Nonlinear optics the first 50 years

Contemporary Physics ISSN: 0010-7514 (Print) 1366-5812 (Online) Journal homepage: http://www.tandfonline.com/loi/tcph20 Nonlinear optics: the first 50 years G H.C New To cite this article: G H.C New (2011) Nonlinear optics: the first 50 years, Contemporary Physics, 52:4, 281-292, DOI: 10.1080/00107514.2011.588485 To link to this article: http://dx.doi.org/10.1080/00107514.2011.588485 Published online: 11 Jul 2011 Submit your article to this journal Article views: 813 View related articles Citing articles: View citing articles Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tcph20 Download by: [Vietnam National University Ho Chi Minh] Date: 23 March 2016, At: 03:37 Contemporary Physics Vol 52, No 4, July–August 2011, 281–292 Nonlinear optics: the first 50 years G.H.C New* Quantum Optics & Laser Science Group, Blackett Laboratory, Imperial College London, London SW7 2AZ, UK Downloaded by [Vietnam National University Ho Chi Minh] at 03:37 23 March 2016 (Received 18 March 2011; final version received 12 May 2011) In the summer of 1961, a landmark experiment was performed at the University of Michigan in which optical second harmonic generation was observed for the first time This event 50 years ago marked the birth of modern nonlinear optics, and this article celebrates the first half century of what is now a vast and vibrant field at the cutting edge of laser technology The focus is mainly on nonlinear optics in the 1960s partly because it is appropriate in this anniversary year to remember the genesis of the field, but also because such remarkable progress was made in the first few years However, a brief review of where things stand at present is included, and one aspect of the field today (high harmonic generation) is taken as a representative example of an area of nonlinear optics that lies at the current frontier of knowledge Keywords: nonlinear optics Introduction This year (2011) is the 50th anniversary of a landmark 1961 experiment in which optical second harmonic generation was observed for the first time This marked the birth of modern nonlinear optics Theodore Maiman had demonstrated the first ruby laser the year before; as we shall see, it was the intense optical frequency electric fields delivered by the laser that made the harmonic generation experiment possible, and that powered the rapid growth of nonlinear optics in the years that followed Strong DC electric fields were of course available to scientists in the nineteenth century, and this enabled two early experiments in nonlinear optics to be performed In 1875, the Rev John Kerr of the Free Church Training College in Glasgow UK showed that the refractive indices of various liquids and dielectric materials were slightly altered in the presence of a high DC field [1] The index change varied as the square of the field, and the process is now known as the DC Kerr effect Then in 1893, Friedrich Pockels at the University of Go¨ttingen published a paper on what we now know as the Pockels effect,1 in which the refractive index of a non-centrosymmetric crystal changes in direct proportion to the strength of an applied DC field [2] The origin of these (and many other) nonlinear optical phenomena can be identified if the optical frequency polarisation P is expanded as a power series in the electric field namely2 P ¼ e0 ðwð1Þ E þ wð2Þ E2 þ wð3Þ E3 þ Þ: *Email: g.new@imperial.ac.uk ISSN 0010-7514 print/ISSN 1366-5812 online Ó 2011 Taylor & Francis DOI: 10.1080/00107514.2011.588485 http://www.informaworld.com ð1Þ Writing E ¼ Edc þ Eocosot, and substituting the expression into Equation (1) yields (among other terms) P ¼ e0 ðwð1Þ þ 2wð2Þ Edc þ 3wð3Þ E2dc ÞEo cos ot: ð2Þ It is well known that the linear susceptibility w(1) is linked to refractive index through the equation n ¼ (1 þw(1))1/2, and it therefore becomes clear from Equation (2) that the terms in w(2) and w(3) represent modifications to n that are respectively proportional to Edc and Edc2; these respectively represent the Pockels effect and the DC Kerr effect For nonlinear optics to progress further, a source of intense optical frequency radiation was needed, and this is what the laser provided 1950–1960 and the invention of the laser The 1950s had been a difficult decade The world was still in the shadow of the Second World War, and economic conditions were still marked by austerity In these circumstances, the dawn of the new decade in 1960 held a peculiar promise: 1945 now lay in the decade before last, and suddenly it was time to look to the future rather than to the past In physics, the 1950s had seen the development of the maser, the microwave device based on stimulated emission that predated the laser By the end of that decade, competition to extend the maser principle into the visible part of the spectrum was intense When Downloaded by [Vietnam National University Ho Chi Minh] at 03:37 23 March 2016 282 G.H.C New Theodore Maiman observed laser action for the first time on Monday 16 May 1960, he discovered the power source that supports most of today’s optical technology Maiman’s initial experiments on the ruby laser provided only indirect evidence of laser action; what he actually observed was a change in the relative populations of two energy levels in his ruby sample, which he took as evidence that a laser was feeding off one of them But by the time of the press conference announcing the achievement on July, he had seen the characteristic pencil beam at 694.3 nm, at the far red end of the visible spectrum His first paper on the laser appeared in Nature [3] on August 1960.3 By the end of the year, Ali Javan at Bell Telephone Laboratories had also observed laser action, this time using a helium–neon mixture pumped by an electrical discharge; this was the world’s first gas laser and the first continuous (CW) laser too [4] From then on, new lasers came thick and fast Laser development proceeded amazingly rapidly, and one of the first applications was in nonlinear optics, where the intense and highly-directional nature of laser radiation was precisely what was needed to get the new field off the ground The world of the 1960s It is worth pausing at this point to reflect on how much has changed in the last 50 years Even people who lived through the 1960s find it hard to remember what a different world it was then There were for example no pocket calculators, so unless one had access to a mainframe computer, calculations had to be done using a slide rule or log tables It is striking