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Nonlinear and Quantum Optics
at Poznan University

Ryszard Tanas

(Nauka, No 3, p.156 (1997) )

Once again, a revolution is taking place in physics: this time optics is revealing a new visage. And though brought dose to perfection many years ago, it is now reaching a still higher degree of complereness and might. Thus wrote Arkadiusz Piekara in 1966, in the Preface to the first editon of his book "The new face of optics" [1]. The 1960-ies, after the first laser had become operative in 1960, witnessed a period of rapid in nonlnear optics --- a new branch of the physics of light which, if generated by a laser and projected onto a [Cmedium consisting of atoms or molecules, leads to the induction of nonlinear polarisation by the strong electric and magnetic fields at optical frequencies unknown in linear optics. The nonlinear response of the medium to an externally applied electric field had been a subject of interest to Professor Piekara in the inter war period when, working in ths laboratory in Rydzyna, he discovered the effect of inverse dielectric saturation. He continued research along these lines after the war as Professor of Poznan Unversity. In 1956, A.D.Buckingham [2] proposed a theory of the orientation of molecules in the strong electric field of a light wave: in an optical field of the order of 10 Hz the permanent dipole moments of the molecules fail to Keep pace with the field, and the mechanism that can couse reorientation of the molecules in such field resides in interaction of the field with a dipole moment in duced in the molecule. The theory of this effect was developed during the 1950-ies, prior to the coming of laser, by Arkadiiusz Piekara and Stanislaw Kielich, the latter then in the first stage of his scentific career. Consequently, nonlinear optics came into existence in Poznan even before the first laser, and because a speciality of our scientific endeavors.

When the earliest lasers and masers made their apperance, Professor Piekara decided that Poznan should become a modern centre for their experimental all radiospectroscopic and nonlinear optical research. Work on the construction of a aser started in 1960 under the guidance of (now) professor Jan Stankowski whereas work on a laser began ander that of (now) professor Franciszek Kaczmarek. These attempts proved successful ---- the first laser bbecame operative in Poznan Unversity on December 5, 1963, and the first maser on January 2, 1964, under dramatic circumstances a more detailed description of which is to be found in a publicaton under the heading On 25 years of the laser [3]. Briefly, Poznan became the leading center for radiospetroscopy and quantum electronics. It was in Poznan that the handbook Interoduction to Quantum Electronics by Jan Stankowski and Andrzej Graja [4], as well as the handbook Introduction to Laur Physics by Franciszek Kaczmarek [5] appeared. In 1964, the first REK Biennial Conference on Radiospectroscopy and Quantum Electronics took place in Poznan, to split in 1974 inyo two thematically independent Conferences: RAMIS - on Radiospectroscopy, and EKON - on Quantum Electronics and Nonlinear Optics. The Organisational Connittee of the latter of these Conferences was headed by professor Stanislaw Kielich. A Kalendarium of these Conferences is to be found in Ref. [3]. The last EKON took place in 1980. The next, which should have taken place in 1982, had to be postponed sine die because martial law was introduced in Poland. EKON was disconntinued, its traditional formula requiring some modernisation. However, though this interruption is to be regretted, research in the field of quantum electronics and nonlinear optics is proceeding unninnterrupted.

As a personal contribution to this short occount of the workk carried out in Poznan, I would like to say, something about the structure and organisational aspect of oue nonlinear and quantum optical studies and, in particular the memorable worh of professor Stanislaw Kielich and his pupils: To start with, nonlinear optics is at present studied in the folllowing laboratorien: Quantum Electronics --- headed by professor Franciszek Kaczmarek Optics --- headed by professor Marian Surma Molecular Biophysics--- headed by professor Andrzej Dobek Nonlinear Optics --- headed by professor Stanislaw Kielich, deceased in 1993, and since then by the present author.

I have the honour pf being a pupilll of professor Stanislaw Kielich. I nave worked under his guidance for more than a quuarter of a century in a field of research that opened up at a rate that gave us the fealing that we belonged to the leading group of researchers. In fact, this was a source of great satisfaction. I regret considerations of space do not permit me to give a picture of all thge results we obtained during these 30 years of inspired actiivity --- a detailed presentationn is to be found in the monographs of professor Piekara [1] and professor Kielich [6] as well as the recently published three volumes od Modern Nonlinear Optics [7] edited by M.Evans and S.Kielich, volumme 1 of contains several review articles presenting the achievements of the "Poznan School" of nonlinear optics. Here, I shalll mention but some results which, I believe, have exerted a decive influence on our understanding of nonlinear optical effects.

