# Exactly 100 years ago, Albert Einstein came up with the idea of stimulated emission, also known as the laser principle.

I train broadband technicians about the laser, and it hit me that it’s exactly 100 years since Einstein came up with the idea of stimulated emission. He wrote in a letter to his friend Besso on November the 18th, 1916: "A splendid light has dawned on me about the absorption and emission of radiation". Einstein's reasoning can be found in three papers, two from 1916 and the third from early 1917 Verh. Deutsch. Phys. Ges. 18, 318 (1916), Mitt. Phys. Ges. Zurich 16, 47 (1916) and Phys. Zeitschr. 18, 121 (1917), according to the scientific Einstein biography "Subtle is the Lord ..." (pg. 405) by Abraham Pais, a physicist who collaborated with Einstein in the United States. The title is part of an Einstein quote: "Raffiniert ist der Herr Gott, aber boshaft ist er nicht." (“Subtle is the Lord, but malicious He is not.”) The paper in Physikalische Zeitschrift 18, 121-128 (1917) with the title "Zur Quantentheorie der Strahlung" (“The Quantum Theory of Radiation”) was received the 3^{rd} of March 1917.

Th. Maiman succeeded in making the first laser, a ruby laser, in 1960. I remember the first laser I saw in 1965 when I went to high school in Östersund. A new teacher came from FOI (the Swedish Defence Research Agency) with a ruby laser that shot holes in an inflated balloon. He told us that the FOI had a powerful laser that could burst a balloon right across a hall, which military visitors greatly appreciated! The laser nearly was "a solution that searched for its problem." Now there are lasers everywhere and I've worked at Ericsson with semiconductor lasers for fiber optic communication (the fiber idea is 50 years old, it was published in 1966 by Kao, who won the Nobel Prize in 2009). Without the laser, there would be no Internet, just to mention one of the most important uses that is not so visible. It all began 100 years ago with Einstein’s idea. It is one of his ideas with the most technical applications. We also use the principle of Einstein's theory of light absorption, the photoelectric effect (for which he received the Nobel Prize), in light detectors in receivers in fiber optics. He proposed a "simplest picture" for the photoelectric effect: a light-quantum gives all its energy to a single electron (Pais pg. 380). So, it is Einstein's ideas at both ends of a fiber cable!

There are lots of pictures of stimulated emission when you google it. The normal energy level diagrams are abstract for the students. A picture of atoms is more concrete. The fiber amplifier uses the stimulated emission in erbium atoms, which are doped into the fiber core.

Although a 100th anniversary is not so newsworthy, the combination of Einstein and the laser is something that people would like to know more about. In Walter Isaacson's great biography "Einstein" from 2007 is a quotation from the letter to Michele Besso: "A brilliant idea struck me regarding the emission and absorption of radiation. You will find it interesting. An amazingly simple derivation, I would say the *right* derivation of Planck's formula. A fully quantized thing.” But it’s dated the 11^{th} of August 1916, (and the 18^{th} of November in Pais' book). Professor Huldt, my professor at KTH used the derivation, with Einstein’s A and B coefficients, in the semiconductor physics course, because it is so 'educationally valuable’, as he said, and it is impressively short and elegant, just a few lines long.

Einstein (and physics) had one of their most creative periods from the autumn of 1915 to the spring of 1917, when he succeeded in formulating both general relativity (the "human mind's greatest achievement concerning Nature, the most amazing combination of philosophical depth, physical intuition and mathematical skill "(according to Max Born, Nobel Prize 1954), Isaacson pg. 246), predicted gravitational waves in 1916 and came upon this stimulated emission. Furthermore, he realized that light quanta have momentum. In February 1917, he got a new idea of a four-dimensional universe that curves back into itself like a four-dimensional sphere. ("This assumption of a finite yet boundless space is one of the most fantastic ideas about the nature of the universe ever conceived" (Max Born) Isaacson pg. 275). He came up with all of this even though he was overworked, had abdominal pain, severe problems with his wife and longed for his two boys in Switzerland.

"In five months, Einstein in 1905 turned physics upside down with his ideas about light quanta, special relativity, and statistical methods to demonstrate that atoms exist. Now he had just finished a more protracted creative work, from the autumn of 1915 to the spring of 1917, described by Dennis Overbye (science journalist) as 'possibly the most stunning example of continuous brilliance from a single person in the entire history of physics'. ... From the smallest possible things, quanta, to the largest, the cosmos itself, Einstein had proved to be a champion. "(Isaacson pg. 277). After the work on Bose-Einstein statistics, in the mid-1920s, his creativity slowed down.

This was said to set the stimulated emission in context within the history of science. Now when we can remove unwanted tattoos with lasers, ordinary people are aware of the importance and usefulness of laser technology. The photoelectric effect (Einstein’s only revolutionary idea, according to Pais) is also one of the most frequently used, for example in all digital mobile cameras.

