# Modulation of light fields¶

In principle, all parameters of a light field can be modulated. This section describes the modulation of the amplitude, phase and frequency of the light. Any sinusoidal modulation of amplitude or phase generates new field components that are shifted in frequency with respect to the initial field. Basically, light power is shifted from one frequency component, the carrier, to several others, the sidebands. The relative amplitudes and phases of these sidebands differ for different types of modulation and different modulation strenghts.

## Phase Modulation¶

Phase modulation can create a large number of sidebands. The number of sidebands with noticeable power depends on the modulation strength (or depth) given by the modulation index $$m$$. Assuming an input field

$E_{\mathrm{in}}=E_0~\exp{\left(\mathrm{i}\,\omega_0\,t\right)},$

a sinusoidal phase modulation of the field can be described as

$E=E_0~\exp\Bigl(\mathrm{i}\,(\omega_0\,t + m \cos{\left(\omega_m\,t\right)})\Bigr).$

This equation can be expanded using the Bessel functions of the first kind $$J_k(m)$$. We can write

(1)$E=E_0~\exp{\left(\mathrm{i}\,\omega_0\,t\right)}~\sum_{k=-\infty}^{\infty}\mathrm{i}\,^{\,k}~J_k(m)~\exp{\left(\mathrm{i}\, k \omega_m\,t\right)}.$

The field for $$k=0$$, oscillating with the frequency of the input field $$\omega_0$$, represents the carrier. The sidebands can be divided into upper ($$k>0$$) and lower ($$k<0$$) sidebands. These sidebands are light fields that have been shifted in frequency by $$k\, \omega_m$$. The upper and lower sidebands with the same absolute value of $$k$$ are called a pair of sidebands of order $$k$$.

Equation (1) shows that the carrier is surrounded by an infinite number of sidebands. However, for small modulation indices ($$m<1$$) the Bessel functions rapidly decrease with increasing $$k$$, so we can use the approximation:

$J_k(m)~=\frac{1}{k!}\left(\frac{m}{2}\right)^k+O\left(m^{k+2}\right).$

In this case, only a few sidebands have to be taken into account. For $$m\ll1$$ we can write

$\begin{array}{lcl} E&=&E_0~\exp{\left(\mathrm{i}\,\omega_0\,t\right)}\\ & & \times\Bigl(J_0(m)-\mathrm{i}\, J_{-1}(m)~\exp{\left(-\mathrm{i}\, \omega_m\,t\right)}+\mathrm{i}\, J_{1}(m)~\exp{\left(\mathrm{i}\, \omega_m\,t\right)}\Bigr), \end{array}$

and with

$J_{-k}(m)=(-1)^kJ_k(m),$

we obtain

$E=E_0~\exp{\left(\mathrm{i}\,\omega_0\,t\right)}~\left(1+\mathrm{i}\,\frac{m}{2}\Bigl(\exp{\left(-\mathrm{i}\, \omega_m\,t\right)}+\exp{\left(\mathrm{i}\, \omega_m\,t\right)}\Bigr)\right),$

as the first-order approximation in $$m$$. In the above equation the carrier field remains unchanged by the modulation, therefore this approximation is not the most intuitive. It is clearer if the approximation up to the second order in $$m$$ is given:

$E=E_0~\exp{\left(\mathrm{i}\,\omega_0\,t\right)}~\left(1-\frac{m^2}{4}+\mathrm{i}\,\frac{m}{2}\Bigl(\exp{\left(-\mathrm{i}\, \omega_m\,t\right)}+\exp{\left(\mathrm{i}\, \omega_m\,t\right)}\Bigr)\right),$

which shows that power is transferred from the carrier to the sideband fields.

Higher-order expansions in $$m$$ can be performed simply by specifying the highest order of Bessel function, which is to be used in the sum in Equation (1), i.e.

$E=E_0~\exp{\left(\mathrm{i}\,\omega_0\,t\right)}~\sum_{k=-order}^{order}i^{\,k}~J_k(m)~\exp{\left(\mathrm{i}\, k \omega_m\,t\right)}.$

## Frequency modulation¶

For small modulation, indices, phase modulation and frequency modulation can be understood as different descriptions of the same effect [14]. With the frequency defined as $$f = d\varphi/dt$$ a sinusoidal frequency modulation can be written as:

$E=E_0~\mEx{\mathrm{i}\,\left(\omega_0\,t + \frac{\Delta\omega}{\omega_m} \cos{\left(\omega_m\,t\right)}\right)},$

with $$\Delta\omega$$ as the frequency swing (how far the frequency is shifted by the modulation) and $$\omega_m$$ the modulation frequency (how fast the frequency is shifted). The modulation index is defined as:

$m = \frac{\Delta\omega}{\omega_m}$

## Amplitude modulation¶

In contrast to phase modulation, (sinusoidal) amplitude modulation always generates exactly two sidebands. Furthermore, a natural maximum modulation index exists: the modulation index is defined to be one ($$m=1$$) when the amplitude is modulated between zero and the amplitude of the unmodulated field.

If the amplitude modulation is performed by an active element, for example by modulating the current of a laser diode, the following equation can be used to describe the output field:

$\begin{array}{lcl} E&=&E_0~\mEx{\mathrm{i}\,\omega_0\,t}~\Bigl(1+m\cos{\left(\omega_m\,t\right)}\Bigr)\\ &=&E_0~\mEx{\mathrm{i}\,\omega_0\,t}~\Bigl(1+\frac{m}{2}~\mEx{\mathrm{i}\, \omega_m \,t}+\frac{m}{2}~\mEx{-\mathrm{i}\, \omega_m\,t}\Bigr). \end{array}$

However, passive amplitude modulators (like acousto-optic modulators or electro-optic modulators with polarisers) can only reduce the amplitude. In these cases, the following equation is more useful:

$\begin{array}{lcl} E&=&E_0~\mEx{\mathrm{i}\,\omega_0\,t}~\left(1-\frac{m}{2}\Bigl(1-\cos{\left(\omega_m \,t\right)}\Bigr)\right)\\ &=&E_0~\mEx{\mathrm{i}\,\omega_0\,t}~\Bigl(1-\frac{m}{2}+\frac{m}{4}~\mEx{\mathrm{i}\, \omega_m \,t}+\frac{m}{4}~\mEx{-\mathrm{i}\, \omega_m\,t}\Bigr). \end{array}$