Continuously tunable true-time delays with ultra-low settling time

Romain Bonjour; Simon A. Gebrewold; David Hillerkuss; Christian Hafner; Juerg Leuthold

doi:10.1364/OE.23.006952

1. Introduction

Ultra-fast, continuously tunable true-time delays (TTDs) are a missing building block in many subsystems such as phased array feeder networks [1–4] in microwave photonics [5,6]; for clock synchronization, or synchronization of time division multiplexing tributaries in optical communication systems [7]; as a tunable delay element in coherence tomography [8], or light detection and ranging (LIDAR) [9]. An ideal TTD should be continuously tunable, have a large operating bandwidth, and offer small settling times. In addition, TTDs should provide the largest possible transparency – i.e. they should operate across a large wavelength range, work with any modulation format and any protocol.

Today, many implementations of tunable TTDs exist for various applications. Such devices are often based on ring-resonators [10–13], spatial light modulators (SLM) [14–18], switched delays [3,19], dispersive fibers [20–25], and fiber gratings [26–29]. All these schemes have different features: Ring-resonator based time delays, for instance, offer continuous tunability and low footprint [10–13]. They rely on the group delay appearing at resonant frequencies but are tuned using slow thermal effects and still have large losses. Spatial light modulators (SLM) can mimic true-time delays by generating an arbitrary filter shape [14–18]. However, the application range is constrained by the limited speed. Other existing solutions are based on switches [3,19,30]. Yet, they only offer discrete time delay steps [31]. Highly dispersive fibers [20–25] or fiber-grating based structures [26–29] can also provide tunable delays even for very large delays. However, as the delay is adjusted by tuning the laser frequency, precisely tuned lasers are needed, which makes these schemes costly if it should be scaled for larger number of units [11]. So far, continuous, fast and transparent tuning has been demonstrated using constant phase shifts rather than implementing true-time delays [32]. Yet, schemes applying a constant phase shift on all spectral components only allow for either beam steering of narrow band signals or steering in a narrow angle in phased array antenna (PAA) [11]. Thus, there is a need for a scheme that offers large bandwidth, ultra-low settling times, transparent operation, and low loss simultaneously.

In this paper, we introduce and demonstrate an ultrafast, continuously tunable true-time delay with a simple configuration that operates across a large bandwidth. The settling time of our approach is only limited by the speed of standard optical phase-modulators and could thus be on the order of tens of picoseconds. Here, we show that true-time delays of up to a quarter-symbol duration can be implemented with little degradation of the signal quality.

The paper is organized as follow. In a first section, we clarify the main differences between phase shifters and true-time delays, we introduce the concept and the theory of complementary phase shifted spectra (CPSS) – our new approach to mimic the behavior of true-time delays. Finally, we use simulations and experimental demonstrations to support the claims.

2. Phase shifters and true-time delays

In various applications, true-time delays (TTD) are approximated by phase shifters (PS). Such phase shifter (PS) schemes have for instance been used to implement phased array feeders [33]. To explain the new concept more clearly we first work out the main differences between a TTD approximated by a PS scheme and an ideal TTD. We also discuss the resulting errors in the frequency response of a phase-shifter scheme with respect to a TTD.

The effect of a PS on a signal is to add a phase offset $φ$ in the frequency domain. The resulting output signal $a_{φ}$ then is

a_{φ} (t) = \int_{- \infty}^{\infty} {\hat{A}}_{0} (f) \cdot e^{j φ} \cdot e^{j 2 π f t} d f,

where

{\hat{A}}_{0}

is the complex spectrum of an incoming signal

a_{0} (t)

, t is the time and f is the frequency. The frequency response

H_{φ}

of the phase shifter in Eq. (1) is

H_{φ} (f) = e^{j φ} .

This can also be rewritten as a magnitude

M (f)

and phase response

ϕ (f)

,

M (f) = 1 and ϕ (f) = φ = c o n s t .

The important aspect is that the phase shift is constant in the frequency domain such that each frequency component experiences the same phase shift

φ

, i.e. there is no frequency dependence in Eq. (2).

