Townes' contribution to nonlinear optics

In honour of the Fiftieth Anniversary of the Nobel Prize in Physics, this talk introduced the contributions of Nicholas Basov and Alexei Prokhorov, who shared the prize with Charles Townes. The talk then detailed the quantum electronics research of Townes, particularly at MIT, which was related to nonlinear optics. The years from 1961 to 1968 were particularly exciting, as the ruby laser enabled a wide variety of new physics to be discovered and explored.


Introduction
The 1964 Nobel Prize in Physics was awarded "for the invention of the maser and the laser" to Charles Hard Townes, Nicolai Basov and Aleksandr Prokhorov (figure 1). At the ceremony it was stated, "Drs. Townes, Basov and Prokhorov: By your ingenious studies of fundamental aspects of the interaction between matter and radiation you have made atoms work for us in a new and most remarkable way." [1] The Nobel Committee went on to say, "The first papers about the maser were published ten years ago as a result of investigations carried out simultaneously and independently by Townes and co-workers at Columbia University in New York and by Basov and Prokhorov at the Lebedev Institute in Moscow. …Masers work as extremely sensitive receivers for short radiowaves. They are of great importance in radio astronomy and are being used in space research for recording the radio signals from satellites." The term "quantum" ties the field to physics and "electronics" ties the field to electrical engineering. The USSR researchers were motivated by the development of synchrotron sources for spectroscopy. Townes was motivated by his work on radar sources during World War II. The state first to analyse theoretically the oscillation condition. Their numerical evaluation of a potassium vapour "optical maser" found the idea to be feasible (although it turned out to be experimentally very difficult).
The Schawlow-Townes analysis excited the quantum electronics community and the race was on to demonstrate a working laser. Their ground-breaking paper defined the laser as an inverted population of atoms/molecules within a large highly-multimode optical cavity that amplifies spontaneous emission noise to generate an output in the infrared or visible. For a directional output, they specified two parallel mirrors (the Fabry-Perot geometry). Townes called this an "optical maser" to affirm that the maser concept at optical frequencies is not different, in principle, from the microwave system. This time period (mid-1950's) was during the Cold War, a mutual military build-up between the USA and the USSR. The US military asked Townes for technical advice about the future of the new field of quantum electronics. He agreed to chair two important committees: one was "to create interest in mm waves" and the other was "to urge continued support for research in the infrared." These commitments were unpaid and after-hours, but he took them very seriously. Thus he was solicited to take a leave of absence from Columbia as Vice President and Director of Research for the Institute for Defense Analysis in Washington D. C., a position he held from 1959-1961. As he said in an oral interview 25 years later, "I felt that there just were not enough good scientists in Washington, and we had a pressing problem with the Russian missiles and other things coming on, and it was just a part of my duty." [16] In the fall of 1961, Townes accepted an offer to become Provost at Massachusetts Institute of Technology, a powerful position that reports directly to the President. Research in lasers became his after-hours unpaid activity. He took me on as his first PhD student; six months later he added Raymond Chiao. We were his only graduate students, but we had an active group that included post-docs and visiting scientists. During this time Townes also became Chairman of the Advisory Committee to NASA's Project Apollo (the man-to-the-moon mission). His responsibility was to secure support from the larger scientific community and to ensure that the moon flight would yield maximum benefits in scientific research. This was a highly political committee that put him in conflict with some at MIT and he was denied the open position of MIT president. So in 1967 he moved to the University of California in Berkeley to search for stimulated emission in astronomy.
Why did Townes leave laser research? He could see that the quantum electronics field was brimming over with eager and talented scientists/engineers. He told me that he wanted to work in unexplored research areas, where he thought his contribution would be greatest. Stimulated emission in astronomy was a wide-open field, eagerly awaiting his arrival. Indeed, he has since made many significant contributions to astronomy (outlined later).
The purpose of this paper is to focus on Townes' contributions to nonlinear optics (NLO), which mostly resulted from the years he was at MIT: 1961-1967.

