## Abstract

The future information infrastructure will be affected by limited bandwidth of optical networks, high energy consumption, heterogeneity of network segments, and security issues. As a solution to all problems, we advocate the use of both electrical basis functions (orthogonal prolate spheroidal basis functions) and optical basis functions, implemented as FBGs with orthogonal impulse response in addition to spatial modes. We design the Bragg gratings with orthogonal impulse responses by means of discrete layer peeling algorithm. The target impulse responses belong to the class of discrete prolate spheroidal sequences, which are mutually orthogonal regardless of the sequence order, while occupying the fixed bandwidth. We then design the corresponding encoders and decoders suitable for all-optical encryption, optical CDMA, optical steganography, and orthogonal-division multiplexing (ODM). Finally, we propose the spectral multiplexing-ODM-spatial multiplexing scheme enabling beyond 10 Pb/s serial optical transport networks.

© 2014 Optical Society of America

## 1. Introduction

The exponential internet traffic growth projections place enormous transmission rate demands on the underlying information infrastructure at every level, ranging from the core to access networks [1]. The 100 Gb/s Ethernet (100 GbE) standard has been adopted recently (IEEE 802.3ba), and 400 GbE and 1 Tb/s Ethernet (1 TbE) are considered by many authors as next natural steps. Terabit optical Ethernet technologies will be affected not only by limited bandwidth of information-infrastructure, but also by its energy consumption [2]. Additionally, future optical networks will be heterogeneous in nature. Another interesting problem not sufficiently addressed in current literature is related to security of the dense wavelength division multiplexing (DWDM) networks. To address the security issues of optical communication systems, the quantum key distribution (QKD) [3] and chaotic cryptography [4] have been proposed recently.

Most research efforts today in QKD have focused on two-dimensional QKD, such as the polarization state of photons. Moreover, the data rates for quantum key exchange are very low, and transmission distance is limited. On the other hand, in chaos cryptography the dynamics of the laser is prescribed to follow a given trajectory depending on the information to be transmitted. To decode the prescribed trajectory, the synchronism between transmitter and receiver is required. To avoid the high cost of QKD, the properly designed fiber Bragg gratings (FBGs) as optical encryption devices have been advocated in [5,6]. In particular, in [6] a super-structured Bragg gratings (SSBGs) approach to all-optical encryption was proposed. The security has been provided by the transformation of the transmitted signal into noise-like patterns in the optical domain, hiding any data signal structure to the non-authorized users. However, since the impulse responses of these encoders are quasi-orthogonal, while the designed method is heuristic, these systems suffer from limited cardinality of corresponding optical encryption signal set.

The purpose of this paper is to address all four problems of the next generation optical networks; namely, limited bandwidth of information infrastructure, high power consumption, heterogeneity, and security problems; in a simultaneous manner. The key idea of the paper is to design the Bragg gratings to have orthogonal impulse responses by means of discrete layer peeling algorithm. The target impulse responses belong to the class of discrete prolate spheroidal sequences, which are mutually orthogonal regardless of the sequence order, while occupying the fixed bandwidth. We then design the corresponding encoders and decoders suitable for all-optical encryption, optical CDMA, optical steganography, and orthogonal-division multiplexing (ODM). To solve for the limited bandwidth infrastructure and high energy consumption problems, we advocate the use of both electrical basis functions (orthogonal prolate spheroidal basis functions) and optical basis functions (ODM and spatial modes), in similar fashion as described in [2].

The paper is organized as follows. In subsection 2.1, we describe the discrete prolate spheroidal sequences (DPSS), which are used as impulse responses for target fiber Bragg grating design. The corresponding FBG design by discrete layer peeling algorithm is described in subsection 2.2. Given that the impulse responses of designed FBGs corresponding to DPSS are strongly dependent on the pulse laser emitted signal pulsewidth, we discuss the cardinality of basis functions set as the function of cross-correlation coefficient in subsection 2.3. In Section 3, we describe different applications of proposed DPSS-FBG family including optical encryption, optical CDMA, optical steganography, and orthogonal-division multiplexing. Finally, some important concluding remarks are provided in Section 4.

