## 2 Theory
### Application of the Lock-in Amplifier  
As mentioned, the LAnTeRN measurement concept enables the use of pure sinusoidal modulations with standard lock-in techniques even in the presence of intermediate scattering media, due to the fact that the measurement is made in transmission. In order to describe the LAnTeRN measurement we will first describe how the signals are encoded, combined prior to transmission, simultaneously received, and then separated through a digital lock-in approach. The time varying output signal amplitude for wavelength \(\lambda_{m}\), is represented mathematically by:  
\[S_{\lambda_{m}}(t)=A_{m}\times\cos(2\pi f_{m}t+\theta_{m}) \tag{1}\]  
where \(A_{m}\) is the signal amplitude, \(f_{m}\) is the modulation frequency, and \(\theta_{m}\) is the initial phase. Note that this is an amplitude modulation and that the wavelength \(\lambda_{m}\) is held fixed. After applying a unique amplitude modulation scheme to each wavelength in the system, the signals are then combined using fiber combiners prior to transmission into the atmosphere. This is made possible by the fact that both the laser and the modulator are telecom-style fiber coupled components. The resulting output of the system is the sum of \(N\) waveforms, each with a unique wavelength of light and unique amplitude modulation frequency as shown in Equation (2). \[S(t)=\sum_{m=1}^{N}S_{\lambda_{m}}(t) \tag{2}\]  
An example of the amplitude of the output signal with \(N=3\) is shown in Figure 1, neglecting noise. For the sake of the discussion we will ignore the effects of the atmosphere and instrument at this time and focus on the lock-in technique for separating the simultaneously transmitted/received wavelengths. The received optical signal, the transmitted power attenuated by the atmospheric state over the path between the transmit and receive telescopes, is converted into a voltage and recorded using a high-quality Analog-to-Digital Converter (ADC). The amplitude of the received signal at wavelength \(\lambda_{m}\) can then be determined through a standard lock-in approach [16] based on the knowledge of the frequencies used to modulate the waveforms. A brief description of the well-known lock-in approach is given here for clarity, but, for simplicity, we are not including the discrete nature of the signal after the ADC and the digitally-generated local oscillator signal. The difference in the discrete case is that  
Figure 1: Example of the transmitted signal amplitude for a combination of a three-wavelength system, using three distinct sinusoidal amplitude modulations, one for each wavelength. the lock-in is performed on a sample-by-sample basis by multiplying the sampled signal with a discrete version of the local oscillator signal used to generate the amplitude-modulated output. To separate out the relative amplitudes of each of the transmitted wavelengths, the received signal is multiplied by both a sine wave and a cosine wave at frequency \(f_{m}\) to determine the in-phase, \(I\), and out-of-phase, \(Q\) components. Determining both of these components and adding them in quadrature renders the system insensitive to phase variations between the local oscillator signal and the received signal. \[I_{\lambda_{m}}(t) = S(t)\times A_{L}\times\sin(2\pi f_{m}t) \tag{3}\] \[Q_{\lambda_{m}}(t) = S(t)\times A_{L}\times\cos(2\pi f_{m}t) \tag{4}\]  
The \(I\) and \(Q\) components consist of two AC signals, which oscillate at both the sum and the difference of the signal frequencies and the local oscillator frequency. Taking the time average of these AC signals comes out to zero unless the signal contains a frequency component at the local oscillator frequency \(f_{m}\). In that case, the time average results in a DC component proportional to the signal amplitude at frequency \(f_{m}\) as the difference frequency approaches zero, and essentially eliminates any signal at frequencies that are not very close to \(f_{m}\).