electro – optic – magneto optic and acousto – optic effects (1)

7/29/2019 Electro optic Magneto Optic and Acousto Optic effects (1)

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Chapter 3

Magneto-, acousto-, and

electro-optical effect

References * Eugene Hecht, "Optics", Addison-Wesley, Reading 1987

* NewFocus Application Note 02: Practical Uses and Applications of Electro-Optic Modulators http://www.newfocus.com/

* AAoptoelectronic: Application notes http://www.aaoptoelectronic.com/

Optical Activity A rotation in the plane of oscillation of a linearly polarizedbeam upon passage through a medium. It occurs in asymmetrical molecules wherethe electrons are more easily accelerated in one orientation than another. Therandom orientation of molecules in the medium does not cancel the effect, sinceflipped molecules retain the same handedness (just as a right-handed helix remainsright-handed upon turning it upside down). Optical activity is much more prominentin nature than would be normally expected due to the fact that almost all aminoacids (the building blocks of life) are left-handed.

The angle of rotation is defined as positive when the electric field is rotatedclockwise,

(3.1) =d

0(nl nr)

where d is the thickness of the medium, 0is the vacuum wavelength,nl isthe index of refraction for parallel polarization and nris the index of refraction forperpendicular polarization. A medium in which nl > nr is called d-rotary, and amedium in which nl < nr is called l-rotary.

51


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52CHAPTER 3. MAGNETO-, ACOUSTO-, AND ELECTRO-OPTICAL EFFECT

A quantity called specific rotary power is defined as

(3.2) [specificrotarypower] =

d

3.1 Magneto-optic effect

A magneto-optic effect is any one of a number of phenomena in which an electromag-netic wave propagates through a medium that has been altered by the presence of aquasistatic magnetic field. In such a material, which is also called gyrotropic or gy-romagnetic, left- and right-rotating elliptical polarizations can propagate at differentspeeds, leading to a number of important phenomena. When light is transmitted

through a layer of magneto-optic material, the result is called the Faraday effect:the plane of polarization can be rotated, forming a Faraday rotator. The resultsof reflection from a magneto-optic material are known as the magneto-optic Kerreffect (not to be confused with the nonlinear Kerr effect).

In general, magneto-optic effects break time reversal symmetry locally (i.e.when only the propagation of light, and not the source of the magnetic field, isconsidered), which is a necessary condition to construct devices such as opticalisolators (through which light passes in one direction but not the other). (Theother, less useful, way to break time reversal symmetry is to rely upon absorptionloss.)

3.1.1 Faraday effect

The Faraday effect or Faraday rotation is a magneto-optical phenomenon, or aninteraction between light and a magnetic field. The rotation of the plane of polar-ization is proportional to the intensity of the component of the magnetic field in thedirection of the beam of light.

The Faraday effect, a type of magneto-optic effect, discovered by Michael Fara-day in 1845, was the first experimental evidence that light and magnetism are re-

lated. The theoretical basis for that relation, now called electromagnetic radiation,was developed by James Clerk Maxwell in the 1860s and 1870s. This effect occursin most optically transparent dielectric materials (including liquids) when they aresubject to strong magnetic fields.

In the quantum mechanical description, it occurs because imposition of a mag-netic field alters the energy levels (cf. Zeeman effect).

The Faraday effect is a result of ferromagnetic resonance when the perme-ability of a material is represented by a tensor. This resonance causes waves to be


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3.1. MAGNETO-OPTIC EFFECT 53

decomposed into two circularly polarized rays which propagate at different speeds, aproperty known as circular birefringence. The rays can be considered to re-combine

upon emergence from the medium, however owing to the difference in propagationspeed they do so with a net phase offset, resulting in a rotation of the angle of linearpolarization.

There are a few applications of Faraday rotation in measuring instruments.For instance, the Faraday effect has been used to measure optical rotatory power,for amplitude modulation of light, and for remote sensing of magnetic fields.

The relation between the angle of rotation of the polarization and the magneticfield in a diamagnetic material is: Polarization rotation due to the Faraday effect

(3.3) = VBd

where is the angle of rotation (in radians), B is the magnetic flux density inthe direction of propagation (in teslas), d is the length of the path (in meters) wherethe light and magnetic field interact, V is the Verdet constant for the material. Thisempirical proportionality constant (in units of radians per tesla per metre) varieswith wavelength and temperature and is tabulated for various materials.