## 2. Discrete prolate spheroidal sequences (DPSS) and discrete layer peeling algorithm for FBG design

#### 2.1 Discrete prolate spheroidal sequences (DPSS)

As we mentioned in Introduction, for optical encryption applications and related fields, it is of high importance to ensure that impulse responses of corresponding encoders are orthogonal to each other. As indicated in [2], the modified orthogonal polynomials can be used as impulse responses for encoders. Unfortunately, the modified orthogonal polynomials offer a limited flexibility in terms of time × bandwidth product. On the other hand, the orthogonal prolate spheroidal wave (OPSW) functions are simultaneously time-limited to symbol duration and bandwidth-limited to target band and as such are extremely suitable for optical communication. Here we are restrict our attention to their discrete version, known as discrete prolate spheroidal sequences (DPSS) [7], which are a frequency band-limited sequences to [-*W*, *W*], while at the same time almost entirely time-limited to length *N*. The DPS sequences $\left\{{u}_{n}^{\left(j\right)}\left(N,W\right)\right\}$ of the *j*-th order are determined as a real-valued solution to the following system of discrete equations [7]:

*i*and

*n*denote the particular sample in each DPSS, while

*j*= 0,1,2, …. denotes the order of particular DPSS out of the set of sequences.

*N*denotes the sequence length of each DPSS,

*W*is a discrete bandwidth, and the shaping factors ${\mu}_{j}\left(N,W\right)$ are ordered eigenvalues of the system of Eq. (1) corresponding to the concentration of each DPSS within the desired time interval of length

*N.*Therefore, the eigenvalues fall within the range $0<{\mu}_{j}\le 1$, with 1 corresponding to the case when the sequence energy is entirely included in the desired time interval. The small values of ${\mu}_{j}$ imply that most of the sequence energy is outside the desired time interval.

After this introduction of DPSS, we turn our attention to the use of these sequences as impulse responses of corresponding Bragg gratings to be employed as encoders for optical encryption and other areas.

#### 2.2 Discrete layer peeling algorithm (DLPA) in DPSS-FBG design

Discrete layer peeling algorithm was proven to be efficient tool for designing fiber Bragg gratings with a desired transfer function [8]. The basic idea is to divide the grating to a series of *J* complex uniform dielectric reflectors/layers, as illustrated in Fig. 1. In Fig. 1, for the *j*-th layer both the forward (*u _{f}*) and backward (

*u*) scattered signals are illustrated.

_{b}By using the causality argument, i.e. the fact that the FBG impulse response at *t* = 0 depends only on the value of the reflection coefficient of the first reflector, the grating is “peeled off” layer by layer until all the required coupling coefficients for the resulting grating are calculated [8,9]. The corresponding algorithm is described by flow-chart shown in Fig. 2, where *M* denotes the number of wavelengths observed. The *r*(*δ*) denotes the target complex reflection spectrum, obtained from corresponding DPSS. The ρ* _{j}* denotes the complex reflection coefficient of the

*j*-th section, and

*δ*is the detuning coefficient defined as $\delta \left(\lambda \right)=(2\pi /\lambda ){n}_{eff}-\pi /\Lambda ,$ where

*n*is the effective refractive index and

_{eff}*Λ*is the grating period. Finally, the

*q*denotes the coupling of the

_{j}*j*-th section with magnitude and phase determined as shown in Fig. 2. The evaluation starts by calculation of the reflection coefficient for the first layer, then the fields are propagated forward to find all required reflection coefficients until the end of all gratings, followed by the determination of the coupling coefficient.