Nonlinear Optics at MIT
It pays to be brilliant and active as a field is just beginning. In the space of 6 years Townes, along with his colleagues: 1) Proved that coherent molecular vibrations drive higher order stimulated Raman scattering (SRS) emission; 2) Identified SRS emission from individual filaments in Ramanactive liquids; 3) Introduced stimulated Brillouin scattering (SBS) and observed it in crystals and liquids; 4) Introduced and theoretically analysed spatial solitons (SS), at that time called selftrapped optical beams; 5) Experimentally demonstrated self-focusing filament-formation in Kerr liquids; 6) Demonstrated small-scale filamentary beam break-up; 7) Introduced the concept of self-phase modulation (SPM) and 8) Introduced the concept of weak-wave retardation and gain in four-wave mixing (FWM).
It was my job to set up the Townes laser laboratory at MIT with the second commercially sold ruby laser (from a company called Trion -the first went to Peter Franken at University of Michigan). The first year was spent just getting the laser to work -the ruby had to be cooled to liquid nitrogen temperatures while excited by a flash lamp operating at kilovolts. Electrical shorts from condensation on the Dewar were a major problem! We evaporated a silver mirror on one end of the ruby rod and a partially silvered mirror on the other. Technological progress toward Qswitching was rapid: our Q-switch was a retro-reflecting prism mounted on a high-speed air-driven turbine (developed at Lincoln Laboratories); the ruby crystal was put into an external cavity; and dielectric mirrors replaced silver (which blew off with the high-power Q-switch). Now we were ready for high power experiments, which are described here.

Coherent Molecular Vibrations in Stimulated Raman Scattering
Townes' first contribution was to describe the stimulated Raman scattering process as introducing coherent molecular oscillations [17]. He had been inspired to delve into nonlinear optics at the Third Quantum Electronics Conference in Paris in Feb. 1963. He had heard Zeiger and Tannenwald's theoretical paper on optical phonons. He had heard that experiments on stimulated Raman scattering at Hughes Research Laboratories demonstrated Stokes frequencies shifted by up to three orders of vibrational frequency. He also heard about the intense anti-Stokes emission seen by both Terhune and Stoicheff. Because Stoicheff became a visitor at MIT, we had early access to his data. All this led Townes to realize that coherent laser light could drive coherent optical phonons (which we called molecular oscillations).
The Raman process was modelled as an ensemble of simple vibrating diatomic molecules with natural resonance frequency Ω r [18]. Townes' innovation was to realize that coherent light interacting with matter transferred that phase coherence to the molecules. Periodic vibrations introduced into the medium by scattering into Stokes frequencies could be transferred back to the light wave through a coherent process, generating anti-Stokes light. This was a classic resonant parametric process.
The anti-Stokes could be explained by the requirement of phase coherence over the interacting length, a process called phase-matching, first described in second harmonic generation. Molecular vibration driven by Stokes generation from a laser photon has a wave vector . A second laser photon scatters off to produce anti-Stokes. Phase-matching means anti-Stokes generation is conical at an angle given by closing the wave-vector diagram (figure 2); i.e. conservation of momentum.
The theoretical paper was followed by an experiment with a Q-switched ruby laser transmitted through calcite that proved that the parametric process did, indeed, occur -that anti-Stokes light depleted a cone of stokes light at the correct phase-matching angle. Figure 3 shows the bright ring of anti-Stokes radiation along with a negative of the Stokes radiation pattern, with the arrow pointing to a weak ring showing depletion at the phase-matching angle [19].   [19].
My research effort was to study SRS in Raman-active liquids using the Q-switched ruby laser [20]; now theory and experiment did not agree, as they did in calcite. This turned out to be due to filament-formation and self-phase modulation in liquids with nonlinear refractive indices.