to recall that during the famous aborted NASA moon shot of April 1970, technicians at Mission Control in Houston used slide rules to work out how to bring the astronauts in the crippled spacecraft back to Earth.4 When the first pocket calculator (the HP-35) appeared in 1972, it sold in the US for a staggering $395, while the UK price was £198 at a time when £1500 was considered a good annual salary But demand was insatiable, and the first batch sold out almost instantly By 1976, one could buy an Apple for $666.66, but this was really a piece of kit for hobbyists, and the personal computer was still in the future Optical fibre communications were of course unknown The concept was mooted by Kao and Hockham in 1966 [5], but the suggestion was not taken seriously at first because everyone thought that absorption in glass made it entirely impracticable Indeed, one of Charles Kao’s many contributions was to demonstrate that the absorption was due almost exclusively to impurities (rather than to inhomogeneities which was the prevailing view at the time), and that fused silica would be an ideal fibre material if the impurities could be removed.5 The first low-loss fibre was produced by Corning in 1970, and the field has of course not looked back since! With so many key areas of technology in their infancy, it is strange to recall that manned moon landings were becoming regular events as the 1960s drew to a close Neil Armstrong had walked on the moon on 21 July 1969, and there was a second landing four months later After the aborted Apollo 13 mission in 1970, four more successful landings followed, two in 1971 and two in 1972 No one has been back since Second harmonic generation We now return to the main storyline and to the situation in early 1961 in particular Remarkably, less than a year after Maiman’s pioneering work, a small US company called Trion was already selling ruby lasers, and the strong optical frequency field in the ruby laser beam was precisely what nonlinear optics was waiting for The link between laser intensity I (¼ power per unit area) and electric field amplitude Eo is I ¼ nce0 E2o : ð3Þ By modern standards, the optical power available from the Trion laser was very modest but, if the beam was tightly focused, the field strength Eo was sufficient for a team led by Peter Franken at the University of Michigan at Ann Arbor to observe optical second harmonic generation (SHG) for the first time in the summer of 1961 [6] In the key experiment, a schematic diagram of which is shown in Figure 1, a ruby laser at 694.3 nm was focused into a thin quartz crystal, and the output beam analysed in a prism spectrograph for evidence of a second harmonic component at 347.15 nm Despite a minuscule conversion efficiency Figure Schematic diagram of the first second harmonic generation experiment at the University of Michigan in1961 (Reproduced from [21] with permission Copyright (2011), Cambridge University Press.) 283 Contemporary Physics Downloaded by [Vietnam National University Ho Chi Minh] at 03:37 23 March 2016 of around in 108, a weak component at the harmonic frequency was detected on a photographic plate.6 Peter Franken (Figure 2) was a larger-than-life figure, brimming with novel ideas, and a brilliant lecturer too In 1973, he moved from Michigan to become Director of the Optical Sciences Center at the University of Arizona, and he continued to live in Tucson until his death in 1999 A fascinating transcript of an extended interview with him covering the early history of nonlinear optics can be found in [7] Phase matching Although the observation of second harmonic generation was genuinely ground-breaking, the low harmonic conversion efficiency meant that it was a curiosity rather than a serious way of generating coherent ultraviolet light Why was the efficiency so poor? The reason was that, due to dispersion, the fields at the fundamental and harmonic frequencies travelled at different phase velocities in the nonlinear crystal, and so quickly got out of step This point can be appreciated by including spatial dependence in the expression for the ruby laser field by writing E ¼ Eo cos{ot k1z}, where k1 ¼ Figure Professor Peter Franken (1928–1999) (Optical Society of America, courtesy of the Emilio Segre` Visual Archives, Gallery of Member Society Presidents.) n1o/c and n1 is the refractive index at o This leads through Equation (1) to a second-order term in the polarisation of the form Pð2Þ ¼ e0 wð2Þ E2o cos ðot À k1 zg ¼ e0 wð2Þ E2o ½1 þ cos f2ot À 2k1 zgŠ: ð4Þ But the space–time dependence of the second harmonic field is cos{2ot k2z}, where k2 ¼ n22o/c, and this does not quite match the final term in Equation (4) In fact, it is easy to see that the two waves are p radians out of step when (k2 2k1)z ¼ p, which provides a natural definition of coherence length7 Lcoh ¼ p l ¼ : jk2 À 2k1 j 4jn2 À n1 j ð5Þ In typical optical materials, Lcoh is *10–20 mm, so only a small fraction of the quartz crystal in the University of Michigan experiment was participating usefully in the SHG process This was why the conversion efficiency was so tiny The problem of limited coherence length was quickly solved Independent papers from Joe Giordmaine at Bell Telephone Labs and a group under Robert Terhune8 at the Ford Motor Company’s research laboratories in Dearborn Michigan appeared together in the New Year’s Day 1962 issue of Physical Review Letters [8] The trick in both cases was to exploit the birefringence of anisotropic crystals, by making the fundamental beam an ordinary wave, and the harmonic beam an extraordinary wave And because the refractive index of an extraordinary wave is dependent on the direction of propagation in the crystal, the angle was adjusted to ensure that n2ext(y) ¼ n1ord Figure illustrates the dramatic improvement in SHG efficiency that phase matching delivers The lower solid line (non-PM) that barely manages to lift itself off the bottom axis follows the second harmonic intensity when phase matching is not achieved By contrast, the upper solid line (PM) shows the harmonic intensity increasing as the square of the distance under phase-matched conditions So, if the coherence length is increased from (say) 10 mm to mm, the intensity rises by *104, and this was what changed nonlinear harmonic generation from a curiosity into a practical proposition On 10 April 1962, Armstrong, Bloembergen, Ducuing and Pershan (ABDP) of Harvard University submitted a paper to Physical Review that is truly remarkable for the breadth and depth