The transfer matrix method is used to analyze the synthesized gratings’ spectra of corresponding encoders. For simplicity of implementation, we assume that the number of sections in the layer peeling algorithm is the same as the number of reflectors previously chosen in the design. In Fig. 3, we provide the flow-chart illustrating that overall grating can be considered as a concatenation of *J* sections, each described by the transfer matrix *T*^{(}^{j}^{)}. The elements of the transfer matrix of the *j*-th section (*j* = 1,2,…,*J*) at the wavelength ${\lambda}_{k},\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1,2,\cdots ,M$ are given as follows:

The set of DPS sequences used as prototype is generated up to the order of 197, and the length of grating was set to 1 cm (corresponding to 10 Gb/s rate). Two particular transfer function instances, corresponding to the 10-th and 100-th orders are shown in the Fig. 4. In the wavelength region of interest, pretty much nice agreement between the target and actual transfer functions have been obtained. The comparison of corresponding impulse responses is provided in Fig. 5. The discrepancy comes from the fact that the narrow Gaussian pulse of width 1 ps has been observed as the input to the grating instead of the unit-sample (delta) function. Notice that the bandwidth occupied is dictated by the number of DPS sequences and target symbol rate, while the central wavelength can be located to be in the wavelength region different from commercially available FBGs, including the wavelength region considered in reference [10].

#### 2.3 Cardinality of DPSS-FBGs

Since the narrow Gaussian pulse has been used as the input to the grating instead of unit-sample (delta) function, the impulse responses orthogonality of DPSS-FBGs will be violated. This restriction has been dictated by currently existing pulse lasers and fabrication limitations. In order to investigate the reduction of cardinality of DPSS set due to practical restrictions, we study the cardinality as the function of maximum tolerable cross-correlation among the impulse responses in the actual set. The corresponding results are summarized in Figs. 6 and 7.

Clearly, for theoretical DPS sequences, the full set of 150 is obtained when input Gaussian pulse width is smaller than or equal to 0.1 ps. Otherwise, for 10% of tolerable cross-correlation, the cardinality drops down to about 60 for 1 ps pulsewidths. On the other hand, for the designs obtained by DLPA, for 1 ps pulse widths the cardinality drops down to 51 for the same level of cross-correlation (10%). This additional degradation is contributed to the DLPA suboptimal design along with the practical fabrication constraints [9].

Since the proposed DPSS-based FBGs can be described using the transfer matrix method given by Eq. (2) and Fig. 3, conventional FBG fabrication methods, including ultra-violet laser writing, are applicable here.

## 3. DPSS-FBGs in optical encryption, optical CDMA, optical steganography, and orthogonal-division multiplexing

Given the description of proposed DPSS-FBGs, in this section we study the different applications of interest including all-optical encryption, optical CDMA, optical steganography, orthogonal-division multiplexing, to name few.

#### 3.1 All-optical encryption

In our information centric society Internet became the cornerstone of the universal information exchange. By year 2015, internet traffic in the United States alone is projected to be 50 times larger than its 2008 level with an exponential growth trend that will continue in upcoming decades. Although there are many proposals how to cope with the incoming bandwidth crunch [1,2], the security of future optical networks appears to fall well behind. Given the fact that by tapping out the portion of DWDM signal, the huge amount of data can be compromised, the security is becoming one of the major issues for future optical networks to be addressed sooner rather than later.

In this section, we describe how the DPSS-FBGs can be used in all-optical encryption, which represents an interesting solution to enhance security of existing optical networks. Here we are concerned with encoders/decoders for optical encryption that can potentially be implemented on a single-chip in integrated optics. Given the fact that the basic building blocks are Bragg gratings and optical circulators, which are already fabricated on a single-chip [11], the proposed design represents a promising low-cost solution to all-optical encryption.

The encoders and decoders for all optical encryption are shown in Fig. 8. The encoder for all-optical encryption is composed of two stages: encryption stage, the upper section of the encoder, and the masking stage, the lower section of the encoder. The same pulse laser is used for both stages. The PRBS sequence is by means of either Mach-Zehnder (MZ) or phase modulator converted into optical domain. The 1:*K* optical switch is used to randomly select one out of *K* available DPSS-FBGs to be used as an encryption device. On the other hand, an arbitrary sequence is used in masking stage to generate orthogonal noise sequence. This sequence is by MZ or phase modulator converted into optical domain. One of *L* available DPSS-FBGs is selected at random by control input of 1:*L* optical switch. Both DPSS-FBGs of encryption and masking sections are derived from the same class of DPS sequences of cardinality ≥*L* + *K*.

