OPTOELEKTRONIKA: April 2015

Kerr effect

The Kerr effect, also called the quadratic electro-optic effect (QEO effect), is a change in the refractive index of a material in response to an applied electric field. The Kerr effect is distinct from the Pockels effect in that the induced index change is directly proportional to the square of the electric field instead of varying linearly with it. All materials show a Kerr effect, but certain liquids display it more strongly than others. The Kerr effect was discovered in 1875 by John Kerr, a Scottish physicist.

Two special cases of the Kerr effect are normally considered, these being the Kerr electro-optic effect, or DC Kerr effect, and the optical Kerr effect, or AC Kerr effect.

Kerr electro-optic effect

The Kerr electro-optic effect, or DC Kerr effect, is the special case in which a slowly varying external electric field is applied by, for instance, a voltage on electrodes across the sample material. Under this influence, the sample becomes birefringent, with different indices of refraction for light polarized parallel to or perpendicular to the applied field. The difference in index of refraction, Δn, is given by

$\Delta n = \lambda K E^2,\$

where λ is the wavelength of the light, K is the Kerr constant, and E is the strength of the electric field. This difference in index of refraction causes the material to act like a waveplate when light is incident on it in a direction perpendicular to the electric field. If the material is placed between two "crossed" (perpendicular) linear polarizers, no light will be transmitted when the electric field is turned off, while nearly all of the light will be transmitted for some optimum value of the electric field. Higher values of the Kerr constant allow complete transmission to be achieved with a smaller applied electric field.

Some polar liquids, such as nitrotoluene (C₇H₇NO₂) and nitrobenzene (C₆H₅NO₂) exhibit very large Kerr constants. A glass cell filled with one of these liquids is called a Kerr cell. These are frequently used to modulate light, since the Kerr effect responds very quickly to changes in electric field. Light can be modulated with these devices at frequencies as high as 10 GHz. Because the Kerr effect is relatively weak, a typical Kerr cell may require voltages as high as 30 kV to achieve complete transparency. This is in contrast to Pockels cells, which can operate at much lower voltages. Another disadvantage of Kerr cells is that the best available material, nitrobenzene, is poisonous. Some transparent crystals have also been used for Kerr modulation, although they have smaller Kerr constants.

In media that lack inversion symmetry, the Kerr effect is generally masked by the much stronger Pockels effect. The Kerr effect is still present, however, and in many cases can be detected independently of Pockels effect contributions.

Optical Kerr effect

The optical Kerr effect, or AC Kerr effect is the case in which the electric field is due to the light itself. This causes a variation in index of refraction which is proportional to the local irradiance of the light. This refractive index variation is responsible for the nonlinear optical effects of self-focusing, self-phase modulation and modulational instability, and is the basis for Kerr-lens modelocking. This effect only becomes significant with very intense beams such as those from lasers.

Magneto-optic Kerr effect

Main article: Magneto-optic Kerr effect

The magneto-optic Kerr effect (MOKE) is the phenomenon that the light reflected from a magnetized material has a slightly rotated plane of polarization. It is similar to the Faraday effect where the plane of polarization of the transmitted light is rotated.

Theory

DC Kerr effect

For a nonlinear material, the electric polarization field P will depend on the electric field E:

\mathbf{P} = \varepsilon_0 \chi^{(1)} : \mathbf{E} + \varepsilon_0 \chi^{(2)} : \mathbf{E E} + \varepsilon_0 \chi^{(3)} : \mathbf{E E E} + \cdots

where ε₀ is the vacuum permittivity and χ⁽ⁿ⁾ is the n-th order component of the electric susceptibility of the medium. The ":" symbol represents the scalar product between matrices. We can write that relationship explicitly; the i-th component for the vector P can be expressed as:

P_i = \varepsilon_0 \sum_{j=1}^{3} \chi^{(1)}_{i j} E_j + \varepsilon_0 \sum_{j=1}^{3} \sum_{k=1}^{3} \chi^{(2)}_{i j k} E_j E_k + \varepsilon_0 \sum_{j=1}^{3} \sum_{k=1}^{3} \sum_{l=1}^{3} \chi^{(3)}_{i j k l} E_j E_k E_l + \cdots

where

i = 1,2,3

. It is often assumed that

P_1 = P_x

, i.e. the component parallel to x of the polarization field;

E_2 = E_y

and so on.