of its coverage of nonlinear optics at such an early date [9] One of the hidden gems in this seminal article is a suggestion for Downloaded by [Vietnam National University Ho Chi Minh] at 03:37 23 March 2016 284 G.H.C New an alternative method of phase matching, one that would take around a quarter of a century to come to full realisation The essence of the idea is reproduced in Figure 4, and the technique, now known as ‘quasiphase matching’, is regarded by many researchers as the preferred method of phase matching They may indeed see it as ‘the best thing since sliced bread’, a particularly appropriate aphorism given that Figure looks very like a sliced loaf However, the slices consist of nonlinear crystal, and are far thinner than anything on the breakfast table! The idea at least is simple At the end of the first coherence length of the harmonic generation process, the harmonic field has reached a maximum, its phase relationship with the harmonic polarisation has slipped by 1808, and the interaction is just about to go into reverse gear So, to keep things on track, we reverse the sign of the nonlinear polarisation by turning the next slice of crystal upside down This enables the harmonic to continue to grow for another coherence length, after which we turn the crystal right way up again And so the process goes on The trouble is that the coherence length is typically only 10–20 mm, so crystal wafers as thin as one hundredth of a millimetre are needed, and that is why it took so long for quasi-phase matching to be reliably implemented New crystal growth techniques had to be devised and, in particular, a technique known as periodic poling in which an electric field is used to force the growth of crystal structure to reverse direction Only then did the process envisaged in ABDP9 become a reality The dotted line (QPM) in Figure compares quasiphase matching to phase-matching based on birefringence, and shows the SHG intensity rising in steps The step height increases with distance, but this is purely because optical intensity goes as the square of the electric field; see Equation (3) If field rather than intensity were plotted, the steps would be of uniform height Figure Second harmonic intensity as a function of distance under non-phased-matched and phase-matched conditions (solid lines) The corresponding curve for quasiphase matching is shown dotted (Reproduced from [21] with permission Copyright (2011), Cambridge University Press.) Figure The early growth of nonlinear optics The Harvard paper set the scene for the gold-rush period in nonlinear optics that occurred in the mid1960s Many new nonlinear phenomena were demonstrated between 1962 and 1965, including optical rectification [10], sum and difference frequency generation [11], third harmonic generation [12], optical parametric amplification [9,13], the optical Kerr effect [14,15], and stimulated Raman and Brillouin scattering [16,17] Laser technology was of course progressing very rapidly at the same time In particular, 1962 saw the development of laser Q-switching [18], which enabled laser pulses in the 20–50 ns range with peak powers of megawatts to be generated The attendant increase in peak power had an immediate impact on the efficiency of nonlinear interactions, and the crossfertilisation between laser physics and nonlinear optics that began at that time continues to this day Schematic of a quasi-phase matched material; as envisaged in Figure 10 of [9] Contemporary Physics Downloaded by [Vietnam National University Ho Chi Minh] at 03:37 23 March 2016 6.1 Stimulated Raman scattering It was in fact during experiments on Q-switching using a nitrobenzene Kerr cell that Woodbury and Ng chanced on the first observation of stimulated Raman scattering [16] They found that laser action was occurring at 765.8 nm (391.5 THz) as well as at 694.3 nm (431.8 THz), and spotted that the 40.3 THz frequency difference corresponded exactly to a vibrational resonance of nitrobenzene They called the device a Raman laser, and the principle is now in widespread use, especially in fibre Raman systems Stimulated Raman scattering is the stimulated counterpart of spontaneous Raman scattering, a process that was first observed in the 1920s using conventional light sources In those earlier experiments, the scattered frequency component was called the Stokes wave, and this terminology has been retained for the stimulated process.10 In practice, the process is frequently based on vibrational resonances, although rotational or electronic states may also be involved A further possibility is that the laser and Stokes waves interact via an acoustic wave; a process that is called stimulated Brillouin scattering [17] 6.2 285 Figure Optical rectification recorded by placing a noncentrosymmetric crystal between capacitor plates (Reproduced from [21] with permission Copyright (2011), Cambridge University Press.) Optical rectification Another process first demonstrated in 1962 was optical rectification [10] The possibility of rectification is evidenced by the presence in Equation (4) of the DC term Pdc ¼ e0 wð2Þ E2o ¼ wð2Þ Io =nc; ð6Þ where the final step follows from Equation (3) The observation of optical rectification (OR) involves what must surely be the simplest experiment in nonlinear optics All one has to is to place a suitable nonlinear crystal between a pair of capacitor plates (see Figure 5), and a voltage appears between the plates in proportion to the laser intensity in the medium, as shown in Figure [19] Readers with a feeling for symmetry principles will immediately ask what determines the sign of the voltage or, to put it another way, what determines the difference between up and down in this experiment? The answer is that there must exist an inherent ‘one-wayness’ in the nonlinear crystal or, to put it more formally, the crystal must ‘lack inversion symmetry’, or ‘be non-centrosymmetric’ Indeed, 21 of the 32 crystal symmetry classes possess this property, so it cannot be considered unusual 6.3 Symmetry considerations This symmetry principle can be put on a sound mathematical footing by considering the term P ¼ Figure Optical rectification signal (upper trace); laser monitor (lower trace) (Reproduced from [21] with permission Copyright (2011), Cambridge University Press.) E2 in the expansion of Equation (1) In a medium possessing inversion symmetry, the sign of P must clearly reverse if the sign of E is reversed But E squared is positive irrespective of the sign of E, and the conclusion is that