Optical Properties of Ferroelectrics and Measurement Procedures

It is well known that the optical properties of ferroelectric materials find wide ranging appli‐ cations in laser devices. Particularly in the recent years, there has been tremendous interest in the investigation of the nonlinear optical properties of ferroelectric thin films [1-5] for pla‐ nar waveguide and integrated –optic devices. A new class of thin film waveguides has been developed using BaTiO3 thin films deposited on MgO substrates [6]. Barium strontium tita‐ nate Ba1-xSrxTiO3 (BST) is one of the most interesting thin film ferroelectric materials due to its high dielectric constant, composition dependent Curie temperature and high optical non‐ linearity. The composition dependent Tc enables a maximum infrared response to be ob‐ tained at room temperature. The BST thin films in the paraelectric phase, have characteristics such as good chemical and thermal stability and good insulating properties, due to this nature they are often considered the most suitable capacitor dielectrics for suc‐ cessful fabrication of high density Giga bit (Gbit) scale dynamic random access memories (DRAMs). Compositionally graded ferroelectric films have exhibited properties not previ‐ ously observed in conventional ferroelectric materials. The most notable property of the graded ferroelectric devices or graded Functionally Devices (GFDs) is the large DC polariza‐ tion offset they develop when driven by an alternating electric field. Such GFDs can find ap‐ plications as tunable multilayer capacitors, waveguide phase shifters and filters [7]. Recently, BST thin films were used in the formation of graded ferroelectric devices by de‐ positing successive layers of BST with different Ba/Sr ratios [8].


Introduction
It is well known that the optical properties of ferroelectric materials find wide ranging applications in laser devices. Particularly in the recent years, there has been tremendous interest in the investigation of the nonlinear optical properties of ferroelectric thin films [1][2][3][4][5] for planar waveguide and integrated -optic devices. A new class of thin film waveguides has been developed using BaTiO 3 thin films deposited on MgO substrates [6]. Barium strontium titanate Ba 1-x Sr x TiO 3 (BST) is one of the most interesting thin film ferroelectric materials due to its high dielectric constant, composition dependent Curie temperature and high optical nonlinearity. The composition dependent T c enables a maximum infrared response to be obtained at room temperature. The BST thin films in the paraelectric phase, have characteristics such as good chemical and thermal stability and good insulating properties, due to this nature they are often considered the most suitable capacitor dielectrics for successful fabrication of high density Giga bit (Gbit) scale dynamic random access memories (DRAMs). Compositionally graded ferroelectric films have exhibited properties not previously observed in conventional ferroelectric materials. The most notable property of the graded ferroelectric devices or graded Functionally Devices (GFDs) is the large DC polarization offset they develop when driven by an alternating electric field. Such GFDs can find applications as tunable multilayer capacitors, waveguide phase shifters and filters [7]. Recently, BST thin films were used in the formation of graded ferroelectric devices by depositing successive layers of BST with different Ba/Sr ratios [8].
In our work, the Barium strontium titanate (Ba 0.05 Sr 0.95 TiO 3 ) ferroelectric thin films were prepared on single crystal [001] MgO substrates using the pulsed laser deposition method. The refractive index of BST (Ba 0.05 Sr 0.95 TiO 3 ) thin films is determined in the wavelength range between 1450-1580 nm at the room temperature. The dispersion curve is found to decrease gradually with increasing wavelength. The average value of the refractive index is found to be 1.985 in the wavelength range between 1450-1580 nm which is important for optoelectronic device applications [9].  (LGO) is regarded as a weak ferroelectric and its curie point T c is 283.5K [10,11]. Due to its intermediate behaviour between order-disorder and displacive types in a conventional grouping of ferroelectric materials LGO remains a subject of interest from both the theoretical and the application point of view. The paraelectric phase above T c is orthorhombic D 14 2h ~ pbcn and below T c the ferroelectric phase is C 5 2v ~ pbc2 1 with four formula units in a unit cell in both the phases. Below T c LGO shows dielectric hysteresis loop and the permittivity shows a sharp peak at T c [10][11][12]. Below T c the spontaneous polarization appears along the c-axis. Many interesting physical properties of LGO such as birefringence [13], elastic behaviour [14], thermal expansion [11], dielectric susceptibility [12,15,16] and photoluminescence [17] exhibit strong anomalies around T c . The optical properties, however vary only to such a small degree that the transition could not be detected with the aid of a standard polarization microscope [13]. Employing a high resolution polarization device, Kaminsky and HaussÜhl [13] studied the birefringence in LGO near T c and observed anomalies at the phase transition.