For a linear medium, only the first term of this equation is significant and the polarization varies linearly with the electric field.

For materials exhibiting a non-negligible Kerr effect, the third, χ⁽³⁾ term is significant, with the even-order terms typically dropping out due to inversion symmetry of the Kerr medium. Consider the net electric field E produced by a light wave of frequency ω together with an external electric field E₀:

\mathbf{E} = \mathbf{E}_0 + \mathbf{E}_\omega \cos(\omega t),

where E_ω is the vector amplitude of the wave.

Combining these two equations produces a complex expression for P. For the DC Kerr effect, we can neglect all except the linear terms and those in

\chi^{(3)}|\mathbf{E}_0|^2 \mathbf{E}_\omega

\mathbf{P} \simeq \varepsilon_0 \left( \chi^{(1)} + 3 \chi^{(3)} |\mathbf{E}_0|^2 \right) \mathbf{E}_\omega \cos(\omega t),

which is similar to the linear relationship between polarization and an electric field of a wave, with an additional non-linear susceptibility term proportional to the square of the amplitude of the external field.

For non-symmetric media (e.g. liquids), this induced change of susceptibility produces a change in refractive index in the direction of the electric field:

\Delta n = \lambda_0 K |\mathbf{E}_0|^2,

where λ₀ is the vacuum wavelength and K is the Kerr constant for the medium. The applied field induces birefringence in the medium in the direction of the field. A Kerr cell with a transverse field can thus act as a switchable wave plate, rotating the plane of polarization of a wave travelling through it. In combination with polarizers, it can be used as a shutter or modulator.

$\mathbf{P} \simeq \varepsilon_0 \left( \chi^{(1)} + \frac{3}{4} \chi^{(3)} |\mathbf{E}_\omega|^2 \right) \mathbf{E}_\omega \cos(\omega t).$ The values of K depend on the medium and are about 9.4×10⁻¹⁴ m V^{−2[citation
needed]} for water, and 4.4×10⁻¹² m V^{−2[citation
needed]} for nitrobenzene.

For crystals, the susceptibility of the medium will in general be a tensor, and the Kerr effect produces a modification of this tensor.

AC Kerr effect

In the optical or AC Kerr effect, an intense beam of light in a medium can itself provide the modulating electric field, without the need for an external field to be applied. In this case, the electric field is given by:

\mathbf{E} = \mathbf{E}_\omega \cos(\omega t),

where E_ω is the amplitude of the wave as before.

Combining this with the equation for the polarization, and taking only linear terms and those in χ⁽³⁾|E_ω|³:^:81–82

As before, this looks like a linear susceptibility with an additional non-linear term:

\mathbf{P} \simeq \varepsilon_0 \left( \chi^{(1)} + \frac{3}{4} \chi^{(3)} |\mathbf{E}_\omega|^2 \right) \mathbf{E}_\omega \cos(\omega t).

and since:

n = (1 + \chi)^{1/2} = \left( 1+\chi_{\mathrm{LIN}} + \chi_{\mathrm{NL}} \right)^{1/2} \simeq n_0 \left( 1 + \frac{1}{2 {n_0}^2} \chi_{\mathrm{NL}} \right)

where n₀=(1+χ_LIN)^1/2 is the linear refractive index. Using a Taylor expansion since χ_NL << n₀², this gives an intensity dependent refractive index (IDRI) of:

n = n_0 + \frac{3\chi^{(3)}}{8 n_0} |\mathbf{E}_{\omega}|^2 = n_0 + n_2 I

where n₂ is the second-order nonlinear refractive index, and I is the intensity of the wave. The refractive index change is thus proportional to the intensity of the light travelling through the medium.

