Nonlinear and Quantum Optics
at Poznan University
(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 close to
perfection many years ago, it is now reaching a still higher
degree of completeness and might". Thus wrote Arkadiusz
Piekara in 1966, in the Preface to the first editon of his
book "The new face of optics" . The 1960-ies, after the
first laser had become operative in 1960, witnessed a period
of rapid progress in nonlinear optics --- a new branch of the physics
of light which, if generated by a laser and projected onto a
medium 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 University. In 1956, A.D.Buckingham
 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 10E+15 Hz the permanent dipole moments of the
molecules fail to keep pace with the field, and the mechanism
that can cause reorientation of the molecules in such fields
resides in interaction of the field with a dipole moment induced
in the molecule. The theory of this effect was
developed during the 1950-ies, prior to the coming of
lasers, by Arkadiusz Piekara and Stanislaw Kielich, the
latter then in the first stage of his scientific career.
Consequently, nonlinear optics came into existence in
Poznan even before the first laser, and became 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
nonlinear optical research. Work on the construction of a maser started in 1960
under the guidance of (now) professor Jan Stankowski whereas work on a laser
began under that of (now) professor Franciszek Kaczmarek. These attempts
proved successful ---- the first laser became 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" . 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 , as well as the handbook Introduction
to Laser Physics by Franciszek Kaczmarek  appeared. In 1964, the first REK
Biennial Conference on Radiospectroscopy and Quantum Electronics took place
in Poznan, to split in 1974 into two thematically independent Conferences:
RAMIS - on Radiospectroscopy, and EKON - on Quantum Electronics and
Nonlinear Optics. The Organisational Committee of the latter of these
Conferences was headed by professor Stanislaw Kielich. A calendarium
of these Conferences is to be found in Ref. . 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
As a personal contribution to this short account of the work carried
out in Poznan, I would like to say, something about the structure and
organisational aspect of our nonlinear and quantum optical studies and, in
particular the memorable work of professor Stanislaw Kielich and his
pupils: to start with, nonlinear optics is at present studied in the following
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 of being a pupill of professor Stanislaw Kielich.
I have worked under his guidance for more than a quarter of a century
in a field of research that opened up at a rate that gave us the feeling 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 the results we obtained during these 30 years of
inspired actiivity --- a detailed presentation is to be found in the
monographs of professor Piekara  and professor Kielich  as well as the
recently published three volumes "Modern Nonlinear Optics"  edited
by M.Evans and S.Kielich, volume 1 of which contains several review articles
presenting the achievements of the "Poznan School" of nonlinear optics.
Here, I shall 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 Poznan 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 at 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 his paper  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 observation of nonlinear interaction of
light and atoms. So we started from theory. 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 shiefly in cooperation
with professor Piekara, and also independently, others in cooperation with
A. Chelkowski, concerning dielectric saturation, Kerr's 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, intrinsic electric
moment, hyperpolarizability). 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 they appeared in the liquids, mostly solutions
of molecules with will defined properties, e.g. a permanent electric moment, in
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. . These papers are, in fact, the earliest
on nonlinear optics carried out in Poznan. Since that time, optical fields
became a permanent factor of our studies.
Light scattering was to become, for many years, a subject of interest
to Kielich and, later, to his pupils. In 1960, Kielich proposed  a
general molecular-statistical theory of light scattering by isotropic media
composed of polar and anisotropic molecules. In his theory, Kielich effects
a separation beetwen the isotropic and anisotropic parts of scattered light and
shows that the anisotropic part is dependent on the angular correlations
between anisotropic molecules. Likewise to effects of saturation, the chief
motivation in the study of 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"  did appear in 1963, it
concerned essentially linear (from the point of view of the optical field)
light scattering by a medium acted on by a strong 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"  in which he proposes a nonlinear (from the viewpoint
of the optical field) theory of light scattering, where there appear
components of light scattered at frequencies 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 double that of the incident light
beam. This 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 
showed that liquids with spherically 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 
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 . For the description of molecular reorientation in such fields
he introduced generalized Langevin functions, now often referred to as
Langevin-Kielich functions. His theory provides a correct description
of light scattering by solutions of macromolecules and colloidal
particles acted on by strong constant as 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. ), T.Bancewicz
and Z.Ozgo (see. eg. ) to multiharmonic light scattering as well as the
study of the spectral properties of nonlinearly i.e. hyper Rayleigh and
hyper Raman scattered light.