In contrast, when applying a true-time delay $Δ t$ , the output signal $a_{T T D} (t)$ is

a_{T T D} (t) = a_{0} (t - Δ t),

which can be expressed using a Fourier transform as

ℱ [a_{0} (t - Δ t)] = {\hat{A}}_{0} (f) \cdot e^{- j 2 π Δ t \cdot f} .

The frequency response of the TTD can be identified in Eq. (5) and is

H_{TTD} (f) = e^{- j 2 π Δ t \cdot f},

or

M (f) = 1 with ϕ (f) = - 2 π Δ t \cdot f .

An ideal TTD thus has a unitary magnitude response

M (f) = 1

. In the phase response

ϕ (f)

, the TTD introduces a linear slope, directly related to the delay time

Δ t

.

An ideal true-time delay unit together with its frequency response is plotted in Fig. 1. In all our illustrative plots, we display input and output signals in a time domain representation, in a frequency domain representation (with the magnitude and phase spectrum), and in a phasor representation. For the sake of simplicity, we have chosen signals with a rectangular shaped spectrum and thus sinc shaped time signals. The initial relative phases are zero for all spectral components of the input signal. In Fig. 1, the sinc shaped input signal is fed into the ideal TTD of Eq. (6). After passing the unit, the signal is delayed in time with respect to the input, see the insets of the output in Fig. 1. The magnitude spectrum of the output signal is unchanged whereas its phase spectrum has picked up a linearly increasing phase shift. The phasor representation of the output also displays the effect of a TTD on three different spectral components f₁, f₂, and f₃. It shows how they are offset by an equispaced phase after the TTD.

Fig. 1 – Effect of an ideal TTD onto a signal. The input signal, depicted in the time domain, in the frequency domain (with its magnitude and phase spectrum), and with a phasor representation, is delayed by a TTD and generates an output signal (also shown by its time, frequency and phasor representation on the right hand side). The distinct features of the TTD are the flat magnitude response and the perfect linear phase response according to Eq. (6) and (7). Both of which are shown at the bottom. Assuming a signal bandwidth of $Δ f = 10 GHz$ , the delay applied in this example corresponds to $Δ t = - Δ φ / (2 π Δ f) = - π / (4 π 10 e^{9}) s = - 25 ps$ . A set of simulated frequency responses of a TTD are plotted in the inset (i) for delay varying between 0 ps and −50 ps.

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When a TTD is approximated by a PS, a phase response error occurs. We now estimate the error by comparing the phase response of a PS from Eq. (2) with the phase response of a TTD given in Eq. (6). We compare the error for a signal operating at a carrier frequency $f_{c}$ where all spectral components are phase shifted by $φ_{o f f}$ with a PS to a signal delayed by a TTD by $Δ t = - φ_{o f f} / 2 π f_{c}$ . The values have been chosen such that the carrier frequency $f_{c}$ experiences the same delay in both cases. The phase error $ε (f)$ versus frequency then is

ε (f) = \arg [H_{φ} (f)] - \arg [H_{TTD} (f)] = φ_{o f f} + 2 π Δ t \cdot f .

Replacing the delay

Δ t

by

- φ_{o f f} / 2 π f_{c}

leads to

ε (f) = φ_{o f f} + 2 π \frac{- φ_{o f f}}{2 π f_{c}} \cdot f = φ_{o f f} \cdot (1 - \frac{f}{f_{c}}) .

The total phase error that a signal with a bandwidth B will experience can be computed by comparing the error for the lowest frequency

f_{m i n} = f_{c} - B / 2

and the highest frequency

f_{m i n} = f_{c} + B / 2

ε_{t o t} (f) = | ε (f_{m a x}) - ε (f_{m i n}) | = | φ_{o f f} | \cdot (\frac{| f_{m a x} - f_{m i n} |}{f_{c}}) = | φ_{o f f} | \cdot F_{B},

where

F_{B}

is the fractional bandwidth, i.e. the ratio between the signal bandwidth

B = f_{m a x} - f_{m i n}

and the carrier frequency

f_{c}

.

