1. Introduction

Recently, there is increasing interest in microwave photonics (MWP)-based signal processing which connects microwave domain with optical domain [2, 3]. The MWP technique enables efficient generation and processing of microwave signals using photonic technology [4, 5]. The main inherent features of MWP are immunity to electromagnetic interference, low loss, wide-bandwidth operation, compactness, and light weight [6, 7]. Further, MWP can offer tunability and reconfigurability for the whole system, which is hard to realize using conventional microwave technologies [8, 9].

One of the main applications of MWP is beam steering and beam forming of microwave and millimeter wave phase array antennas (PAAs) [10, 5]. This application has been demonstrated successfully for wireless communications [11-13] and radar imaging systems [14]. The steering operation of traditional PAAs relies on microwave phase shifters which are used to control the phase of each radiating elements (sub-antenna) in the PAA. The interference of electromagnetic waves from these radiating elements control the beam pattern in the free space. Microwave photonic phase shifters suffer from beam-squint problem which limits the operation bandwidth [5]. This problem arises due to the dependence of beam steering angle on the microwave frequency which leads to the divergence of beam directions of different microwave frequencies at increasing steering angles. This problem can be solved by using a true time delay line (TTDL) which offers a linear phase shift-frequency characteristics and hence ensures a frequency-independent steering angle. Photonic delay line has been emerged as promising candidates for TTDL beam steering networks in terms of loss and bandwidth beside its suitability for implementation in MWP platform [15, 16]. In photonic beam steering network, the microwave signal is mapped into an optical carrier. The modulated optical carrier is then launched into the photonic TTDL to get the required delay and the resultant optical signal is photodetector to get a delayed version of the microwave signal.

It is worth to mention here that photonic-based beam steering networks cannot be in general realized using conventional photonic transmission lines (optical fibers or optical waveguides). These photonic lines offer time delay which is almost independent of the laser wavelength. The time delay offered by photonic line of length is given by , where is the speed of light in free space and is the refractive index of propagation medium. Therefore, the phase shift between successive radiating elements cannot be varied by tuning the laser wavelength. In contrasts photonic TTDL implemented using fiber Brag grating (FBG) technology offers a time delay which is a function of the laser wavelength [6]. The corresponding phase shift where is the time period of a microwave signal of frequency (or wavelength ). Hence the steering angle [17] becomes equal to , where is the radiating elements spacing. In this case is independent of the microwave signal spectral contents and can be controlled by tuning the laser wavelength The individual radiating element are frequently space about a half–wavelength apart (i.e., ), where is taken as the center wavelength of the microwave signal spectrum. This makes Therefore, to obtain a fixed value of the time delay should be adapted according to the operating microwave frequency At low microwave frequencies, the required value of can not be obtained by one FBG and cascaded configuration of photonic TTDLs is required. This issue is addressed in this paper where an adaptive photonic TTDL is proposed using FBGAs and optical switches. The concepts of this work may be applied in base stations of wireless communication systems which will deal with PAAs having different values of and number of radiating elements. Therefore, it is essential to design smart beam-steering networks for these base stations and this subject will be addressed in the accompanying paper [1].

2. Background on Apodized Linearly Chirped Fiber Bragg Grating

This section introduces the main concepts of FBG and its linearly chirped version. The principles of refractive index-apodization profiles are briefly illustrated.

2.1. Concepts of Fiber Bragg Grating

Fiber Bragg gratings (FBGs) are formed by constructing a periodic or aquasi-periodic modulation of refractive index inside the core of an optical fiber, see Figure 1. This change in index of refraction is typically created by exposing the fiber core to an intense interference pattern of UV energy. The exposure produces a permanent increase in the refractive index of the fiber's core, creating a fixed index modulation according to the exposure pattern. This fixed index modulation is called a grating [18]. A small amount of light is reflected at each period. All the reflected light signals combine coherently to one large reflection at a particular wavelength. This is referred to as the Bragg condition, and the wavelength at which this reflection occurs is called the Bragg wavelength. Only those wavelengths that satisfy the Bragg condition are affected and strongly back reflected through the same core of the fiber. Uniform FBGs are based on a periodical modulation of the refractive index in the core of an optical fiber Figure 1, and they are the most popular grating-based type of technology.