all second-order nonlinear effects are forbidden under these circumstances A non-centrosymmetric medium is therefore essential for w(2) to be non-zero On the other hand, no such restriction applies for n ¼ 3, so third-order effects based on w(3) exist in all optical materials, irrespective of their symmetry properties 0w (2) 6.4 Sum and difference frequency generation Sum-frequency generation (SFG) [11] refers to the process o1þ o2 ¼ o3, of which SHG is the special case where o1 ¼ o2 ¼ o and o3 ¼ 2o As an aid to understanding, it is helpful to multiply all terms in the SFG formula by h, which then reads  ho1 þ ho2 ¼ ho3 This highlights the fact that, for each photon gained at o3, one is lost at the two lower frequencies But there are other possibilities too If only the waves at o3 and o2 were present initially, what about ho3 ho2 ¼ ho1 or, if we started with o3 286 G.H.C New Downloaded by [Vietnam National University Ho Chi Minh] at 03:37 23 March 2016 and o1, we could have  ho3  ho1 ¼  ho2 In fact, in a comprehensive analysis of sum frequency generation, the two difference-frequency generation (DFG) processes need to be considered as well, and all three processes work together to ensure that the energy in the radiation field is conserved It is the phase relationship between the three waves that determines whether the direction of energy flow is according to ho þ   ho2 )  ho3 or to  ho )  ho þ  ho2 6.5 Optical parametric amplification and the optical parametric oscillator A fascinating possibility now presents itself If the input to the nonlinear medium were a single wave at o3, could this be enough to initiate the process ho3 ) ho þ   ho2? One could for instance argue that energy at o1 and o2 will certainly be present in the background noise spectrum But the puzzling question remains: what determines how the o3 photon divides into two if there is no specific input at either o1 or o2 to get the interaction off the ground? After all, there is an infinity number of ways of cutting a cake into two so, if the process we are discussing is going to work, what determines the split? The answer to this conundrum lies in a process of natural selection governed by the phase-matching conditions Think of the waves exploring all possible division of ho3 into  ho1 and  ho2, and it is the one that is phase matched that will win out Indeed, although not mentioned at the time, it is the critical importance of phase matching in nonlinear optics that allowed us to focus on the SFG combination o1þ o2 ¼ o3, and to ignore all the other possibilities (like 2o1 and o1 o2) The process ho3 ) ho1 þ  ho2 is called optical parametric amplification (OPA) [9] The highest frequency o3 is called the pump, and the other two are called the signal and the idler.11 In practice, a weak initial signal beam is normally used as a seed, to get the process started Lots of interesting phenomena occur in OPA, and one can have lots of fun simulating the process A typical result is shown in Figure where transverse intensity profiles of pump (top), signal (centre) and idler (bottom) are shown at the end of the interaction in a mm crystal of lithium triborate (LBO) [20, 21] The three beams have been separated purely for display purposes; in reality, they lie on top of each other, and the cross-hairs define the common centre line LBO is a biaxial nonlinear crystal, which means that it exhibits the more complicated of the two types of birefringence In this case, however, the geometry is as simple as possible; the signal and idler beams are in effect ordinary waves, while the pump is an extraordinary wave Extraordinary waves exhibit strange Figure Simulation of optical parametric amplification in a mm sample of lithium triborate (LBO) showing the transverse energy profiles of pump, signal, and idler at the end of the interaction See text for more details about walkoff and the non-collinear beam geometry (A black and white version of this picture appeared in [21].) Reproduced with permission Copyright (2011), Cambridge University Press propagation properties; for example, the direction of energy flow is not perpendicular to the wavefront This feature is evidenced in the figure by the fact that the pump has ‘walked-off’ roughly 60 mm to the left On the other hand, to optimise the bandwidth, the initial signal seed has been deliberately angled 1.38 to the right, which causes the idler to be angled 2.18 to the left to satisfy the phase-matching condition Interestingly, these angles would imply respective sideways shifts of around 150 and 250 mm for signal and idler over the mm length of the crystal, far larger than what is seen in the figure But this ignores the fact that the three fields need to overlap for the interaction to proceed, so Downloaded by [Vietnam National University Ho Chi Minh] at 03:37 23 March 2016 Contemporary Physics signal and idler have to cling together (and to the pump) in order to survive and grow The pump profile has clearly been depleted fairly uniformly, which suggests that the pump energy has been used quite efficiently in this geometry In a particularly exciting extension of optical parametric amplification, the nonlinear crystal is located between high reflectivity mirrors at the signal frequency to create an optical parametric oscillator (OPO) Since the signal frequency is determined by the phase-matching conditions, and can therefore be controlled, an OPO solves one of the most vexing questions of laser technology: how to generate coherent radiation at an arbitrary frequency While some lasers have broad bandwidths that offer limited tunability, laser sources are still to a large extent restricted to the energy levels that nature has given us But OPOs offer freedom from this fundamental constraint Like quasi-phase matching, OPOs went through a long gestation period before becoming standard components in the well-found laser laboratory, as they are today As for QPM, the problems were in materials technology The first OPO was demonstrated in 1965 [13], but it was not until the 1980s that nonlinear crystals of the exceptional optical quality required for an efficient and reliable device became available 6.6 Third-order nonlinear effects Nonlinear optics in the 1960s was not restricted to processes based on the second-order term of Equation (1); indeed, several third-order processes have already been mentioned in this paper [12,14–17] To see the possibilities, we extend Equations (2) and (4) by inserting E ¼ Edc þ Eo cos{ot k1z} into Equation (1) Including