**4. Photonic true-time delay**

Many current applications, such as RF imaging systems and wireless communications, are required to exhibit improved resolution, wider angular scans, and wide bandwidths. To obtain an optimal radiation pattern for broadband transmission, the signals received by or transmitted from the antenna array must be accurately time-compensated via true-time RF delay generation (Meijerink *et al.*, 2010; Jiang *et al.*, 2005). Currently electronic phase shift for phased-array antenna features the advantages of flexibility and reconfigurability. However, they are limited in their processing speed and bandwidth. In addition, the electronic phase shift control is accomplished through metallic coaxial cables or waveguide feeds which are heavy, bulky, lossy, and susceptible to electromagnetic interference (EMI) and crosstalk. The technology of digital signal processor (DSP) is currently thwarted either by the limited

The unique capability of the proposed tunable filter structure is its ability to continuously tune the time delay increment. This is accomplished by changing the spacing between adjacent pixel blocks (i.e., the pixel block size and center). Figures 7(a), 7(d), and 7(g) show the phase holograms applied to the Opto-VLSI processor to synthesize a normalized weight profile of [0.6, 0.8, 1.0, 0.8, 0.6] but different time delay increments. The corresponding wavebands are shown in Figures 7(b), 7(e), and 7(h). For these three cases, the sizes of the pixel blocks were 200, 300, and 400 pixels, respectively, corresponding to waveband separations of 1.72 nm, 2.60 nm, and 3.60 nm, respectively. Note that by optimizing the phase hologram for each pixel block, a normalized weight profile [0.6, 0.8, 1, 0.8, 0.6] was maintained in all cases. Figures 7(c), 7(f), and 7(i) show the corresponding measured (solid) and simulated (dashed) filter responses. It is evident from Fig. 7 that the tuning of the waveband separation (ie. the time delay increment) controls the free-spectral-range (FSR) as well as the bandwidth of the filter. For example, in Fig. 7, the FSR was reduced from 1.52 GHz, to 1.01 GHz, and then to 722MHz, and the filter bandwidth dropped from 216 MHz to 142 MHz and then to 86 MHz, when the time-delay increment was increased from 0.66 ns, to 0.99 ns, and then to 1.38 ns, respectively. Note that a good agreement between the simulated

By investigating the filter responses shown in Figures 6(c, f, i) and Figures 7(c, f, i), one can see that the spectral response fades remarkably as the RF frequency increases. This is due to the interaction between the dispersion of the HDF and the nonzero optical bandwidth of each waveband (Taylor *et al.*, 2007; Pastor *et al.*, 2003a). This limitation is inherent to all filters that use spectral slicing. The bandwidth of the sliced wavebands in Fig. 6 and Fig. 7 was about 0.5 nm. Preliminary experimental results have shown that a narrower waveband bandwidth can be achieved using a higher dispersion grating plate and/or a larger distance

Even though the optical tap generation involves fiber to free-space and free-space to fiber coupling, standard passive micro-assembly can be used to realise low-cost, robust and stable alignment without the need of automated high- precision stages (Baxter, 2006). Note that the maximum number of optical taps that can be generated depends on the bandwidth of the ASE source, the size of the active window of the Opto-VLSI processor, and the waveband separation. For our current experimental system, which based on the 1-D Opto-VLSI processor, up to 12 optical taps with the waveband separation of 2.60 nm can be generated. Note that, by using a 2-D Opto-VLSI processor, one can increase the number of optical taps

Many current applications, such as RF imaging systems and wireless communications, are required to exhibit improved resolution, wider angular scans, and wide bandwidths. To obtain an optimal radiation pattern for broadband transmission, the signals received by or transmitted from the antenna array must be accurately time-compensated via true-time RF delay generation (Meijerink *et al.*, 2010; Jiang *et al.*, 2005). Currently electronic phase shift for phased-array antenna features the advantages of flexibility and reconfigurability. However, they are limited in their processing speed and bandwidth. In addition, the electronic phase shift control is accomplished through metallic coaxial cables or waveguide feeds which are heavy, bulky, lossy, and susceptible to electromagnetic interference (EMI) and crosstalk. The technology of digital signal processor (DSP) is currently thwarted either by the limited

and measured filter responses is displayed in Fig. 7.

between the grating and the Opto-VLSI processor.

significantly.

**4. Photonic true-time delay** 

resolution and the narrow bandwidth of analogue-to-digital converter, or by high power consumption of broadband analogue-to-digital converter.

