22 Introduction to Nonlinear Optics
Dr. Ajit K. Mahapatro
Learning Objectives
From this module students may get to know about the following
i. The fundamentals of light propagation.
ii. Polarization of light for constraining the components of the unpolarized light to propagate to have component in a specified direction.
iii. Understanding of various processes and techniques to observe linearly, circularly and elliptically polarized lights.
1. Polarization: Microscopic and Macroscopic
1.1 Unpolarized Light
A light is a wave that propagates in all directions. Photon is the quanta of light and exhibits electromagnetic wave that vibrates in variety of directions. The light propagated in a given direction consists of independent waves whose planes of vibration are randomly oriented perpendicularly to the direction of propagation. Light propagation in more than one plane is termed as unpolarized light. This is schematized in Figure 1. Light emitted by the sun, by a lamp or by a candle flame are the examples of unpolarized light. The arrows in Figure 1 represent the direction of propagation of light waves. The randomly orientated plane waves produce symmetry about the direction of propagation.
Figure 1: Unpolarised light and the direction of the components of light.
1.2 Polarized Light
Polarization is a property of waves that propagates in a specifically defined orientation. Radio waves and radar waves are also electromagnetic (EM) waves, which are resulting from a surging dipole charge up and down. These kinds of EM waves are not randomly generated, not symmetric , and prefer non-symmetric orientations of the electric and magnetic oscillations relative to the direction of propagation. Such kinds of waves are referred as polarized. Randomly oriented waves could be polarized by placing a polarizer on the way of the unpolarized light, as schematized in Figure 2.
Most sources of light are incoherent and unpolarized or only “partially polarized” because they consist of a random mixture of waves having different frequencies, wavelengths, phases, and polarization states. The waves that inherently oscillate in the direction of propagation such as sound waves in a gas or liquid could not be polarized, because sound waves are longitudinal waves.
Figure 2: Polarized light and the direction of propagation.
Origin of Nonlinear Optics (NLO)
When light wave propagates through an optical medium, the oscillating electromagnetic field exerts a polarizing force on all the electrons in a medium. Since the inner electrons of the atoms are tightly bound to the nuclei, the major polarizing effect is exerted on the outer or valence electrons. With the excitation of ordinary light sources through a medium, the radiation fields created by the incident lights are much smaller than the fields that bind the electrons to the atoms (~ 108 V/m) of the medium. Hence the radiation acts as a small perturbation. This produces a polarization that is proportional to the electric field of the light wave (P ∝ E). The light wave with such a low intensity can not affect the atomic fields to an extent that could change the optical parameters (viz., refractive index, intensity, absorption coefficient, frequency, etc…). However, if the radiation field is comparable with the atomic fields (e.g. laser radiation), then the relation between the polarization
When light wave propagates through an optical medium, the oscillating electromagnetic field exerts a polarizing force on all the electrons in a medium. Since the inner electrons of the atoms are tightly bound to the nuclei, the major polarizing effect is exerted on the outer or valence electrons. With the excitation of ordinary light sources through a medium, the radiation fields created by the incident lights are much smaller than the fields that bind the electrons to the atoms (~ 108 V/m) of the medium. Hence the radiation acts as a small perturbation. This produces a polarization that is proportional to the electric field of the light wave (P µ E). The light wave with such a low intensity can not affect the atomic fields to an extent that could change the optical parameters (viz., refractive index, intensity, absorption coefficient, frequency, etc…). However, if the radiation fie ld is comparable with the atomic fields (e.g. laser radiation), then the relation between the polarization (P) and radiation field (E) is no longer linear (P ≠ εoE).
Nonline aroptics (NLO) is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dilectric polarization P responds nonlinearly to the electric field E of the light. This nonlinearity is typically observed at very high light intensities (values of the electric field comparable to interatomic electric fields, typically 108 V/m) such as those provided by lasers.
2. Polarization State of Light
Polarization is a phenomenon of waves that allows the waves to vibrate in a direction perpendicular to their direction of propagation, i. e., referred as transverse waves. Light is a transverse electromagnetic wave. In an electromagnetic wave, both the electric and magnetic field oscillates in different directions. Light propagates as a plane wave in free space and as a transverse wave in an isotropic medium. Both the electric and magnetic fields are perpendicular to the wave’s direction of travel, and are the transverse waves. The oscillation of these fields can be in any direction, it may be in a single direction ( linear polarization), or it may rotate at the optical frequency (circular or elliptical polarization).