selected terms up to the third-order term yields   P ¼ e0 wð1Þ þ 2wð2Þ Edc þ 3wð3Þ ðE2dc þ E2o Þ Â Eo cos fot À k1 zg   ð2Þ ð3Þ w þ w Edc E2o cos f2ot À 2k1 zg þ 2  þ wð3Þ E3o cos f3ot À 3k1 zg þ : ð7Þ The term in the bottom line is third harmonic generation which is analogous to second harmonic generation except that it can occur in a centrosymmetric medium [12] The first term in the third line is of course second harmonic generation, but a term involving w(3) now appears as well SHG normally 287 requires a non-centrosymmetric medium, but the presence of a DC field provides a preferred direction in a centrosymmetric environment 6.7 Intensity-dependent refractive index The first three terms in the first line of Equation (7) appeared in Equation (2), but the fourth represents a new effect: the dependence of the refractive index on the square of the optical frequency field or (through Equation (3)) on the optical intensity This is the basis of intensity-dependent refractive index (IDRI), an enormously important process with a wide range of practical applications [14,15] The associated coefficient is normally positive, which means that refractive index almost always increases with intensity This has the unfortunate and potentially catastrophic consequence that the attendant wavelength reduction along the axis of an intense optical beam leads it to collapse upon itself [14], causing irreversible damage to expensive optical components Drastic steps have to be taken in large laser systems to avoid the calamitous consequences On the other hand, the effect of IDRI on an optical pulse is to cause the peak central region to travel slower than the leading and trailing wings This affects the carrier wave structure, causing what is known as self-phase modulation (SPM) where the local frequency is lowered ahead of the peak and raised behind it The result is that the carrier frequency rises through the pulse, a condition known as an ‘up-chirp’ If IDRI becomes strong, the pulse envelope is distorted as well, causing the trailing edge gradient to rise as the peak suffers increasing delay, with the potential formation of a rear-end optical shock This selfsteepening effect was analysed in detail in 1967 [22] although, by a strange quirk of history, the analogous steepening of the trailing edges of the optical carrier waves had been considered two years earlier by Rosen [23] Like so many things published in the 1960s, Rosen’s 1965 paper was rediscovered in the 1990s and, today, ‘carrier wave steepening’ even has potential applications [24] Self-phase modulation (SPM) has numerous important uses Its most significant characteristic is the consequent increase in the spectral bandwidth [25] and, whenever bandwidth goes up, the laser physicist immediately thinks ‘and that means that the potential pulse duration goes down, if the chirp can be removed’ The idea of imposing strong SPM and following it with negative group velocity dispersion (GVD), goes back as far as a 1969 paper by Fisher, Kelley, and Gustafson [26]; the operation is now routinely used for optical pulse compression A more esoteric outcome occurs if SPM and negative GVD occur simultaneously in an optical Downloaded by [Vietnam National University Ho Chi Minh] at 03:37 23 March 2016 288 G.H.C New fibre, which leads potentially to the formation of optical solitons, unique self-contained solutions of the nonlinear wave equation In 1965, Zabusky and Kruskal [27] performed numerical simulations of soliton pulse propagation, but this was in the pre-fibre era It was not until 1973 that Hasegawa and Tappert [28] suggested that optical fibres were ideal media for soliton propagation, and it was not until the 1980s, that the concept was actually demonstrated In principle, solitons prevent the dispersive spreading that ultimately increases the bit error rate in optical fibre communications Unfortunately, long-distance soliton transmission is difficult to implement, and soliton-based fibre systems are a rarity, having been overtaken by advances in conventional fibre technology 6.8 Theoretical foundations The approach to nonlinear optics adopted in this short article is very simplistic, and would have been regarded as such from the outset All we have done is to plug expressions such as E ¼ Edc þ Eocos{ot k1z} for the electric field into the hypothetical power series expansion of Equation (1) The procedure has certainly revealed a number of important nonlinear processes, but it has provided no explanation for how the nonlinearity originates and it has led to some misleading conclusions too A rigorous quantum mechanical treatment based on time-dependent perturbation theory (TDPT) provides broad justification for Equation (1), with successive terms in the expansion corresponding to different orders of perturbation The complicated TDPT expressions for the nonlinear coefficients that are generated turn out to be dependent on all the frequencies participating in a given nonlinear interaction Hence, the second harmonic generation coefficient is (for example) not identical to the coefficient governing optical rectification as Equation (4) suggests, and the coefficients for the DC and optical frequency Kerr effect are not directly related as Equation (7) seems to imply As explained in Section 6.3 above, non-centrosymmetric crystals are needed to observe second-order nonlinear phenomena Many of these are also optically anisotropic, and the associated birefringence is of course exploited in birefringent phase matching In a complete analysis of nonlinear interactions, the fields and the polarisation are vector quantities, and the coefficients w(n) will be (n þ 1)th rank tensors These features add extra layers of complexity to the theory of nonlinear optics, when one delves into the subject for real Was nonlinear optics all done in the 1960s? So far, this article has dwelt almost exclusively on the early years of nonlinear optics One reason for this is that one naturally focuses on early work in an anniversary year, but it is also appropriate given that such amazing progress really was made in that period I have in fact often teased my research students by telling them that nonlinear optics was all done in the 1960s! As it stands, this statement is of course ridiculous, but to say that most of the basic principles of the subject were established in that first decade is arguably true There really are remarkably few fundamental ideas in the nonlinear optics of today that were not known (albeit in basic form) by 