Figure 1. Concept of fiber Bragg grating. Illustrative spectra of the incident, reflected and transmitted waves are given

The Bragg wavelength equals to [19]

(1)

where is the period of the refractive index modulation, and is the effective refractive index of the fiber core. In a chirped FBG, periodicity of the modulation is not constant, but it changes along the propagation axis z; the function defines the chirp pattern. This implies that each different section of the grating reflects a different Bragg wavelength, and the overall spectrum of the FBG results from the spectrum of each section of the grating.

Consider an uniform FBG extended along the z-axis from z=0 to z=L. Assume that the periodic variation of the refractive index over the grating axial direction (z–axis) takes the following form [20]

(2)

where is the axial average of the refractive index, represents the peak of the refractive index variation, and denotes the grating pitch which represents the period of the z-dependent refractive index variation as shown in Figure 2. The typical value of varies in the range to . Under weak coupling (i.e., , the fundamental mode of the grating wavelength can be expressed as [18]

Figure 2. Refractive index change in optical fiber

(3)

where and represent forward- and backward-propagating modes, respectively, and is the propagation constant

(4)

Here is the effective refractive index of the fundamental mode whose wavelength is

The relation between the forward mode and the backward mode is governed by the following coupled equations [18]

(5a)

(5b)

where is the coupling coefficient

(5c)

(5d)

Equations 5a and 5b should be solved as two simultaneous equations to yield and . Taking the derivative of both sided of equ. 3 with respect to z yields

(6a)

Subsisting equ. 5a into equ. 6a gives

(6b)

(7)

The solution that will incorporate the boundary condition

(8)

where

Using equ. 5b yields

(9)

Using the boundary condition gives

(10)

The amplitude reflection coefficient is given by

(11)

(12)

2.2. Linearly Chirped FBG

The chirped FBG has nonuniform grating period along its length z. When varies linearly with z, the structure is called linearly chirped (LC) FBG. The LCFBG is characterized by a wider reflection spectrum since each wavelength component is reflected at different grating position. This type of FBG can be fabricated by varying the peak refractive index variation (see equ. 2) with the axial length z. For an FBG extending from z = 0 to z = L, the linearly chirped period can be expressed as [20]

(13a)

where is the center period of the grating and is the chirp parameter. The maximum and minimum grating periods occur at z = 0 and z = L, respectively

(13b)

(13c)

Note that and is the total chirp. Recall that the Bragg wavelength is related to the grating period and effective refractive index by Then for a LCFBG, the minimum and maximum value of the Bragg wavelength can be computed from

(14)

(15)

The dispersion parameter of the FBG is defined as the variation of the time delay associated with reflection transfer function with wavelength

(16)

For a LCFBG, the dispersion parameter can be calculated approximately by

(17)

where which corresponds to the grating period at the axial center of the FBG. Note that is constant over the spectral range of interest and this ensures a linear variation of time delay with This feature is very important to design TTDL where the time delay can be varied linearly with the laser wavelength.

2.3. Apodization Profiles

Refractive index-apodization profile refers to the case when the peak of the refractive index variation is not uniform along the FBG axis (see equ. 2). In this case is z-dependent and can be expressed as

(18)

where corresponds to the uniform apodization profile where the peak of the refractive index is uniform across the length of the FBG and T(z) is the apodization profile or function. Note that T(z) is similar to a spatial window used in signal processing.

It is well known from the literature that using apodization and linear chirping in Bragg gratings reduces the side lobe level in the reflectivity response and also the group time delay response ripple [21]. However, different apodization profiles having different impact and it would be useful to choose the profile that is more suitable for the problem under investigation. In this work, the apodization profile is chosen to give large time delay difference across the LCFBG spectrum. The investigation done in this work reveals that the best results are obtained by using two types of apodization profiles

(i) Modified Gaussian Apodization (MGA) [22]

(19)

where n is the order of MGA and is the taper parameter.