2.1 Linear Polarization
In electrodynamics, linear polarization or plane polarization of electromagnetic radiat ion is a confinement of the electric field vector or magnetic field vector to a given plane along the direction of propagation. When the original unpolarized light passes through a linear polarizer, the system allows light oscillates in one direction.
Figure 3: Linear polarization of electromagnetic radiation
2.2 Circular Polarization
In electrodynamics, circular polarization of an electromagnetic wave is the confinement in which the electric field of the passing wave does not change strength but only changes the direction in a rotating manner.
In electrodynamics, the strength and direction of an electric field is defined in the electric field vector. In circularly polarized wave, at a given point in space, the tip of the electric field vector, describes a circle as time progresses (as shown in the Figure 4). If the wave is frozen in time, the electric field vector of the wave describes a helix along the direction of propagation.
Figure 4: Circular polarization of electromagnetic radiation.
2.3 Elliptical Polarization
In electrodynamics, elliptical polarization is the confinement of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An elliptically polarized wave may be resolved into two linearly polarized waves in phase quadrature with their polarization planes at right angles to each other. Since the electric field can rotate clockwise or counterclockwise as it propagates, elliptically polarized waves exhibitchirality.
Figure 5. Elliptical polarization of electromagnetic radiation
3. Techniques for Production of Polarized Light
3.1 The Wire Grid Polarizer and the Polaroid
The wire grid polarizer essentially consists of a large number of thin copper wires placed parallel to each other. When an unpolarized electromagnetic wave is incident on it then the component of the electric vector along the length of the wire is absorbed. This is due to the energy associated with the electric field is lost in the Joule heating of the wires.
Figure 6. Polarization of light by wire grid polarizer
Since it is extremely difficult to fabricate a wire grid polarizer which would be effective for visible light, one may employ long chain polymer molecules with high conductivity a long the length of the chain alignedal most parallel to each other. The high conductivity provided by atoms (generally iodine is used) in the molecules, the electric field parallel to the molecules get absorbed. A sheet containing such long chain polymer molecules is known as polaroid.
3.2 Polarization by Reflection
Let us consider the incidence of a plane wave on a dielectric. Assume that the electric field vector associated with the incident wave lies in the plane of incidence. The Brewste r’s Law defines the Brewster’s angle:
θp = tan-1 (n1/n2) (1)
where, n1 and n2 are the refractive indices of the two different media. At this angle of incidence, the reflected and the transmitted rays are at right angles to each other, and the angle θp is known as the polarizer or Brewster angle.
If the angle of incidence (θ) is equal to θp then the reflection coefficient is zero. If an unpolarized beam is incident at an angle θ = θp, the reflected beam will be linearly polarized with its electric vector perpendicular to the plane of the incidence at the interface of the two different mediums with refractive indices of n1 and n2.
Example: At the air-glass interface, n1 = 1 and n2 = 1.5, giving θp = 57º. The transmitted beam is partially polarized and using a large number of reflecting surfaces would provide an almost plane polarized transmitted beam.
Figure 7. Reflection of light at the interface of two different media
3.3 Polarization by Double Refraction
When an unpolarized light beam is incident on a calcite crystal, it usually splits up into two linearly polarized beams. The beam which travels undeviated is known as the ordinary ray (o-ray) and obeys Snell’s Laws of refraction. On the other hand, the second one, which in general, does not obey Snell’s Law is known as the extra-ordinary ray (e-ray). The Figure 8 represents the schematic diagram for the polarization of light by double refraction. The appearance of two beams is due to double refraction and a crystal like calcite is referred as a double refracting crystal.
Figure 8. Polarization of light by double refraction
The velocity of the o-ray is same in a ll directions, the velocity of e -ray is different in different directions. A substance (like quartz, calcite) that exhibits different properties in different directions is called anisotropic substance. The axis along which the two velocities are equal is known as the optic axis of crystal. In a crystal like calcite, two rays have the same speed only a long one direction, such crystals are known as uniaxial crystals.