It can be seen that for a low fractional bandwidth signal ( $F_{B} ≪ 1$ ), the phase error remains small. Therefore, the maximum error is negligible and consequently the approximation may be considered correct. This is the case in optical communication as the laser frequencies are orders of magnitude higher than the signal bandwidths. But for radio-over-fiber applications with large fractional bandwidths, this assumption is no longer valid. Any device aiming to mimic the effect of a TTD in application with large fractional bandwidth should thus have a frequency response as close as possible to the ideal true-time delay given by Eq. (6), Fig. 1.

3. Complementary phase shifted spectra for true time delays

In our proposed scheme, we mimic the characteristics of TTD by a new method that we call complementary phase shifted spectra (CPSS). In this scheme depicted in Fig. 2, the input signal is first split by means of a filter into two complementary spectra. After adding a relative phase offset on one arm, the two spectra are recombined. We will show that the frequency response obtained by this technique is a good approximation to a TTD while the tuning speed is fast and only limited by the speed of the built-in phase modulator.

Fig. 2 – New true-time delay (TTD) scheme by means of a complementary phased shifted spectra (CPSS). In this method an input signal (shown in the time, frequency and phasor representation on the left) is first guided into a first filter stage that creates the two complementary signals shown in subfigures (i) and (ii). While one of the signal is unchanged (i), the second one (ii) is phase shifted by a phase modulator. By combining (i) and (ii) in the coupler using complex addition, each frequency components interferes. This results in a new output signal shown in the time, frequency and phasor representation on the right hand side. By comparing this output signal with the output signal of an ideal TTD, Fig. 1, one can see that the CPSS based delay module provides a good approximation of a true-time delay. The frequency dependent phase shifts obtained with this method is indeed almost linear in frequency. The inset (iii) shows the magnitude and phase responses of this configuration for different CPSS phases.

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The true-time delay adapted in the CPSS method comprises of three stages: First, complementary spectra are generated by a filter. In our example shown in Fig. 2, we use filters with triangular spectral shapes. The amplitude spectra of the lower and upper arms are shown as insets (i) and (ii) in Fig. 2, respectively. In the second stage, the signal on the lower arm is guided through a phase modulator and picks up a phase offset that we will call the CPSS phase. In the phasor representation, such a phase shift corresponds to a rotation of all the phasors by e.g. π/2. Finally, the two spectral parts are recombined in a coupler. The resulting phase for the individual spectral components can now be determined by adding the phasors from the two arms in the complex plane. The output signal after combining the fields in the coupler is plotted in time, in frequency (magnitude and phase spectrum), and in a phasor representations. The phase spectrum inherits its frequency dependence from the frequency dependent splitting ratio between the two spectra. By comparing this result with the output of an ideal time delay, Fig. 1, it is clear that the proposed method is a good approximation to an ideal TTD.If another CPSS phase offset is applied in the phase-modulator section of the CPSS module, the phasors in Fig. 2 (ii) will have a different angle. Therefore, the addition of (i) and (ii) in the complex plane will lead to another delay, i.e. there will be a lower phase difference between f₁ and f₃.

In Fig. 2(iii), a set of simulated magnitude and phase response of the CPSS delay module are plotted for the triangular shaped filters of Fig. 2 for different CPSS phases. Figure 2 also shows some frequency dependence in the magnitude response. Such non-ideal response functions can be mitigated by using more optimized filters for the generation of the two complementary spectra or by using multiple instances of CPSS.

An advantage of the suggested scheme is that the tuning speed may be in the GHz because it is only limited by the frequency response of the modulator. Another advantage of this scheme is that any time delay in the working range of the module ( ± π/2) can be generated, since the phase offset can be controlled continuously using the phase modulator. If no phase shift is applied on the second signal, the device will not delay the signal while a negative phase shift will lead to a delay in the opposite direction.

To avoid destructive interferences, the CPSS phase should preferably not be tuned above ± π/2. Consequently, the maximum delay that can be generated by the CPSS method is $Δ t_{m a x} = \pm 1 / 4 B$ , as explained below.