(ii) Gaussian Apodization (GA) [22]

(20)

Note that the two profiles are symmetric around the center of the grating and normalized such that

3. Proposed Beam-steering Network

3.1. System Configuration

Figure 3 illustrates the main concepts of the configurable photonic TTDL-based beam-steering network investigated in this work. For N-element PAA, the outputs of N-tunable semiconductor lasers (TSLs) are multiplexed and the resultant waveform is then modulated by the microwave signal. The modulated multiplexed signal is launched into the configurable TTDL and the output is applied to 1:N demultiplexer. Each of the demultiplexer output is detected by a photodiode (PD) to recover a delayed version of the microwave signal which is then amplified by a low-noise amplifier (LNA) before feeding the radiating element. The configurable photonic delay line consists of K cascaded identical FBGs. Successive FBGs are separated by electrically controlled optical switch (OS) (see Figure 3(b-d)). The OS operates in either bar state or cross state depending whether the state of the control signal is ON or OFF, respectively. When the OS operates in the bar state, the switch passes the input optical signal to next FBG for extra delay. In the cross state, the input signal does not need extra time delay and therefore passes directly to the demultiplexer input.

Figure 3. Proposed photonic TTDL-based beam steering. (a) System configuration; (b) Configurable photonic delay line; (c) Subsystem A; (d) Subsystem B

Few remarks related to the configurable photonic delay line are given in the following

(i) The proposed TTDL is quite general and can be used with different PAAs having their own operating microwave frequencies and number of radiating elements.

(ii) The first FBG is always in operation independent of the states of the OSs.

(iii) The FBGs operate in the reflection mode. Therefore, the reflection transfer functions of the FBGs are used to determine the associated time delays.

(iv) The proposed configurable delay line can be redesigned to use the transmission functions of the FBGs.

(v) Nonidentical FBGs can be used to design the photonic line if needed for certain applications.

3.2. Design Guidelines

The performance of the photonic TTDL depends mainly on the reflection characteristics of the employed FBGs. The design of the FBG should satisfy the following requirements

(i) High-peak and wideband optical power reflection spectrum centered at a given wavelength The wavelength of the used TSLs should distributed around the center wavelength and see almost the same power reflection. According to that, -0.5dB bandwidth measure is used here to estimate the bandwidth of the power reflection spectrum This ensures 0dB (i.e., 100%) power reflection at the center wavelengths and -0.5dB (i.e., 89%) at the lower and higher cutoff wavelengths. The slight variations in the power reflections seen by the semiconductor lasers can be equalized at the receiver end by adapting the gains of the LNAs.

(ii) Linear variation of the time delay associated with FBG amplitude (field) reflection transfer function across the spectrum and without ripple. This is useful to ensure a linear variation of the time delay with TSLs wavelength spacing.

(iii) Large time delay across the spectrum and it is preferred to yield high dispersion parameter

A linearly chirped FBG (LCFBG) is used in this delay line to yield a linear time delay-wavelength characteristics. The Bragg wavelength varies linearly along the FBG axis and yields a time delay that decreases linearly with the wavelength in the reflection spectrum bandwidth. Figure 4 shows an ideal time delay characteristics of a LCFBG across the power reflection bandwidth Here and are, respectively, the upper and low cutoff wavelengths of the spectrum. The chirp parameter of the FBG is a positive quantity and computed from

(21)

Figure 4. Time delay characteristics of a LCFBG across the power reflection bandwidth

The time delay at a given wavelength within the reflection bandwidth can be expressed as

(22)

where and are the time delays associated with and the reference wavelength , respectively.

The time delay difference between two adjacent TSLs whose wavelengths differs by is given by

(23)

where with

The TSLs operate in continuous-wave (CW) mode with equal wavelength spacing The wavelengths of the TSLs () are assumed to be distributed symmetrically around the spectrum center wavelength. Figure 5 illustrates the wavelengths corresponding to a 4-element PAA.

Figure 5. Wavelengths distribution of four TSLs used in the 4-element of PAA

When the PAA is designed with N even, the wavelengths of the unmodulated (CW) tunable lasers can be calculated using the following formula

(24)

According to equ. (24), and

The maximum wavelength spacing is set to to ensure that the wavelengths of the first and last lasers (i.e., and ) are not located at the spectrum edges. According to that and even when approaches it maximum value. Setting to is useful to tolerate FBG fabrication error. In other words, this setting ensures that the wavelengths of the TSLs are laying within the FBG spectrum. When tends to its maximum value then and approach their minimum and maximum values, respectively.

