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EAL603-2013
Dr. Salah Hassab Elnaby
Lecture 9
Electro Optic and Photorefractive effects
Electro Optic effect
The electro optic effect is the change in refractive index of a material induced
by the presence of a static (or low-frequency) electric field.
1- Pockels effect.
In some materials, the change in refractive index depends linearly on the
strength of the applied electric field Since the linear electro optic effect can be
described by a second-order nonlinear susceptibility, it follows that a linear
electro optic effect can occur only for materials that are noncentrosymmetric.
The relation between the electric field strength and the electric displacement
vector
Let us define
Then we get
For uniaxial crystal
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Dr. Salah Hassab Elnaby
The change in the refractive index depend on the direction of polarization of
the light and the direction of the applied static (or low frequency) this should
be calculated from :
For example for KDP and ADP this coefficients are
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Dr. Salah Hassab Elnaby
Longitudinal electro_optic modulation
For KDP for example if the applied field is in the Z-direction then the indicatrix
ellipsoid becomes
To find the new principle coordinate system we define
Substituting we get
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(
(
So we get
Dr. Salah Hassab Elnaby
)
)
This means that the XY section of the ellipsoid before the applied voltage was a
circle. Now applying a voltage V in the Z direction will produce a field EZ= V/L
where L is the length of the crystal.
A wave incident in the Z-direction polarized in the X=direction shall have two
components in the x and y directions. The phase difference between the is
nx-nycL=
Note that in this case the modulation angle doesn’t depend on the length of the
crystal.
As a home work study the case of transverse electro_optic modulation
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Dr. Salah Hassab Elnaby
Photo refractive effect
The photorefractive effect often leads to a very strong nonlinear response. This
response usually cannot be described in terms of a χ(3) (or an n2) nonlinear
susceptibility, because the nonlinear polarization does not depend on the
applied field strength in the same manner as the other mechanisms listed.
The propagation of an optical wave in insulating or semi-insulating electrooptical crystals induces a charge transfer. The new distribution of charges
induces in turn an electric field which produces a variation of the refraction
index.
The main characteristics of this effect are the following:
(1) Sensibility to energy (and not to the electric field),
(2) Nonlocal effect (charge distributions and the electric field are not located at
the same position),
(3) Inertia (charges need a certain time to move),
(4) Memory and reversibility (in the dark the space charge, and therefore the
index variation, is persistent but an uniform light redistributes uniformly all
charges —
this yields applications to holography).
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Dr. Salah Hassab Elnaby
Photorefractive materials exhibit photoconductive and electro-optic
behavior, and have the ability to detect and store spatial distributions of
optical intensity in the form of spatial patterns of altered refractive index.
Photoinduced charges create a space-charge distribution that produces an
internal electric field, which, in turn, alters the refractive index by means of
the electro-optic effect. Ordinary photoconductive materials are often good
insulators in the dark. Upon illumination, photons are absorbed, free charge
carriers (electron-hole pairs) are generated, and the conductivity of the
material increases. When the light is removed, the process of charge
photogeneration ceases, and the conductivity returns to its dark value as the
excess electrons and holes recombine.
In photorefractive materials the refractive index can be modified by illumination with light. The
photorefractive effect can show very high light sensitivity in appropriate material thus permitting the
observation of such effects at very low light power(< μW). There are many different effects which can
lead to photorefraction as for example photochemical effects, photoinduced reorientation of
molecules, Kerr effect at high intensity, or photoinduced thermal effects. We will restrict the
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definition of the photorefractive effect to the light induced refractive index change due to charge
transport in electro-optic materials. This definition has become common in literature. Photorefraction
gives rise to many interesting effects as for example light induced wave-guiding, phase conjugation ,
beam amplification , and four wave mixing.
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EAL603-2013
Dr. Salah Hassab Elnaby
Conventional Model
The so-called conventional model was elaborated by Kukhtarev et al. [7,8,9]
considering only photo-excitation and recombination of one species of
charge carriers between a single donor level and the corresponding
conduction band. This single-level band scheme, along with the involved
physical mechanisms, is depicted in Fig. 1.3. Note that while this simplified
model describes satisfactorily the processes in a large number of materials
showing the photorefractive effect, there exist several crystals where details
of the charge transport mechanism are better described by considering
additional defect levels [10, 11]. Another important limitation is that the
photon energies of the light illumination must be smaller than the band-gap
energy.
The involved processes can be described by the following set of equations
where the photoexcited charges are assumed to be electrons. The symbols
in the above equations are:
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Equation (1) is the rate equation for the concentration of the ionized donors.