The study of piezo-optic dispersion of LGO (un-irradiated and x-irradiated) in the visible region of the spectrum of light at room temperature (RT=298 K) shows an optical zone/ window in between 5400Å and 6200Å with an enhanced piezo-optical behavior [18]. The temperature dependence of the photoelastic coefficients of the ferroelectric crystals Li 2 Ge 7 O 15 (both un-irradiated and x-irradiated) in a cooling and a heating cycle between room temperature and 273K shows an interesting observation including the lowering of the T c under uniaxial stress contrary to the increase of T c under hydrostatic pressure and observation of thermal photoelastic hysteresis similar to dielectric behavior [19]. The study of fluorescence spectra of the crystals Li 2 Ge 7 O 15 :Cr 3+ in the temperature interval 77-320 K shows the sharply decrease of intensities of the R 1 and R 2 lines (corresponding to the Cr 3+ ions of types I and II) during cooling process near the temperature T c = 283.5 K [16].
The present chapter includes optical properties of the ferroelectric BST thin films and the Lithium heptagermanate (Li 2 Ge 7 O 15 ) single crystals, fabrication methods, measurement procedures of the refractive index of BST thin films on MgO substrates, the fluorescence spectra and the photoelastic coefficients of LGO single crystals (un-irradiated and x-irradiated) at different wave lengths and temperatures around the phase transition temperature T c . The potential of these materials for practical applications in the opto-electronic devices will also be discussed.

Optical property of Barium strontium titanate thin films
The Barium strontium titanate (Ba 0.05 Sr 0.95 TiO 3 ) ferroelectric thin films were prepared on single crystal MgO substrates using the pulsed laser deposition (PLD) method at a substrate temperature of 780 o C and then annealed at 650 o C for 55 min. The x-ray diffraction (XRD) analysis revealed that the films are oriented with [001] parallel to the substrate [001] axis and thus normal to the plane of the films [9]. The films were grown to a thickness of 430 nm.  The He-Ne laser beam is used as a source of light to setup the alignment of the reflected beam of light from the samples to the detector. The incident beam is allowed to pass through a polarizer onto the sample. The reflected light is then passed through the same polarizing beam splitter oriented at 45 o relative to the incident light and finally allowed to fall on the detector that is at 90 o to the reflected/incident beam of light. The reflectivity measurement of the black metal, mirror, MgO substrate and Ba 0.05 Sr 0.95 TiO 3 thin films were carried using the Agilent 8164A Light Wave Measurement system in the wavelength region of 1450-1580 nm at room temperature.
The refractive index of substrate MgO is taken to be 1.7 [9]. The reflectivity of Ba 0.05 Sr 0.95 TiO 3 film is then normalized with respect to the mirror. The value of refractive index is derived from model described in ref. [20]. The fitting is done with the calculated data of the reflectivity of Ba 0.05 Sr 0.95 TiO 3 in the wavelength range between 1450-1580 nm. Figure 2 shows the refractive index of Ba 0.05 Sr 0.95 TiO 3 thin films as a function of wavelength at room temperature.
The refractive index of Ba 0.05 Sr 0.95 TiO 3 with Ba x Sr 1-x TiO 3 (BST) and other materials of ferroelectric thin films at different wavelengths are presented in table 1. As shown in Figure 2, the dispersion curve decreases gradually with increasing wavelength. The average value of the refractive index is found to be ~ 1.985 in the wavelength range of 1450-1580 nm which is important for optoelectronic device (optical waveguide) applications. The variation of refractive index is attributed predominantly to the changes of electronic structure associated with the larger lattice parameter and variations in atomic co-ordination [27] that is local relaxations.

Growth and structure of Li 2 Ge 7 O 15 Crystals
Single crystals of Li 2 Ge 7 O 15 are grown in an ambient atmosphere by Czochralski method from stoichiometric melt, employing a resistance heated furnace. Stoichiometric mixture of powdered Li 2 CO 3 and GeO 2 in the ratio of 1.03 and 7.0 respectively was heated at 1100 K for 24 hours to complete the solid state reaction for the raw material for the crystal growth. The crystals were grown by rotating the seed at the rate of 50 rpm with a pulling rate of 1.2 mm/ hour. The cooling rate of temperature in the process of growth was 0.8-1.2 K/hour. The crystals grown were colourless, fully transparent and of optical quality. The crystal axes were determined by x-ray and optical methods. The desired impurities such as Cr +3 , Mn +2 , Bi +2 , Cu 2+ and Eu +2 etc are also introduced in desired concentration by mixing the appropriate amount of the desired anion salt in the growth mixture. The crystal structure of LGO above T c is orthorhombic (psedohexagonal) with the space group D 14 2h (Pbcn). The cell parameters are a: 7.406 Å, b: 16.696 Å, c: 9.610 Å, Z = 4 and b~√3c. Below T c a small value of spontaneous polarization occurs along c-axis and the ferroelectric phase belongs to C 5 2v (Pbc2 1 ) space group. The crystal structure contains strongly packed layers of GeO 4 tetrahedra linked by GeO 6 -octahedra to form a three dimensionally bridged frame work in which Li atoms occupy the positions in the vacant channels extending three dimensionally [14,28,29]. The size of the unit cell (Z = 4) does not change at the phase transition and ferroelectric phase transition is associated with a relaxational mode as well as the soft phonon [30]. Activation of the pure crystals with impurity ions will demand charge compensating mechanism through additional defects in the pure lattice.

