1. Introduction

Two characteristics of great importance in LASER detection and ranging (LADAR) systems are long range performance and fine range resolution. The long range performance implies a good carrier-to-noise ratio; therefore matched filters are generally used during detection. The point target matched filter response for an arbitrary waveform is the autocorrelation function (ACF) which forms a Fourier transform pair with the energy spectrum of the signal. However, most real world targets have multiple reflection points, and the sidelobes generated by the matched filter for any one reflection point will contribute to the noise for all the other reflection points. Therefore, reducing the waveform’s ACF sidelobe levels can greatly improve the overall carrier-to-noise ratio and thus the long range performance.

Fine range resolution stems from the ability to temporally distinguish reflections generated by adjacent points. Therefore, reducing the temporal width of the mainlobe of a waveform’s ACF will improve fine range resolution. The temporal width is measured at the half-maximum points, aka the full width half maximum (FWHM) time.

With this in mind it becomes clear that choosing the correct waveform is of great importance in a LADAR system. Over the years, many different waveforms have been proposed including linear frequency modulation (LFM) chirped waveforms [1], pseudo-random phase modulated waveforms [2], poly-phase (P4) waveforms [3], and others. Although pulse carving a waveform’s amplitude window is the most common method used for reducing the sidelobes of its ACF, sidelobe reduction can be accomplished by distorting the frequency modulation function, i.e. create a non-linear frequency modulated (NLFM) chirp [4–7], which avoids a costly loss of carrier-to-noise ratio. Not much literature exists about hardware implementations of NLFM chirped waveforms for LADAR or otherwise. In [8] the authors propose a non-linearly stepped frequency waveform, mimicking an NLFM chirped waveform. In [9] the authors show how a series of integrators can create a polynomial phase chirp. In this paper we propose a new method for implementing an NLFM chirped optical waveform using integrated optical ring resonators.

2. Integrated optical ring resonator simulation

2.1 Basic setup

The primary tool to generate NLFM chirped waveforms for consideration here is the tunable integrated optical ring resonator in an all-pass configuration (see Fig. 1a ) [10]. Of the three defining quantities of a ring – the power coupling constant κ, round trip loss factor γ, feedback delay time T – the value most easily adjusted in a dynamic manner is the feedback delay time. Surprisingly little change is needed in the ring waveguide’s refractive index to create a significant change in the spectral response. Each notch and phase curve will shift along the wavelength axis (λ) or frequency axis (f) according to Eq. (1) and Eq. (2):

 

Fig. 1 a) A basic integrated optical ring resonator to modulate the phase of an input signal. The modulators change the refractive index of the waveguide, the one internal to the ring creates a non-linear frequency chirp and the one outside the ring to create an approximately linear chirp b) The ring power coupling constant κ equates to the through and cross port transmission coefficients ${\cos }^2(\theta )$and ${\sin }^2(\theta )$respectively. c) The theoretical phase response of an optical ring resonator over one FSR (50GHz FSR, $0.2\pi$ transmission coefficient). Rapidly changing the refractive index of the ring will shift the phase curve. If a narrow-band optical wave with a wavelength of $1549.75$nm (shown in black) were input to the ring, its phase would change correspondingly.

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The phase response of a ring in an all-pass configuration changes very rapidly around resonance. By shifting the ring’s frequency response around the carrier, non-linear phase shifts can be introduced on the narrowband carrier which will then in turn lead to non-linear frequency chirps. For example, a $16.67$MHz cosine wave modulation introduced to a ring’s refractive index will yield a NLFM chirped waveform as shown in Fig. 2a. As you can see the frequency chirps are evenly spaced, and these correspond to the time at which the filter’s resonance point passes over the carrier wavelength.

 

Fig. 2 a) The time and frequency output of a ring resonator with a coupling constant of $\kappa =0.75$ $(\theta =0.33\pi )$whose refractive index is modulated with a $16.67$MHz cosine wave. Here a $110$MHz carrier was used as the narrow-band ring input for graphical purposes. The blue or solid line shows the output wave, and the red or dashed line shows the instantaneous frequency of the output wave. b) The output wave and instantaneous frequency of a straight waveguide refractive index modulator, modulated with the $16.67$ MHz cosine wave.