A positive Verdet constant corresponds to L-rotation (anticlockwise) when thedirection of propagation is parallel to the magnetic field and to R-rotation (clock-wise) when the direction of propagation is anti-parallel. Thus, if a ray of light is

passed through a material and reflected back through it, the rotation doubles.Some materials, such as terbium gallium garnet (TGG) have extremely high

Verdet constants ( 40radT1m1). By placing a rod of this material in a strongmagnetic field, Faraday rotation angles of over 0.78 rad (45) can be achieved. Thisallows the construction of Faraday rotators, which are the principal component ofFaraday isolators, devices which transmit light in only one direction.

Similar isolators are constructed for microwave systems by using ferrite rodsin a waveguide with a surrounding magnetic field.

Faraday rotation in the interstellar medium The Faraday effect is imposedon light over the course of its propagation from its origin to the Earth, through theinterstellar medium. Here, the effect is caused by free electrons and can be char-acterized as a difference in the refractive index seen by the two circularly polarizedpropagation modes. Hence, in contrast to the Faraday effect in solids or liquids,interstellar Faraday rotation has a simple dependence on the wavelength of light(), namely: = RM2, where the overall strength of the effect is characterized byRM, the rotation measure.


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Faraday rotation is an important tool in astronomy for the measurement ofmagnetic fields, which can be estimated from rotation measures given a knowledge

of the electron number density.

Two gyrotropic materials with reversed rotation directions of the two principalpolarizations, corresponding to complex-conjugate tensors for lossless media, arecalled optical isomers.

3.1.2 Gyrotropic permittivity

In particular, in a magneto-optic material the presence of a magnetic field (either

externally applied or because the material itself is ferromagnetic) can cause a changein the permittivity tensor of the material. The becomes anisotropic, a 3 3matrix, with complex off-diagonal components, depending of course on the frequency of incident light. If the absorption losses can be neglected, is a Hermitian matrix.The resulting principal axes become complex as well, corresponding to elliptically-polarized light where left- and right-rotating polarizations can travel at differentspeeds (analogous to birefringence).

More specifically, for the case where absorption losses can be neglected, themost general form of Hermitian is:

(3.4) =

xx xy + igz xz igyxy igz yy yz + igxxz + igy

yz igx

zz

or equivalently the relationship between the displacement field D and the electricfield E is:

(3.5) D = E = E+ iE g

where is a real symmetric matrix and g = (gx, gy, gz) is a real pseudovector called

the gyration vector, whose magnitude is generally small compared to the eigenvaluesof. The direction ofg is called the axis of gyration of the material. To first order,g is proportional to the applied magnetic field:

(3.6) g = 0(m)H

where (m) is the magneto-optical susceptibility (a scalar in isotropic media, butmore generally a tensor). If this susceptibility itself depends upon the electric field,one can obtain a nonlinear optical effect of magneto-optical parametric generation


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3.1. MAGNETO-OPTIC EFFECT 55

(somewhat analogous to a Pockels effect whose strength is controlled by the appliedmagnetic field).

The simplest case to analyze is the one in which g is a principal axis (eigen-vector) of , and the other two eigenvalues of are identical. Then, if we let g liein the z direction for simplicity, the tensor simplifies to the form:

(3.7) =

1 +igz 0igz 1 0

0 0 2

Most commonly, one considers light propagating in the z direction (parallel to g).In this case the solutions are elliptically polarized electromagnetic waves with phasevelocities 1/(1 gz)(where is the magnetic permeability). This difference inphase velocities leads to the Faraday effect.

For light propagating purely perpendicular to the axis of gyration, the prop-erties are known as the Cotton-Mouton effect and used for a Circulator.

Cotton-Mouton effect The production of birefringence when a constant mag-netic field is applied to a transparent medium perpendicular to the direction ofpropagation. It occurs in liquids, and is much stronger than the Voigt effect. It isthe magnetic analog of the Kerr effect.

(3.8) n B2

3.1.3 Magneto-optic Kerr effect

Magneto-optic Kerr effect (MOKE) is one of the magneto-optic effects. It describeschanges of the reflections from the magnetized media. It is similar to the Faradayeffect that describes the light passing through the media. The light that is reflectedfrom a magnetized surface, changes in polarization.