Study of fluorescence spectra of Li 2 Ge 7 O 15 :Cr 3+ crystals
The fluorescence spectra of the crystals Li 2 Ge 7 O 15 :Cr 3+ were studied in the temperature interval 77-320 K including the phase transition temperature T c = 283.5 K. The experimental set to record the fluorescence spectra of the crystals Li 2 Ge 7 O 15 :Cr 3+ is shown in fig.3. A laser with the pair of mode (λ 1 =510.6 nm, λ 2 =578.2 nm) was used as a source of excitation of the crystal sample. The recording of fluorescence spectra were carried out by the optical multichannel analyzer in combination with the polychromator. The radiation beam was initially polarized with glen prism. The plane Polaroid was used as an analyzer that was placed before the input aperture of the polychromator.
The fluorescence spectra consist of narrow intensity lines referred to R 1 and R 2 with frequencies γ 1~1 4348 cm -1 and γ 2 ~14572 cm -1 . These lines split further into two components each R 1 / and R 2 / respectively at lowering the temperature towards 77 K. Besides this, a wide long wavelength region/zone is observed in the spectra. It may be related with the effect of electron-photon interaction. It is known that Cr 3+ doping ions in the structure of Li 2 Ge 7 O 15 :Cr 3+ crystals substitute the Ge 4+ host ions within oxygen octahedral (GeO 6 ) complexes [31][32][33][34][35][36]. The optical spectra of Cr 3+ ions shows the existence of two types of Cr 3+ centre (type I and II with different values of effective g-factor) as observed in EPR (Electron Paramagnetic Resonance) spectra of Cr 3+ ions in ferroelectric phase of the crystals Li 2 Ge 7 O 15 :Cr 3+ [32,33]. Two pair of R lines 4A 2 -2 E (at T=77 K, its positions are R 1 =14348 cm -1 , R 1 / =14402 cm -1 , R 2 =14572 cm -1 and R 2 / =14593 cm -1 ) are observed at low temperature region (T<190 K) in the optical spectra of the crystals Li 2 Ge 7 O 15 :Cr 3+ as shown in Fig. 4. Actually the two different types of Cr 3+ centers (R and R / ) with pretty different positions below Ē and above 2Ᾱ levels of the excited E 2 level are duplicate [37,38] and conform to the EPR observations. The intensity of fluorescence of the R 1 and R 2 lines of the crystals Li 2 Ge 7 O 15 :Cr 3+ were studied near the phase transition temperature T c at the direction E┴[001]. It is observed that the intensity of R 1 and R 2 lines are decreased sharply near the phase transition temperature T c but at high temperature region (T>T c ) the intensity again increases as shown in figure 5. Such nature of suppression of R 1 and R 2 lines was not observed previously and it may be related with the mechanism of interaction of excitation spectra of light in the crystals Li 2 Ge 7 O 15 :Cr 3+ near the phase transition temperature T c [17].
The Crystals doped with chromium impurity give EPR (Electron Paramagnetic Resonance) signals characteristics of the trivalent chromium ions [ Fig.6]. It is known that impurity Cr 3+ ions substitute the Ge 4+ host ions within oxygen octahedral in the basic structure of (LGO) crystal [31][32][33][34][35][36]. Incorporation of tri-positive chromium ions into GeO 6 -octahedra changes the local symmetry of the lattice site from monoclinic C 2 group to triclinic C 1 group. The local symmetry lowering is attributed to the effect of the additional Li + defect required for compensating the charge misfit of Cr 3+ ion at the Ge 4+ site. Taking into account a weak coupling of lithium ions with the germanium -oxygen lattice framework, the interstitial Li + is considered to be the most probable charge compensating defect, located within the structural cavity near the octahedral CrO 6 complex ( Fig.7). Subsequent measurements of optical spectra have confirmed the model of Cr 3+ -Li + pair centers in the LGO crystal structure [32,33]. The available data make it possible to assume that electric dipole moments of Cr 3+ -Li + pairs are directed along the crystal axis "a" of the crystal. Interstitial Li + ions locally break the symmetry axis C 2 of the sites within the oxygen octahedral complexes [34]. As a result, there are two equivalent configurations of the pair centers which are conjugated by broken C 2 axis and have dipole moments with opposite orientations. It may be assumed that pair centers can reorient due to thermal activation. Reorientation of the pair centers should be accompanied by: i) shortening of the configuration life time and ii) switching of defect dipole moments [35]. This is reflected in the typical temperature dependence of the imaginary part of dielectric permittivity of chromium doped LGO single crystals [35] along the a-axis of the crystal shown in fig.8.  (2); 5 kHz (3); 10 kHz (4). Distinct peak is observed in ε'' in the temperature range 350-450 K and the peak is observed to shift to higher temperature for higher frequencies. In contrast no such peak is observed for the real part ε' of the permittivity. The typical behavior is observed only along crystal a -axis which incidentally coincides with the proposed Cr 3+ -Li + centers.