The processing of radio frequency (RF) and microwave signals in the optical domain is an attractive approach to overcome the bottlenecks encountered in conventional electronic signal processing systems (Capmany *et al.*, 2005). A wide range of emerging RF signal processing applications require specifically high resolution, wide-range tunability, and fast reconfigurability. These requirements are difficult to achieve using conventional allelectronic processing, but feasible with photonics-based signal processing.

The use of photonics-based true time delay units has extensively been investigated in the last decade for applications ranging from modern microwave radar to wireless communication systems. In particular, broadband microwave phased-array antennas require the generation of variable true-time delays at each antenna element to realize beam or null steering, and optical fibers have been the best candidates for true-time delay synthesis. Compared with all-electrical techniques, optical true-time delay generation offers the advantages of broader bandwidth, lower insertion loss, higher phase stability, smaller size, lighter weight, and excellent immunity to both electromagnetic interference and crosstalk (Frigyes&Seeds, 1995; Italia *et al.*, 2005b; Y. Chen&Chen, 2002; Rideout *et al.*, 2007). Several approaches have been adopted to realise tunable true-time delay units, including the use of in-fiber chirped Bragg gratings (FBGs) (Italia *et al.*, 2005b), white cells or fiber delay lines in conjunction with MEMS (Mital *et al.*, 2006b; Anderson *et al.*, 2006; Vidal *et al.*, 2006), integrated optical waveguides (C. M. Chen *et al.*, 2010), optically-switched fiber delay structures (Tong&Wu, 1998), dispersion-enhanced photonic-crystal fibers (Jiang *et al.*, 2005), and higher-order mode dispersive multi-mode fibers (Raz *et al.*, 2004). However, most of these reported true-time delay architectures have mainly been used for realising beam steering in phased array antennas, and therefore, they do not have the flexibility to simultaneously generate multiple arbitrary true-time delays. In addition, such architectures can only generate discrete true-time delays, making them impractical for broadband null steering (Zmuda *et al.*, 2000). Zmuda et al. have reported a few adaptive true-time delay architectures based on the use of multiple tunable lasers in conjunction with high-dispersion fibres for the implementation of broadband nulling in microwave phased arrays (Zmuda *et al.*, 2000; Zmuda *et al.*, 1998). However, the use of multiple tunable lasers requiring continuous calibration makes the system implementation very expensive and impractical.

Recently, a novel true-time delay unit has been demonstrated through uploading appropriate holograms onto an Opto-VLSI processor to synthesize multiple arbitrary time delays (Juswardy et al., 2009). This true-time-delay unit, which consists of a broadband optical source using Amplified Spontaneous Emission (ASE) and high dispersion fibers, has the capability to generate multiple true-time delays for several antenna elements simultaneously, making it attractive for broadband null-steering in phased array antennas.

The principle of the phased-array antenna architecture shown in Fig. 8 is demonstrated using the experimental setup illustrated in Fig. 4. In this setup, a broadband ASE source was modulated, via a JDS Uniphase electro-optical modulator (EOM) with a half-wave voltage of 6 V, by an RF signal which was generated using a 20 GHz network analyzer. The RFmodulated optical signal was amplified by an EDFA and collimated at 1-mm diameter and then launched onto a diffractive grating plate. The latter demultiplexed the collimated ASE beam into multiple RF-modulated wavebands, which were then mapped onto the active window of a 256-phase-level 1×4096-pixel Opto-VLSI processor of 1-µm pixel size and 0.8 µm dead spacing between adjacent pixels.

Photonic Microwave Signal Processing Based on Opto-VLSI Technology 371

free spectral range of the transfer function is given by the following equation (Capmany et

Note that, from Eq. 7, the time delay, τ, depends on the dispersion coefficient of the dispersion medium, therefore, a higher dispersion medium results in a longer maximum

The RF insertion losses of the whole tunable true-time delay system, defined as the RF power ratio between input and output of the RF signal, can be approximately expressed as

*RFout opt opt*

Where Z0 is the effective EOM RF input impedance or resistance of the EOM electrode, V<sup>π</sup> is the voltage for a π-radian optical phase shift, R (A/W) is the photodetector responsivity, Popt is the input continuous wave (CW) optical power to the EOM, and Topt is the optical power transmission parameter that embraces all the optical losses and/or gain in the optical processing including the EOM insertion losses. In this experiment, the RF insertion loss is mainly due to the free-space optical system including the fibre collimator, the diffraction grating and the Opto-VLSI processor, which contributes around 12.5dB loss. Furthermore, the high dispersion fibre (HDF) and the EOM have insertion loss 4.6dB and 3.8dB, respectively. The total optical insertion loss of the entire system is around 21dB, whereas the EDFA provides a low gain of about 10dB due to saturation. The overall RF insertion loss in the experiment was about 25dB. However, this RF insertion loss can be compensated for by the use of an optical amplifier

