Chapter 2

2.1. Reverse Saturable Absorption

In the mid 1960s shortly after the invention of the laser, many researchers were investigating dyes for potential application to Q-switching of the laser cavity. For this application, dyes were sought that would bleach to transparency under intense illumination (saturable absorbers). Guiliano and Hess [2a] in 1967 were investigating vat dyes and their modified cousins and noted some examples that not only did not bleach to transparency but instead darkened at high intensities. This was the first recognition of the property of reverse saturable absorption (RSA).

Reverse saturable absorption generally arises in a molecular system when the excited state absorption cross section is larger than the ground state cross section. The process can be understood by considering a system that is modeled using three vibronically broadened electronic energy levels, as shown in figure 2.1. The cross section for absorption from the ground state 1 is ₁. ₂ is the cross section for absorption from the first excited state 2 to the second excited state 3. The lifetime of the first excited state is ₂ (seconds).

Figure 2.1: Three level and Four level models for RSA

As light is absorbed by the material, the first excited state begins to become populated and contributes to the total absorption cross section. If ₂ is smaller than ₁, then the material becomes more transparent or ‘bleaches’ i.e. it is a saturable absorber.

If ₂ is larger than _1, then the total absorption increases, and the material is known as a reverse saturable absorber. This behavior is shown in figure 2.2

Figure 2.2: Plot of the incident intensity versus the transmitted intensity of a typical three level RSA material.

The change in intensity of a beam as it propagates through the material is:

, (2.1)

Where z is the direction traversed, N_T is the total number of active molecules per area in the slice dz, N₂ is the population of level 2 and the population of level 3 has been neglected.

Initially, the material obeys Beer’s law when 2 is unpopulated, and the transmission is constant as the incident fluence is increased. The slope is given by. At a sufficiently high fluence, however, the first excited state 2 becomes substantially populated and in the limit of complete ground state depletion the slope again becomes constant at the new value of. The optical limiting action is not truly limiting, as the fluence, which is transmitted, is still increasing with increasing incident fluence, but it does so more slowly. If the ratio ₂/₁, is sufficiently large, however, the new transmission will be small and in a properly designed system the dynamic range of the sensor will be greatly extended. The three level diagrams describe the simplest case for RSA materials but can generally only be applied for subnanosecond pulses and under circumstances such that transitions from the second excited state are negligible.

The energy states involved in three level materials usually consists of singlet states and the transitions are all allowed. The transition cross sections are therefore large, but a disadvantage is that de-excitation is rapid (₂ is small). This necessitates larger intensities for long pulses to activate the nonlinearity through populating the excited electronic state. Fortunately, on longer timescales in some systems, significant intersystem crossing to other states can occur from the first excited state. In this case the five level diagrams shown in figure 2.1 is applicable. The excited state 4 is usually a triplet or other long-lived state, and for long pulses it can act as a metastable state that accumulates population during the pulse. The lifetime of 4 gives an indication of the maximum pulse width for which the material is efficient to act as an optical limiter. Pulses with duration longer than the metastable state allow some of the metastable molecules generated by the leading edge of the pulse to decay to the ground state before the trailing edge have passed, thereby reducing the RSA. In most systems, ₃ and ₅ are very small and significant populations of 3 and 5 do not accumulate. Therefore, N₃ and N₅ can be set to zero, considerably simplifying the dynamical equations describing. The equations representing the full five level models are given below by:

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)

and

(2.8)

Where h is the energy per photon, I is the intensity of the pulse and stimulated emission has been neglected. The latter assumes that optical coupling to the excited states is well above the bottom of the vibronic manifolds and that relaxation from the optically-coupled states to the bottom of the manifolds occurs on a time scale that is much shorter than the pulse duration. To completely understand the response of an RSA device, these equations must be solved as the pulse propagates through the material. The material parameters necessary to solve the equations are ₁, ₂, ₄, ₂, ₄ and ₂₄. For optimum optical limiting performance, certain parameters need to be maximized. The ratio of the excited state absorption to the ground state, ₂/₁, ₄/₁ should be large to minimize the transmission of the limiter at high incident intensity.