Principle of Photoelasticity
If a rectangular parallelepiped with edges parallel to x[100], y[010] and z[001] axes is stressed along z-axis and observation is made along y-axis, as shown in Fig.9, then the path retardation δ zy introduced per unit length due the stress introduced birefringence is given by δ zy = (Δn z -Δn x ) = C zy P zz (1) where Δn z and Δn x are the changes in the corresponding refractive indices, (Δn z -Δn x ) is the corresponding stress induced birefringence, P zz is the stress along z-axis and C zy is a constant called the Brewster constant or the relative photoelastic coefficient. In general the Brewster constant is related to the stress optical and strain optical tensors of forth rank [39] and is a measure of the stress induced (piezo-optic) birefringence. It is conveniently expressed in the unit of 10 -13 cm 2 /dyne per cm thickness along the direction of observation is called a Brewster [39].

Measurement procedure of photoelastic constants
To study the piezo-optical birefringence the experimental set up consists of a source of light (S), a lens (L) to render the rays parallel, a polarizer (P), an analyzer Polaroid (A), a Babinet compensator (B) and a detector (D), as shown in Fig.10. The P and A combination are adjusted for optimal rejection of light. The sample with stressing arrangement and a Babinet compensator are placed between P and A. A monochromator and a gas flow temperature controlling device are used to obtain the piezo-optic coefficients (C λ ) at different wavelengths and temperature. The subscript λ in the symbol C λ denotes that the piezo-optic coef-ficient depends on the wavelength of light used to measure it. The experiments are carried out for different wavelengths using white light and a monochromator and the monochromatic sodium yellow light. An appropriate stress along a desired direction of the sample is applied with the help of a stressing apparatus comprising a mechanical lever and load.  To start with, the Babinet compensator is calibrated and the fringe width is determined for different wavelengths of light in the visible region. The crystal specimen is placed on the stressing system so that the stress could be applied along vertical axis and observation made along horizontal axis. A load on the crystal shifts the fringe in the Babinet compensator and this shift is a measure of the piezo-optic behavior. The piezo-optic coefficients (C λ ) are now calculated using the calibration of the Babinet compensator. The experiment is repeated for other orientations of the crystals and the results are obtained.
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Piezo-optic Dispersion of Li 2 Ge 7 O 15 Crystals
The experimental procedure for the piezo-optic measurements is described in section 1.5. The polished optical quality samples worked out to dimensions i) 5.9 mm, 9.4 mm and 5.0 mm; ii) 3.17 mm, 5.88 mm and 6.7 mm, along the crystallographic a, b and c axes respectively. The stress was applied with an effective load of ~23 kg in each case [40].
The values of C λ thus obtained at different wavelengths are given in Table 2 and the results are plotted in Fig. 11. Here C pq is the piezo-optic coefficient with the stress direction being p and observation direction being q. The results show an interesting piezo-optic behavior. A survey of literature indicates that the piezo-optic behavior of materials studied till now shows a reduction of C λ with increasing wavelength in the visible region [39]. In the present case, C λ decreases with wavelength up to a certain wavelength as in other normal materials and then suddenly shows a peak and later on the usual behavior of reduction in the values of piezo-optic coefficients is observed. To the best knowledge of the authors this behavior is unique to the LGO crystals. For the sake of convenience we denote C λ measured at λ = 5890 Å as C 5890 and so on. The results show that sometimes the value of C 5890 is even higher than that at C 4400 , the value of piezooptic coefficient obtained at the lowest wavelength studied here. This is the case with C xy , C zx and C xz . For other orientations the value is lower than that at 4400 Å. Further, C λ is found to have increased to more than 50% in the case of stress along [001] and observation along [100]. Also, it is interesting to note that the value of C 6140 , is less than that of C 5390 , in tune with usual observation of piezo-optic dispersion. Thus one can see an "optical window" in between 5400 Å and 6200 Å. The height of this optical window is different for various orientations, though the width seems approximately the same. The maximum height of about 1.5 Brewster was found for C zx followed by C xz with about 0.9 Brewster. It should be noted here that z-axis is the ferroelectric axis for LGO. It is also interesting to note that the change in height is more in the former while the actual value of C λ , is less compared to that of the latter. The percentage dispersion also is different for various orientations. It is very high, as high as 25% for C zy , while it is just 10% for C xy .  Figure 12 shows the variation of C zx (λ) at the temperatures ranging from 298K to 283K on cooling process of the sample LGO. It is clear from the figure that the distinct peak of C zx (λ) appears only at the sodium yellow wavelength of 5890 Å for the whole range of temperatures (298 K-283 K) investigated. It is also interesting to note that a temperature anomaly is also observed around 283 K.