Different phase holograms, which are depicted in Fig. 9(a), were applied to the Opto-VLSI processor to generate five equally-separated RF-modulated optical wavebands with different wavelength separations as shown in Fig. 9(b). Fig. 9(c) shows the measured RF responses for seven true-time delay generation scenarios, corresponding to wavelength separations of 1.74nm, 2.64nm, 3.66nm, 4.32nm, 5.16nm, 5.88nm and 6.84nm, respectively. The measured free spectral ranges of the various RF responses shown in Fig. 9(c) were used to calculate the true time delays, using Eq. 6. In addition, the measured waveband spacings (Fig. 9(b)) were also used to calculate the time delays for each scenario using Eq. 7. Table 1 summarizes the free spectral ranges of the various measured RF responses, the measured waveband separations, and their corresponding time delays calculated using Eq. 6 and Eq. 7, for the different scenarios. Excellent agreement between the true-time delays calculated

*P PTZ T R P V*

π = =

2

π

2 0

*FSR f*

The time delay, τ, can also be expressed in terms of the dispersion of the HDF as

denotes the dispersion coefficient of the HDF, and Δ

*RFin*

separation between the centers of two adjacent wavebands.

*RF*

of 12.5dB gain placed after the HDF (before photodetection).

using Eq. 6 and Eq. 7 is displayed in Table 1.

τ α= ⋅Δλ

1

τ

<sup>=</sup> (6)

(7)

λ

is the wavelength

(8)

al., 2005):

where

α

attainable time delay.

(Capmany et al., 2006)

Fig. 8. Experimental setup used to demonstrate tunable time delay generation.

Labview software was specially developed to generate optimised phase holograms that couple the specific RF-modulated wavebands back into the fiber collimator, and at the same time equalize their intensities by changing the maximum phase levels applied to the different pixel blocks. The selected wavebands were then routed via a circulator to a 22-km high dispersion fiber (HDF) of dispersion coefficient 382.5 ps/nm and insertion loss 4.6 dB. An optical spectrum analyzer (OSA) was used to monitor the spectrum detected by a photodiode built in the Network Analyzer, as illustrated in Fig. 4.

One of the attractive features of the Opto-VLSI-based tunable time delay architecture is its ability to generate multiple RF delays without the need for RF splitters. Furthermore, the amplitude weight of each generated RF delay sample can simultaneously be controlled. This architecture offers excellent flexibility in applications such as phased-array null steering because multiple true-time RF delays for each antenna element can simultaneously be synthesized using computer generated holograms.

In order to measure the true-time-delay between the wavebands, the network analyzer was set to measure the RF response produced after the photodetection of the delayed RFmodulated wavebands. The transfer function that results from detecting M wavebands can be described as (Capmany et al., 2005):

$$H(f) = \sum\_{r=0}^{M} a\_r \exp[-j2\pi rf\tau] \tag{5}$$

Where f is the RF frequency, M is the number of the detected RF-modulated wavebands, *<sup>r</sup> a* is the rth tap weight, which is proportional to the optical power of the rth waveband, and τ is the time delay between adjacent wavebands introduced by the high dispersion fiber. The

Circulator

HDF

Collimator

**1**

**2**

**N**

Opto-VLSI PD

Grating

**1 2 N**

**Σ**

Fig. 8. Experimental setup used to demonstrate tunable time delay generation.

Labview software was specially developed to generate optimised phase holograms that couple the specific RF-modulated wavebands back into the fiber collimator, and at the same time equalize their intensities by changing the maximum phase levels applied to the different pixel blocks. The selected wavebands were then routed via a circulator to a 22-km high dispersion fiber (HDF) of dispersion coefficient 382.5 ps/nm and insertion loss 4.6 dB. An optical spectrum analyzer (OSA) was used to monitor the spectrum detected by a

One of the attractive features of the Opto-VLSI-based tunable time delay architecture is its ability to generate multiple RF delays without the need for RF splitters. Furthermore, the amplitude weight of each generated RF delay sample can simultaneously be controlled. This architecture offers excellent flexibility in applications such as phased-array null steering because multiple true-time RF delays for each antenna element can simultaneously be