As I stated above, nonlinear optics started in Poznann even before the apperance of the first lasers, at the impulse of professor Piekara whose interest in the nonlinear dielectric effect or, as we called it st the time, dielectric saturation, extended rather naturally from electric fields constant in time (or slowly varying ar radio frequencies) to fields at optical frequencies. In 1956, when Buckingham published has paper [2] on the reorientation of molecules in optical fields, professor Piekara immediately included the latter into the subject matter of our research. However, at the time, sufficiently strong sources of light ---- lasers ---- were still not available, such as would permit the experimental abservation of nonlinear interaction of light and atoms. So we started from theorz. The problem was approached by Stanislaw Kielich then a young assistant professor Piekara who has taken his M.Sci. degree in 1955. His earliest papers were written shefly in cooperation with [rpfessor Piekara, and also independently, others in cooperation with A. Chelkowski, concerning dielectric saturation, Kerrs effect, and Cotton-Mouton effect in dielectric liquids. The theoretical description of liquids composed of polar molecules was based on classical statistical physics permitting the connection between the macroscopic properties (the molar constants for the above named effects) and the properties of the individual molecules (their polarizability, inntrinsic electric moment, huperpolarizability). Moreover, the use of statistical physics enabled the separation of the contributions from the intermolecular interactions, so highly essential in liquids. This approach, disclosing the various contribution to the effect under investigation, led to a description of the factors responsible for the effects as thez appeared in the liquids, mostly solutions of molecules wiyh will defined properties, e.g. a permanent electric moment, in a nonlinear a nondipolar solvent. The earliest papers of Piekara and Kielich involving a strong optical field as a factor of molecular reorientation in dielectric liquids appeared in 1958 [8,9]. A complete description of the respective mechanism is to be found in Ref. [9]. These papers are,, in fact, the earliest on nonlinnear optics sarried out in Poznan. Since that time,, optical fields becane a permanent factor of our studies.

