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: AmnNm + BmnNmr = BnmNnr.
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:
Nn pn em– en ——— — = — e . Nm pm k T
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: em– en = 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:
Amn 8πhv3 — = ——— .
Bmn c3
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
Einstein’s 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.
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 validation
Experimental verification of Einstein’s
concept of stimulated emission came
decades after Einstein theoretically predicted it. In 1954, Charles Townes and
Arthur Schawlow invented 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 con-
cept 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 inver-
sion and thus achieve laser action.
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. t
Barry R. Masters ( brmail2001@yahoo.com) 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.
Member
[ 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).