(25a)

(25b)

Additional design guidelines related to single- and multi-FBG steering networks along with the used OS are given in the following subsections.

a- Single-FBG Steering Network

Consider now a uniform linear PAA having N radiating elements and operates with a microwave frequency with its beam-steering network uses a single LCFBG. The phase shift between two successive radiating elements is related to the corresponding time delay difference by

(26)

where and is the period and frequency of the microwave signal used to modulate the lasers optical carriers, respectively. Then the steering angle is given by

(27)

The argument of the function should not exceed 1. This yields the following threshold condition which determines the upper bound of the wavelength spacing,

(28)

If the network operates with then the maximum achievable steering angle is given by

(29)

According to equ. (29), the maximum number of radiating elements of the antenna whose beam can be steered toward 90° using one FBG-based network is computed from

(30a)

(30b)

where denotes the maximum integer in the argument.

b- Multi FBG-Based Beam-Steering Network

Consider the case when the beam steering is achieved by using number of FBGs, from the K cascaded FBG available in the photonic line. The used optical circulators isolate the incident field from the reflected field for each used FBG. Therefore, the isolator acts as a buffer stage between two successive FBGs. Hence the total reflection transfer function of the cascaded gratings can be computed as the product of the transfer functions of gratings. The word “isolated” is used here to identify that the reflected field of the FBG does not affect the operation of the previous FBG in the cascaded network. According to this discussion, the total reflection transfer function of the cascaded gratings can be expressed as

(31a)

where is the field reflection transfer function of the FBG involves in the operation and is the optical frequency. Further, the magnitude and the phase of the total field reflection transfer function are expressed, respectively as

(31b)

(31c)

The total time delay seen by used cascaded FBGs is computed as

(32)

where represents the time delay associated with the reflection transfer function of the FBG evaluated at the optical frequency Note that

(i) The total time delay is the accumulation of the individual time delays of the used FBGs calculated under isolation condition.

(ii) When the used FBGs are identical, then and where and correspond to a single FBG.

(iii) If each FBG is designed to have 100% power reflection at the center frequency and -0.5dB bandwidth consideration, then the total power reflection evaluated at the FBG spectrum edges is 89.%, 79.4%, 70.8%, and 63.1% when = 1, 2, 3, and 4, respectively, This requires careful consideration to adapt the gains of the LNAs to compensate for the power reflection variation seen by the TSLs.

(iv) For identical gratings, the steering angle becomes

(33a)

(33b)

Then the minimum number of cascaded FBGs required to achieve maximum steering of 90° angle can be computed from

(33c)

(33d)

where denotes ceil function which rounds up the argument to the next integer.

(v) Using element spacing equals yields

(34a)

(34b)

To achieve the required wavelength difference can be estimated the following expression

(35a)

(35b)

c- Optical Switch

The optical switch can be designed using Mach-Zehnder (MZ) configuration and implemented using silicon photonic platform. Figure 6 shows a schematic of MZ switch which consists of an input optical coupler, interferometric MZ region, and an output optical coupler. The control voltage is applied across one of the MZ arms to change its refractive index and hence introducing propagation phase difference with respect to the other arm [23].

Figure 6. Schematic of the Mach-Zhender switch used in the proposed photonic delay line

The two electric fields at the optical coupler output are related to the two input fields by the following transfer matrix [24]

(36)

where and are, respectively, the bar and cross transmission coefficients of the coupler. The MZ interferometer (MZI) is assumed to have lossless arm waveguides and therefore, the transfer matrix which relates the output fields to the input fields will be determined by the propagation phases of the two arms

(37)

where and are, respectively, the phases gained by the electric fields at the end of the upper and lower waveguides.

If one assumes identical input and output couplers, the switch output fields and are related to its input fields and by

(38a)

where is the transfer function of the switch

(38b)

(38c)

Using equs. (36) and (37) into equ. (38b) yields

(39a)

(39b)

(39c)

Consider the practical case of which corresponds to a 3dB-directionl coupler. Introducing and as the phase average and difference, respectively, leads to

(40a)

(40b)

(40c)

Note that the condition leads to . Recall that and this is independent of the condition . Consider now the following special operating states

(i) Case I: No applied voltage

In this case, (assuming equal-length MZ arms). Then and This corresponds to cross-state operation. Since the optical power is proportional to the absolute-value square if the electric field, then and

(ii) Case II: Voltage is applied to yield

In this case and

This indicates bar-state operation which yields and The insertion loss of the switch, , can be Compared from

For the two special cases mentioned above, . Note that the loss of the directional coupler and this equals when . For lossless coupler (i.e., ) then . For lossy coupler, Note further that since the MZ waveguide arms are assumed lossless.