Obs
LGO undergoes a second order phase transition at 283.5 K from the high temperature paraelectric phase to the low temperature ferroelectric phase. So this anomaly is related to this phase transition of the LGO crystal.
The observed peculiarity of piezo-optic behavior could be due to many factors, viz., i) anomalous behavior of refractive index or birefringence ii) anomalous ferroelastic transformation at some stage of loading iii) shift of absorption edge due to loading. The following have been done to identify the reasons for this peculiar behaviour.
Birefringence dispersion has been investigated in the visible region and no anomalies in its behavior has been observed. This rules out the first of the reasons mentioned. The reason due to ferroelastic behavior also is ruled out since the effect would be uniform over all the wavelengths investigated. It was not possible to investigate the effect of load on the absorption edge. Hence an indirect experiment has been performed. If there is a shift in the absorption edge due to loading the sample, the peak observed now at sodium yellow light would shift with load. No clear shift of the peak could be observed within the experimental limits. Another interesting experiment was done to identify the source of the anomaly. It is well known that T c of LGO changes under uniaxial stress. The measurements were made near T c under different stress (loads). Although T c was found to shift a little with load the dispersion peak did not show any discernible shift. No particular reason could be established as to why a dispersion peak appears around sodium yellow region. Another interesting work in this direction is on Gd 2 (Mo0 4 ) 3 -where an anomalous peak was recorded in spontaneous birefringence at 334.7 nm [41], an observation made for the first time. It is well known that the photoelasticity in crystals arises due to change in number of oscillators, effective electric field due to strain and the polarisability of the ions. In the present case, as the wavelength approaches around 5400 Å, the ionic polarisability seems to be changing enormously. There is no optical dispersion data available on LGO. We have conducted an experiment on transmission spectra of LGO along x, y and z-axes, which shows a strong absorption around 5400 Å. The observed anomaly in the piezo-optic dispersion may be attributed to the absorption edge falling in this region. This explanation needs further investigation in this direction. It is also known that the strain optical dispersion arises due to the shift in absorption frequencies and a change in the oscillator strength caused by the physical strain in the crystal.

Irradiation Effect on Piezo-optic Dispersion of Li 2 Ge 7 O 15 Crystals
The ferroelectric single crystals Li 2 Ge 7 O 15 was irradiated by x-ray for one hour and the experimental processes described in section 1.5 were repeated for the crystal (irradiated) LGO in order to understand the radiation effect on piezo-optical birefringence dispersion [18].
The values of C λ of the crystal (irradiated) LGO thus obtained at different wavelengths are given in Table 3 and the results are plotted in Fig. 13. Some interesting results are obtained in the case of irradiated crystal LGO. The peak value of C / zx has decreased about 18% and that of C / zy has increased about 25% at the wave length λ= 5890 Å. Also, it is interesting to note that the value of C 6140 , is less than that of C 5390 for the un-irradiated and irradiated sample of LGO crystal, in tune with usual observation of piezooptic dispersion.

Obs
Irradiation of crystals can change physical properties of the crystals. Irradiation brings about many effects in the crystal such as creating defects, internal stress and electric fields etc.
These irradiation effects in turn are supposed to affect the physical properties of the irradiated crystal as compared to un-irradiated crystal. While there was no appreciable change in the lattice parameters, a significant drop in the value of dielectric constant and tan δ was observed upon x-irradiation of ferroelectric glycine phosphate. An appreciable shift in the phase transition temperature towards the lower temperature was observed. These changes are attributed to the defects produced in it by irradiation [42]. The studies of triglycine sulphate (TGS) showed that very small doses of x-irradiation can give large changes of the ferroelectric properties. The direct evidence of domain clamping by defects was obtained from optical studies. With increasing dosage the dielectric constant peak and polarization curve broaden and move to lower temperatures. In our present studies, the x-irradiation is believed to produce internal stress and electric fields inside the crystals LGO due to defects that can change the values of piezo-optic constants [43].