The first term describes the photoionization process (seI) and the thermal
excitation of electrons from the donor level. The second term takes into
account the recombination of the electrons (e) into traps, in our case ionized
donors. The second equation (2) is the continuity equation for the electron
density. The additional term with respect to Eq. (1) describes the divergence
of the electron current density. Eq. (3) describes the different contributions
to the electron current density. The first term gives the drift current in the
total electric field E = Esc+E0, where E0 is the applied field. The second
term describes the diffusion process of the electrons generated by the
electron concentration gradient. Thus, this term only gives a contribution for
inhomogeneous illumination. The last term gives the photo galvanic current,
if present. The last equation (4) is the Poisson equation for the electric field.
It describes the spatially modulated part of the electric field generated by the
nonuniform distribution of the charge carriers in the crystal.
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Dr. Salah Hassab Elnaby
Applications of the Photorefractive Effect
An image I(x, y) may be stored in a photorefractive crystal in the form of a
refractive-index distribution n(x, y). The image can be read by using the
crystal as a spatial-phase modulator to encode the information on a uniform
optical plane wave acting as a probe. Phase modulation may be converted
into intensity modulation by placing the cell in an interferometer, for xample.
Because of their capability to record images, photorefractive materials are
attractive for use in real-time holography. An object wave is holographically
recorded by mixing it with a reference wave. The intensity of the sum of two
such waves forms a sinusoidal interference pattern, which is recorded in the
photorefractive crystal in the form of a refractive-index variation. The crystal
then serves as a volume phase hologram. To reconstruct the stored object
wave, the crystal is illuminated with the reference wave. Acting as a volume
diffraction grating, the crystal reflects the reference wave and reproduces
the object wave.
Since the recording process is relatively fast, the processes of recording
and reconstruction can be carried out simultaneously. The object and
reference waves travel together in the medium and exchange energy via
reflection from the created grating. This process is called two-wave mixing.
Waves 1 and 2 interfere and form a volume grating. Wave 1 reflects from
the grating and adds to wave 2; wave 2 reflects from the grating and adds to
wave 1. Thus the two waves are coupled together by the grating they create
in the medium. Consequently, the transmission of wave 1 through the
medium is controlled by the presence of wave 2, and vice versa. For
example, wave 1 may be amplified at the expense of wave 2.
Home Work
1- A GaAs crystal with refractive index n = 3.6 and electro-optic coefficient
r = 1.6 pm/V is used as an electro-optic phase modulator operating at
= 1.3 pm in the longitudinal configuration. The crystal is 3 cm long and
has a 1-cm2 cross-sectional area. Determine the half-wave voltage V
the transit time of light through the crystal, and the electric capacitance
of the device (the dielectric constant of GaAs is o = 13.5). The voltage
is applied using a source with 50 resistance. Which factor limits the
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speed of the device, the transit time of the light through the crystal or the
response time of the electric circuit?
2- In a KDP crystal an electric field is applied along the optic axis of this
uniaxial crystal, it changes into a biaxial crystal. The new principal axes
are the original axes rotated by 45o about the optic axis. Assume a
longitudinal modulator configuration .(d/L = 1) in which the wave travels
along the optic axis. The two normal modes have refractive indices
given in the lecture. Calculate the half wave voltage.
3- Cascaded Phase Modulators. (a) A KDP crystal (r41 = 8 pm/V, r63 = 11
pm/V; no = 1.507, ne = 1.467 at o = 633 nm) is used as a longitudinal
phase modulator. The ray is incident in the z direction and polarization
as in the preveous problem.
Determine the half-wave voltage V at o = 633 nm.
(b) An electro - optic phase modulator consists of 9 KDP crystals
separated by electrodes that are biased as shown below. How should
the plates be oriented relative to each other so that the total phase
modulation is maximized?
Calculate V for the composite modulator.
4- Double Refraction in an Electra-Optic Crystal. (a) An unpolarized He-Ne
laser beam (o = 633 nm) is transmitted through a l-cm-thick LiNbO3
plate (ne = 2.17, no = 2.29, r33 = 30.9 pm/V, r13 = 8.6 pm/V). The beam is
orthogonal to the plate and the optic axis lies in the plane of incidence of
the light at 45o with the beam. The beam is double refracted. Determine
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Dr. Salah Hassab Elnaby
the lateral displacement and the retardation between the ordinary and
extraordinary beams. (b) If an electric field E = 30 V/m is applied in a
direction parallel to the optic axis, what is the effect on the transmitted
beams? What are possible applications of this device?