Filament formation in SRS emission
Typical images of the radiating cones of anti-Stokes SRS generated in acetone are shown in figure  4a. The bright ring, always observed, violated the theory described above. It is the weak inner ring that corresponds to the above phase-matching angle. Further analysis showed that the bright ring at a larger cone angle corresponds to a theory describing phase-matching only along the direction of propagation. The reason was shown later to be due to the formation of self-focusing filaments. Because the Raman Stokes emission was strongly forward-directed, there was not enough Stokes emission at the angle required for anti-Stokes to be generated at the volume phase-matching angle. The measured anti-Stokes cone angle phase-matched in the direction of propagation only, and corresponded to a parametric interaction with the intense forward-going Stokes wave. The lack of -- lateral phase-matching was acceptable because this Raman interaction took place only in narrow filaments. This case can be called Cerenkov phase-matching, as opposed to volume phasematching. Several additional experimental results verified this discovery. When the 10-cm liquid cell was tilted off-axis, reflection from its glass faces provided feedback into an off-axis cavity. This produced a Raman Stokes laser with an intensity 10 4 times larger than that without feedback. The Stokes radiation at the required volume phase-matching angle (figure 4c) was now bright enough to parametrically generate anti-Stokes radiation at the expected angle (figure 4d, e). Proof of the parametric process is the decreased Stokes at the volume phase-matching angle is shown in figure  4f).  [21,22].
Independent proof that the anti-Stokes radiating into the large angle cones came from filaments (and thus thought of as Cerenkov radiation) is shown in figure 5a. This is the first direct evidence of self-trapped filaments emitting SRS. Further proof can be seen by comparing elliptical focussed volume phase-matching anti-Stokes in calcite (which has no self-trapped filaments) in figure 5b with anti-Stokes from benzene, where the same elliptical focusing geometry generated mostly circular emission from self-trapped filaments.
(a) (b) c) Figure 5. Angular distribution of stimulated Raman anti-Stokes radiation; (a) Separate radiation from two different filaments caused side-by-side cones of anti-Stokes. A single anti-Stokes cone from volume phase-matching is also seen; (b) Elliptical volume phasematching in calcite produced with a cylindrical lens; (c) Circular Cerenkov phasematching from self-trapped filaments produced with the same conical lens [22,23].
When the laser output was multimode, the anti-Stokes emission had a broad wavelength spectrum.
Later it was realized that this was due to self-phase modulation in the nonlinear organic liquids. With a slit and an imaging spectrometer, the wavelength dependence of the output angles differs markedly between volume emission and filament emission ( figure 6). With a single-mode ruby laser, the anti-Stokes output was single-frequency, both with volume and surface emission. The spectrometer nicely separated out the different orders of anti-Stokes emission, although the lack of dynamic range of photographic film made it difficult to see several orders simultaneously.

Stimulated Brillouin Scattering
Townes was the first to predict stimulated Brillouin scattering (SBS) and his group was the first to demonstrate it. He understood that "Stimulated Brillouin scattering of an intense optical maser beam involves coherent amplification of a hypersonic lattice vibration and a scattered light wave. … It is analogous to Raman maser action, but with the molecular vibration replaced by an acoustic wave with frequency near 30 GHz…Both the acoustic and scattered light waves are emitted in specific directions" [24]. Figure 6. Angular-image spectra for stimulated anti-Stokes radiation excited by multimode ruby laser, with spectrum broadened by self-phase modulation. Sharp lines are from reference light source. Higher orders have successively more phase-modulation, with Cerenkov phase-matching changing much less in angle than volume phasematching. Small white triangles indicate expected location of single-frequency Raman line [22,23].
The Brillouin wave has a Stokes frequency shift Ω s that depends on its emission angle θ as Ω s = 2ω o (v ac /v ph )sin(θ/2), where v ac is the velocity of the acoustic wave and v ph is the velocity of the photon. For retro-reflected waves, which are the most intense, the Stokes down-shift is Ω s = 2ω o (v ac /v ph ). Stimulated Brillouin scattering can be considered parametric generation of an acoustic wave and a scattered light wave from an initial laser wave. With such a small frequency shift, SBS is most easily observed with a Fabry-Perot interferometer.
With solids quartz and sapphire, SBS occurred just as predicted [24]. This first observation of SBS in fused silica validated a process that was shown later to strongly limit the power that can be transmitted through fibers. Figure 7a shows typical Fabry-Perot spectra in these solids. Brillouin-shifted retro-reflected beam (B); (b) Single-frequency laser light illumination; (c) SBS from a ruby laser focused in water [24,25].
In liquids, a series of several orders of Stoke Brillouin shifts was observed, as shown in Fig. 7c  compared to Fig. 7b, which could not be explained by the SBS theory. It turned out that the Stokes Brillouin wave was retro-reflecting into the inhomogeneously broadened ruby laser, where it found additional gain because it was at a new frequency [25]. It then came out of the laser as a new Brillouin-shifted frequency and entered the liquid cell where SBS caused another retro-reflected Brillouin shift. And so on. Many years later it was determined that the ability of the SBS Stokes light to go exactly back into the laser had occurred because of phase conjugation [26]. In 1964, while I was still a graduate student, I did not think sufficiently about why the retro-reflecting light should follow such an exact backward path through the lens, the mode selector and the imperfect ruby rod. If I had, I would have been the first to identify phase conjugation!