1970 What has characterised the development of the subject in the intervening years has been not so much the establishment of new principles (although there are some), but the amazing rate of advance in the technology available to exploit the original principles, and to so in new ways, in new materials, in new combinations, in new environments, and especially on increasingly challenging time and distance scales Whether the issue is quasi-phase matching, optical parametric oscillators, optical solitons, or countless other examples, the idea was probably there in the 1960s, and its application may even have been demonstrated in some rudimentary form But it was probably many years (or even decades) before the idea came to full fruition Several further observations can be made about the state of nonlinear optics in 2011 Firstly, laser physics and nonlinear optics have always been closely connected, and the relationship is now closer than ever Not only are lasers inherently nonlinear devices, but virtually all laser systems these days exploit nonlinear optics in one way or another, either as their basis of operation (e.g Kerr-lens mode-locking [29]), or in the ancillary systems they drive, or at least in a number of key components Could one perhaps argue that nonlinear optics is the wider field, and see laser physics as a compartment within it? Secondly, while new principles may be few in number, a glance at the session headings of a leading international conference on lasers and photonics today (e.g CLEO, or CLEO Europe) is revealing Typical topics under nonlinear optics are likely to include nanostructures, photonic crystal fibres, metamaterials, high-harmonic generation, attosecond science, spatial solitons, electromagnetically-induced transparency (EMIT), slow light, remote sensing, among others Some of these terms would have been meaningful 40 years ago but others would not EMIT for example represents a dramatic extensions of ideas that were wellknown by 1970 since it is related to self-induced transparency, which was studied in detail by McCall 289 Downloaded by [Vietnam National University Ho Chi Minh] at 03:37 23 March 2016 Contemporary Physics and Hahn [30] in the late 1960s However, the transparency in EMIT is now mediated by a strong field on a connected transition [31] The primary resonance now lies in a region of strong normal dispersion in which the group velocity of an optical signal slows down to walking pace; hence ‘slow light’ [32] Thirdly, despite all the achievements of the first golden decade, some things in nonlinear optics 50 years down the line really are truly novel For example, the appearance of photonic crystals and metamaterials in the list signals an important current trend towards artificial ‘designer’ media, which seem likely to figure prominently in the future of the subject; this area was virtually unknown in the 1960s.12 And, while highharmonic generation would certainly have been a meaningful term in 1970, it is highly unlikely that anyone at that time had the idea of ionising an atom at optical intensities of *1014 W cm72, allowing the field to accelerate the freed electron, and driving it back to collide with the parent ion to create a broadband harmonic spectrum Yet HHG is now an extremely active area at the cutting edge of current research, and, in the section that follows, we take it as a representative example of a truly novel field that has developed within nonlinear optics in recent years model [36] In step 1, an atom is ionised in the optical field, creating a free electron In step 2, the electron released moves almost freely in the field and, if the conditions are right, it is accelerated first away, and then back towards the parent ion In the ‘recollision’ that ensues (step 3), the electron returns to the parent ion, and the accumulated kinetic energy combined with the energy of ionisation is released in a harmonic photon Sophisticated quantum mechanical techniques are needed to justice to steps and 3, but a simple onedimensional classical model serves well for step 2, which turns out to play a crucial role in the overall process Figure 8, which requires nothing more than first-year physics to produce, shows the kind of results that one obtains The time variation of the driving field is shown at the top, while the curves below are (in grey) a set of electron trajectories tracing the electron displacement and (in black) the corresponding electron velocity The twelve different curves in each set correspond to different times of ionisation within the first quarter-cycle of the field All curves end at the point of recollision (X ¼ 0), and the length of the verticals to the abscissa from the black curves represent the recollision velocity, from which the recollision kinetic energy UK can be calculated It is easy to show that the maximum value of UK is High harmonic generation High harmonic generation (HHG) grew from work on harmonic generation in gases and metal vapours in the 1960s and 1970s Equation (7) includes a third harmonic term arising from the E3 term in Equation (1), and third harmonic generation itself was studied in detail in the 1960s (see e.g [12,33]) Experiments on third and fifth harmonic generation continued in the 1970s and early 1980s,13 but they remained within the perturbative regime, where the series expansion of Equation (1) remains valid, and the conversion efficiency for successive harmonic orders drops off sharply By the late 1980s, however, much higher laser intensities were available, and some remarkable experimental results were achieved, from which it was evident that a new strongfield regime was being entered [34] The conversion efficiency still fell away for the lower harmonics but, after about the seventh or ninth, the efficiency remained essentially constant across a broad plateau that ended in a fairly abrupt high-frequency cut-off The cut-off could be extended to higher frequencies by increasing the laser intensity, up to a saturation intensity beyond which no further extension of the plateau was possible By the early 1990s, harmonic orders well into the 100s were generated in neon [35] The simplest picture of this high harmonic generation (HHG) process is based on the so-called three-step Umax ¼ 3:17UP ; K ð8Þ where UP is the ponderomotive energy defined by   e2 ð9Þ UP ¼ Il2 : 8p2 e0 c3 m Figure Set of classical electron trajectories in HHG showing the free electron displacement X (grey lines) and the velocity V (black lines) Each member of the set is for a