3.4 Polarization by Scattering
Polarization also occurs when light is scattered while traveling through a medium. When light strikes the atoms of a material, it will often set the electrons of those atoms into vibration. The vibrating electrons then produce their own electromagnetic wave that is radiated outward in all directions. This newly generated wave strikes neighboring atoms, forcing their electrons into vibrations at the same frequency. These vibrating electrons produce another electromagnetic wave that is once more radiated outward in all directions. This absorption and remission of light waves causes the light to be scattered about the medium. This scattered light is partially polarized.
Example: Polarization by scattering is observed when light passes through the atmosphere. The scattered light often produces a glare in the skies (as shown in Figure 9). Photographers know that this partial polarization of scattered light leads to photographs characterized by a washed-out sky. The problem can easily be corrected by the use of apolaroid filter. As the filter is rotated, the partially polarized light is blocked and the glare is reduced. The photographic secret of capturing a vivid blue sky as the backdrop of a beautiful foreground lies in the physics of polarization and Polaroid filters. Also, if an unpolarized beam is allowed to fa ll on a gas, then the beam scattered at 90o to the incident beam is linearly polarized.
Figure 9. Polarization due to scattering of light
4. Malus’ Law
Malus’ law (Étienne-Louis Malus) states that when a perfect polarizer is placed in a polarized beam of light, the intensity (I) of the light that passes through, is given by
where, I0 is the initial intensity, and θi is the angle between the light’s initial polarization direction and the axis of the polarizer. A beam of unpolarized light can be thought of as containing a uniform mixture of linear polarizations at all possible angles. Since the average value of cos2 is 1/2, the transmission coefficient becomes
In practice, some light is lost in the polarizer and the actual transmission of unpolarized light will be somewhat lower than this, around 38% for Polaroid-type polarizers but considerably higher (>49.9%) for some birefringent prism types.
If two polarizers are placed one after another (the second polarizer is generally called an analyzer), the mutual angle between their polarizing axes gives the value of θ in Malus’ law. If the two axes are orthogonal (θ), the polarizers are crossed and makes cos θ = 0, indicating no light is transmitted. Practically, no polarizer is perfect and the transmission is not exactly zero (for example, crossed polaroid sheets appear slightly blue in color). If a transparent object is placed between the crossed polarizers, any polarization effects present in the sample (such as birefringence) will be observed through increase in transmission. This effect is used in polarimetry to measure the optical activity of a sample.
Real polarizers are also not perfect blockers of the polarization orthogonal to their polarization axis; the ratio of the transmission of the unwanted component to the wanted component is called the extinction ratio, and varies from around 1:500 for polaroid to about 1:106 for Glan–Taylor prism polarizers.
In X-ray the Malus’ law (relativistic form):
where, f0 is the frequency of the polarized radiation falling on the polarizer, f is the frequency of the radiation passes through polarizer, λ is the Compton wavelength of electron, and c is the speed of light in vacuum.
5. Application of Polarized Lights
i. Polaroids are used in sun-glasses. They reduce the intensity and decrease the glare by cutting down the horizontally polarized light.
ii. Polarized light is useful to determine size and shape of viruses.
iii. Polarization is used in infrared spectroscopy.
iv. Polarization of cosmic microwave background is being used to study the physics of the early universe .
v. All radio transmitting and receiving antennas are polarized which is used in radar. AM and FM radios uses vertical polarization while television uses horizontal polarization.
vi. Polarization of light is used in Opthalmic instruments to eliminate strong reflection from the patient’s cornea by using 90o polarizer.
Summary
This lecture summarizes the fundamentals of light propagation. The unpolarized light has components of light in every direction and could be polarized to have component in a specified direction. Various processes and techniques to observe polarized lights have been discussed. This includes linearly, circularly and elliptically polarized lights.
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References
• Ajoy Ghatak, Optics, The McGraw-Hill Companies,4th Edition,2009.2.
• N.Subramanyam, Brij Lal, M.N.Avadhanulu, A Textbook of Optics, S.C hand, 2012.
• Jianquan Yao, Yuyue Wang, Non linear Optics and Solid State Lasers, Springer series in Optical Sciences,2012.
• R.W.Boyd, Nonlinear Optics (Academic Press, Boston, 1992)