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In addition to the ring resonator a phase modulator external to the ring is added such that approximately linear chirps can be added to the non-linear frequency modulated chirped waveform generated by the ring. This simulation assumes a sinusoidal input to the external phase modulator, which actually generates sinusoidal frequency modulated chirps (see Fig. 2b). This is acceptable though since the chirps are approximately linear over the region of interest, i.e. their ramp falls off at the same time that the rings frequency ramp increases dramatically.

2.2 Optimized autocorrelation function of fixed bandwidth ring generated frequency chirps

As mentioned previously the metrics of interest for the generated waveform are the sidelobe levels and FWHM time of the waveform’s ACF. It turns out that one ring cannot generate enough NLFM chirp to create a waveform with an ACF with multiple sidelobes; therefore the following simulations assume two or three lossless rings with identical coupling constants and FSR-s. Although a full analysis of the effects of ring loss on the generated waveforms is beyond the scope of this paper, suffice it to say that increasing loss tends to increase FWHM times and raise sidelobe levels.

Previous research [11] focused on achieving minimum ACF mainlobe width and sidelobe level for a fixed bandwidth through optimization of the ring coupling constant only. One important factor that was neglected was the analysis of chirps generated by rings with different round-trip phases. Phase offset can come not only due to fabrication processes, but also by application of a constant offset to the ring round-trip modulator.

As it turns out, variations in ring round-trip phase can lead to much better optimized performance of an NLFM chirp with a fixed bandwidth. Figures 3 a and b show the mainlobe widths and sidelobe levels of ring generated NLFM chirped waveforms under a wide variety of settings, all with a fixed chirp bandwidth $=10/T$. As can be seen, NLFM chirps generated with rings with identical coupling constants but different phase offsets exhibit lower ACF mainlobe widths and sidelobe levels.

 

Fig. 3 Scatter plots of the mainlobe widths and sidelobe levels of simulated ACF’s of NLFM chirped waveforms generated by rings with a variety of coupling constants and phase offsets. In all cases bandwidth$=10/T$ a) Green circles: two different rings $\theta =0.3\pi \mathrm{...0.47}\pi$ $\partial \varphi =\mathrm{0...}\pi$; blue diamonds: two identical rings $\theta =0.335\pi \mathrm{...0.365}\pi$, and $\partial \varphi =\mathrm{0...0.7}\pi$; red circles: two different rings $\theta =0.35\pi \mathrm{...0.42}\pi$and no phase variation. b) Green circles: three different rings $\theta =0.3\pi \mathrm{...0.47}\pi$ $\partial \varphi =\mathrm{0...0.75}\pi$; blue diamonds: three identical rings $\theta =0.3775\pi \mathrm{...0.4025}\pi$and $\partial \varphi =\mathrm{0...0.45}\pi$; red circles three different rings $\theta =0.415\pi \mathrm{...0.44}\pi$and no phase variation.

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2.3 Post-fabrication optimization

Of prime interest for the fabrication of optical ring resonators is the parameter tolerance, in this case the coupling constant tolerance. The ring coupling constant is usually determined during fabrication and generally isn’t adjustable thereafter. In contrast the ring round-trip phase can come by application of a constant offset to the ring modulator. Therefore phase offset can be easily tuned after fabrication is complete. By adjusting the phase offset and linear chirp contribution, the ACF sidelobe level and mainlobe width of the ring generated NLFM chirp are fairly well tunable after fabrication across a range of coupling constants.

Scatter plots showing the simulated ACF mainlobe widths and sidelobe levels of NLFM chirp generated by two ring resonators with a coupling constants of 0.65 and 0.85 are shown in Figs. 4 a and b respectively. For the case where the ring coupling$\kappa =0.65$ sidelobe levels decreased by about $-5db$per $0.01*T$FWHM increase, up to $-24db$(as shown in Fig. 4 a); whereas for the case where the ring coupling$\kappa =0.85$ sidelobe levels decreased by about $-3db$per $0.01*T$FWHM increase, up to $-31db$(as shown in Fig. 4 b).