4. Performance Evaluation of Linearly Chirped FBG

The time delay-wavelength characteristics of the proposed photonic delay line dependents on the reflection response of the used LCFBG. This section presents parametric study to characterize the time delay spectrum associated with the reflection transfer function of LCFBG operating with apodization profile and 1550 nm reference wavelength. The simulation results are obtained using the commercial software package "OptiGrating ver. 14-1". Initial simulation tests are performed to select the apodization profile suitable for efficient design of photonic.

TTDL-based beam steering. The target is to get

(i) Large time delay difference across the FBG spectrum bandwidth (which is denoted here by ).

(ii) The time delay varies linearly across the spectrum bandwidth without ripple.

The initial simulation tests performed in this work have shown that modified Gaussian apodization (MGA) designed with, n = 4 and s = 0.7 offers the best time delay spectrum performance. The next apodization profile to be used is the Gaussian (GA) designed with s = 0.5. Therefore, the investigation in this section focuses only on these two apodization types. In the following, results are reported for MGZ-based FBG. Results related to Gaussian profile are given in the Appendix.

The FBG structure parameters used in the simulation are chosen to match those of a standard single-mode fiber (SSMF), see Table 1. This is useful to achieve high coupling efficiency at the FBG / SSMF interface. The of the FBG is estimated to be 2.24 at nm using the expression [32]. Here is the core diameter, is the core refractive index, and is the cladding refractive index. Note that is lower than 2.405 which is the upper bound for single-mode operation. Unless otherwise started, the following parameters are used in the simulation: refractive index modulation total chirp and FBG length L= 80 mm.

Table 1. Parameters values of the LCFBG used in the simulation

Figure 7 illustrates the main characteristics of a 80 mm-LCFBG designed with MGA Parts a and b of this figure describe the apodization profile and chirp profile, respectively. The variation of the grating refractive index modulation and grating period along the FBG length is displayed in Figure 7c. Note that the grating period decreases linearly with z (from nm at z = 0 to nm at z = L). The total chirp is and the chirp parameter nm/mm. Part d of Figure 7 shows the spectrum of both power reflection and transmission transfer functions. The reflectivity spectrum has a bandwidth nm at -0.5dB level and 0dB peak level at the center wavelength nm. The variation of the time delay across the reflectivity spectrum is presented in Figure 7e which shows a linear decrease with wavelength. The maximum time delay difference ps which yields a dispersion parameter ps/nm.

Figure 7. Characteristics of 80 mm-LCFBG. (a) Apodization profile (b) Chirp profile. (c) Grating profile. (d) Spectrum of both power reflection and transmission transfer functions. (e) Spectrum of time delay

The effect of FBG length on its power reflectivity and time delay spectra is illustrated in Figures 8(a–e) for L = 40, 60, 80, 100, and 120 nm, respectively. The main conclusions drawn from these figures are

(i) For L = 40, 60, and 80 mm, the time delay decreases linearly with wavelength and characterized by and ps over the reflectivity bandwidth and respectively. The corresponding dispersion parameters are 69, 103.85 and 138.11 ps/nm, respectively.

(ii) Increasing L to 100 and 120 nm destroys the linear characteristics and leads to two and three sections, respectively.