Light scattering was to become, for many years, a subject of innterest to Kielich and, later, to his pupils. In 1960, Kielich proposed [11] a general molecular-statistical theory of light scattering by isotropic media composed of polar and anisotropic molecules. In his theory, Kielich effects a separation the isotropic and anisotropic parts of scattered light and shows that the anisotropic part is dependent on the angular correlation between anisotropic molecules. Likewise to effectts of saturation, the chief motivation in the studz oof light scattering resides in the gaining of information on the medium (on the properties of the component molecules and their interactions) on the basis of macroscopic measurements, also concerning light scattered by the medium. From the viewpoint of the modern classification of nonlinear optical effects, the earliest papers deal with linear Rayleigh scattering and thus can hardly be counted as nonlinear. Although a paper On Non-Linear Light Scattering in Gases [12] did appear in 1963, it concerned essentially linear (from the point of view of the optical field) light scattering bya medium acted on by a stronmg constant electric or magnetic field. This approach was continued in later papers, and provided a good starting point for the development of the theory of nonlinear light scattering. In 1964, Kielich publishes a work on Light Scattering by an Intense Light Beam [13] in which he proposes a nonlinear (from the viewpoint of the optical field) theory of light scattering, where there appear components oflight scattered at freqiencies doubled and tripled in relation to the frequency of the strong incident beam. His papers on multiharmonic light scattering represent pioneering contributions to this field of research. Experimentally, second-harmonic light scattering has been observed in several liquids by Terhune et al in 1965. The liquids consisted of molecules without a centre of symmetry, so that even a single molecule could act as a dipole oscillating at a frequency dpuble that of the incident light beam. Thes could not be the case for centrosymmetric molecules --- a mechanism of this kind could not give second-harmonic scattering by liquids with centrosymmetric molecules. However, in 1967, Kielich [15] showed that liquids with sphericallly symmetric molecules could in fact exhibit second-harmonic light scattering due to interaction between the molecules. Later, scattering of this kind was observed and referred to by Kielich in cooperation with French physicists at Bordeaux [16] in 1971; reorientation of the molecules in an intense electric or optical field modified the properties of the ordered molecular system. In 1970, Kielich proposes a theory of these modifications due to optical saturation i.e. to reorientation of anisotropic molecules in an intense optical field [17]. For the description oof molecular reorientation in such fields he introduced generalized Langevin functions, non often farred to as Langevin-Kielich functions. His theory provides a correct description of light scattering by solutions of macromolecules and colloidal particles acted on by constantas well as optical reorienting fields. Kielich succeeded in proving that the study of optical saturation provides not only the magnitude but moreover the sign of the optical anisotropy of the molecules. The work of Kielich was later extended in cooperation with his pupilds: M.Kozierowski (see e.g. [18]), T.Bancewicz and Z.Ozgo (see. eg. [19]) to multiharmonic light scattering as well as the study of the spectral properties of nonlinearlz i.e. hyper Rayleigh and hyper Raman scattered light.
< BR > The phenomenon of optical reorientation of molecules --- a subject of intrest to the researchers at Poznan from the very start of nonlinear optics --- can cause electric and magnetic anisotropy in a medium isotropic in the absence of an optical field. Anisotropy of this kind leads to so-called inverse effects: inverse Kerr effect, inverse Faraday effect, inverse Cotton-Moutonn effect, in which the roles of the measuring fields and the field polarizing the medium are inversed. The role of a measuring fields in taken over by a constant weak or slowly varying electric or magnetic field whereas an optical field assumes the role of a strong field, polarizing the medium linearly. The statistical theory of electric anisotropic induced in an isotropic medium by a strong laser beam was proposed by Kielich in 1967 [20]. Also in 1967 there appears a thermodynamical theory of electric and magnetic anisotropy taking into account contributions from opticastriction and the opticocaloric effect [21] and, in 1969, a molecular-statistical theory of nonlinear magneto-optical effects in colloids [22]. Again in 1969, Kielich pointed to [23] the possibility of nonlinear changes in optical actiivity in liquids. Such changes in optical activity were observed by Vlasov and Zaitsev [24] in 1971. Molecular reorientation in opticall fields is also on of the mechanisms of a Kerr effect residing in the induction of brefringence in a medium by a strong optical field. More generally, the refractive index of the medium can be said to be a nonlinnear function of the intensity of the light polarizing the medium. Such a nonlinear dependence of the index leads to self-focussing and self-collimation of light. The problem was dealt with and discussed in full detail by professor Piekara [1] deawing attention to the various mechanism conntributing to the effect, in particular the contribution from radial correlations of the molecules: the latter can appear in addition to contributions from molecular reorientation and deformation and differs from zero also for spherically symmetric molecules. Radial correlations were introduction by Kielich as early as in 1960 [11] and their contribution to the nonlinear part of the refractive undex was considered detail by Kielich and Wozniak [25].

An isotropic medium (gaseous or liquid) becomes anisotropic whenn acted on by an electric or a magnetic field. The anisotropy induced by the external field permits the ebservation of effects that cannot take place in isotropic media. Second-harmonic generation is an effect of this kind. Kielich [26] proposed a molecular- -statistical theory of the last named effect in liquids subjected to a constant electric field. If the measuring or polarizing fields are non-constant in time, the respective processes of molecular orientatiuon have to be described kinetically. The result of the action of the fields depends on whether the molecules keep pace with the variations of the field, or not. That is to say that we deal with relaxational processes. Kielich proposed a relaxational theory of optically induced birefringence in 1966 [27]. The theory was further developed by S.Kielich, B.Kasprowicz-Kielich, W.Alexiewicz and J.Buchert (see. e.g. [8]). Kielich with R.Zawodny worked out a theory of nonlinear effects in magnetized crzstals and isotropic solids [29] (ee also [30]). Baside isotropic media like gases and liquids, crystals became on object of investigation; their symmetries naturally suggest the recourse to group theory for the finding the nonzero and idependent tensor components of the nonlinear susceptibilities.