3.1 Frequency response of CPSS

In this section, the frequency response $H_{CPSS}$ of the CPSS module described above is derived by applying a transfer matrix for each of the three stages in the CPSS filter

H_{CPSS} = T_{Coupler} \cdot T_{φ} \cdot T_{Filter},

The first element $T_{Filter}$ can be described by a set of two filters with a linear magnitude response (triangular shaped spectrum). The filter response functions $H_{1} (f)$ and $H_{2} (f)$ generating the complementary spectra are defined by

H_{1} (f) = 1 - \frac{f - f_{c} + B / 2}{B} and H_{2} (f) = \frac{f - f_{c} + B / 2}{B},

where B is the bandwidth of the incoming signal,

f_{c}

the carrier frequency, and

f

the frequency (

f

takes any values between

f_{c} - B / 2

and

f_{c} + B / 2

leading to outputs between 0 and 1).

Using the functions from Eq. (12) to describe the filter response $T_{Filter}$ , Eq. (2) to describe the phase-shifter section $T_{φ}$ together with ideal values for a standard 50:50 coupler $T_{Coupler}$ we find

\begin{matrix} H_{CPSS} (f, φ) = [\begin{matrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{matrix}] \cdot [\begin{matrix} 1 & 0 \\ 0 & e^{- j φ} \end{matrix}] \cdot [\begin{matrix} 1 - \frac{f - f_{c} + B / 2}{B} \\ \frac{f - f_{c} + B / 2}{B} \end{matrix}] \\ = \frac{1}{\sqrt{2}} \cdot (1 - \frac{f - f_{c} + B / 2}{B}) + \frac{1}{\sqrt{2}} \cdot e^{j φ} \cdot \frac{f - f_{c} + B / 2}{B} . \end{matrix}

The magnitude and phase responses of this module are plotted in Fig. 2(iii). This somewhat simplistic CPSS filter provides a very good match to the ideal filter response of a TTD. Deviations from the TTD response can only be seen in a drop of the magnitude in the center and some small modulation in the phase response. While not perfect, the phase response has a strong linear component which results in a true-time delay in the spirit of Fig. 1. The slope of the phase response can be controlled by setting the CPSS phase in the phase-shifter. The larger the slope, the larger the true-time delay.

4. Implementation using delay interferometer

So far, the proposed solution for an implementation of a TTD was explained with linear amplitude filters. However, such filters are difficult to produce. Yet, many filters can be used that generate complementary spectra and therefore might be suitable for an implementation of TTD by means of the CPSS method. For instance, a delay interferometer (DI) is a simple and reasonable implementation of such a filter. A DI is a standard component that can be implemented in many ways. E.g. as a free space solution element (e.g [34].), as an integrated photonic chip (e.g [35].) or by optical fibers (e.g [36].) depending on the specific purpose of the application. A DI has two outputs with a sine frequency response on one of them and a cosine frequency response on the second one [34]. By adjusting the frequency response of the DI to its 50:50 operation point, two complementary spectra are generated. A CPSS implementation with a DI (CPSS-DI) is depicted in Fig. 3. The signals at the two outputs of the DI are plotted as insets (i) and (ii). Their amplitude responses are complementary. A phase offset is then applied to one of the output signals of the DI (ii) before the signals are recombined in the last stage. Simulated frequency responses of this implementation are plotted as insets (iii) of Fig. 3 for different CPSS phases.

Fig. 3 – Implementation of a TTD by a CPSS module based on a delay interferometer (DI). A signal is split into two complementary spectra (i) and (ii) by a DI. The DI is tuned to its 50:50 operation point. A phase-offset is then added to one of the DI outputs by means of a phase modulator. The two signals in the arms are then recombined in a coupler. The inset (iii) shows the magnitude and phase responses of this configuration for different CPSS phases. Using a DI as input filter, the phase response is close to be perfectly linear as needed for a time delay line.