Figure 8. Effect of FBG length on its power reflectivity and time delay spectrum (a) L= 40 mm (b) L= 60 mm (c) L= 80 mm (d) L= 100 mm (e) L= 120 mm. Delay Reflection

The dependence of and on FBG length is illustrated in Figures 9(a-c), respectively. At L = 95 nm, the characteristics becomes discontinuous and consists of two linear sections. The discontinuous multisession behavior remains for L 95 nm. Therefore, the results in Figure 9 are presented for L 90 mm. Note that spectrum bandwidth increases sublinearly with FBG length. In contrast, the parameter increases almost linearly with L and reaches a maximum value of 562.12 ps at L 80 mm. Then decreases with L as L increases toward 90 mm. In the region of L 80 mm, the relation can be fitted by the linear characteristics where and L are measured in ps and mm, respectively. Further, ps and In other words, increases by about 7.8 ps for each 1 mm increase in the FBG length when L 80 mm. Note further that the sublinear and linear variations of and respectively, with L (for L 80 mm region) makes the dispersion parameter an increasing function of L here. At L 80 mm, approaches its maximum value of 562.12ps and equals 4.07 nm. This leads to ps/nm which represents the maximum dispersion parameter in the region of L 80mm. Therefore, 80 mm-LCFBGs will be used later to design the beam-steering network. It is worth to mention here that the peak power reflectivity equals (85%) (98%), and 0dB (100%) at L 10, 20, and 30 mm, respectively.

Figure 9. Effect of FBG length on the (a) bandwidth (b) time delay (c) dispersion

The simulation is extended further to investigate the effect of total chirp on the characteristic of the 80 mm-LCFBG and the results are presented in Figure 10. The performance parameters are recorded when the total chirp is varied from 1.8 to 2.3 nm. The main conclusions drawn for this figure are

(i) The bandwidth increases linearly with the total chirp (slope 1.74).

(ii) The time delay difference decreases slightly and almost linearly with the total chirp (slope -50 ps/nm).

(iii) The dispersion parameter decreases almost linearly with total chirp (slope -71 ps/nm).

Figure 10. Effective of total chirp of 80 mm-LFBG on the (a) bandwidth. (b) time delay. (c) dispersion

According to the above conclusions, the following question needs to be answered. Which is the suitable value of the total chirp that can be used to design the LCFBG for the beam-steering network?. Using FBG with large bandwidth makes its suitable for use with large N-PAA. Further, FBG with large time delay difference reduces the lower bound of the microwave frequency to be steered. In this work, a total chirp of 2 is used as a trade off between the two mentioned performance parameters, and

Table 2 lists performance comparison between two LCFBGs, one is designed with MGA and the other is designed with GA The comparison results are presented for three FBG lengths (40, 60 and 80 mm). Other structure parameters used in the simulation are identical for both FBGs as given in Table 1. The simulation reveals that the peak power reflectivity is 0dB for all the cases considered here and therefore not listed in Table 2. Investigation the results in Table 2 highlights the following findings. Using MGA enhances both spectrum bandwidth and time delay almost by the same factor over GA. This leads to approximately the same dispersion parameter for the two apodization profiles. This enhancement factor is about 1.30, 1.22, and 1.18 for L = 40, 60, and 80 mm, respectively. From engineering point of view, one can say that MGA offers about 20-30% increase in both and over GA and this leads to the same dispersion parameter

Table 2. Performance comparison between MGA- and MA-LCFBGs

5. Conclusions

A Configurable photonic true time delay line (TTDL) consisting of cascaded linearly chirped fiber Bragg grating (LCFBGs) has been proposed to support the design of PAA beam-steering networks. Parametric study of the designed LCFBG has been performed and the simulation results reveal that designing the LCFBG with modified Gaussian apodization (MGA) offers high-performance time delay characteristics. A LCFBG of 80mm length offers 562.12 ps time delay difference across the 4.07 nm spectrum spectral bandwidth. The length of the LCFBG should be chosen carefully to yield a linear time delay-optical wavelength characteristics. Designing the LCFBG with L>80 mm will destroys this linear relation.

Appendix Performance Parameters of LCFBG Design with Gaussian Apodization

Figure A1. Characteristics of 80 mm-LCFBG. (a) Apodization profile. (b) Chirp profile. (c) Grating profile. (d) Spectrum of both power reflection and transmission transfer functions

Figure A2. Effect of FBG length on its power reflectivity and time delay spectra (a) L= 40 mm (b) L= 60 mm (c) L= 80 mm (d) L= 100 mm (e) L= 120 mm. Delay Reflection

Figure A3. Effect of FBG Length on the (a) bandwidth (b) time delay and (c) dispersion

Figure A4. Effective of total chirp of 80 mm-LFBG on the (a) bandwidth (b) time delay and (c) dispersion