# Calculating Bandwidth for RF/Photonic Components based on Velocity mismatch

The bandwidth of a device such as a modulator or photodetector is an important figure. When designing a modulator or photodetector for high frequencies, much attention is paid to matching the velocity of the optical waves and the RF waves.

By finding the propagation time difference between the optical and RF waves, we model this in the time domain as a rect function. Note that for the rect function, the difference in propagation time is the tau variable. Performing the Fourier transform on the rect function will give us a sinc function. The 3dB cutoff point of this sinc function in the frequency domain gives us the device bandwidth. Note the MATLAB algorithm used below. The 3dB bandwidth is calculated using a simple manipulation of the frequency vector indices.

v_optical = ; %simulated optical velocity [define]

v_RF = ; %simulated RF velocity [define]

l_device = ; %device length [define]

f_max = ; %max frequency of vector (should be higher than bandwidth) [define]

f_num = ; %number of frequencies in vector [define]

tau = abs((l_device/v_optical)-(l_device/v_RF)) ; %propagation time difference

W = linspace(0,f_max,f_num); %frequency vector

S = tau*sinc(W*tau/2); %sinc function in frequency domain

Qs = find(20*log10(S)<=(20*log10(S(1))-3)); %intermediate calculation for index of 3db cutoff

BW_3dB= f_max*(Qs(1))/f_num %This is the result

# The Pockels Effect and the Kerr Effect

The Electro-optic effect essentially describes the phenomena that, with an applied voltage, the refractive index of a material can be altered. The electro-optic effect lays the ground for many optical and photonic devices. One such application would be the electro-optic modulator.

If we consider a waveguide or even a lens, such as demonstrated through problems in geometrical optics, we know that the refractive index can alter the direction of propagation of a transmitted beam. A change in refractive index also changes the speed of the wave. The change of light propagation speed in a waveguide acts as phase modulation. The applied voltage is the modulated information and light is the carrier signal.

The electro-optic effect is comprised of both a linear and non-linear component. The full form of the electro-optic effect equation is as follows: The above formula means that, with an applied voltage E, the resultant change in refractive index is comprised of the linear Pockels Effect rE and a non-linear Kerr Effect PE^2.

The Pockels Effect is dependent on the crystal structure and symmetry of the material, along with the direction of the electric field and light wave. # Introduction to Electro-Optic Modulators

Electro-optics is a branch or topic in photonics that deals with the modulation, switching and redirection of optical signals. These functions are produced through the application of an electric field, which alters the optical properties of a material, such as the refractive index. The refractive index refers to the speed of light propagation in a medium relative to the speed of light in a vacuum.

Modulators vs. Switches

In a number of situations, the same device may function as both a modulator and a switch. One dependent factor on whether the device would be suitable or not for a switch as opposed to a modulator would be the strength of the effect that an electric field may have on the device. If the device’s primary role is to impress information onto a light wave signal through temporary varying of the signal, then it is referred to as a modulator. A switch on the other hand either changes the direction or spatial position of light or turns it off completely. # Theory of Operation

Electro-optic Effect

The electro-optic effect presumes the dependence of the refractive index on the the applied electric field. The change in refractive index, although small allows for various applications. For instance, a lens may be applied an electric field and depending on the material and the applied field, the focal length of the lens can change. Other optical instruments that utilize this effect may also see use, such as a prism. A very small adjustment to the refractive index may still produce a delay in the signal, still large enough to detect and, if information was implied by the delay that was produced on the signal, the delay can be phase demodulated at the receiving end.

Electroabsorption

Electroabsorption is also another effect that is used to modify the optical properties of a material by the application of an electric field. An applied electrical field may increase the bandgap of the optical semiconductor material, turning the material from optically transparent to optically opaque. This process is useful for making modulators and switches.

Kerr Effect and Pockels Effect

The Pockels Effect and the Kerr Effect both account for the change in refractive index through the application of an electric field. The Kerr Effect states that this effect is nonlinear, while the Pockels Effect states that the effect is linear. Although the Pockels Effect is more pronounced in Electro-optical modulator design, both are applied in many situations. The linear electro-optic effect exists only in crystals without inversion symmetry. The design of electro-optic modulators or switches requires special attention to the waveguide material and how the electric field reacts with the material. Common materials (also maintaining large Pockels coefficients) are GaAs, GaP, LiNbO3, LiTaO3 and quartz. The Kerr Effect is relatively weak in commonly used waveguide materials.

# Properties of the Electro-Optic Modulator

Modulation Depth

Important for both modulators and switches is the modulation depth, also known as the modulation index. Modulation depth has applications for the several types of optical modulators, such as intensity modulators, phase modulators and interference modulators. The modulation depth may be conceptually understood as the ratio of effect that is applied to the signal. In other words, is the modulation very noticeable? Is it a strong modulation or is it a weak modulation?

Bandwidth

The bandwidth of the modulator is critically important as it determines what range of signal frequencies may be modulated onto the optical signal. Switching time or switching speed may be equally applied to an optical switch.

Insertion Loss

Insertion loss of optical modulators and switches is a form of optical power loss and is expressed in dB. However, the result of insertion loss often results in the system requiring more electrical power and would not explicitly reduce performance of the modulation or switching function of the device.

Power Consumption

In distinction from the electric field, a modulator or switch also needs a power supply for itself. The amount of power required increases with modulation frequency. A common figure of merit is the drive power per unit bandwidth, typically expressed in milliwatts per megahertz.

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