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4.1 Frequency response of CPSS-DI

The frequency response of the CPSS-DI module can be computed by replacing the filter response $T_{Filter}$ in Eq. (11) by the matrix transfer function of the delay interferometer i.e. by replacing $T_{Filter}$ by a sequence of an input splitter $T_{A}$ , a delay line $T_{B}$ and a 2x2 output coupler $T_{C}$

\begin{matrix} H_{CPSS - DI} = T_{Coupler} \cdot T_{φ} \cdot T_{Filter} = T_{Coupler} \cdot T_{φ} \cdot T_{C} \cdot T_{B} \cdot T_{A} \\ = [\begin{matrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{matrix}] \cdot [\begin{matrix} 1 & 0 \\ 0 & e^{j φ} \end{matrix}] \cdot [\begin{matrix} \frac{- j}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{- j}{\sqrt{2}} \end{matrix}] \cdot [\begin{matrix} 1 & 0 \\ 0 & e^{- j 2 π Δ t f - j π / 2} \end{matrix}] \cdot [\begin{matrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{matrix}] . \end{matrix}

In Eq. (14), a phase shift of -π/2 is added to the delay line

T_{B}

in order to align the frequency response of the DI to its 50:50 operation point. After replacing the fixed delay

Δ t

by the free spectral range (FSR),

F S R = 1 / Δ t

we obtain for Eq. (14),

H_{CPSS - DI} (f, φ) = \frac{1}{2 \sqrt{2}} (- j + e^{- \frac{j 2 π f}{F S R} - \frac{j π}{2}}) + \frac{e^{j φ}}{2 \sqrt{2}} (1 - j \cdot e^{- \frac{j 2 π f}{F S R} - \frac{j π}{2}}) .

The frequency response of a CPSS based on a delay interferometer, Eq. (15), is plotted in Fig. 3(iii) for different CPSS phase shifts. It can be seen that the phase response is close to perfectly linear and corresponds to an ideal true-time delay. On the other hand the magnitude response is not perfect. This leads to two unwanted effects: First the average power may change by up to 1.5 dB when tuning the delay, acting as a small intensity modulation. However, the intensity modulation of 1.5 dB is lower than what is offered by comparable approaches based on ring resonator filters [37] which have already proven their effectiveness. Second, the magnitude response is not flat for all the delays. The outer frequency components may be attenuated by up to 3 dB compared to the center frequency. It should however be noticed that the delayed signal is undistorted, whereas the undelayed signal is slightly bandpass filtered. If such small distortions should be an issue, active compensation by an intensity modulator could be used to mitigate these effects.

4.2 Simulation

To confirm the working principle of the CPSS-DI implementation, simulations were performed with VPI Transmission Maker ©. An optimal FSR of 20 GHz, corresponding to twice the bandwidth of a 10 Gbit/s signal, was used along with a 20 GHz Gaussian band pass filter (2nd order). Figure 4 depicts the resulting eye diagrams for the maximum and minimum CPSS phase shift on the phase modulator ( ± π/2) - covering therefore the whole delay range. The simulation shows how a symbol can be delayed across a 25% of its symbol duration with good signal quality.

Fig. 4 – Simulation of a true-time delay of 25 ps induced by changing the phase between ± π/2 in the CPSS filter module. The simulations have been performed for an OOK at 10 Gbit/s with a SNR of 11 dB. The pulse is delayed by one quarter of the symbol duration when detuning the phase from the minimum (-π/2) to the maximum phase shift (π/2). The simulation has been performed with VPI Transmission Maker ©.