Piezo-optic Birefringence in Li 2 Ge 7 O 15 Crystals
The temperature dependence of the photoelastic coefficients of the ferroelectric crystals Li 2 Ge 7 O 15 in a cooling and heating cycle between 298 K and 273 K was carried out with the experimental procedure described in section 1.5 [19]. A special arrangement was made to vary the temperature of the sample. The temperature was recorded with a digital temperature indicator and a thermocouple sensor in contact with the sample.  The temperature dependence of piezo-optic coefficients C pq of the crystals Li 2 Ge 7 O 15 between 298 K and 273 K were determined and are shown in Fig. 14 and Fig. 15. The values of C pq at  [44] and it was observed that there were large changes in the values of C zy and C yz at 278 K and 291 K as compared to other components and C zy did not show a peak in its temperature dependence between 291K and 278 K. Figure 15. Anomalous temperature dependence of the piezo-optic coefficients C zx and C zy of the crystals LGO in a cooling (0) and heating (Δ) cycle.

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Here in contrast we observed a peak in the temperature dependence of both C zy and C zx at 279 K. The temperature dependence of C pq are quite interesting, for example the piezo-optic coefficients C yz , C yx and C xz have negative temperature derivatives but C xy has a positive temperature derivative. In complete contrast both C zy and C zx have both positive and negative temperature derivatives at different temperature intervals between 298 K and 273 K (Table: 4). Besides a clear thermal hysteresis is observed in C zy and C zx in a complete cooling and heating cycle (Fig. 15) whereas no discernible hysteresis is observed in rest of the piezooptic coefficients (Fig. 14). The two distinct anomalies in the temperature dependence of C zy and C zx are characterized by a valley at T m (∼289 K) and a peak at T c (∼279 K). Anomalous temperature dependence of C zx at different wave lengths is also shown in Fig. 16. The tem-perature dependence of the dielectric permittivity along the c-axis of LGO shows a sharp peak at T c (283.5 K) and the Curie-Weiss law holds only for a narrow range of temperature (T c ± 4 K) [11,15,16]. The peak for piezo-optic coefficient is attributed to the paraelectric to ferroelectric phase transition of LGO at T c . To check the curie-Weiss law like dependence near T c the following relation is used.
Where C T pq and C 0 pq denote the value of the corresponding piezo-optic coefficients at temperature T and 273 K respectively and K pq is a constant. Fig. 17 shows the (C T pq -C 0 pq ) −1 vs (T−T c ) curve for C zx and C zy . It is clear from these curves that like dielectric constant the relation fits well only within a narrow range of temperature near T c (T c ± 4 K). The solid lines denote the theoretical curves with the following values K zx = 1.05; K zy = 0.92 for T > T c , K zx = −0.40; K zy = −0.34 for T < T c and T c = 279 K.  Furthermore the magnitudes of the ratio of the temperature derivatives below and above T m and T c are given in Table 4 and we can see that the ratio near T c comes out to be about 2. Therefore it satisfies the law of two for the ratio of such derivatives of quantities which are coupled with the spontaneous polarization in second order ferroelectric phase transition such as in the case of triglycine sulphate [45] and LGO. Therefore the peak around T c is [ 13,15,16] attributed to the paraelectric to ferroelectric phase transition of LGO. The smallness of K pq and the applicability of relation (2) above only in a narrow range of temperature suggest that LGO may be an improper ferroelectric. The law of two does not hold for the ratio at T m (Table 4). Therefore this anomaly is not related to the spontaneous polarization.
From the behaviour that only C zx and C zy show anomalous it is obvious that birefringence (Δn z -Δn y ) and (Δn z -Δn x ) show steep increase around T c and below T c show a (T-T c ) 1/2 behaviour correlated to the spontaneous polarization which is parallel to the z-axis (crystallographic c-axis). From the behaviour of C xy and C yx which do not show any temperature anomaly we may say that only n z is responsible for the anomaly in accordance with the behaviour of the dielectric properties where only ε 33 is strongly affected by the phase transition. These observations are in accordance with the results of Faraday effect and birefringence in LGO [13]. As mentioned by Lines and Glass [43], under an external pressure T c of a ferroelectric phase transition may be shifted. This shift may be to the higher or the lower side of normal T c . Wada et al. [46] studied the pressure effect on the ferroelectric phase transition in LGO through the dielectric and Raman scattering measurements and found a positive pressure coefficient dT c /dp = 14.6 K/kbar. Preu and HaussÜhl [12] studied the dependences of dielectric constants on hydrostatic and uniaxial pressure as well as temperature. They observed a shift of T c at a rate of 14.02 K/kbar for the hydrostatic pressure and ∼7 K/kbar for the uniaxial pressure. In the present case the position of the peak of C zy is found to depend on the stress applied. If the peak position is believed to represent the T c , it appears to shift to the lower side under the uniaxial stress. To see whether T c shifts linearly with uniaxial stress similar to the earlier observations [12,46], we used different stresses within the elastic limits of LGO for C zx and found a linear relationship (Fig. 18). However, a negative stress coefficient dT c /dp ∼ −22 K/kbar is obtained in this case which agrees only in magnitude with the hydrostatic pressure coefficient. The linear curve (Fig. 18) extrapolates to a T c = 281.5 K in the unstressed state instead of 283.5K as determined by dielectric measurements [11,15,16]. This may be due to a non linear dependence of shift of T c under stress near 283.5 K.  Now we turn to the anomaly around T m . Morioka et al. [47] proposed that there is an interaction between the soft phonon mode and a relaxational mode in the paraelectric phase in the temperature interval 300 K to T c . The critical slowing down of the relaxational mode near T c is expected to cause the increase of the fluctuation of the spatially homogeneous polarization and thereby the increase of the fluctuation of the hyperpolarizability with k c = 0. Wada et al. [48] measured the soft phonon mode with the help of their newly designed FR-IR spectrometer and proposed that as T c is approached from above soft phonon mode becomes over damped and transforms to a relaxational mode.