In order to measure the true-time-delay between the wavebands, the network analyzer was set to measure the RF response produced after the photodetection of the delayed RFmodulated wavebands. The transfer function that results from detecting M wavebands can

( ) exp[ 2 ]

π τ

= − (5)

*H f a j rf*

Where f is the RF frequency, M is the number of the detected RF-modulated wavebands, *<sup>r</sup> a* is the rth tap weight, which is proportional to the optical power of the rth waveband, and τ is the time delay between adjacent wavebands introduced by the high dispersion fiber. The

0

=

*M r r*

**N 2 1**

RF out

photodiode built in the Network Analyzer, as illustrated in Fig. 4.

synthesized using computer generated holograms.

be described as (Capmany et al., 2005):

ASE

RF in

**2**

**1**

EOM EDFA

**N**

free spectral range of the transfer function is given by the following equation (Capmany et al., 2005):

$$f\_{FSR} = \frac{1}{\pi} \tag{6}$$

The time delay, τ, can also be expressed in terms of the dispersion of the HDF as

$$
\boldsymbol{\pi} = \boldsymbol{\alpha} \cdot \boldsymbol{\Delta \mathcal{I}} \tag{7}
$$

where α denotes the dispersion coefficient of the HDF, and Δλ is the wavelength separation between the centers of two adjacent wavebands.

Note that, from Eq. 7, the time delay, τ, depends on the dispersion coefficient of the dispersion medium, therefore, a higher dispersion medium results in a longer maximum attainable time delay.

The RF insertion losses of the whole tunable true-time delay system, defined as the RF power ratio between input and output of the RF signal, can be approximately expressed as (Capmany et al., 2006)

$$T\_{RF} = \frac{P\_{RFout}}{P\_{RFin}} = \left(\frac{\pi P\_{opt} T\_{opt} Z\_0}{2V\_\pi} R\right)^2 \tag{8}$$

Where Z0 is the effective EOM RF input impedance or resistance of the EOM electrode, V<sup>π</sup> is the voltage for a π-radian optical phase shift, R (A/W) is the photodetector responsivity, Popt is the input continuous wave (CW) optical power to the EOM, and Topt is the optical power transmission parameter that embraces all the optical losses and/or gain in the optical processing including the EOM insertion losses. In this experiment, the RF insertion loss is mainly due to the free-space optical system including the fibre collimator, the diffraction grating and the Opto-VLSI processor, which contributes around 12.5dB loss. Furthermore, the high dispersion fibre (HDF) and the EOM have insertion loss 4.6dB and 3.8dB, respectively. The total optical insertion loss of the entire system is around 21dB, whereas the EDFA provides a low gain of about 10dB due to saturation. The overall RF insertion loss in the experiment was about 25dB. However, this RF insertion loss can be compensated for by the use of an optical amplifier of 12.5dB gain placed after the HDF (before photodetection).

Different phase holograms, which are depicted in Fig. 9(a), were applied to the Opto-VLSI processor to generate five equally-separated RF-modulated optical wavebands with different wavelength separations as shown in Fig. 9(b). Fig. 9(c) shows the measured RF responses for seven true-time delay generation scenarios, corresponding to wavelength separations of 1.74nm, 2.64nm, 3.66nm, 4.32nm, 5.16nm, 5.88nm and 6.84nm, respectively.

The measured free spectral ranges of the various RF responses shown in Fig. 9(c) were used to calculate the true time delays, using Eq. 6. In addition, the measured waveband spacings (Fig. 9(b)) were also used to calculate the time delays for each scenario using Eq. 7. Table 1 summarizes the free spectral ranges of the various measured RF responses, the measured waveband separations, and their corresponding time delays calculated using Eq. 6 and Eq. 7, for the different scenarios. Excellent agreement between the true-time delays calculated using Eq. 6 and Eq. 7 is displayed in Table 1.