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5. Experimental setup

In order to perform a quick and simple demonstration of the CPSS method, we have built a fiber-based setup (as depicted in Fig. 5) [32,38,39]. An on-off-keying (OOK) signal is first generated in a 10G transmitter using a PRBS sequence generator and a Mach-Zehnder optical modulator. The input signal then enters the first stage of the CPSS module which is implemented by a polarization maintaining fiber (PMF) based delay interferometer. In such a DI, the linearly polarized input is split with a polarization controller (A) onto the fast and slow axis of the PMF (B). At the output of the PMF, a second polarization controller (C) is used to combine the fast and slow signals that are delayed with respect to each other. The alignment of the two output signals is then arranged such that one of the signals is aligned to the main axis of the subsequent LiNbO₃ phase modulator. As the second axis of the modulator is also slightly active, the applied voltage has to be larger than the specifications in order to produce the desired CPSS phase offset between the complementary signals. The signals then enter the third stage of the CPSS unit i.e. the coupler which is realized with a polarizer aligned at 45 ° between the phase modulator axes. Thus the output signal detected in a receiver is a combination of the two complementary phase shifted spectra. As a polarizer aligned at 45° filters out half of the power on both polarization, 3 dB losses are added to the system. These additional losses are however not intrinsic to the CPSS method and could be removed in another implementation.

Fig. 5 – Fiber based TTD implemented by a CPSS-DI module relying on polarization diversity for implementing a delay interferometer. In the transmitter, an optical signal is encoded with a 10 Gbits/s OOK. The signal is then launched into a polarization maintaining fiber based delay interferometer i.e. the filter creating the complementary phase-shifted spectra. In the interferometer, the input signal is first split into two polarizations (A). Then, one polarization is delayed with respect to the other by the birefringence of the fiber (B). Finally, the slow and fast signals are recombined (C) to produce the two complementary outputs in different polarizations. The two complementary signals are then fed into a LiNbO3 phase modulator that induces a certain phase offset onto the signal in the active axis only. Finally, the signals are recombined in the third stage and fed into a receiver for characterization.

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The FSR of the DI generating the complementary spectra is a critical parameter as it defines the maximum delay that a CPSS setup can produce, see section below. In our experimental setup, the length L_PMF of the PMF defines the FSR of the delay interferometer and is computed using

F S R = \frac{c}{L_{P M F} \cdot Δ n},

where c is the speed of light and

Δ n

the refractive index difference between the polarizations. With

Δ n = λ / L_{B}

[40] where L_B is the beat length (3.5 mm for SM.15-P-8/125-UV/UV-400) and

λ

the wavelength (1550 nm). The PMF should thus be 33.9 m long to match the signal bandwidth of 10 GHz (FSR = 20 GHz).

5.1 Results

Using the fiber based implementation described in the previous section, measurements were performed in order to demonstrate the TTD concept. Due to the availability of a fiber with 40 m rather than 33.9 m only, we performed the experiment with a FSR of about 17 GHz rather than 20 GHz. This means that the resulting delay range will be slightly higher (29.5 ps) than the expected 25% of the symbol duration (25 ps @ 10 Gbit/s) but also that the signal quality will decrease for the largest CPSS phases.

Figure 6 shows the received eye diagrams (a) to (c) of a 10 Gbit/s on-off keying signal for CPSS phases of -π/2, 0 and + π/2, respectively. A total delay range of 31.9 ps is found when varying the phase shift between ± π/2, corresponding to the expected value of 29.5 ps.

Fig. 6 – Experimental results showing a TTD of up to 31.9 ps only detuned by shifting the phase in the CPSS-DI module. (a-c) Eye diagrams of a 10 GBit/s OOK signal with CPSS phases of -π/2, 0, and π/2, respectively.

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Fast tunability is demonstrated in Fig. 7 by switching between two CPSS phase shifts ( ± π/4) every 33 µs. The switching time is limited by the 30 MHz function generator that was used to drive the phase modulator (Keysight 33520B). The resulting delay range of 18.3 ps is slightly larger than the expected 14.8 ps (half of the 29.5 ps calculated above). This is mainly due to the impedance mismatch between the phase modulator at 30 MHz and the output impedance of the function generator. As the two eye diagrams can be distinguished, it can be concluded that the proposed method not only allows to delay signals but that the settling time is only limited by the phase modulator and its driving electronics.

Fig. 7 – Demonstration of fast true-time delay tunability by means of a CPSS-DI fiber based delay line. The CPSS phase is switched at a rate of 30 MHz between ± π/4. The resulting delay range is 18.3 ps for this configuration (FSR = 17 GHz).