Spatial Solitons
In 1964, Dr. Townes became aware of the fact that when Mike Hercher at University of Rochester focused a powerful Q-switched ruby laser into a glass block, it left a trail of damage that did not appear to spread with distance by diffraction [27]. That led Townes to think about how optical nonlinearities could overcome diffraction and to our working out the theory, which we submitted in September and was published in October, 1964. We described the reported experimental data this way: "Thin threads of damage in glass and other materials …fairly easily demonstrated in glass by focusing a ruby-laser beam greater than a few megawatts inside good optical glass. Usually, though not always, there is extensive damage near the focal point and beyond the focal point a long straight filament of small bubbles and damage along the lens axis, accompanied by ionization. This filament may be as long as several centimeters and at the same time have a diameter of only a few wavelengths. This diameter is in some cases two orders of magnitude smaller than the focal diameter, assuming linear optics." [28] The idea was as follows: "an electro-magnetic beam can produce its own dielectric waveguide and propagate without spreading. This may occur in materials whose dielectric constant increases with field intensity, but which are homogeneous in the absence of the electromagnetic wave." [28] An estimate was obtained by considering diffraction of a circular optical beam of uniform intensity across diameter D in material for which the index of refraction may be quadratic in field with a coefficient n 2 . A self-consistent waveguide will occur when the divergence angle of the finite beam, θ D = 1.22λ/nD, is set equal to the critical angle for total internal reflection, for a refractive index discontinuity of Δn = n 2 |E| 2 . This gives a threshold power in the laser beam of P = (1.22λ) 2 c/64n 2 , a total power independent of beam diameter. Inserting numbers, the beam should self-trap at approximately 1 Megawatt.

Exact solutions for confined beams
When lateral variation occurs only in one dimension (a slab-shaped beam), the nonlinear Helmholtz equation for the transverse field is given by E yy -Г 2 E y + (ε 2 /2)k o 2 E 2 E = 0, where the y subscript means the derivative in the y direction [28]. The analytic solution for the transverse field dependence is stable: E(y) = E o sech(Гy), where Г = ½ ε 2 1/2 k o E(0). Such self-trapped beams have, indeed, been generated in waveguides.
In two dimensions with a cylindrical beam, the Helmholtz equation (in dimensionless units) is There is no analytic solution, but Figure 8 shows the "Townes Profile," which was calculated by computer. When compared to the Gaussian beam, it can be seen to be close. Integration of this solution gives the critical power: P = 5.763 λ 2 cn eff /8π 3 n 2 n 0 , very close to the initial estimate. We observed self-trapping experimentally by inserting microscope slide cover slips along the beam path and reflecting successive weak images as the beam traveled through carbon disulfide [29]. Figure 10 shows the near field of the beam at successive locations for (a) low power, with (b) and (c) representing successively higher powers. The beam is seen to not diffract but to maintain a constant beam diameter. At higher powers a higher order spatial soliton can be seen. It should be pointed out, however, that this early data was not at all definitive because other nonlinearities such as SRS and SBS were undoubtedly occurring at the same time. It would take considerable additional work before these effects began to be separately studied.  The self-trapped beam that was predicted is now known as a "spatial soliton" and has been seen in a number of situations. Today, an important application for self-trapped beams is in plasmas, particularly those created by ultra-short laser beams in air. Self-focusing in plasmas was, in fact, independently predicted before self-focusing in liquids or solids [30].