Photonic Microwave Signal Processing Based on Opto-VLSI Technology 373

Note that the spectral response fades as the RF frequency increases, as shown in Fig. 9(c). This is due to the interaction between the dispersion of the HDF and the nonzero optical bandwidth of each waveband (Taylor *et al.*, 2007; Pastor *et al.*, 2003b). The bandwidth of the sliced wavebands in Fig. 9(b) was about 0.5 nm, and this caused about 4dB spectral walk-off at 3GHz. This limits the practical bandwidth of the proposed true-time delay system. However if the sliced wavebands are narrow, the walk off effect becomes insignificant as reported in (Pastor *et al.*, 2003b). In addition, our preliminary experimental results have shown that a narrower waveband can be achieved using a higher dispersion grating plate

Note that the number of multiple time delays that can be generated simultaneously depend on (i) the spectral width of the ASE source, (ii) the maximum delay time (i.e., the maximum wavebands separation), and (iii) the size of the active window of the Opto-VLSI processor. The larger the size of the active window of the Opto-VLSI processor, the larger the number of time delays that can be generated. On the other hand, the more wavebands are required (more nulls), the smaller the maximum waveband separation that could be achieved, thus limiting the null angle which can be synthesized. Note, however, that for a certain number of nulls, the required number of wavebands for each antenna element is fixed. In this case the ASE source

Finally, the experimental results shown in Fig. 9 and Table 1 confirm the ability of the Opto-VLSI-based true time delay unit to adaptively generate arbitrary RF delays for broadband null steering of phased array antennas. In order to generate multiple RF true-time delays for each antenna element, the Opto-VLSI processor was driven by optimised phase holograms that select and couple appropriate RF-modulated wavebands into a HDF that simultaneously delays the selected wavebands. To measure the generated time delays simultaneously, the selected wavebands were detected by a single photodetector to generate a microwave transversal filter response whose FSR and shape factor (measured by the network analyzer)

and/or a larger distance between the grating and the Opto-VLSI processor.

should have sufficient spectral width to ensure the synthesis of arbitrary null angles.

can be used to calculate the amplitudes and delay times of the wavebands.

Scenario 3

Scenario 4

1.51 0.98 0.72 0.60 0.50 0.43 0.39

0.66 1.02 1.39 1.67 2.00 2.32 2.58

1.74 2.64 3.66 4.32 5.16 5.88 6.84

0.67 1.02 1.41 1.66 1.99 2.26 2.63

Table. 1. Measured free spectral ranges, waveband separations, and their corresponding

Scenario 5

Scenario 6

Scenario 7

Scenario 2

Scenario 1

time delays calculated by Eq. 6 and Eq. 7.

Measured FSR (GHz)

Time delay (ns) calculated using Eq.6

Measured waveband separation (nm)

Time delay (ns) calculated using Eq.7

Fig. 9. (a) Opto-VLSI hologram, (b) Optical spectrum of 5 RF-modulated wavebands, (c) Measured RF responses due to the photodetection of the RF-modulated wavebands displayed in (b).

Scenario 1

Scenario 2

Scenario 3

Scenario 4

Scenario 5

Scenario 6

**Optical field (μw)**

**Phase (2**

displayed in (b).

**π)**

**(a) Pixel number (b) Wavelength (μm) (c) Frequency (GHz)**

Scenario 7

Fig. 9. (a) Opto-VLSI hologram, (b) Optical spectrum of 5 RF-modulated wavebands, (c) Measured RF responses due to the photodetection of the RF-modulated wavebands

**Response (dB)**

Note that the spectral response fades as the RF frequency increases, as shown in Fig. 9(c). This is due to the interaction between the dispersion of the HDF and the nonzero optical bandwidth of each waveband (Taylor *et al.*, 2007; Pastor *et al.*, 2003b). The bandwidth of the sliced wavebands in Fig. 9(b) was about 0.5 nm, and this caused about 4dB spectral walk-off at 3GHz. This limits the practical bandwidth of the proposed true-time delay system. However if the sliced wavebands are narrow, the walk off effect becomes insignificant as reported in (Pastor *et al.*, 2003b). In addition, our preliminary experimental results have shown that a narrower waveband can be achieved using a higher dispersion grating plate and/or a larger distance between the grating and the Opto-VLSI processor.

Note that the number of multiple time delays that can be generated simultaneously depend on (i) the spectral width of the ASE source, (ii) the maximum delay time (i.e., the maximum wavebands separation), and (iii) the size of the active window of the Opto-VLSI processor. The larger the size of the active window of the Opto-VLSI processor, the larger the number of time delays that can be generated. On the other hand, the more wavebands are required (more nulls), the smaller the maximum waveband separation that could be achieved, thus limiting the null angle which can be synthesized. Note, however, that for a certain number of nulls, the required number of wavebands for each antenna element is fixed. In this case the ASE source should have sufficient spectral width to ensure the synthesis of arbitrary null angles.