Einstein's reasoning is presented like this in A. Pais' scientific biography "Subtle is the Lord ... (I have shortened this a little):

Einstein's method is based on general hypotheses about the interaction between matter and radiation. He considered a system in thermal equilibrium. *N _{m}* and

*N*are the number of atoms in the energy levels

_{n}*E*and

_{m}*E*where

_{n}*E*>

_{m}*E*and have a distribution:

_{n}

*N _{m}* =

*q*exp (–

_{m}*E*/

_{m }*kT*) where

*q*is a weight factor.

_{m}

Einstein's new hypothesis is that the total number of transitions *dW* per time interval *dt* is given as:

*dW _{mn}= N_{m}*

*(*

_{ }*A*+

_{mn}*pB*)

_{mn}*dt*for downwards transitions,

*m*to

*n*

*dW _{nm}= N_{n}*

_{ }*pB*

_{nm}*dt*for upwards transitions,

*n*to

*m*

where the *A _{mn}* -coefficient corresponds to spontaneous transitions from

*m*to

*n*, which occurs with a probability that is independent of the radiation density

*p*, while the

*B*-coefficients correspond to stimulated absorption

*B*and stimulated emission

_{nm}*B*.

_{nm}Microscopic reversibility implies that:

*N _{n}*

_{ }*pB*=

_{nm}*N*

_{m}*(*

_{ }*A*+

_{mn}*pB*)

_{mn}i.e.

The number of transitions of stimulated absorption is equal to the number of transitions of spontaneous emission and stimulated emission.

Now one shall solve the radiation density *p* and get first:

*A _{mn}*

*q*=

_{m}*p*[

*B*

_{nm}*q*exp[(

_{n}*E*–

_{m}*E*)/

_{n}*kT*] –

*B*

_{mn}

*q*]

_{m}

Note that the second term on the right corresponds to stimulated emission. The term is necessary to get Planck's radiation law. If there was no stimulated emission you would get Wien’s radiation law, which is wrong. If the temperature *T* rises to high values you get on the other hand Rayleigh-Jeans’ radiation law, which means that: *B _{nm}*

*q*=

_{n}*B*

_{mn}*q*which finally gives:

_{m}*p* = (*A _{mn}*/

*B*) / [exp[(

_{mn}*E*–

_{m}*E*)/

_{n}*kT*] – 1]

The end of the derivation in Pais’ book.

Abracadabra, we have received Planck's radiation law! The term –1 in the denominator gives Planck's radiation law and it comes from the term for the stimulated emission. I still remember my surprise when professor Huldt showed this short derivation in 1970 and I couldn’t help but ask how it was that Einstein made these assumptions. He could only reply with "it's because it agrees". Genius and flashes of inspiration are not easy to explain ...

Einstein was unaware that 40 years later the idea of the stimulated emission would lead to the invention of the laser (which produced many Nobel Prizes) and 100 years later it would have lots of applications. He was "just" happy about the new simple derivation of the correct radiation law. In addition, he noted that for the above expressions for the number of transitions *dW* to lead to Planck's radiation law, it is necessary that the transitions between *m* and *n* are accompanied by a single monochromatic radiation quantum: *E _{m}* –

*E*

_{n}_{ }=

*hf*. By this reasoning, Einstein established a bridge between blackbody radiation and Bohr's theory of spectra!

Einstein wrote about the assumptions for this derivation: "The simplicity of the hypotheses makes it seem probable to me that these will become the basis of the future theoretical description." That turned out to be true. His papers contained another result, one which Einstein himself considered far more important than his derivation of the radiation law, light-quanta carry a momentum: *hf */*c.*

For physicists all of this is, of course, impressive, with simple ideas and derivations that led to great results, and it should be highlighted to the public. The equations fit more in a physics journal, but the equilibrium equation describes the new physics in the derivation, and expressed with words:

the number of transitions of stimulated absorption = the number of transitions of spontaneous emission + stimulated emission along with pictures of the transitions, this would perhaps be accessible and usable in a popular scientific poster. The subject is fascinating, and gives an insight into the mind of a genius.

Energy diagram for stimulated emission from Wikipedia:

About Bernt Sundström

Born in Hammerdal, Jämtland, Sweden in 1946. M.Sc. in Engineering Physics, Ph.D. in Solid State Electron Physics, KTH. Development Engineer for Fiber optics and lasers at Ericsson for 20 years. M.Ed. Qualified Upper Secondary School Teacher in Maths, Physics and Technology. Researcher and Associate Professor at Telecom Systems Lab, KTH Royal Institute of Technology. Fiber optics and lasers Teacher, at ICT Education, Hudiksvall.