Instabilities in self-trapped beams
Paul Kelley pointed out that the same nonlinearity that creates a self-trapped beam of just the right power will self-focus beams of powers higher than the critical threshold [31]. The nonlinearity overpowers diffraction, creating spherical surfaces of constant phase that focus the beam as shown in figure 9. The self-focusing process will cease when higher-order nonlinearities take over.
In order to achieve the results shown in figure 10, it was necessary to ensure that the ruby laser operated in a single spatial mode. With multi-spatial modes, large-scale trapping was not observed. In both cases, however, close observation showed that nonuniformities in the beam caused it to break up into many small-scale filaments on the order of only a few microns.
More careful analysis showed that superimposed on the large scale trapping were many smallscale filaments, only a few μm in size, apparently introduced by instabilities [32]. Such instabilities in χ 3 materials had been predicted by Talanov [33].
Later experiments working with Brewer from IBM demonstrated that these filaments last only ~ 10 -10 sec, after which they "blow up" [34], presumably because heating expands the liquid. With measured diameters of 4 μm in CS 2 and 12 μm in nitrobenzene, they were estimated to contain about 10kW peak power. This power level agreed with theory that assumed the Kerr effect is the principal mechanism for initiating filaments. For powers above the trapping threshold, filaments were seen to decrease in size and increase in intensity until higher-order terms in the electric field caused saturation of the nonlinear refractive index. Measured diameters were 10  larger than theoretical limiting diameters based on Kerr effect saturation, which indicated that some other mechanism determined filament size. We measured lower than expected Raman gain in most of the filament length, which was explained by frequency broadening and dispersion.
Self-focusing and self-trapping have remained a very important factor in a number high-power laser experiments and many of the interrelated phenomena have been carefully disambiguated. A recent review offers an excellent and up-to-date review of the status today of these filaments [35]. While self-trapped filaments tend to be unstable in a material with a χ 3 nonlinearity, they are stable in photorefractive χ 2 materials. Figure 11 shows an example in photorefractive barium titanate.
Figure11. Photograph of green laser beam through photorefractive barium titanate: (a) normal diffraction at lower power; (b) self-trapped beam at higher power.

Stimulated four-photon light scattering
Influenced by the realization of stimulated scattering from Raman and Brillouin effects, Townes' group also analyzed stimulated Rayleigh scattering [36]. They assumed a stationary, or nearstationary, periodic variation in refractive index change would occur as a result of interfering light beams and a nonlinear refractive index. Stimulated Rayleigh-wing scattering discussions had not yet included the influence of stimulated light-by-light scattering and its associated weak-wave retardation. With both effects having large gains in the forward direction, the proper analysis required coupled waves. The paper analyzed the interaction between three waves: undepleted laser field E o , weak wave E 1 at a small angle (or frequency difference) and amplified weak wave E 2 . We were the first to show that weak wave E 2 experienced a gain per unit length of g = 4πΔn/λ o . In nonlinear media, g = πε 2 |E o | 2 /n o λ o (cgs units). Since this is a parametric process, the weak wave E 1 experiences a loss per unit length equal to the gain of E 2 per unit length. When the waves experience 1/e loss in lengths L 1 and L 2 , there will be a threshold for observation of this parametric amplification given by 4πΔn = πε 2 |E o | 2 /n o > λ o /(L 1 L 2 ) 1/2 . Because the weak waves are retarded relative to the strong wave (whose wave vector is extended by the nonlinearity), phase-matching gives rise to small angles for E 1 and E 2 with respect to E o . The gain was shown to be strongly peaked at the phase-matching angle. Just as in SRS, transverse momentum conservation requires that if wave E 1 travels with angle θ with respect to the laser beam, wave E 2 travels at -θ, both angles being very small. It is simple to show that this angle is θ = ± (2Δn/n o ) 1/2 . This is easiest seen through picturing that interference causes stationary periodic layers in the medium, which can produce reflections in a manner similar to the Raman-Nath effect.
The assumption was local nonlinearity, so that degenerate gratings in the material are unshifted with respect to the optical interference fringes. Even so, this analysis showed that steady-state amplification was possible, via the parametric four-wave interaction. The phase-matching condition was met only for a strict angular window, and only there could net gain be achieved. An experiment was carried out with the Q-switched ruby laser in carbon disulfide to demonstrate that a weak wave did, indeed, experience parametric gain at the appropriate angle [37].
Considering only a weak and a strong wave, the nonlinear shift in the index of refraction induced by the pump wave on the probe wave was shown to be double that compared to the pump wave itself, which implies that the probe-wave vector is longer than the pump-wave vector. This additional lengthening of the weak-wave propagation vector had lasting impact on the thinking about two-wave, three-wave and four-wave mixing, particularly in photo-refractive media. It also provided early understanding of how instabilities grow and cause filamentation of high power beams traveling through nonlinear media.