Finally, the experimental results shown in Fig. 9 and Table 1 confirm the ability of the Opto-VLSI-based true time delay unit to adaptively generate arbitrary RF delays for broadband null steering of phased array antennas. In order to generate multiple RF true-time delays for each antenna element, the Opto-VLSI processor was driven by optimised phase holograms that select and couple appropriate RF-modulated wavebands into a HDF that simultaneously delays the selected wavebands. To measure the generated time delays simultaneously, the selected wavebands were detected by a single photodetector to generate a microwave transversal filter response whose FSR and shape factor (measured by the network analyzer) can be used to calculate the amplitudes and delay times of the wavebands.


Table. 1. Measured free spectral ranges, waveband separations, and their corresponding time delays calculated by Eq. 6 and Eq. 7.

Photonic Microwave Signal Processing Based on Opto-VLSI Technology 375

fibre chirped Bragg gratings (FBGs) (Italia *et al.*, 2005a), free-space in conjunction with white cells (Mital et al., 2006a), integrated optical waveguides (Flamand et al., 2000), opticallyswitched fibre delay structures (Tong&Wu, 1998). However, none of these reported photonicsbased true-time delay units has the flexibility to either tune the true-time delay continuously or generate multiple tunable true-time delays for each antenna element simultaneously. Furthermore, the limited flexibility, reconfigurability, and tunability of current photonic

Broadband null-steering beamformers are much more difficult to realise than beam-steering beamformers. Theoretical analysis of broadband null steering of phased-array antennas shows multiple variable true-time delays are needed for each antenna element, while only one variable true-time delay for an antenna element is required for broadband beam steering. An N-element smart antenna can synthesise (N–1) nulls only, and this requires the beamformer to simultaneously generate (2N-1–1) delayed versions of the RF signal received

Fig. 10(a) shows a typical N-element phased-array antenna architecture, whose array factor

( ) 1 1

*N nm n m AF x x W x*

= x (θn) is a zero of the polynomial AFN corresponding to an antenna null at the angular coordinate θn. Note that a change of even one zero affects all the weights, Wm. Note also that with N antenna elements, the phased-array antenna can synthesize only (N–1) nulls, as

Without loss of generality, considering a 4-element phased array antenna, with its main lobe at an angle θ and nulls located along angular coordinates, θ1, θ2, and θ3, the array factor

( ) *jmkd jkd jkd jkd jkd jkd jkd*

4( ) *jjj jjj j AF x x e e e x e e e e*

From Eq. (11), it can be observed that for a 4-element phased array antennas, 24-1–1 = 7 delay taps need to be generated by the true time delay unit in order to synthesis three nulls, and

11 1 2 12 1 3 13 2 3

 θ

sin sin , sin sin , sin sin

= + =+ =+

 θ

 ωτ

*AF W e e e e e e e*

θθ

 ωτ

( ) ( ) ( )

 θτ

*ddd ccc*

() () ( )

θθθ

01 1 2 3

sin sin sin

= ++

21 1 22 2 23 3

*ddd ccc*

> θτ

sin , sin , sin

ωτ

1 0

= =

− −

*<sup>N</sup> <sup>N</sup> <sup>m</sup>*

, d is the antenna element spacing, k = wave number = ω/c, and xn

( ) ( ) ( ) <sup>123</sup>

θθ

 ωτ  ωτ

> θ

 ωτ

θθ

> θ

(12)

sin( ) sin( ) sin( ) sin( ) sin( ) sin( ) sin( )

= =− − − (10)

( ) ( ) 3 2 <sup>21</sup> <sup>22</sup> <sup>23</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>01</sup>

() () ( ) ( ) ( ) ( )

 θτ

ωτ

=− + + + + + − (11)

= −= ∏ (9)

beamformers make them impractical for realising broadband null steering.

( )

θ

(or directional response) is given by (Zmuda *et al.*, 1998):

θ

θ

that the time delays required to be synthesized are:

===

by the antenna.

where *x jkd* = exp sin ( )

evident from Equation (9).

3

*m*

0

=

*d c*

4

θ

τ

τ

τ

takes the form (Zmuda *et al.*, 2000):

*m*

By expanding Eq. (10), we obtain

θ

 θτ

> θ

This experiment demonstrated that arbitrary single or multiple true-time delays could be synthesized by slicing an RF-modulated broadband optical source and routing arbitrary sliced wavebands, through upload a phase hologram onto an Opto-VLSI processor, to a high-dispersion fiber where they experience RF delays that depend on their centre wavelengths.