 

Fig. 4 The simulated ACF mainlobe widths and sidelobe levels of NLFM chirped waveforms generated by two lossless rings a) with identical coupling constants $\kappa =0.65$, $\partial \varphi =\mathrm{0...0.25}\pi$ phase variation, and $1/T\mathrm{...}10/T$ bandwidth of additional linear chirp b) with identical coupling constants $\kappa =0.85$, $\partial \varphi =\mathrm{0...0.25}\pi$ phase variation, and $1/T\mathrm{...}10/T$ bandwidth of additional linear chirp.

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The above results were all computed for a single isolated chirp; a series of chirps will generate ACF’s with higher sidelobes. If the chirp series follows a sawtooth pattern, each successive chirp being identical to the last, the chirp ACF’s sidelobes will only be slightly higher (≈3db). On the other hand if the chirps have a triangular pattern, alternating between frequency increasing chirps and frequency decreasing chirps (see Fig. 2 a), the chirp ACF’s sidelobes will be significantly higher (≈10db). If the modulators are given a sine wave input (see Fig. 1 a) the system will generate chirps in a triangular pattern, however a fully rectified sine wave input will generate chirps in a sawtooth pattern.

3. Proof of concept experiment

This experiment is designed to simply prove that tunable integrated optical ring resonators will indeed produce chirps when their refractive index is tuned sufficiently rapidly. An off-the-shelf silicon-nitride ring designed for dispersion compensation will be used. This ring has been equipped with a thermal controller such that the refractive index can be controlled by the thermo-optic effect. For most real-world applications a faster index changing effect than the thermo-optic effect will be desired (such as charge carrier injection, or the electro-optic effect). Because the refractive index is directly related to temperature in this case, the refractive index can only be dynamically tuned on an order of milliseconds. This means that at best only kilohertz frequency chirps will be possible, but such will suffice for this simple proof of concept experiment.

Furthermore, these rings’ thermal controllers have been designed to simply ramp up or down to a given temperature. Therefore, the refractive index can only be modulated by a ramp function as opposed to the cosine wave function used in the preceding simulations. Therefore only half of the NLFM chirp can be generated with this setup. Despite all of these draw-backs, several optical frequency chirped waveforms were obtained. The instantaneous frequency of one thermally controlled ring was able to achieve about + 28 kHz of chirp as shown in Fig. 5 . Comparison of Fig. 5 with Fig. 2a shows that this chirp is similar in form to half of one simulated NLFM waveform.

 

Fig. 5 Instantaneous frequency chirp created by the Si3N4 tunable integrated ring resonator, reaching a peak frequency of 28kHz.

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4. Conclusions

This paper first showed that a tunable optical ring resonator in series with a phase modulator is able to produce NLFM chirped waveforms. Ring round trip phase was shown to have a profound effect on the chirp ACF mainlobe width and sidelobe levels. For a bandwidth $=10/T$ the optimal settings for two rings were found to be $\theta =0.335\pi \mathrm{...0.365}\pi$ and $\partial \varphi =\mathrm{0...0.7}\pi$. Surprisingly the addition of an additional ring didn’t improve results in the fixed bandwidth case.

The post-fabrication tunability of the system was then explored. For a system employing two lossless rings with coupling constants between $\kappa =\mathrm{0.65...0.85}$, it was found that by changing the ring round-trip phase and linear chirp contribution, a tradeoff between ACF mainlobe width and sidelobe level was achieved at a rate of about $-3db$ to $-5db$ per $0.01*T$FWHM increase. This yielded sidelobe levels $0-15db$ lower and mainlobe widths of $0-50\%$ wider as compared to the ACF of an equivalent LFM signal Results concur well with previous publications [8,9].

Lastly a proof of concept experiment was performed to verify that tunable integrated optical ring resonators will indeed yield a NLFM chirped waveform when their refractive index is changed. Despite serious hardware drawbacks, a 28kHz chirp from a single thermally tuned silicon-nitride ring was obtained. Many improvements remain to be gained by using a series of integrated optical ring resonators that employ a faster index tuning effect [12].