Agile photonic fractional Fourier transformation of optical and RF signals: supplementary material

This document provides supplementary information to “Agile photonic fractional Fourier transformation of optical and RF signals,” https://doi.org/10.1364/OPTICA.4.000907 . In the first part, we detail the calculation of the output field of a frequency shifting loop (FSL), and show that this output signal maps in the time domain, the (cid:2235) (cid:2202)(cid:2190) Fractional Fourier transform (FrFT) of the input signal. The second part is dedicated to the frequency resolution of our FrFT technique. In the third part, we demonstrate the capability of FSLs for the measurement of RF chirp rates. Finally, the last part provides information about the experimental setup. © 2017 Optical America


Generation of the FrFT in a frequency shifting loop (FSL) Electric field in the FSL
We consider a FSL described by = 1/ , its roundtrip-time, and = 2 , the angular frequency shift per roundtrip. Suppose that the FSL is seeded with an input light field ( ). We define as the time duration of the input signal. For > , the whole input signal has entered the FSL, and the electric field at the output of the FSL writes [1]: ( ) corresponds to the net gain of the amplitude of the input light after roundtrips. Because of the finite duration of , all terms in the sum vanish, except for = ⁄ consecutive values of the integer . For simplicity, we assume that ( ) = 1 for any integer belonging to the afore-mentioned range Notice that in practice, the number of roundtrips in the FSL is limited by a spectral bandpass filter of bandwidth Δ . In the case where is larger than Δ /( ), only a truncation of the input signal of duration Δ /( ) is loaded in the FSL. The following calculations remain valid, but is then equal to: = Δ / . In all cases, can be defined as the number of frequency shifted replicas of the input signal travelling simultaneously in the FSL.
We study the particular case where can be written as = / + ∆ , where is an integer, and Δ = Δ /2 ≪ 1/ . We define ψ′ = 2 Δ , = − , and a dimensionless variable = , where is an arbitrary time scaling factor. We also define the function ( ) = ( ). The output electric field writes:

Expression of the output field as a FrFT
Based on the definition of the FrFT, we express in terms of its FrFT: Choosing α such as cot = − = −2 ∆ leads to: Invoking the Poisson summation formula gives: , which also writes: Then the output optical signal is proportional to delayed replicas of the FrFT of order of , mapped in the time domain, and multiplied by a chirp term. The period of the output signal is / . The system enables to measure in real-time the FrFT of the input light field, provided that the different contributions arising in the sum are separated along time. In this case, the photocurrent generated by the photodiode at the output of the FSL is proportional to: which shows that the output time trace consists of periodic waveforms, proportional to the square modulus of the FrFT of the input signal .

Discussion
-An interesting case occurs when the product = is a multiple of 2 . Then ′ = 0 and the order of the FrFT is /2. In this case, the output field maps the Fourier transform (FT) of the input signal. This capability of the system to provide, in real-time, the FT of an input optical waveform, corresponds to the results reported in [2].
-When the input optical signal is a CW laser at an angular frequency modulated by an input RF signal ( ), we can write ( ) = ( ) and ( ) = ( ) . From [3], we have: ( ) rewrites as: In that case, the output signal is proportional to the FrFT of the slowly varying envelope of the input RF modulation signal. This property demonstrates the practical interest of the FSL for realtime processing of RF analog signals.

Frequency resolution of the FrFT based on a FSL
In order to determine the frequency resolution of the FrFT as implemented by our technique, recall that the FrFT can be simply seen as the projection on a continuous basis of infinite, linearly frequency-modulated (LFM) signals, having all the same chirp rate , but with different offset frequencies : ℎ ( ) = [4]. The frequency resolution of the FrFT can be defined as the minimum frequency shift between two LFM components that can be detected with our setup. From (eq. S8), we can see that a change on the input frequency Δ produces a time shift of the output trace equal to: Δ = = Δ /( ). This value has to be compared to the shortest temporal signal that can be generated at the output of the FSL. The latter is simply equal to the inverse of the spectral bandwidth of the system . Therefore the angular frequency resolution of the technique is Δ = ∆ = ∆ . To give a reference, the current parameters of the FSL ( ≈ = 8.6 MHz, ∆ ≈ 10 GHz) enable a frequency resolution close to 70 kHz. It is interesting to notice that this value is independent from the order of the FrFT. As expected, this expression is also identical to the frequency resolution of the real-time FT in a FSL, as described in [2].

Electric output field for an input chirped signal
We now consider the case where the input signal is modulated by a linearly chirped signal. Without loss of generality, we write ( ) = ( ) , and ( ) = ( ) . The output electric field writes: where ′ = 1 + − . We replace ( ) by its FrFT of order , the latter being chosen such as: cot = −2 ∆ + The same reasoning as before leads to:

Measurement of the chirp rate
Suppose now that is the electric field of a monochromatic wave of duration . Without loss of generality, we assume that the amplitude of is constant over . The system implements the real-time FT of the field when = , i.e. when 2 = − . In this case the output signal intensity consists of periodic sinc -like waveforms with maximum peak power. This property of the system leads to a simple way of determining the chirp rate : we sweep the value of around the value / ; for a given value of Δ = − / , the peak intensity of the output waveforms is maximum.