< BR >
The phenomenon of optical reorientation of molecules --- a subject of
interest 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-Mouton effect, in which the roles of the measuring fields and the
field polarizing the medium are inversed. The role of a measuring field
is 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 anisotropy
induced in an isotropic medium by a strong laser beam was proposed by
Kielich in 1967 . Also in 1967 there appears a thermodynamical theory
of electric and magnetic anisotropy taking into account contributions from
opticostriction and the opticocaloric effect  and, in 1969, a
molecular-statistical theory of nonlinear magneto-optical effects in
colloids . Again in 1969, Kielich pointed to  the possibility of
nonlinear changes in optical activity in liquids. Such changes in optical
activity were observed by Vlasov and Zaitsev  in 1971. Molecular
reorientation in optical fields is also on of the mechanisms of a Kerr
effect residing in the induction of birefringence 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 
drawing attention to the various mechanisms contributing 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  and their
contribution to the nonlinear part of the refractive index was
considered in detail by Kielich and Wozniak .
An isotropic medium (gaseous or liquid) becomes anisotropic
where acted on by an electric or a magnetic field. The anisotropy
induced by the external field permits the observation of effects
that cannot take place in isotropic media. Second-harmonic generation
is an effect of this kind. Kielich  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
orientation 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 proposes a relaxational
theory of optically induced birefringence in 1966 . The theory
was further developed by S.Kielich, B.Kasprowicz-Kielich,
W.Alexiewicz and J.Buchert (see. e.g. ). Kielich in cooperation with R.Zawodny
worked out a theory of nonlinear effects in magnetized crystals
and isotropic solids  (ee also ). Beside 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 independent tensor components of the
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 in 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 usually 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 evidence 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 statistics 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  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
thg effect of nonlinear optical activity can lead to the evolution
of photon anticorrelation. The method is sometimes referred to as
"the method 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
 photon antibunching can occur in second-harmonic generation and,
in cooperation with Kielich, we generalized this result to higher
harmonics . These results met with widespread acknowledgement in
various places. Suffice it to say that Mandel and Wolf, in their
modern book on optics , 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 interest 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 below the level defined by
the state of photon vacuum is highly promising, 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 research group we studied i.a. harmonics generation ,
fluorescence at resonance  and light propagation in a Kerr medium
. 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
, 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 .
As I mention 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 arrived to Rydzyna (if was
I who was to read our joint report) I was told that professor Piekara
wished to see me. I found him in the park surrounding the palace.
So 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 is? 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 formulae, "what that squeezing was".
I had but recently obtained my degree of Dr. Sci., so you can
imagine the impression the conversation made on me, especially in Rydzyna,
and especially with professor Piekara. There and then, I felt as if "was again
revealing 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 , quantum beating , the
production of quantum states of the field as superpositions of
macroscopically discernable coherent states (so-called Schrodinger
cats) , collective vanishing and revivals of Rabi oscillations
, Jaynes-Commings models , classical and quantum analysis
of chaos in nonlinear dynamics , and the feasibility of
producing one-photon states . 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 recently been invited to present a review of our,
though not only our results for Progress in Optics  with a quantum
discussion of the phase of the optical field produced in nonlinear
At present, beside research in quantum optics, work in our
Laboratory of Nonlinear Optics proceeds along lines initiated by
professor S.Kielich: the theory of dielectric relaxation 
is developed, nonlinear effects in optically active liquids  are studied,
as well as light scattering spectra.
These are but some recent examples. Earlier results have been
published in Ref. .
As mentioned above, the Laboratory of Nonlinear Optics is not
the only laboratory where nonlinear and quantum optical studies are
persumed. However, I do not feel competent 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. ) 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
concentrating 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.
 A.H.Piekara, The new visage of optics (in Polish) (PWN,
 A.D.Buckingham, Proc.Phys.Soc. B69, 344 (1956).
 On the 25-th anniversary of lasers (in Polish) No.55
in Fizyka, edited by F.Kaczmarek (Wydawnictwo Naukowe UAM,
 J.Stankowski and A.Graja, An introduction to quantum
electronics (in Polish) (Wydawnictwa Komunikacji i Lacznosci
 F.Kaczmarek, An introduction to the physics of lasers (PWN,
 S.Kielich Molecular nonlinear optics (Russian edition,
completed) (PWN, Warszawa, 1977), (Nauka, Moskwa, 1981).
