Mode Theory of 3D Hologram

ISSN 0030󰀎400X, Optics and Spectroscopy, 2012, Vol. 112, No. 2, pp. 305–311. © Pleiades Publishing, Ltd., 2012.Original Russian Text © V.G. Sidorovich, 2012, published in Optika i Spektroskopiya, 2012, Vol. 112, No. 2, pp. 335–342.

3051. STATEMENT OF THE PROBLEMApproximately for 15 years after Yu.N. Denisiukdiscovered the effect of light field reproduction by 3Dhologram in 1962 [1], the only available method ofquantitative analysis of 3D holograms was the so󰀎called kinematic theory, which can be used at low lightscattering efficiency of a hologram. Simultaneously,researchers used the theory of X󰀎ray scattering in crys󰀎tals (developed by P. Ewald and M. Laue as early as inthe beginning of the 20th century [2]), which made itpossible to analyze the high󰀎efficiency transformationof a probe X󰀎ray wave into scattered radiation in thesimplest particular cases. This theory was referred to asdynamic. The dynamic theory was successfullyapplied by G. Kogelnik to analyze the diffraction effi󰀎ciency of the simplest hologram, recorded using twoplane light waves and composed of a single sinusoidalspatial refractive index grating [3].The early failures in the generalization of thedynamic theory of light scattering in a hologram fromone sinusoidal grating to a set of such gratings, whichcoexist inside a hologram, were caused by the extraor󰀎dinary complexity of the system of equations describ󰀎ing the scattering from a gratings assembly.

2. APPROACH TO SOLUTION OF THE PROBLEMA careful consideration of the system of equationsdetermining the light propagation in a hologram [4–6]reveals that the entire variety of the terms in the equa󰀎tion describing the propagation of each individualplane wave in the hologram is divided into two non󰀎equivalent groups.The first group corresponds to the scattering of aplane wave from all gratings in recording of which thiswave was involved when forming a hologram by inter󰀎fering with other plane waves (below, they will bereferred to as “intrinsic” gratings). For each planereading wave, the number of intrinsic gratings is N – 1(Nis the number of plane components of the lightrecorded in the hologram). Although this number ismany billions in cases of practical importance, it issmall in comparison with the total number of gratings:N(N – 1). However, under certain conditions, thisparticular group (despite its “small” number) makes adecisive contribution to the transformation of lightpropagating in a hologram, because, obviously (andaccording to the equations), scattering from intrinsicgratings satisfies the Bragg condition.

The second group of terms is much more numer󰀎ous, because it includes the contributions from theplane󰀎wave scattering from all (except for intrinsic)gratings of the hologram formed during its recording.However, most of the polarization waves induced inthem are not synchronized with light and make a smallcontribution to the light transformation in hologramin cases of practical importance. Some of the polariza󰀎tion waves induced in such “alien” gratings can besynchronized with light; however, these polarizationwaves have, generally speaking, random phases mak󰀎ing their contribution to the light conversion in a holo󰀎gram much smaller.

The conditions under which the second󰀎groupterms can be neglected were found in [5, 6]; note thatthese conditions are not artificial constructions, butspecifically the conditions under which a hologramreproduces the recorded radiation without distortionsduring reading (i.e., when it appears to be a hologramin the generally accepted understanding of the term).HOLOGRAPHY

Mode Theory of 3D Hologram

V. G. Sidoroviche󰀎mail: sidor@opten.ruReceived September 7, 2011

Abstract—The fundamentals of the theory of 3D hologram, which was developed by the author many yearsago and has been applied in many fields of holography and nonlinear optics, are briefly described. In Decem󰀎ber 2010, the author reported the mode theory in Readings in the Memory of Academician D.S. Rozh󰀎destvensky (these readings are performed annually by the Vavilov State Optical Institute and the Rozh󰀎destvensky Optical Society in St. Petersburg). In September 2011, the author was invited to report the funda󰀎mentals of the theory at the conference Goloekspo󰀎2011 in Minsk. The response to these reports showed aninterest in the mode theory. This interest prompted this publication, the purpose of which is to prepare thereader for comprehending the original studies.DOI: 10.1134/S0030400X12020245306

OPTICS AND SPECTROSCOPY Vol. 112 No. 2

2012SIDOROVICH

3. LIGHT FIELD RECORDING

In this study, the mode theory is set forth as appliedto a transmission hologram [4–6]. The mode theory ofa reflection hologram was developed by the author andLeshchev and published in [9]. Figure 1 shows a sche󰀎matic diagram of hologram recording. Note that theseparation of the recording wave into object and refer󰀎ence beams, as shown in the figure, is reflected in noway in the equations. However, an important feature isthe paraxiality of the recording and reading radiation,due to which one can neglect the inclination of thecomponents of the angular spectrum to the light beamaxis when describing its interaction with the hologramduring reading.