In the early years of nonlinear optics, nonlinear optical processes were a source of novel information bearing on the nonlinear media. The intense optical fields, polarizing the media, were dealt with clasically and the changes brought about is the fields as such by the nonlinear interaction with the medium were mostly left unconsidered. In particular, with regard to the enormous number of photons in strong laser beams, is was usuallyy assumed that the quantum nature of the field has no essential bearing on description of the nonlinear effects. The 1970-ies, however, were a time of very intense interest in quantum optics, that is, in the properties of light itself and its grain-like quantum structure. The nonlinear transformation of the light involved in a nonlinear process will modify its properties essentially bringing to eviidence its nonclassical nature. The subject matter of investigation extends to the statistical properties of light as described by higher order field correlation functions, determined from intensity and photon strisrics correlation measurements. This type of studies reaches Poznan. The earliest paper on photon statistics: On Nonlinear Optical Activity and Photon Statistics, by the present author, appeared in 1974 [31] and passed almost completely unnticed. It contained an approximate method of calculating the correlation of a quantum field propagating in a nonlinear medium used to prove that thge effect of nonlinear optical rectivity can lead to the evolution of photon anticorrelation. The method is sometimes referred to as "the of short optical paths" (or "the method of short paths" for evolution in time) and has been widely applied for the calculation of quantum correlation functions in other nonlinear processes. In cooperation with M.Kozierowski we showed for the first time that [32] photon antibunching can occur in second-harmonic generation and, in cooperation with Kielich, we generalized this result to higher harmonics [33]. These results met with widespread acknowledgement in various places. Suffice it to say that Mandel and Wolf, in their modern book on optics [34], make the following statement when dealing with quantum effects in second-harmonic generation: "We shall largely adopt the approach of Kielich and his colloborators (Kozierowski and Tanas, 1977, Kielich, Kozierowski and Tanas, 1978)". Photon anticorrelation, or sub Poissonian photon statistics are univocal proof of the quantum nature of the field and justify the high interest they give rise to. Essential for the obtaining of these nonclassical effects is the nonlinear transformation of the field that occurs in nonlinear processes. In this way, traditional nonlinear optics went over into nonlinear quantum optics. It appears that almost any nonlinear effect can be the source of a field with nonclassical properties. The search for such nonclassical properties was, for a time, an immportant object of our studies shared, in addition to the above named, by Z.Ficek and P.Szlachetka. Another problem of intetest concerning the quantum nature of fields which we dealt with in Poznan concerned the possibility of obtaining and investigating squeezed states. Originally, in English, the word stands for the squeezing of fluctuations of photon vacuum. The feasibility of reducing quantum noise is below the level defined by the state of photon vacuum. Highly propising, also from the point of view of optical communication and the enhancement of the sensitivity of optical instruments. Similarly to photon anticorrelation, squeezed states of the field can intervene in any nonlinear processes. In our esearch group we studied i.a. harmonics generation [35], fluorescence at resonance [36] and light propagation in a Kerr medium [37]. In the last case we laid a bridge, as it were,between the pioneering work concerning the optical Kerr effect which paved the way for nonlinear optics in Poznan on the one hand, and modern quantum optics on the other. Light propagating in a Kerr medium with intensity-dependent refractive index can be described simply on the model of an anharmonic oscillator, strictly solvable and leading to squeezed states of the field, with highly interesting properties (38), which became the object of intense studies. It may be worth noting that this model predicts a high degree of quantum noise attenuation (98%), thus a strong nonclassical state of the field, with a great number of photon in the light beam, a fact at variance with the common belief that fields with a great number of photons can be described classically. The results obtained in Poznan concerning photon anticorrelation and squeezed states of the field met with a most favourable reception. Our review article on the subject was published in a special volume of Optica Acta edited on the occasion of the 25-th anniversary of lasers [39].

As I mertion squeezed states, this may be the moment for a short digresion. In 1982, an International Optical Conference was taking place in Rydzyna at which I read a report on squeezed states of the field jointly with professor Kielich. The Conference was also attended by professor Piekara with a report. When I aeeived to Rydzyna (if was I who to read our joint report) I was told that professor Piekara wished to see me. I found him in the park surrounding the palace. S I approached him and asked him what were his wishes. He said: "Now I see you are to read a report on squeezing. Could you please explain to me what that squeezing in? But no formulae, please!" We found a bench in the park, sat down, and for half an hour or more I sat there eye to ego with Professor Piekara doing my best to explain to him, avoiding firmulae, "what that squeezing was". I had but recently obtained my degree of Dr. Sci., so you can (magine the impression the conversation made on me, especially with professor Piekara. There and then, I felt as if "was again nevealing a new face to me".