For maximum efficiency, the lifetime of the triplet state (₂) and the intersystem crossing rate l/₂₄ should be large to populate the triplet state and maintain the population throughout the pulse. By the mechanism of RSA we get better performance on optical limiting.

2.2. Two-Photon Absorption (TPA):

Two-photon absorption (TPA) can also be used in a manner similar to RSA to construct optical limiters. In contrast with reverse saturable absorption, TPA is an instantaneous nonlinearity that involves the absorption of photon from the field to promote an electron from its initial state to a virtual intermediate state, followed by the absorption of a second photon that takes the electron to its final state. Since the intermediate state for such transitions is virtual, energy need not be conserved in the intermediate state but only in the final state. The mechanism of TPA can be thought of in terms of the three level RSA model for the case where the lifetime of the intermediate state approaches zero and the ground state absorption is extremely low (highly transparent). The intensity of the beam as it traverses the material is:

(2.9)

Where z is the linear absorption coefficient and  is the TPA coefficient which is related to the imaginary part of ⁽³⁾ by the equation (SI units):

(2.10)

Here,  is the circular frequency of the optical field, n₀ is the linear index of refraction, and c is the speed of light in vacuum. The solution to the propagation equation for = 0 (transparent material at low intensities) is given by

(2.11)

Where L is the sample length. This clearly demonstrates that the output intensity decreases as the input intensity increases, exactly the behavior that is desired for an optical limiter. The strength of this reduction is explicitly dependent on the TPA coefficient, the incident intensity and the sample thickness.

For TPA, the material response is of the order of an optical cycle and is, therefore, independent of the optical pulse length for a fixed intensity. The device will respond virtually instantaneously to the pulse. On the other hand, because of the limited magnitude of  in existing materials, high intensities are required to realize significant TPA. Since the intensity is essentially the energy density divided by the pulse duration, short pulses are required to achieve limiting with TPA for energy densities that may be high enough to damage an optical sensor.

2.3. Free-Carrier Absorption:

This type of limiting occurs in semiconductor materials. Once carriers are optically generated in a semiconductor, whether by single photon or two-photon absorption, these electrons (holes) can be promoted to states higher (lower) in the conduction (valence) band by absorbing additional photons. This process is often phonon assisted, although depending on the details of the band structure and the frequency of the optical excitation, it may also be direct. The phonon assisted phenomenon is referred to as free-carrier absorption, and it is analogous to excited-state absorption in a molecular system. It is clearly an accumulative nonlinearity, since it depends on the buildup of carrier population in the bands as the incident optical pulse energy is absorbed. Free-carrier absorption always plays some role in the operation of a semiconductor limiter, if the excitation process results in the generation of significant free carrier populations in the bands. While it certainly contributes to the limiter performance and its inclusion is important in the precise modeling of the response of such devices, just as in the case of TPA, its importance typically pales in comparison with nonlinear refractive effects, whether the carriers are generated by single photon or two photon transitions.

2.4. Nonlinear Refraction

Optical limiters based on self focusing and defocusing form another class of promising devices. The mechanism for these devices may arise from nonlinear refraction associated with carrier generation by either linear or two photon absorption in a semiconductor. Both self focusing and defocusing devices operate by refracting light away from the sensor as opposed to simply absorbing the incident radiation. Compared to strictly absorbing devices, these limiters can, therefore, potentially yield a larger dynamic range before damage to the limiter itself.