On the other hand there may exist a relaxational mode with an independent degree of freedom as well as the soft phonon mode and the character of the softening transfers from the phonon to the relaxational mode. This is an important problem in determining the dynamics of the peculiar ferroelectric phase transition of LGO, where both the dielectric critical slowing down characteristic of the order-disorder phase transition and the soft phonon mode characteristic of the displacive phase transition are observed [11,14]. In the light of the above discussion we may say that the change up to T m is caused by the softening of mode and the softening character transforms to the relaxations mode near T m causing a change in the trend below T m and near T c the relaxational mode becomes dominant. The valley around T m is perhaps caused by the interplay between the competitive relaxational mode and the soft phonon mode. It has been observed that softening of the velocity and rise of the damping of acoustic phonon occur in the paraelectric phase of LGO even quite far from T c , i.e.(T-T c ) ∼ 30 K and the effect is attributed to the fluctuation induced contributions [49]. Another interesting aspect is the observation of a significant thermal photoelastic hysteresis (Fig. 15). Although the peak position does not shift in the heating cycle the values of the photoelastic constants get reduced significantly in the heating cycle as compared to the corresponding values in a cooling cycle. A similar kind of hysteresis was observed in the dielectric behaviour of LGO and the appearance of the dielectric hysteresis is attributed to the internal space charge (electrets state) effects which produce an internal electric field in LGO on heating from the ferroelectric phase [15][16][17]. It was possible to compensate the internal electric field effects in dielectric measurements by an external electric field [15][16][17]. It is suspected that the photoelastic hysteresis also occurs due to similar effects. Although it was not possible to try to compensate the electric field effects in the present investigation, it is possi-ble to attempt experiment under the simultaneous application of a suitable electric field and stress along z-direction.  Table 6. Piezo-optic coefficients c pq (in Brewsters) for some ferroelectric crystals in their paraelectric (PE) phases.
The Stress optical coefficients C pq of the crystals Li 2 Ge 7 O 15 at paraelectric phase (RT = 298 K) and at T c = 279 K are presented in Table 5. It is important to compare the values of C pq for Li 2 Ge 7 O 15 with other ferroelectric crystals given in Table 6 particularly with Rochelle-salt (RS) which belongs to the orthorhombic class like LGO [44]. The values of C pq are significantly higher for LGO as compared to these ferroelectric systems. So, the large photoelastic coefficients and the other properties like good mechanical strength, a transition temperature close to room temperature and stability in ambient environment favour LGO as a potential candidate for photoelastic applications.
The EPR (Electron Paramagnetic Resonance) spectroscopy of the transition metal ion doped crystals of LGO (Mn 2+ , Cr 3+ ) has also been studied both in Paraelectric (PE) and ferroelectric (FE) phases in the temperature interval from 298 K to 279 K during cooling and heating cycles [17,36,53]. It is observed that on approaching T c in a cooling cycle, the EPR lines are slightly shifted to the high field direction and undergo substantial broadening. At the temperature T c ( ≈ 283.4 K), the EPR lines are splitted into two components which are shifted to the higher and lower field directions progressively as a result of cooling the sample below T c as shown in Fig.19.
During heating cycle (i.e. approaching T c from below), the phenomena occurred were just opposite to the above processes observed in the cooling cycle. However, the EPR line width (peak to peak ∆H pp ) for H║c, H ┴ a was found to decrease to about one third of its value at T c in a heating cycle as compared to its value in the cooling cycle. The shape of the EPR resonance lines far from T c has a dominant Lorentzian character (a Lorentzian line shape) but very near to T c , the line shape has been described mainly by Gaussian form of distribution (a Gaussian line shape). All the peculiarities observed are attributed to the PE ↔ FE phase transition of the LGO crystals. The line width reduction near T c is attributed to the internal space charge (electret state) effects which produce an internal electric field inside the crystals  Figure 20. Temperature dependence of photoelastic coefficients C xy , C xz , C yz, C yx, C zx and C zy of the crystal (x-irradiated) LGO in a cooling (0) and heating (∆) cycle.