The values of n₂ are relatively small for most materials, on the order of 10⁻²⁰ m² W⁻¹ for typical glasses. Therefore beam intensities (irradiances) on the order of 1 GW cm⁻² (such as those produced by lasers) are necessary to produce significant variations in refractive index via the AC Kerr effect.

The optical Kerr effect manifests itself temporally as self-phase modulation, a self-induced phase- and frequency-shift of a pulse of light as it travels through a medium. This process, along with dispersion, can produce optical solitons.

Spatially, an intense beam of light in a medium will produce a change in the medium's refractive index that mimics the transverse intensity pattern of the beam. For example, a Gaussian beam results in a Gaussian refractive index profile, similar to that of a gradient-index lens. This causes the beam to focus itself, a phenomenon known as self-focusing.

As the beam self-focuses, the peak intensity increases which, in turn, causes more self-focusing to occur. The beam is prevented from self-focusing indefinitely by nonlinear effects such as multiphoton ionization, which become important when the intensity becomes very high. As the intensity of the self-focused spot increases beyond a certain value, the medium is ionized by the high local optical field. This lowers the refractive index, defocusing the propagating light beam. Propagation then proceeds in a series of repeated focusing and defocusing steps.

SOURCE :
http://en.wikipedia.org/wiki/Kerr_effect

0 komentar |

Pockels effect

The Pockels effect (after Friedrich Carl Alwin Pockels who studied the effect in 1893), or Pockels electro-optic effect, produces birefringence in an optical medium induced by a constant or varying electric field. It is distinguished from the Kerr effect by the fact that the birefringence is proportional to the electric field, whereas in the Kerr effect it is quadratic to the field. The Pockels effect occurs only in crystals that lack inversion symmetry, such as lithium niobate or gallium arsenide and in other noncentrosymmetric media such as electric-field poled polymers or glasses.

Pockels Cells

Pockels Cells are voltage-controlled wave plates. The Pockels effect is the basis of Pockels Cells operation. Pockels Cells may be used to rotate the polarization of a passing beam. See Applications below for uses.

A transverse Pockels Cell comprises two crystals in opposite orientation, which give a zero order wave plate when voltage is turned off. This is often not perfect and drifts with temperature. But the mechanical alignment of the crystal axis is not so critical and is often done by hand without screws; while misalignment leads to some energy in the wrong ray (either e or o – for example, horizontal or vertical), in contrast to the longitudinal case, the loss is not amplified through the length of the crystal.

The electric field can be applied to the crystal medium either longitudinally or transversely to the light beam. Longitudinal Pockels Cells need transparent or ring electrodes. Transverse voltage requirements can be reduced by lengthening the crystal.

Alignment of the crystal axis with the ray axis is critical. Misalignment leads to birefringence and to a large phase shift across the long crystal. This leads to polarization rotation if the alignment is not exactly parallel or perpendicular to the polarization.

Dynamics within the cell

Because of the high relative dielectric constant of ε_r ≈ 36 inside the crystal, changes in the electric field propagate at a speed of only c/6. Fast non-fiber optic cells are thus embedded into a matched transmission line. Putting it at the end of a transmission line leads to reflections and doubled switching time. The signal from the driver is split into parallel lines which lead to both ends of the crystal. When they meet in the crystal their voltages add up. Pockels Cells for fibre optics may employ a traveling wave design to reduce current requirements and increase speed.

Usable crystals also exhibit the piezoelectric effect to some degree^[1] (RTP has the lowest, BBO and lithium niobate are high). After a voltage change sound waves start propagating from the sides of the crystal to the middle. This is important not for pulse pickers, but for boxcar windows. Guard space between the light and the faces of the crystals needs to be larger for longer holding times. Behind the sound wave the crystal stays deformed in the equilibrium position for the high electric field. This increases the polarization. Due to the growing of the polarized volume the electric field in the crystal in front of the wave increases linearly, or the driver has to provide a constant current leakage.