Photon statistics and squeezed states of the fields have been and are topics of strong interest for our group. In recent years our interest has extended to comprise yet other topics, such as laser-induced autoionisation [40], quantum beating [41], the production of quantum states of the field as superpositions of macroscopically discernable coherent states (so-called Schrodinger cats) [42] , collective vanishing and revivals of Rabi oscillations [43] Jaynes-Commings [18] models, classical and quantum analysis of chaos in nonlinear dynamics [44], and the feasibility of producing one-photon states [45]. Recently, much attention has been devoted to the quantum description of the phase of optical field produced in nonlinear optical processes. The quantum description of phase is still controversial, though much has already been done to clarify the problem. We have succeeded in joining in these new studies actively with some 30 widely quoted papers. The results obtained by our group are hardly adapted for a discussion in this Report but we have recetly been invited to present a review of our, though not only our results for Progress in Optics [46] with a quantum discussion of the phase of the optical field produced in nonlinear processes.

At present, beside research in quantum optics, work in our Laboratory of Nonlinear Optics proceeds along lines initial by professor S.Kielich: the theory of dielectric relaxation [47] is developed, nonlinear effects in optically active [48] are studied, as well as light scattering spectra. These are but some recent examples. Earlier results have been published in Ref. [7].

As mentioned above, the Laboratory of Nonlinear Optics is not the ohly laboratory where nonlinear and quantum optical studies are pursued. However, I do not feel componentr to give an account of the work carried out by Colleagues in the other Laboratories. I would like to mention the outstanding results of professor R.Parzynski and Dr Wojcik, of the Quantum Electronics Laboratory (see, e.g. [50]) as well as the monographs of F.Kaczmarek and R.Parzynski published by Adam Mickiewicz University [51, 52].

Considerations of volume forbid a move detailed discussion of the results obtained here during the 40 years of nonlinear optical studies (a list of the titles alone would exceed the limits of the present Report). So I has to give a strict selection consentrating on the most essential facts. As a pupil of Stanislaw Kielich and a "sientific grandson" of Arkadiusz Piekara (these are the words of the latter), I felt it my date to outline the path traversed in the Nonlinear Optics Laboratory of Poznan University since the earliest paper in 1958.