different ionisation time within the first quarter cycle of the driving field (top) (Reproduced from [21] with permission Copyright (2011), Cambridge University Press.) Downloaded by [Vietnam National University Ho Chi Minh] at 03:37 23 March 2016 290 G.H.C New Figure Typical HHG spectrum showing the characteristic broad plateau, and the sharp cut-off at around the 60th harmonic; see text for details (Courtesy of J.W.G Tisch, Imperial College Attosecond Laboratory; reproduced from [21] with permission Copyright (2011), Cambridge University Press.) In this equation, e and m are the electron charge and mass, and l is the wavelength of the fundamental driving field As noted already, the energy released at recollision includes the energy of ionisation UI, so the maximum max harmonic photon energy is (UK þ UI) In the region of interest, the kinetic energy term is the larger of the two, max so dividing UK by the photon energy hc/l gives a rough idea of the harmonic order that can be generated The result can be expressed in the convenient form HK ¼ 2:39  10À13 ðlðmmÞÞ3 IðW cmÀ2 Þ: ð10Þ Clearly, for l ¼ mm, little of interest is likely to happen at intensities below about 1013 W cm72 At 1014W cm72, on the other hand, HK is approaching 50 and, since the contribution HI from UI is around 13 for neon, harmonics up to around 60 can be expected An experimental spectrum obtained at intensities of this order using neon as the parent atom, and excitation pulses a few femtoseconds in duration is presented in Figure It shows a broad plateau of harmonics stretching up to about the 60th harmonic, after which there is the sharp cut-off characteristic of HHG spectra We have already noted how wide bandwidth radiation always hints at the potential for very narrow pulses and, in this case, the bandwidth theorem indicates that, if the phases of the harmonic can be suitably organised, pulses of around 50 as (1 as ¼ 10718 s) would be produced So far, pulses of less than 100 as have been reliably recorded (e.g [37]), but this too lies at the current research frontier Conclusion We have seen how nonlinear optics grew from the landmark SHG experiment of 1961 into an extensive field in its own right by 1970 In the intervening 40 years, it has continued to expand at an astonishing rate to become, in this 50th anniversary year, a vast and vibrant field at the forefront of twenty-first century technology As we have noted, developments in the field have been greatly facilitated by technological advances, particularly in materials fabrication While most of the foundation principles of the subject were established in the early years, there have been notable exceptions to this rule, and high harmonic generation has been taken as an example of a truly novel research frontier that is growing rapidly at the current cuttingedge In conclusion, I am tempted to recycle the old chestnut that nonlinear optics has ‘come of age’ at 50 But that hints at the onset of middle life and an ending in decrepitude, and neither fate is in any way appropriate What we see is rather the maturity of a large and established field of science that retains all the energy of youth, and can only grow and grow in the future With so many real and potential applications, the second half century of nonlinear optics is guaranteed to be at least as exciting as the first Acknowledgements I would like to thank Professor Richard Thompson and Professor Sir Peter Knight FRS for reading the manuscript in draft, and making a number of helpful suggestions I am greatly indebted to Professor John Ward for drawing my attention to reference but, much more importantly, for introducing me to nonlinear optics 46 years ago! Notes The Pockels effect is also known as the electro-optic effect, or sometimes (confusingly) as the linear electrooptics effect, because the refractive index changes are linearly proportional to the applied DC field Polarisation in this sense refers to the polarisation of matter, and should not be confused with the polarisation of light Written P in EM theory, its units are coulombs per square metre This was the author’s 18th birthday, so a career in laser physics was clearly ‘meant’ The scene and the slide rules were faithfully reproduced in the 1995 movie ‘Apollo 13’ Charles Kao was awarded a share in the 2009 Nobel Prize for Physics for his contribution to the development of optical fibre communications No trace of the harmonic can be seen in the figure printed in [6] It seems that someone at the journal removed the harmonic signal with Tippex, thinking it was a blemish! The incident is recorded in [7] Unfortunately there is no generally agreed definition of coherence length in nonlinear optics, and at least three other formulae are quoted in the literature that differ from this one by small numerical factors The same group at Ford under Robert Terhune made many other important contributions to nonlinear optics particularly their work on third harmonic generation [15] and other third-order processes [15] (see Section 5) Contemporary Physics 10 11 Downloaded by [Vietnam National University Ho Chi Minh] at 03:37 23 March 2016 12 13 See Figure 10 of [8] The reference is to Sir George Stokes, who first discussed the change of wavelength of light in the context of fluorescence in a famous paper of 1852 The terminology is increasingly used to describe any downshifted frequency The wave of primary interest is normally named the signal Whether it is labelled wave or 2, and whether its frequency is higher or lower than that of the idler is not defined However, multilayer dielectric stacks are examples of miniature periodic structures in 1D, and their properties were studied in the late nineteenth century Their application in anti-reflection coatings and multilayer dielectric mirrors were well known in the 1960s A selection of citations to this work can be found in [38] Notes on contributor As a postgraduate student at Oxford in the mid-1960s, Professor Geoff New, with his D.Phil supervisor Professor John Ward, performed the first experiment in which frequency conversion (third harmonic generation) was observed in the gas phase After spending six years at Queen’s University Belfast, Geoff New moved to Imperial College London with Professor Dan Bradley’s group in 1973 Since then, he has published well over 100 theoretical and computational papers on laser physics and nonlinear optics with special emphasis on ultrafast phenomena and exotic optical beams He has recently completed a book entitled Introduction