On the receiver side, the conjugate DPSS-FBGs have been used to decipher the transmitted sequence. When the transmitter and receiver use complementary DPSS-FBGs strong autocorrelation peak get generated at every signaling interval, while zero autocorrelation is obtained after non-matched DPSS-FBGs. The decision circuit selects the branch with strongest cross-correlation peak at every symbol interval. Since the masking sequence has been generated by randomly selected DPSS-FBG, generated from the same set of DPS sequences as for the encryption portion of the transmitter there is no need to use de-masking at the receiver side. Therefore, the masking is used to hide any data structure to the eavesdropper in both time- and spectral-domains. As an illustration, in Fig. 9(a), we provide the output of encoder observed at different time intervals. Clearly, the regular structure of the data sequence is not visible to the eavesdropper, as the output appears as Gaussian noise to the eavesdropper. On the other hand, at the receiver side as shown in Fig. 9(b), the transmitted sequence 1 0 0 0 1 1 0 0 1 0 1 0 can easily be detected by simple threshold receiver.

Given the fact that proposed DPSS-based FBGs are particular instances of FBGs, as they satisfy Eq. (2), various methods already in use to tune FBGs, including piezoelectric and thermal tunings, are applicable in tuning DPSS-FBGs as well. This tunability property can be used to simplify the encoder and decoder configurations, described in Fig. 8.

Regarding the security, it is dictated by the cardinality of the DPSS set, as discussed in [6]. Since the pseudo-orthogonal impulse responses have been used in [6], while orthogonal ones in this paper, the cardinality of proposed DPSS set is higher, which leads to better security compared to scheme introduced in [6]. To improve the security further, we propose to use the masking sequences at data rates different from transmitted sequence, as illustrated in Fig. 10. In this particular example, the masking sequences of the same rate, half rate, and twice higher rates than transmitted sequence have been used. The parameters of corresponding DPSS-FGBs for nominal transmitted sequence rate of 10 Gb/s are provided in Table 1. The BER performance of proposed scheme are evaluated in Fig. 11. Clearly, when the same masking rate is used as transmitted data rate there is no any performance degradation compared to the case without encoding. On the other hand, as we increase the number of masks of different rates, there is certain performance degradation. Given the fact that in modern optical communication systems either turbo-product or LDPC codes have been used, for moderate BER threshold rates of 10^{−2}, the BER performance degradation even for different masking rates will be negligible.

#### 3.2 Optical steganography

Another interesting application of the proposed DPSS-FBGs technology is the optical steganography [10,12,13]. In general, the encrypted data can be recorded by eavesdropper regardless of its degree of encryption, so at least the information of ‘data existence’ on line will be leaked, even when the advanced masking proposed in this paper is used. This will allow the eavesdropper to “store and wait” for chosen channels till new technologies emerge for cracking them. On the other hand, the optical steganography will provide the possibility of hiding data within the public channel, by using so called “stealth channels” that are only visible to the authorized recipients. The proposed DPSS-FBGs can also be used for optical steganography applications.

In Fig. 12, we provide the BER performance of stealth channels and compare them against the BER performance of the public channel. The masking data rate is the same as transmitted data rate. Clearly, there is no any performance degradation even for five stealth channels compared to the performance of the public channel.

#### 3.3 Optical CDMA (OCDMA)

The optical code division multiple access (OCDMA) has been a hot research topic during the past two decades [14-17]. The advantages are optical CDMA numerous including privacy and inherent security in transmission, simplified network control (there is no need for the centralized network control), scalability of the network, support to multimedia applications, provision for quality of service (QoS), to mention few. The proposed DPSS-FBGs can also be used in OCDMA. In Fig. 13, we provide the BER results corresponding to different numbers of users, ranging from 1 to 36. In simulations, the laser pulse width is set to 1 ps and data rate of individual users to 10 Gb/s.

The DPSS-FBGs, used in simulations are the same as in Table 1, corresponding to 10 Gb/s. Clearly, there is no any degradation when 36 users are used compared to a single user. This represents a key advantage of the proposed scheme, based on orthogonal impulse responses, compared to conventional OCDMA approaches based on pseudo-orthogonal impulse responses. Although the BPSK is used in this example, arbitrary complex signal constellations are applicable as the decoding is performed in optical domain, which represents another key advantage of the proposed scheme compared to conventional OCDMA techniques [14-17].