Figure 2.3 (a) shows the typical device configuration for a self defocusing limiter, while figure 2.3 (b) shows a similar device based on self focusing. A converging lens is used to focus the incident radiation so it passes through the nonlinear medium. This lens provides optical gain to the system, allowing the device to activate at low incident intensities. The output passes through an aperture before impinging on the detector. At low input levels, the nonlinear medium has little effect on the incident beam, and the aperture blocks an insignificant portion of the beam, thus allowing for a low insertion loss for the device. When nonlinear refraction occurs, however, the nonuniform beam profile within the medium results in the generation of a spatially nonuniform refractive index. This acts as either a negative or positive lens, depending on the sign of the refractive nonlinearity, causing the incident beam to either defocus or focus.

Figure 2.3: (a) Typical self defocusing optical limiter configuration (b) Typical self focusing optical limiter configuration.

In a properly designed system, this self lensing results in significant energy blocked by the system aperture, thereby protecting the sensor. The location of the nonlinear medium is critical to the operation of the refractive limiting device. A self-focusing limiter works best if the nonlinear medium is placed approximately a Rayleigh range before the intermediate focus of the device. When the focusing lens is induced the effective focal length of the device is reduced, and hence a larger beam appears at the exit aperture. For a self-defocusing material, the optimum geometry is approximately one Rayleigh range after the focus. This geometry dependence can be exploited to determine not only the sign of the nonlinear refraction in a given medium, but the magnitude as well. This is the principle behind the so-called Z-scan technique, which has been pioneered by Van Stryland and coworkers [2b,2c].

The technique consists of moving the nonlinear medium through the focal region of a tightly focused beam while measuring the transmittance through an aperture placed in the far field of the focal plane. When the medium is far before the focal plane, no self-lensing occurs. As the medium approaches the focal plane, the high intensity begins to induce a lens in the medium. For a negative nonlinearity, this lens tends to collimate the beam, thereby increasing the transmittance through the aperture. Near the focal plane, even though the intensity is highest, the influence of the induced lens is minimized, resulting in a transmittance comparable to the linear transmittance. This is similar to placing a thin lens at the focus of a beam; this results in minimal effect on the far field beam pattern. As the sample is moved beyond the focal plane, the negative lens tends to increase the beam divergence, resulting in a decrease in the aperture transmittance. As the medium is moved still farther from focus, the intensity again becomes weak enough that the induced lensing is negligible. This sequence results in a change in transmittance with a characteristic peak, followed by a null, followed by a valley as the sample is moved from the input lens, through focus, toward the output lens. For a positive nonlinearity, the pattern consists of a valley, a null, and then a peak. Thus, the sign of the nonlinearity is readily determined. While nonlinear absorption has been neglected in this discussion, if present, it must also be accounted for. This is readily done by removing the aperture in the limiter and collecting all the light transmitted by the nonlinear material. This measurement is then insensitive to nonlinear refraction. The response in this case is a valley symmetrically located about the focal plane. It should be noted that nonlinear absorption and induced scattering cannot be distinguished by this technique. The general shape of the Z-scan for a positive index change, negative index change, and a nonlinear absorber or scatterer is shown in figure 2.4

.

Figure 2.4: Schematic representation of z-scan results for a negative refractive nonlinearity (dashed curve) and a positive refractive nonlinearity (dotted curve). Both curves have been corrected for absorption. The solid curve shows the result of removing the aperture from the measurement apparatus and collecting all the transmitted light, thus isolating the nonlinear absorption [1e].

2.5. Induced Scattering

Scattering roots from interaction of light with small centers which may be physical particles or simple interfaces sandwiched between non-excited and excited molecular groups. The size of the scattering centers determines whether the scattering will be quite directional or reasonably uniform. Transmission of a medium, for a given solid angle, decreases when scattering centers are induced in the medium by an optical signal. Therefore, this phenomenon of scattering induced by optical signal may be applied to manufacture of optical limiters for sensor protection. Optical limiters based on induced scattering are usually focused on liquid media, as the phenomenon is usually reversible in these media. That is to say, the liquid in the excited state can return to equilibrium with ease in the absence of chemical or structural decomposition. However, in solids, usually irreversible decomposition processes generate the scattering centers which can lead to degradation in the device’s linear operation.