to Nonlinear Optics (Cambridge 2011) He was a Rank Prize Fund Fellow from 1975 to 1978, and a Leverhulme Emeritus Fellow from 2007 to 2009 He has been a Fellow of the Optical Society of America since 1984 References [1] J Kerr, A new relation between electricity and light: dielectrified media birefringent, Phil Mag (4th Series) 50 (1875), p 337 [2] F Pockels and Abhandl Gesell, Wiss, Go¨ttingen 39 (1893), p [3] T.H Maiman, Stimulated optical radiation in ruby, Nature 187 (1960), p 493 [4] A Javan, W.R Bennett Jr., and D.R Herriot, Population inversion and continuous optical maser oscillation in a gas discharge containing a HeNe mixture, Phys Rev Lett (1961), p 106 [5] K.C Kao and G.A Hockham, Dielectric-fibre surface waveguides for optical frequencies, Proc IEEE 113 (1966), p 1151 [6] P.A Franken, A.E Hill, C.W Peters, and G Weinreich, Generation of optical harmonics, Phys Rev Lett (1961), p 118 [7] Niels Bohr Library & Archives Available at: http:// www.aip.org/history/ohilist/4612.html [8] J.A Giordmaine, Mixing of light beams in crystals, Phys Rev Lett (1962), p 19; P.D Maker, R.W Terhune, M Nisenoff, and C.M Savage, Effect of dispersion and focusing on the production of optical harmonics, Phys Rev Lett (1962), p 21 [9] J.A Armstrong, N Bloembergen, J Ducuing, and P.S Pershan, Interactions between light waves in a nonlinear dielectric, Phys Rev 127 (1962), p 1918 291 [10] M Bass, P.A Franken, J.F Ward, and G Weinreich, Optical rectification, Phys Rev Lett (1962), p 446 [11] M Bass, P.A Franken, A.E Hill, C.W Peters, and G Weinreich, Optical mixing, Phys Rev Lett (1962), p 18 [12] P.D Maker, R.W Terhune, and C.M Savage, Optical harmonic generation in calcite, Phys Rev Lett (1962), p 404 [13] J.A Giordmaine and R.C Miller, Tunable coherent parametric oscillation in LiNbO3, Phys Rev Lett 14 (1965), p 973 [14] R.Y Chiao, E Garmire, and C.H Townes, Selftrapping of optical beams, Phys Rev Lett 13 (1964), p 479 [15] P.D Maker, R.W Terhune, and C.M Savage, Intensity-dependent changes in the refractive index of liquids, Phys Rev Lett 12 (1964), p 507; P.D Maker, and R.W Terhune, Study of optical effects due to an induced polarization third order in the electric field strength, Phys Rev 137 (1965), p A801 [16] E.J Woodbury and W.K Ng, Ruby laser operation in the near IR, Proc IRE 50 (1962), p 2367 [17] R.Y Chiao, C.H Townes, and B.P Stoicheff, Stimulated Brillouin scattering and coherent generation of intense hypersonic waves, Phys Rev Lett 12 (1964), p 592 [18] F.J McClung and R.W Hellwarth, Giant optical pulsations from ruby, J Appl Phys 33 (1962), p 828 [19] J.F Ward and G.H.C New, Optical rectification in ammonium dihydrogen phosphate, potassium dihydrogen phosphate and quartz, Proc Roy Soc (Lond) A229 (1967), p 238 [20] C.L Tsangaris, Transverse effects in optical cavities and nonlinear optics, Ph.D thesis, Imperial College London, 2005 [21] G.H.C New, Introduction to Nonlinear Optics, Cambridge University Press, Cambridge, 2011 [22] F DeMartini, C.H Townes, T.K Gustafson, and P.L Kelley, Self-steepening of light pulses, Phys Rev 164 (1967), p 312 [23] G Rosen, Electromagnetic shocks and the self-annihilation of intense linearly polarised radiation in an ideal dielectric material, Phys Rev 139 (1965), p A539 [24] S.B.P Radnor, L.E Chipperfield, P Kinsler, and G.H.C New, Carrier-wave self-steepening and applications to high-order harmonic generation, Phys Rev A 77 (2008), p 033806 [25] N Bloembergen and P Lallemand, Complex intensitydependent index of refraction, frequency broadening of stimulated Raman lines, and stimulated Rayleigh scattering, Phys Rev Lett 16 (1966), p 81 [26] R.A Fisher, P.L Kelley, and T.K Gustafson, Subpicosecond pulse generation using the optical Kerr effect, Appl Phys Lett 14 (1969), p 140 [27] N.J Zabusky and M.D Kruskal, Interactions of ‘‘solitons’’ in a collisionless plasma and the recurrence of initial states, Phys Rev Lett 15 (1965), p 240 [28] A Hasegawa and F Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibres, Appl Phys Lett 23 (1973), pp 142–171 [29] D.E Spence, P.N Kean, and W Sibbett, 60-fsec pulse generation from a self-mode-locked Ti:sapphire laser, Opt Lett 16 (1991), p 42 [30] S.L McCall and E.L Hahn, Self-induced transparency by pulsed coherent light, Phys Rev 183 (1969), p 457 Downloaded by [Vietnam National University Ho Chi Minh] at 03:37 23 March 2016 292 G.H.C New [31] S.E Harris, Electromagnetically induced transparency, Phys Today 507 (1997), pp 36–42 [32] Slow Light Special Issue, Nat Photonics (2008), pp 447–509 [33] J.F Ward and G.H.C New, Optical third harmonic generation in gases by a focused laser beam, Phys Rev 185 (1969), p 57 [34] A McPherson, G Gibson, H Jara, U Johann, T.S Luk, I.A McIntyre, K Boyer, and C.K Rhodes, Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases, J Opt Soc Am B (1987), p 595; M Ferray, A L’Huillier, X.F Li, L.A Lompre´, G Mainfray, and C Manus, Multiple-harmonic conversion of 1064 nm radiation in rare gases, J Phys B 21 (1988), p L31 [35] J.J Macklin, J.D Kmetec, and C.L Gordon III, Highorder harmonic generation using intense femtosecond pulses, Phys Rev Lett 70 (1993), p 766; A L’Huillier, and Ph Balcou, High-order harmonic generation in rare gases with a 1-ps 1053-nm laser, Phys Rev Lett 70 (1993), p 774 [36] P.B Corkum, Plasma perspective on strong field multiphoton ionization, Phys Rev Lett 71 (1993), p 1994; K.J Schafer, B Yang, L.F DiMauro, and K.C Kulander, Above threshold ionization beyond the high harmonic cutoff, Phys Rev Lett 70 (1993), p 1599 [37] E Goulielmakis, M Schultze, M Hofstetter, V.S Yakovlev, J Gagnon, M Uiberacker, E.M Gullikson, D.T Attwood, R Kienberger, F Krausz, and U Kleineberg, Single-cycle nonlinear optics, Science 320 (2008), p 1614 [38] J.G Eden, High-order harmonic generation and other intense optical field-matter interactions: review of recent experimental and theoretical advances, Prog Quantum Electron 28 (2004), p 197  ... University Press.) Figure The early growth of nonlinear optics The Harvard paper set the scene for the gold-rush period in nonlinear optics that occurred in the mid1960s Many new nonlinear phenomena... many years (or even decades) before the idea came to full fruition Several further observations can be made about the state of nonlinear optics in 2011 Firstly, laser physics and nonlinear optics. .. physics, the 1950s had seen the development of the maser, the microwave device based on stimulated emission that predated the laser By the end of that decade, competition to extend the maser