Irradiation Effect on piezo-optic Birefringence in Li 2 Ge 7 O 15 Crystals
The photoelastic coefficients C pq of the ferroelectric crystals Li 2 Ge 7 O 15 (x-irradiated) in a cooling and heating cycle between 298 K and 273 K was carried out with the experimental procedure described in section 1.5 and are shown in Fig. 20 [54]. The results show an interesting photoelastic behaviour.
Peaks are observed in the temperature dependence of the photoelastic coefficients C zy and C zx at temperature ~ 279 K in a complete cooling and heating cycle whereas no discernible hysteresis is observed in rest of the photoelastic coefficients. Anomalous temperature dependence of C zx of the crystal (x-irradiated) LGO at different wave lengths are shown in Fig.21.
It is observed that the peak value of C zy has increased about 25% and that of C zx has decreased about 18% at the wave length λ=5890 Å during cooling process of the crystal (Fig.15 and Fig.20). The peak value of C zx of the crystal (un-irradiated and x-irradiated) LGO thus obtained at different wave lengths ( Fig.16 and Fig.21) are given in Table 7 and the results are plotted in Fig.22. It has been observed that the changes in the value of photoelastic coefficients C zy and C zx of the crystal (x-irradiated) LGO in a cooling and heating cycle occur only if the crystal is stressed along the polar axis (z-axis). It is known that the irradiation of crystals can change physical properties of the crystals. LGO at different wave lengths in the cooling and heating cycles. Irradiation brings about many effects in the crystal such as creating defects, internal stress and electric fields etc [43]. In our present studies, the x-irradiation is believed to produce internal stress and electric fields inside the crystals Li 2 Ge 7 O 15 due to defects that can change the values of photoelastic coefficients.

Summary
It is known that the Barium strontium titanate Ba 1-x Sr x TiO 3 (BST) is one of the most interesting thin film ferroelectric materials due to its high dielectric constant, composition dependent curie temperature and high optical nonlinearity. The wavelength dependence of refractive index of BST (Ba 0.05 Sr 0.95 TiO 3 ) thin films has shown a nonlinear dependence in the 1450-1580 nm wavelength range at room temperature as described in section 1.1. The dispersion curve decreases gradually with increasing wavelength. The average value of the refrac-tive index is found to be 1.985 in the 1450-1580 nm wavelength range which is considered to be important for optoelectronic device applications.
The study of fluorescence spectra of the crystals Li 2 Ge 7 O 15 :Cr 3+ in the temperature interval 77-320 K shows the sharply decrease of intensities of the R 1 and R 2 lines (corresponding to the Cr 3+ ions of types I and II) during cooling process near the temperature T c = 283.5 K as described in section 1.3. Such nature of suppression of R 1 and R 2 lines was not observed previously and it may be related with the mechanism of interaction of excitation spectra of light in the crystals Li 2 Ge 7 O 15 :Cr 3+ at the temperature T c . The doping of chromium in LGO is believed to create Cr 3+ -Li + defect pairs in the host LGO lattice at Ge 4+ sites creating dipoles in two conjugate directions. The EPR, optical, dielectric and fluorescence studies conform each other and pose more scope for further studies.
The high optical quality, good mechanical strength and stability in ambient environment and large photoelastic coefficients in comparison with other ferroelectric crystals like Rochelle-salt, KDP and ADP favour the crystals LGO as a potential candidate for photoelastic applications. The piezo-optic dispersion of the crystals (un-irradiated and x-irradiated) LGO in the visible region of the spectrum of light at room temperature (298 K) have been described in sections 1.6 and 1.7. It shows an "optical zone or optical window" in between the wavelengths 5400 Å and 6200 Å with an enhanced piezo-optical behavior. This peculiar optical window can have a technical importance for example this window region can act as an optical switch for acousto-optical devices. From the studies undertaken it may be concluded that LGO is an attractive acousto-optic material which deserves further probe. It may be possible to understand the observed behavior if extensive piezo-optic and refractive index data become available over an extended range of wavelengths.
The temperature dependence of the photoelastic coefficients of the crystals (un-irradiated and x-irradiated) LGO in a cooling and heating cycle between room temperature (298 K) and 273 K have shown an interesting observations: lowering of the T c under uniaxial stress contrary to the increase of T c under hydrostatic pressure and observation of thermal photoelastic hysteresis similar to dielectric hysteresis behavior as described in sections 1.8 and 1.9. In our studies, the x-irradiation is believed to produce internal stress and electric fields inside the crystals LGO due to defects that can change the values of photoelastic coefficients, as described in sections 1.7 and 1.9.