When light is incident on a particle, the electric charges within the particle oscillate due to its interaction with the electric field. Radiations are then caused by the oscillation. In 1899, Lord Rayleigh originally presented the analytic expression and theory of the elastic scattering of light from particles with dimensions smaller than the wavelength of light. Rayleigh scattering is the name given to the phenomenon. This applies only to particles whose dimensions are quite smaller than the wavelength of light or which are non-absorbing. However, in 1908, Mie developed a theory for particles with dimensions comparable to the wavelength of light or greater [2d]. The transmitted intensity equations of the Mie scattering are notably more intricate than of Rayleigh scattering. In Mie scattering, a bigger percentage of the scattered radiation is in forward direction as the size of the scattering particles increases, implying that limiting based on Mie scattering will not be as effective as Rayleigh scattering.

2.6. Photorefraction

Two devices, namely coherent-beam excisor and the beam fanning limiter based on the photorefractive effect are used to limit coherent optical radiation. Materials showing photorefraction should have a nonzero χ⁽²⁾. The traditional photorefractive mechanism is based on the photorefractive crystal which possesses deep levels that can be excited optically to generate free charge in the conduction or valence band. In a material showing photorefraction, when two coherent beams interfere, additional mobile charge are generated at the peaks of the intensity pattern than at the valleys through photoexcitation of the deep levels of the crsytal. These charges which are photoexcited at the peaks diffuse into the valleys ensuing a variation of charge spatially, in correspondence to the material’s interference pattern. These charges results in an electrostatic space-charge field which gives rise to a change in refractive index through the electro-optic effect in a properly oriented crystal. Energy coupling and energy exchange can then be achieved between the two beams through the grating generated, which is 90 degrees phase shifted from the intensity of the photon field.

A high intensity coherent beam when incident singly on a photorefractive crystal, the energy can be coupled into a large amount of low intensity scattered beams. Fields with new wave vectors are generated inside the crystal by the scattering of the incident beam at the crystal imperfections. The photorefractive gratings are then produced by the interference of the incident field with these scattered fields. Optical signal can later be coupled from the incident beam to the scattered beams through diffraction from these gratings. The light gets preferentially scattered to one side of the crystal as there is a preferred direction of energy transfer for photorefractive gratings which is determined by the direction of the c-axis of the crystal and the charge carriers’ sign. This photorefractive beam fanning phenomenon can be quite efficient in reducing the intensity of the transmitted beam. Construction of an optical limiter using this beam fanning process has been demonstrated by Cronin-Golomb and Yariv [2e].

The photorefractive excisor is another device which provides a weak seed beam to interfere with the incident beam. It is assembled to protect the sensor in such a way that the photorefractive grating produced by the interference of the primary beam with the seed beam at high intensities couples energy from the strong incident beam to the weak seed beam. The speed and efficiency of the device is thus improved.

2.7. Summary

All of the nonlinear phenomena discussed above can be used for optical limiting, and figure 2.5 schematically illustrates the application of some of these processes. Figure 2.5 (a) depicts the use of induced absorption, such as reverse saturable absorption, two-photon absorption, and free-carrier absorption. Figures 2.5(b) and 2.5(d) represent, respectively, a self-defocusing limiter, self-focusing limiter, and an induced scattering limiter. Finally, figures 2.5(e) and 2.5(f) illustrate a photorefractive beam fanning limiter and a photorefractive excisor device. While it is often the case that any given material will exhibit multiple nonlinear properties, for simplicity the effects of each individual process have been separately depicted in figure 2.5.

Figure 2.5: Some optical limiters based on different mechanisms (a) an induced absorption limiter (b) Self defocusing limiter (c) Self focusing limiter (d) Induced scattering limiter (e) Beam fanning limiter (f) Photorefractive excisor device [1e].