 Modern Nonlinear Optics, Vol.85 of Advances in Chemical Physics,
edited by M.Evans and S.Kielich, J.Chem.Phys. 29, 1297 (1958).
 A.Piekara and S.Kielich, J.Chem.Phys. 29, 1297 (1958).
 A.Piekara and S.Kielich, Archives des Sciences 11, 304 (1958),
(C.R. du 7-e Colloque Ampere, Paris, 1958).
 S.Kielich, A.Piekara, Acta Phys. Pol. 18, 439 (1959).
 S.Kielich, Acta Phys. Pol. 19, 149 (1960).
 S.Kielich, Acta Phys. Pol. 23, 321 (1963).
 S.Kielich, Acta Phys. Pol. 25, 85 (1964).
 R.W.Terhune, P.D.Maker and C.M.Savage, Phys. Rev. Lett. 14,
 S.Kielich, Acta Phys. Pol. 32, 297 (1967).
 S.Kielich, J.R.Lalanne and F.R.Martin, Phys. Rev. Lett., 26
 S.Kielich, Acta Phys. Pol. A37, 719 (1970).
 M.Kozierowski, Phys. Rev. A47, 723 (1993).
 T.Bancewicz and Z.Ozgo, in Ref. , p.89.
 S.Kielich, Physica 34, 365 (1967).
 S.Kielich, Acta Phys. Pol. 32, 405 (1967).
 S.Kielich, J. Colloid and Interface Sci. 30, 159 (1969).
 S.Kielich, Acta Phys. Pol. 35, 861 (1969).
 D.V.Vlasov and V.P.Zaitsev, JETP Lett. 14, 112 (1971).
 A.Kielich and S.Wozniak, Acta Phys. Pol. A39, 233 (1971).
 S.Kielich, Chem. Phys. Lett. 2, 569 (1968).
 S.Kielich, Acta Phys. Pol. 30, 683 (1966).
 W.Alexiewicz and B.Kasprowicz-Kielich, in. Ref. , p.1.
 S.Kielich and R.Zawodny, Acta Phys. Pol. A43, 579 (1973).
 R.Zawodny, in Ref. , p.307.
 R.Tanas, Optik 40, 109 (1974).
 M.Kozierowski and R.Tanas, Opt.Commun. 21, 229 (1977).
 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.
 L.Mandel and E.Wolf, Optical Coherence and Quantum Optics
(Cambridge University Press, New York, 1995).
 M.Kozierowski and S.Kielich, Phys. Lett. A94, 213 (1983).
 Z.Ficek, R.Tanas and S.Kielich, Opt. Commun. 46, 23 1983).
 R.Tanas and S.Kielich, Opt. Commun. 45, 351 (1983).
 R.Tanas, in Coherence and Quantum Optics V, edited by
L.Mandel and E.Wolf (Plenum, New York, 1984), p.645.
 S.Kielich, M.Kozierowski and R.Tanas, Optica Acta 32, 1023 (1987).
 W.Leonski, R.Tanas and S.Kielich, J.Opt. Soc.Am. B4, 72 (1987).
 Z.Ficek, R.Tanas and S.Kielich, J.Mol.Opt. 35, 81 (1988).
 A.Miranowicz, R.Tanas and S.Kielich, Quantum Opt. 2, 253 (1990).
 M.Kozierowski, S.M.Chumakov, J.Swiatlowski and A.A.Mamedov,
Phys. Rev. A46, 7220 (1992).
 P.Szlachetka, K,Grygiel and J.Bajer, Phys. Rev. E48, 101 (1993).
 W.Leonski and R.Tanas, Phys. Rev. A49, R20 (1994).
 R.Tanas, A.Miranowicz and T.Gantsog in Progress in Optics, edited
by E.Wolf (Elsevier, Amsterdam, 1996), Vol. 35, p.355.
 W.Alexiewicz and H.Derdowska-Zimpel, Physica A214, 9 (1995).
 S.Wozniak and G.Waniere, Opt. Commun. 114, 131 (1995).
 T.Bancewicz, Chem. Phys. Lett. 244, 305 (1995).
 A.Wojcik and R.Parzynski, Phys. Rev. A50, 2475 (1994).
 F.Kaczmarek and R.Parzynski, Laser Physics. Part.I: Introduction
to Quantum Optics (Wydawnictwo Naukowe UAM, Poznan,, 1990).
 F.Kaczmarek, Laser Physics. Part II: Quantum Electronics
(Wydawnictwo Naukowe UAM, Poznan, 1994).