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Comparisons between measured and simulated signal qualities are shown in Fig. 8 for the different resulting delays. Experiments and simulations shows some similar trends but are not perfectly fitted. The mismatch is due to the limited stability of the fiber DI which was detuning the system over time. We are currently implementing a CPSS delay line with active adjustment of the DI. This should then fully mitigate the detuning issue and therefore shows better performances.

Fig. 8 - Comparison of signal quality for various true-time delays for both simulations and experiments. The simulations predict that the signal quality will be maintained throughout the true-time delay tuning range. The experimental results are slightly worse. In our setup we mainly attribute this to the instability of the fiber based DI arrangement.

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6. Discussions

The proposed solution results in a frequency response that is close to an ideal true-time delay. For this reason the scheme can handle any modulation format. A constraint of the proposed method is that the maximum delay time is limited. Indeed, for any specific spectral bandwidth B, the phase difference cannot exceed ± π/2. This follows from the phasor representation in Fig. 2. If the phase of a signal in one arm over the phase in the other arm should become larger then ± π/2, destructive interferences occur. The maximum delay range can thus be computed using Eq. (6)

φ (f) = - 2 π Δ t \cdot f \to Δ t = \frac{- Δ φ}{2 π Δ f},

where

Δ f

is the operation range of the CPSS module. The useable frequency range is about half the FSR of the DI. Ideally, the signals bandwidth

B \leq Δ f

. In the CPSS scheme the maximum phase is

Δ φ ≅ π

( ± π/2). For a signal with a maximum bandwidth of

B = Δ f

that can carry symbols of length

T_{S} = 1 / 2 B

(non-return-to-zero pulse shaping) one can deduce a maximum true-time operation range of

Δ t_{m a x} = \frac{π}{2 π B} = \frac{1}{2 B} = \frac{1}{4} T_{S} .

Thus the effective tunable delay is in the order of 25% of the symbol duration. Better pulse shaping such as Nyquist could increase this range by up to 50% ( $T_{S} = 1 / B$ ). This is a sufficiently large delay for most applications. In fact, in most systems where an ultra-fast tunable delay unit is needed, the requested delay range is related to the carrier frequency and not to the signal bandwidth. Moreover, if a larger delay is required, many systems could be cascaded (after optimization of the filter shapes).

The discussions were made so far with baseband signals. For the case of a pass band signal as in a phased array antenna or other microwave photonics applications, two implementations are possible. First, the free spectral range of the DI can be adapted to cover the full radio-over-fiber bandwidth (the RF carrier and the data). This is an easy to implement approach. However, it will result in quite limited delay range, as can be seen by Eq. (18). Second, a more advanced implementation would phase offset the reference laser by means of a second phase modulator while the CPSS approach is only applied to the laser carrying data. This technique called Separate Carrier Tuning (SCT) [41] is compatible with CPSS and thus can extend the working range of the method.

7. Conclusion

We have introduced a true time delay module based on complementary phase shifted spectra (CPSS). We have shown ultra-fast tunability with settling times only limited by the underlying optical phase modulator technology. To the best of our knowledge, our experimental demonstration is already three orders of magnitude faster over other methods that offer large bandwidth and continuous tunability. The CPSS method should enable settling times in the order of tens of picosecond, i.e. another three order of magnitude improvement over what has been shown in this work. Another advantage of our method is that it can be directly integrated on a photonic chip and thus may allow for an integration of more complex systems. The suggested method may serve new applications in a wide range of fields from beam steering for next generation mobile communication networks to all optical clock recovery and could bring new opportunities for both network provider and users.

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Continuously tunable true-time delays with ultra-low settling time

Abstract

1. Introduction

2. Phase shifters and true-time delays

3. Complementary phase shifted spectra for true time delays

3.1 Frequency response of CPSS

4. Implementation using delay interferometer

4.1 Frequency response of CPSS-DI

4.2 Simulation

5. Experimental setup

5.1 Results

6. Discussions

7. Conclusion

References and links

Cited By

Figures (8)

Equations (18)

Optics Express