MOKE can be further categorized by the direction of the magnetization vectorwith respect to the reflection surface and the plane of incidence. When the magne-tization vector is perpendicular to the reflection surface and parallel to the plane ofincidence, the effect is called the polar Kerr effect. To simplify the analysis, nearnormal incidence is usually employed when doing experiments in the polar geometry.In the longitudinal effect the magnetization vector is parallel to both the reflectionsurface and the plane of incidence. When the magnetization is perpendicular tothe plane of incidence and parallel to the surface it is said to be in the transverseconfiguration.


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Reflection of a beam of linearly polarised light from a magnetised surface causesthe polarisation to become elliptical, with the principal axis rotated with respect to

the incoming light. Usually the amount of rotation (in radians) and of ellipticity(ratio between the minor and major axis of the ellipse) induced in the reflectedbeam is of the order of 1/1000, i.e. relatively small. This phenomenon is knownas the magneto-optic Kerr effect (MOKE). In the case of ultrathin films with totalthickness below about 10 nm, the effect is proportional to the film thickness and thetechnique assumes the name Surface-MOKE (SMOKE). SMOKE is mainly used tomeasure the hysteresis loops of thin magnetic films, by plotting the signal (rotationor ellipticity) as a function of the applied magnetic field. Unfortunately, such asignal is proportional to the magnetisation, but an absolute quantitative estimationof it requires the use of other magnetometry techniques, such as vibrating sample

magnetometry. MOKE can also be used as a scanning microscopy technique, sothat magnetic domain imaging with micrometric resolution becomes possible. Evensubmicrometric resolution can be achieved combining magnetooptic measurementswith a scanning near-field optical microscopy (SNOM) apparatus.

Although a microscopic explanation of magneto-optic effects would have toconsider the coupling between the electrical field of the polarized light and the elec-tron spin within the magnetic medium which occurs through the spin-orbit interac-tion, a simple interpretation is usually achieved using a macroscopic point of view,where magneto-optic effects arise from the antisymmetric, off-diagonal elements inthe dielectric tensor. In particular, the magneto-optical effect can be understood

on the basis of different response of the electrons to left and right polarized elec-tromagnetic waves. In fact, a linearly polarised beam of light can be thought as asuperposition of such two kinds of waves. Electrons will be driven into left circularmotion by the left-polarized wave while the right-polarized wave will drive theminto right circular motion, with equal radii of the orbits. Now, since the dipolemoment is proportional to this radius, an external magnetic field applied to in thepropagation direction will cause an additional Lorentz force to act on each electron.As a consequence, the radius for the left-circular motion will be reduced and theradius for the right circular motion will be increased, so that the emerging light iselliptically (instead of linearly) polarized.

3.2 Electro-optical effect

The electro-optical effect causes a change in the refractive index as a function of anelectric field

* Pockels effect (or linear electro-optic effect): change in the refractive indexlinearly proportional to the electric field. Only certain crystalline solids show the


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3.2. ELECTRO-OPTICAL EFFECT 57

Pockels effect, as it requires inversion asymmetry

* non-linear Kerr effect (or quadratic electro-optic effect, QEO effect): change

in the refractive index proportional to the square of the electric field. All materialsdisplay the Kerr effect, with varying magnitudes, but it is generally much weakerthan the Pockels effect

* electro-gyration: change in the optical activity.

3.2.1 Pockels effect

The Pockels effect, or Pockels electro-optic effect, produces birefringence in an opti-cal 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 electricfield, whereas in the Kerr effect it is quadratic in the field. The Pockels effect occursonly in crystals that lack inversion symmetry, such as lithium niobate or galliumarsenide.

Friedrich Carl Alwin Pockels studied the effect which bears his name in 1893.

Pockels cells The Pockels effect is used to make Pockels cells, which are voltage-controlled wave plates. The electric field can be applied to the crystal medium eitherlongitudinally or transversely to the light beam.

The electric field can be longitudinal or transverse to the light ray. Longitu-dinal Pockels cells need transparent or ring electrodes. Transverse voltage require-ments can be reduced by lengthening the crystal.

