# Imperfectly Phase-Modulated Light / Residual Amplitude Modulation

• A
• Twigg
In summary, the first equation in the paper describes an imperfectly phase modulated optical field from an electro-optic phase modulator that also has residual amplitude modulation. The equation includes factors such as the TEM profile of the beam, carrier frequency, alignment factors, modulation indexes, phase modulation frequency, and phase offsets. The question is how this equation represents amplitude modulation, and the suggestion is to think of the two modulated waves as phasors in a phasor diagram, where their amplitudes add vectorially and their phases are altered. This helps to understand the presence of amplitude modulation as the two modulated waves move out of sync with each other.
Twigg
Gold Member
I'm trying to understand this paper and others on the same topic. I struggle conceptually with their first equation, which is an expression for an imperfectly phase modulated optical field from an electro-optic phase modulator (EOM) that is contaminated with a little bit of amplitude modulation (residual amplitude modulation, or RAM for short): $$E^{PM,RAM}_{inc}(x,y,t) = E(x,y) e^{i\omega t} [ae^{i(\delta_o sin \Omega t + \phi_o)} + be^{i(\delta_e sin \Omega t + \phi_e)} ]$$
where ##E(x,y)## is the TEM profile of the beam, ##\omega## is the beam's carrier frequency, ##a## and ##b## are alignment factors determined by the polarization angles (I think they're the amount of the beam that's aligned with the ordinary or extraordinary axes?), ##\delta_{o,e}## are the modulation indexes in the ordinary and extraordinary axes, ##\Omega## is the phase modulation frequency, and ##\phi_{o,e}## are phase offsets in the ordinary and extraordinary axes. The ordinary and extraordinary axes here are determined by the crystallographic alignment of the EOM crystal. I don't really have a clear picture in my head.

The big question for me is: how does this represent amplitude modulation? I see what looks like phase modulation on the fast and slow axes of the electro-optic crystal, OK. How does the linear combination of two phase modulated signals with different modulation depths turn into an amplitude modulation?

Any input here at all is appreciated. Sorry I couldn't be more use framing my question. Thanks!

Edit: It can be assumed that ##a \approx 1## and ##b << a##.

ergospherical
If we add two waves in a phasor diagram you will see that the ampitudes add vectorially and also that the resultant is altered in phase. If either wave changes in some way then both amplitude and phase of the resultant alter.

ergospherical and Twigg
Oh! Thanks for that suggestion @tech99. Thinking of the two modulated exponentials as phasors really helps! A lot easier to draw the resultant than to do the algebra here. And I catch your point about there being some amplitude modulation as the two modulated phasors move out of sync with each other.

## What is imperfectly phase-modulated light?

Imperfectly phase-modulated light is a type of light that has undergone phase modulation, a process in which the phase of the light wave is altered. However, due to technical limitations or imperfections in the modulation process, the resulting light wave is not perfectly modulated.

## What is residual amplitude modulation?

Residual amplitude modulation (RAM) is a phenomenon that occurs when phase modulation is applied to a light wave. It refers to the presence of small amplitude variations in the modulated light wave, even though the modulation was intended to only alter the phase.

## How does imperfectly phase-modulated light affect optical systems?

Imperfectly phase-modulated light can cause problems in optical systems, as it can introduce additional noise and distortions in the transmitted signal. This can result in decreased performance and accuracy in applications such as optical communication and sensing.

## What are some techniques for reducing residual amplitude modulation?

There are several techniques that can be used to reduce residual amplitude modulation, including using higher quality modulators, optimizing modulation parameters, and implementing feedback control systems to correct for imperfections in real-time.

## What are the potential applications of imperfectly phase-modulated light?

Imperfectly phase-modulated light can be used in various applications, such as in optical communication systems, where it can be used to encode information and transmit data. It can also be used in optical sensing applications, where it can be used to measure changes in phase and amplitude of the light wave to detect environmental changes or disturbances.

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