The driver electronics

The driver must withstand the doubled voltage returned to it. Pockels Cells behave like a capacitor. When switching these to high voltage a high charge is needed; consequently, 3 ns switching requires about 40 A for a 5 mm aperture. Shorter cables reduce the amount of charge wasted in transporting current to the cell.

The driver may employ many transistors connected parallel and serial. The transistors are floating, and need DC isolation for their gates. To do this, the gate signal is connected via optical fiber, or the gates are driven by a large transformer. In this case, careful compensation for feedback is needed to prevent oscillation.

The driver may employ a cascade of transistors and a triode. In a classic, commercial circuit the last transistor is an IRF830 MOSFET and the triode is an Eimac Y690 triode. The setup with a single triode has the lowest capacity; this even justifies turning off the cell by applying the double voltage. A resistor ensures the leakage current needed by the crystal and later to recharge the storage capacitor. The Y690 switches up to 10 kV and the cathode delivers 40 A if the grid is on +400 V. In this case the grid current is 8 A and the input impedance is thus 50 ohms, which matches standard coaxial cables, and the MOSFET can thus be placed remotely. Some of the 50 ohms are spent on an additional resistor which pulls the bias on -100 V. The IRF can switch 500 volts. It can deliver 18 A pulsed. Its leads function as an inductance, a storage capacitor is employed, the 50 ohm coax cable is connected, the MOSFET has an internal resistance, and in the end this is a critically damped RLC circuit, which is fired by a pulse to the gate of the MOSFET.

The gate needs 5 V pulses (range: ±20 V) while provided with 22 nC. Thus the current gain of this transistor is one for 3 ns switching, but it still has voltage gain. Thus it could theoretically also be used in common gate configuration and not in common source configuration. Transistors, which switch 40 V are typically faster, so in the previous stage a current gain is possible.

Applications of Pockels Cells

Pockels Cells are used in a variety of scientific and technical applications:

A Pockels Cell, combined with a polarizer, can be used for a variety of applications. Switching between no optical rotation and 90° rotation creates a fast shutter capable of "opening" and "closing" in nanoseconds. The same technique can be used to impress information on the beam by modulating the rotation between 0° and 90°; the exiting beam's intensity, when viewed through the polarizer, contains an amplitude-modulated signal.
Preventing the feedback of a laser cavity by using a polarizing prism. This prevents optical amplification by directing light of a certain polarization out of the cavity. Because of this, the gain medium is pumped to a highly excited state. When the medium has become saturated by energy, the Pockels Cell is switched, and the intracavity light is allowed to exit. This creates a very fast, high intensity pulse. Q-switching, chirped pulse amplification, and cavity dumping use this technique.
Pockels Cells can be used for quantum key distribution by polarizing photons.
Pockels Cells in conjunction with other EO elements can be combined to form electro-optic probes.

A Pockels Cell was used by MCA Disco-Vision (DiscoVision) engineers in the optical videodisc mastering system. Light from an argon-ion laser was passed through the Pockels Cell to create pulse modulations corresponding to the original FM video and audio signals to be recorded on the master videodisc. MCA used the Pockels Cell in videodisc mastering until the sale to Pioneer Electronics. To increase the quality of the recordings, MCA patented a Pockels Cell stabilizer that reduced the second harmonic distortion that could be created by the Pockels Cell during mastering. MCA used either a DRAW (Direct Read After Write) mastering system or a photoresist system. The DRAW system was originally preferred, since it didn't require clean room conditions during disc recording, and allowed instant quality checking during mastering. The original single-sided test pressings from 1976/77 were mastered with the DRAW system as were the "educational", non-feature titles at the format's release in December 1978.

source :

http://en.wikipedia.org/wiki/Pockels_effect