The crystal axis can be longitudinal or transverse. A longitudinal cell has tobe quite big, as the crystals are somehow inefficient in this mode. Alignment of thecrystal axis with the ray axis is critical as misalignment leads to birefringence and toa large phase shift across the long crystal. This leads to polarization rotation if thealignment is not exactly parallel or perpendicular to the polarization. A transversecell consists of 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 doneby hand without screws; while misalignment leads to some energy in the wrong ray(either e or o), in contrast to the longitudinal case, this is not amplified through thelength of the crystal.

Pockels cells may be used to rotate the polarization of a passing beam.

Because of the high dielectric constant of the crystal, Pockels cells behavelike a capacitor. When switching these to high voltage a high charge is needed;


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consequently, fast switching requires large currents. Pockels cells for fibre opticsmay employ a travelling wave design to reduce current requirements.

Applications of Pockels cells Pockels cells are used in a variety of scientific andtechnical applications:

A Pockels cell, combined with a polarizer, can be used for a variety of applica-tions. Switching between no optical rotation and 90 rotation creates a fast shuttercapable of "opening" and "closing" in nanoseconds. The same technique can beused to impress information on the beam by modulating the rotation between 0

and 90; the exiting beams intensity, when viewed through the polarizer, containsan amplitude-modulated signal.

Preventing the feedback of a laser cavity by using a polarizing prism. Thisprevents optical amplification by directing light of a certain polarization out of thecavity. Because of this, the gain medium is pumped to a highly excited state. Whenthe medium has become saturated by energy, the Pockels cell is switched, and theintracavity 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.

3.2.2 Kerr effect

The Kerr effect or the quadratic electro-optic effect (QEO effect) is a change in therefractive index of a material in response to an electric field. It is distinct from thePockels effect in that the induced index change is directly proportional to the squareof the electric field instead of to the magnitude of the field. All materials show aKerr effect, but certain liquids display the effect more strongly than other materialsdo. The Kerr effect was discovered in 1875 by John Kerr, a Scottish physicist.

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

Kerr electro-optic effect (DC Kerr)

The Kerr electro-optic effect, or DC Kerr effect, is the special case in which theelectric field is a slowly varying external field applied by, for instance, a voltage onelectrodes across the material. Under the influence of the applied field, the materialbecomes birefringent, with different indexes of refraction for light polarized parallelto or perpendicular to the applied field. The difference in index of refraction, n,is given by


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(3.9) n = KE2

,

where is the wavelength of the light, K is the Kerr constant, and E is the amplitudeof the electric field. This difference in index of refraction causes the material to actlike a waveplate when light is incident on it in a direction perpendicular to theelectric field. If the material is placed between two "crossed" (perpendicular) linearpolarizers, no light will be transmitted when the electric field is turned off, whilenearly all of the light will be transmitted for some optimum value of the electricfield. Higher values of the Kerr constant allow complete transmission to be achievedwith a smaller applied electric field.

Some polar liquids, such as nitrotoluene (C5H7NO2) and nitrobenzene (C6H5NO2)exhibit very large Kerr constants. A glass cell filled with one of these liquids is calleda Kerr cell. These are frequently used to modulate light, since the Kerr effect re-sponds very quickly to changes in electric field. Light can be modulated with thesedevices at frequencies as high as 10 GHz. Because the Kerr effect is relativelyweak, a typical Kerr cell may require voltages as high as 30 kV to achieve completetransparency. This is in contrast to Pockels cells, which can operate at much lowervoltages. Another disadvantage of Kerr cells is that the best available material,nitrobenzene, is both poisonous and explosive. Some transparent crystals have alsobeen used for Kerr modulation, although they have smaller Kerr constants.

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

(3.10) P = 0(1)E+ 0

(2)EE+ 0(3)EEE+ . . .

where 0 is the vacuum permittivity and (n) is the n-th order component of the

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

For materials exhibiting a non-negligible Kerr effect, the third, (3) term issignificant. Consider the net electric field E produced by a light wave of frequency together with an external electric field E

0:

(3.11) E = E0 +E cos(t),

where E is the vector amplitude of the wave.

