used these equations to derive Planck’s blackbody distribution law; he noted that the transitions between the two states are mediated by a light quantum of definite frequency v, thus he derived the equation: εm– εn = hv, where h is a constant (Planck’s constant). This equation, which related the energies of two atomic states and the energy of the absorbed or emitted radiation, is the Bohr frequency condition, which previously was only assumed by Bohr.
Einstein showed that the ratio of Amn/Bmn is proportional to v3 and given by:
There are several physical conclusions that follow from his derivation. From this cubed dependence on frequency, we conclude that the larger the energy difference between the two states, the higher the probability for spontaneous emission as compared to stimulated emission.
Assuming thermal equilibrium with the radiation, if hv » kT, then spontaneous emission is much more probable than stimulated emission. Conversely, if hv « kT, then stimulated emission can predominate; for the visible spectrum, this naturally occurs in stars.
Einstein suggested that the Einstein coefficients Amn and Bmn could be calculated if a new version of electrodynamics and mechanics was available that is in agreement with the quantum hypothesis (a new quantum mechanics). In 1927, Dirac used his version of quantum mechanics to give an expression to Einstein’s B coefficients, and in the second paper derive the expression for Einstein’s A coefficient (spontaneous emission).
Experimental verification of Einstein’s concept of stimulated emission came decades after Einstein theoretically predicted it. In 1954, Gordon, Zeıger and Townes ınvented the maser (microwave amplification by stimulated emission of radiation), which operated in the microwave region. And, in 1960, Theodore H. Maiman produced stimulated emission in a ruby crystal as part of the first laser.
Credit is also due to Gordon Gould, who while a graduate student in physics at Columbia University developed the concept of an optical resonator in the form of a Fabry-Pérot interferometer constructed with two mirrors. Gould discussed his work with Townes. Several months later, Townes and Schawlow independently developed the same concept; they called their device the “optical maser.” Gould, who obtained patents on his invention, suggested that the gain media between the mirrors could be pumped (by atomic collisions) to achieve a population inversion and thus achieve laser action.
The 1964 Nobel Prize in Physics was shared by Charles H. Townes, Nicolay G. Basov and Aleksandr M. Prokhorov for their independent work on the laser.
Einstein’s legacy in light
Einstein’s seminal works in optics have transformed our understanding of light as well as our ability to harness it for applications in a broad range of areas, including medicine, telecommunications, electronics, and beyond. His work on stimulated emission contributed to the development of the laser—one of the most important scientific breakthroughs of the past century. His explorations into light-matter interactions in the early 1900s established one of the most shocking ideas in 20th century physics—that we live in a quantum world in which light can behave as both a particle and a wave. His works revealed a universe that is more amazing and complex than anyone could have imagined.
Barry R. Masters is a Fellow of AAAS, OSA and SPIE. He is a visiting scientist with the department of biological engineering, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.
References and Resources
>> A. Einstein. “On a Heuristic Viewpoint Concerning the Production and Transformation of Light,” Annalen der Physik 17, 132 (1905).
>> A. Einstein. “On the Present Status of the Radiation Problem,” Physikalische Zeitschrift 10, 185 (1909).
>> A. Einstein. “Emission and Absorption of Radiation in Quantum Theory,” Deutsche Physikalische Gesellschaft. Verhandlungen 18, 318 (1916).
>> A. Einstein. “On the Quantum Theory of Radiation,” Physikalische Zeitschrift 18, 121 (1916).
>> A. Einstein. Out of my Later Years, Philosophical Library, N.Y. (1950).
>> R.H. Stuewer. The Compton Effect, Turning Point in Physics, Science History Publications, N.Y. (1975).
>> A. Pais. ‘Subtle is the Lord…’ The Science and the Life of Albert Einstein, University Press, N.Y. (1982).
>> Z. Rosenkranz. The Einstein Scrapbook, The Johns Hopkins University Press, Baltimore (2002).
>> J. Stachel. Einstein from ‘B’ to ‘Z’, Birkhäuser, Boston (2002).