Self-steepening of light pulses
Townes and his group predicted a change in temporal shape of light pulses traveling through a medium with an intensity-dependent refractive index [38]. They calculated the time required for a Gaussian pulse to steepen into a temporal optical shock through both analytic and numerical solutions for the pulse as it traveled through the medium. They investigated temporal development of pulses for both zero and nonzero relaxation times of the nonlinear χ 3 refractive index.
The frequency spectrum calculated in the zero-relaxation time limit, found the largest peak intensities on the lower-frequency side ( figure 12). The rate of steepening is modifed when pulse decay time is as short as the nonlinear refractive index; in a dispersionless medium the pulse decay time can become arbitrarily short. Estimates were made for both the thickness of the optical-shock region and the frequency spreading allowed by dispersion when relaxation was added. This phenomenon is now well-known and more familiarly called self-phase-modulation. Figure 12. Self-steepening and frequency-broadening of a Gaussian laser pulse through χ 3 medium. (a) Time dependence of intensity of a sinusoidal input at z=0, z 1 and z 2 ; (b,c) Spectra of a sinusoidal pulse at z 1 and z 2 respectively. At z 2 the most intense peak is ~ 2000 cm -1 below the ruby laser wavelength. Adapted from [35].

Other MIT research in Nonlinear Optics
Professor Townes continued to interact with and inspire more work on self-trapped filaments of light. With Brewer at IBM he looked at standing waves in these filaments [39]; with Sacchi and Lifsitz he investigated anti-Stokes generated in trapped filaments [40]; phase modulation was investigated with Cheung et. al. [41]. As a result of the earlier work, spectral broadening of light in filaments was analysed [42], along with self-trapping in saturating media [43]. Similar studies broadened to new fields, such as thermal self-focusing [44] and coherent excitation of polaritons [45]. This list includes only work done in the field by those who had collaborated directly with Townes while he was at MIT and is not meant to be comprehensive (567 papers in nonlinear optics were published between 1961 through 1969, according to the Information Sciences Index).

Contributions of Townes to Astronomy (1967-2011)
Here is a list of some of the major topics of investigation in which Townes was involved at University of California, Berkeley: • Water in interstellar space • Ammonia in interstellar space • Dust around stars • Mechanisms in the galactic center • Mechanisms of star formation in galaxies • Far-infrared spectroscopy of galaxies • A heterodyne stellar interferometer for the mid-infrared Townes had long been interested in astronomy, ever since his initial work with radar during WWII and had a firm conviction that the nonlinear photon-atom processes that were observed terrestrially with lasers would be observable in space. He was rewarded by finding this so and continuing active research well into his 90's. The last research papers listing him as a co-author were in 2011 when he was 96 years old!

Contributions of Basov and Prokhorov (1967-2011)
This paper focused on Charles Townes, one of the Nobel Prizewinners. It did not include the important contributions that Nobelists Basov and Prokhorov have made to the field. Basov introduced the idea of semiconductor lasers as early as 1958 [46]. Prokhorov ran a laboratory with wide-ranging interests, with developments often proceeding in parallel with those in the US, including Q-switching, new pumping techniques, improving laser coherence and increasing power. Contributions were made in nonlinear optics, fiber and integrated optics, and in a variety of laser applications, especially in fiber-optic communication, laser technologies and the use of lasers in medicine and ecology. An excellent source for English translations of early papers describing quantum electronics research in the USSR can be found on-line [47].