Combining these two equations produces a complex expression for P. For theDC Kerr effect, we can neglect all except the linear terms and those in (3)|E0|

2E:

(3.12) P 0

(1) + 3(3)|E0|2E cos(t),


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which is similar to the linear relationship between polarization and an electric field ofa 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 changed of susceptibility

produces a change in refractive index in the direction of the electric field:

(3.13) n = 0K|E0|2,

where 0 is the vacuum wavelength and K is the Kerr constant for the medium. Theapplied field induces birefringence in the medium in the direction of the field. AKerr cell with a transverse field can thus act as a switchable wave plate, rotating theplane of polarization of a wave travelling through it. In combination with polarizers,it can be used as a shutter or modulator.

The values of K depend on the medium and are about 9.4 1014mV2 forwater, and 4.4 1012mV2 for nitrobenzene.

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

Optical Kerr effect (AC Kerr)

The optical Kerr effect, or AC Kerr effect is the case in which the electric field is dueto 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 forthe nonlinear optical effects of self focusing and self-phase modulation, and is thebasis for Kerr-lens modelocking. This effect only becomes significant with veryintense beams such as those from lasers.

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

(3.14) E = E cos(t),

where E is the amplitude of the wave as before.

Combining this with the equation for the polarization, and taking only linearterms and those in (3)|E|

3:

(3.15) P 0

(1) +

3

4(3)|E|

2

E cos(t).

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

(3.16) = LIN + NL = (1) +

3(3)

4|E|

2,


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and since:

(3.17) n = (1 + )1/2

= (1 + LIN + NL)1/2

n0

1 +

1

2n02NL

where n0 = (1 + LIN)1/2 is the linear refractive index. Using a Taylor expansion

since NL n20, this give an intensity dependent refractive index (IDRI) of:

(3.18) n = n0 +3(3)

8n0|E|

2 = n0 + n2I

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

The values of n2 are relatively small for most materials, on the order of1020m2W1 for typical glasses. Therefore beam intensities (irradiances) on theorder of 1 GW cm2 (such as those produced by lasers) are necessary to producesignificant 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 amedium. This process, along with dispersion, can produce optical solitons.

Spatially, an intense beam of light in a medium will produce a change in themediums refractive index that mimics the transverse intensity pattern of the beam.

For example, a Gaussian beam results in a Gaussian refractive index profile, similarto that of a gradient-index lens. This causes the beam to focus itself, a phenomenonknown as self-focusing.

Kerr-lens modelocking

Kerr-lens modelocking is a method of modelocking lasers via a nonlinear opticalprocess known as the optical Kerr effect. This method allows the generation ofpulses of light with a duration as short as a few femtoseconds.

The optical Kerr effect is a process which results from the nonlinear responseof an optical medium to the electric field of an electromagnetic wave. The refractiveindex the medium is dependent on the field strength .

hard aperture Kerr-lens modelocking principle

Because of the non-uniform power density distribution in a Gaussian beam (asfound in laser resonators) the refractive index changes across the beam profile; therefractive index experienced by the beam is greater in the centre of the beam thanat the edge. Therefore a rod of an active Kerr medium works like a lens for high


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intensity light. This is called self-focusing and in extreme cases leads to materialdestruction. In the laser cavity short bursts of light will then be focused differently

to continuous waves (cw).The favor the pulses over cw, the cavity could be made unstable for cw-

operation, but more often a low stability is a by-product of a cavity design puttingemphasis on aperture effects. Older designs used a hard aperture, that simply cutsoff, while modern designs use a soft aperture, that means the overlap between thepumped region of the gain medium and the pulse. While the effect of a lens on afree laser beam is quite obvious, inside a cavity the whole beam tries to adapts tothis change. The standard cavity with flat mirrors and a thermal lens in the lasercrystal has the smallest beam width on the end-mirrors. With the additional kerrlens the width on the end-mirror gets even smaller. Therefore small end-mirrors

(hard aperture) favor pulses. For a soft aperture consider an infinite laser crystalwith a thermal lens. A laser beam is guided like in a glass fibre. With an additionalkerr lens the beam width gets smaller. In a real laser the crystal is finite, but for asoft aperture laser the cavity mirrors are designed to act as a 1:1 telescope imagingthe light, which exits an end-face, back onto the same end-face. So for the beam,the crystal looks infinite.

The length of the medium used for KLM is limited by group velocity dispersion.KLM is used in Carrier envelope offset control.