[1] A.H.Piekara, The new visage of optics (in Polish) (PWN, Warszawa, 1968).
[2] A.D.Buckingham, Proc.Phys.Soc. B69, 344 (1956).
[3] On the 25-th anniversary of lasers (in Polish) No.55 in Fizyka, edited by F.Kaczmarek (Wydawnictwo Naukowe UAM, Poznan, 1987).
[4] J.Stankowski and A.Graja, An introduction to quantum electronics (in Polish) (Wydawnictwa Komunikacji i Lacznosci Warszawa, 1972).
[5] F.Kaczmarek, An introduction to the physics of lasers (PWN, Warszawa, 1978).
[6] S.Kielich Molecular nonlinear optics (Russian edition, completed) (PWN, Warszawa, 1977), (Nauka, Moskwa, 1981).
[7] Modern Nonlinear Optics, Vol.85 of Advances in Chemical Physics, edited by M.Evans and S.Kielich, J.Chem.Phys. 29, 1297 (1958).
[8] A.Piekara and S.Kielich, J.Chem.Phys. 29, 1297 (1958).
[9] A.Piekara and S.Kielich, Archives des Sciences 11, 304 (1958), (C.R. du 7-e Colloque Ampere, Paris, 1958).
[10] S.Kielich, A.Piekara, Acta Phys. Pol. 18, 439 (1959).
[11] S.Kielich, Acta Phys. Pol. 19, 149 (1960).
[12] S.Kielich, Acta Phys. Pol. 23, 321 (1963).
[13] S.Kielich, Acta Phys. Pol. 25, 85 (1964).
[14] R.W.Terhune, P.D.Maker and C.M.Savage, Phys. Rev. Lett. 14, 681 (1965).
[15] S.Kielich, Acta Phys. Pol. 32, 297 (1967).
[16] S.Kielich, J.R.Lalanne and F.R.Martin, Phys. Rev. Lett., 26 1295 (1971).
[17] S.Kielich, Acta Phys. Pol. A37, 719 (1970).
[18] M.Kozierowski, Phys. Rev. A47, 723 (1993).
[19] T.Bancewicz and Z.Ozgo, in Ref. [7], p.89.
[20] S.Kielich, Physica 34, 365 (1967).
[21] S.Kielich, Acta Phys. Pol. 32, 405 (1967).
[22] S.Kielich, J. Colloid and Interface Sci. 30, 159 (1969).
[23] S.Kielich, Acta Phys. Pol. 35, 861 (1969).
[24] D.V.Vlasov and V.P.Zaitsev, JETP Lett. 14, 112 (1971).
[25] A.Kielich and S.Wozniak, Acta Phys. Pol. A39, 233 (1971).
[26] S.Kielich, Chem. Phys. Lett. 2, 569 (1968).
[27] S.Kielich, Acta Phys. Pol. 30, 683 (1966).
[28] W.Alexiewicz and B.Kasprowicz-Kielich, in. Ref. [7], p.1.
[29] S.Kielich and R.Zawodny, Acta Phys. Pol. A43, 579 (1973).
[30] R.Zawodny, in Ref. [7], p.307.
[31] R.Tanas, Optik 40, 109 (1074).
[32] M.Kozierowski and R.Tanas, Opt.Commun. 21, 229 (1977).
[33] S.Kielich, M.Kozierowski and R.Tanas, in Coherence and Quantum Optics IV, edited by L.Mandel and E.Wolf (Plenum New York, 1978), p.511.
[34] L.Mandel and E.Wolf, Optical Coherence and Quantum Optics (Cambridge UniversityPress, New York, 1995).
[35] M.Kozierowski and S.Kielich, Phys. Lett. A94, 213 (1983).
[36] Z.Ficek, R.Tanas and S.Kielich, Opt. Commun. 46, 23 1983).
[37] R.Tanas and S.Kielich, Opt. Commun. 45, 351 (1983).
[38] R.Tanas, in Coherence and Quantum Optics V, edited by L.Mandel and E.Wolf (Plenum, New York, 1984), p.645.
[39] S.Kielich, M.Kozierowski and R.Tanas, Optica Acta 32, 1023 (1987).
[40] W.Leonski, R.Tanas and S.Kielich, J.Opt. Soc.Am. B4, 72 (1987).
[41] Z.Ficek, R.Tanas and S.Kielich, J.Mol.Opt. 35, 81 (1988).
[42] A.Miranowicz, R.Tanas and S.Kielich, Quantum Opt. 2, 253 (1990).
[43] M.Kozierowski, S.M.Chumakov, J.Swiatlowski and A.A.Mamedov, Phys. Rev. A46, 7220 (1992).
[44] P.Szlachetka, K,Grygiel and J.Bajer, Phys. Rev. E48, 101 (1993).
[45] W.Leonski and R.Tanas, Phys. Rev. A49, R20 (1994).
[46] R.Tanas, A.Miranowicz and T.Gantsog in Progress in Optics, edited by E.Wolf (Elsevier, Amsterdam, 1996), Vol. 35, p.355.
[47] W.Alexiewicz and H.Derdowska-Zimpel, Physica A214, 9 (1995).
[48] S.Wozniak and G.Waniere, Opt. Commun. 114, 131 (1995).
[49] T.Bancewicz, Chem. Phys. Lett. 244, 305 (1995).
[50] A.Wojcik and R.Parzynski, Phys. Rev. A50, 2475 (1994).
[51] F.Kaczmarek and R.Parzynski, Laser Physics. Part.I: Introduction to Quantum Optics (Wydawnictwo Naukowe UAM, Poznan,, 1990).
[52] F.Kaczmarek, Laser Physics. Part II: Quantum Electronics (Wydawnictwo Naukowe UAM, Poznan, 1994).

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