Starting a Kerr-lens modelocked laser Initiation of Kerr-lens modelockingdepends on the strength of the nonlinear effect involved. If the laser field buildsup in a cavity the laser has to overcome the region of cw operation, which often isfavored by the pumping mechanism. This can be achieved by a very strong Kerr-lensing that is strong enough to modelock due to small changes of the laser fieldstrength (laser field build-up or stochastic fluctuations). Modelocking can also bestarted by shifting the optimum focus from the cw-operation to pulsed operationwhile changing the power density by kicking the end mirror of the resonator cavity(though a piezo mounted, syncronous oscilating end-mirror would be more turnkey). Other principles involve different nonlinear effects like saturable absorbers

and saturable Bragg reflectors, which induce pulses short enough to initiate theKerr-lensing process.

Modelocking - evolution of the pulse Intensity changes with lengths of nanosec-onds, are amplified by the Kerr-lensing process and the pulselength further shrinksto achieve higher field strengths in the center of the pulse. This sharpening processis only limited by the bandwidth achievable with the laser material and the cavity-mirrors as well as the dispersion of the cavity. The shortest pulse achievable with a


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given spectrum is called the bandwidth-limited pulse.

Laser media for ultrashort pulses (e.g. Ti:Sapphire) Dispersion management

with prism sequences. Chirped mirror technology allows to compensate timing mis-match of different wavelengths inside the cavity due to material dispersion whilekeeping the stability high and the losses low.

The Kerr effect leads to the Kerr-lens and Self-phase modulation at the sametime. To a first approximation it is possible to consider them as independent effects.

Applications Since Kerr-lens modelocking is an effect that directly reacts on theelectric field, the response time is high enough to produce light pulses in the visibleand near infrared with lengths of less then 5 femtoseconds. Due to the high elec-

trical field strength focused ultrashort laser beams can overcome the threshold of1014W cm2, which surpasses the field strength of the electron-ion bond in atoms.These short pulses open the new field of ultrafast optics, which is a field of nonlinearoptics that gives access to a completely new class of phenomena like measurementof electron movements in an atom (attosecond phenomena), coherent broadbandlight generation (ultrabroad lasers) and thereby gives rise to many new applicationsin optical sensing (e.g. coherent laser radar, ultrahigh resolution optical coherencetomography) material processing and other fields like metrology (extremely exactfrequency and time measurements).

Electro-optic modulator

The lectro-optic modulator (EOM) is an optical device in which a signal-controlledelement displaying electro-optic effect is used to modulate a beam of light. Themodulation may be imposed on the phase, frequency, amplitude, or direction of themodulated beam. Modulation bandwidths extending into the gigahertz range arepossible with the use of laser-controlled modulators.

Generally a nonlinear optical material (organic polymers have the fastest re-sponse rates, and thus are best for this application) with an incident static or lowfrequency optical field will see a modulation of its refractive index.

Types of EOMs

Phase modulation The simplest kind of EOM consists of a crystal, suchas Lithium niobate, whose refractive index is a function of the strength of the localelectric field. That means that if lithium niobate is exposed to an electric field, lightwill travel more slowly through it. But the phase of the light leaving the crystal


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is directly proportional to the length of time it took that light to pass through it.Therefore, the phase of the laser light exiting an EOM can be controlled by changing

the electric field in the crystal.Note that the electric field can be created placing a parallel plate capacitor

across the crystal. Since the field inside a parallel plate capacitor depends linearlyon the potential, the index of refraction depends linearly on the field (for crystalswhere Pockels effect dominates), and the phase depends linearly on the index ofrefraction, the phase modulation must depend linearly on the potential applied tothe EOM.

Amplitude modulation Once one can make a phase modulating EOM, itsa fairly simple matter to turn that into an amplitude modulating EOM by using aMach-Zehnder interferometer. Simply use a beam splitter to divide the laser lightinto two paths, one of which has a phase modulator as described above, and thenrecombine the two beams. By changing the electric field on the phase modulatingpath, one can control whether the two beams constructively or destructively interfereand thereby control the amplitude or intensity of the exiting light.

Applications A very common application of EOMs is for creating sidebandsin a monochromatic laser beam. To see how this works, first imagine that thestrength of a laser beam with frequency leaving the EOM is given by

(3.19) Aeit

Now suppose we apply a sinusoidally varying potential to the EOM with frequency and small amplitude . This adds a time dependent phase to the above expression,

(3.20) Aeit+isin(t).

Since is small, we can use the Taylor expansion for the exponential

(3.21) Aeit (1 + isin(t)) ,

to which we apply a simple identity for sine,

(3.22) Aeit

1 +

2(eit eit)

= A

eit +

2ei(+)t

2ei()t

.

This expression we interpret to mean that we have the original carrier frequencyplus two small sidebands, one at + and another at . Notice however that


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3.3. ACOUSTO-OPTICAL EFFECT 65

we only used the first term in the Taylor expansion - in truth there are an infinitenumber of sidebands. There is a useful identity involving Bessel functions

(3.23) Aeit+isin(t) = Aeit

k=0

Jk()eikt +

k=0

(1)kJk()eikt

,

which gives the amplitudes of all the sidebands. Notice that if one modulatesthe amplitude instead of the phase, one gets only the first set of sidebands,

(3.24) (1 + sin(t)) Aeit = Aeit +A

2i ei(+)t ei()t

.

3.3 Acousto-optical effect

Modification of the refractive index by the oscillating mechanical pressure of a soundwave

An acousto-optic modulator (AOM), also called a Bragg cell, uses the acousto-optic effect to diffract and shift the frequency of light using sound waves (usuallyat radio-frequency). They are used in lasers for Q-switching, telecommunicationsfor signal modulation, and in spectroscopy for frequency control. A piezoelectric

transducer is attached to a material such as glass. An oscillating electric signaldrives the transducer to vibrate, which creates sound waves in the glass. Thesecan be thought of as moving periodic planes of expansion and compression thatchange the index of refraction. Incoming light scatters (Brillouin scattering) offthe resulting periodic index modulation and interference occurs similar to in Braggdiffraction. The interaction can be thought of as four-wave mixing between phononsand photons. The properties of the light exiting the AOM can be controlled in fiveways:

Deflection A diffracted beam emerges at an angle that depends on the wave-

length of the light relative to the wavelength of the sound

(3.25) sin =

m

2

where m = ...-2,-1,0,1,2,... is the order of diffraction. As the AOM get thicker onlyphasematched orders are diffracted, this is called Bragg diffraction. The angulardeflection can range from 1 to 5000 beam widths (the number of resolvable spots).Consequently, the deflection is typically limited to tens of milliradians.


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3.3. ACOUSTO-OPTICAL EFFECT 67

Intensity The amount of light diffracted by the sound wave depends on the in-tensity of the sound. Hence, the intensity of the sound can be used to modulate

the intensity of the light in the diffracted beam. Typically, the intensity that isdiffracted into m=0 order can be varied between 15% to 99% of the input lightintensity. Likewise, the intensity of the m=1 order can be varied between 0% and80%.

Frequency One difference from Bragg diffraction is that the light is scatteringfrom moving planes. A consequence of this is the frequency of the diffracted beam fin order m will be Doppler-shifted by an amount equal to the frequency of the soundwave F.

(3.26) f f + mF

This frequency shift is also required by the fact that energy and momentum (ofthe photons and phonons) are conserved in the process. A typical frequency shiftvaries from 27 MHz, for a less-expensive AOM, to 400 MHz, for a state-of-the-artcommercial device. In some AOMs, two acoustic waves travel in opposite directionsin the material, creating a standing wave. Diffraction from the standing wave doesnot shift the frequency of the diffracted light.

Phase In addition, the phase of the diffracted beam will also be shifted by the

phase of the sound wave. The phase can be changed by an arbitrary amount.

Polarization Collinear transversal acoustic waves or perpendicular longitudinalwaves can change the polarization. The acoustic waves induce a birefringent phase-shift, much like in a Pockels cell. The acousto-optic tunable filter, especially thedazzler, which can generate variable pulse shapes, is based on this principle.

Realization Acousto-optic modulators are much faster than typical mechanicaldevices such as tiltable mirrors. The time it takes an AOM to shift the exiting

beam in is roughly limited to the transit time of the sound wave across the beam(typically 5 to 100 microseconds). This is fast enough to create active modelockingin an ultrafast laser. When faster control is necessary electro-optic modulators areused. However, these require very high voltages (e.g. 10 kilovolts), whereas AOMsoffer more deflection range, simple design, and low power consumption (


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electro – optic – magneto optic and acousto – optic effects (1)

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