#### 3.4 Orthogonal-division multiplexing (ODM)

As a solution to limited bandwidth of information infrastructure, high energy consumption, and heterogeneity problems, the use of all available degrees of freedom for conveyance of the information over fibers supporting spatial-division multiplexing (SDM); such as few-mode fibers (FMFs), few-core fibers (FCFs) or few-core-few-mode fibers (FCFMFs); has been advocated in [2]. The optical degrees of freedom include the polarization and spatial modes in FMFs and FCFs. The electrical degrees of freedom include orthogonal prolate spheroidal wave (OPSW) functions. These degrees of freedom are used as the basis functions for multidimensional signaling. Given the orthogonality of impulse response of proposed DPSS-FBGs, they can be used to provide an additional degree of freedom, in so called ODM, whose principles are illustrated in Fig. 14. Clearly, this scheme represents the generalization of OCDMA scheme discussed in previous section. The pulse laser output is split into *K* branches. Every branch is used as input of an electro-optical (E/O) modulator such as MZ, phase, or I/Q modulator. The output of the *k*-th modulator (*k* = 1,2,…,*K*) is used as the input the *k*-th DPSS-FBG, indicating that independent data streams are imposed on orthogonal impulse responses. The outputs of corresponding DPSS-FBGs are combined by *K*:1 star coupler and transmitted to remote destination over either fiber-optics of free-space optical (FSO) system of interest. On receiver side, as shown in Fig. 14(b), the independent data streams are separated by corresponding conjugate DPSS-FBGs, whose outputs are used as inputs of corresponding coherent detectors. Notice that for FSO applications, the coherent detectors can be replaced by direct detectors. This scheme is applicable to any modulation format including on-off keying, M-PSK, M-QAM, to mention few. Since the orthogonal-division demultiplexing is performed in optical domain, both coherent and direct detections can be used. Finally, the system is compatible with both SMF and SDM fiber applications.

Now we describe how the orthogonal-division multiplexing can be used to enable beyond 10 Pb/s serial optical transport, which represents the generalization of the scheme proposed in [18]. The signal frame shown in Fig. 15 is flexible and envisioned to support bit rates of up to 10 Pb/s. It is organized into 10 band-groups with center frequencies being orthogonal to each other. Each spectral component caries 1 Tb/s Ethernet (1 TbE), while each spectral band group carries 10 TbE traffic. We employ a four-step hierarchical architecture with a building block being 1 Tb/s supperchannel signal. Next, 1 TbE spectral slots are arranged in spectral band-groups to enable up to 10 TbE. By combining two (four) spectral band-groups, the scheme can enable 20 TbE (40 TbE). The second layer is related to spectral-division multiplexing, resulting in 100 Tb/s aggregate data rate per spatial mode, corresponding to 100 TbE. By combining two/four/ten such obtained signals by using the orthogonal-division multiplexer shown in Fig. 14(a), the scheme is compatible with 200 Tb/s/400 Tb/s/1 Pb/s. Finally, the fiber link layer is implemented by combining the signals from spatial modes to achieve 10 Pb/s serial optical transport.

The adaptive software-defined LDPC-coded multiband OFDM with spectral multiplexing, ODM, and SDM that we have proposed is shown in Figs. 16(a). For easier explanation, only a single polarization state is shown, with no details with respect to synchronization and clock recovery circuits. The independent adaptive regular/irregular LDPC-coded data streams are written into *m _{i}* ×

*n*(

*i*∈{

*x*,

*y*}) block-interleaver [see Fig. 16(b)]. The

*m*bits from block-interleaver are taken column-wise and used to select the coordinates of 2

_{i}*M*-dimensional signal constellation (employing 2

*M*electrical basis functions). The configuration of corresponding 2

*M*-dimensional modulator can be found in [18]. The even (odd) coordinates of 2

*M*-dimensional signal-constellation after up-sampling are passed through corresponding discrete-time (DT) pulse-shaping filters of impulse responses

*h*(

_{m}*n*) =

*Φ*(

_{m}*nT*), whose outputs are combined together into a single real (imaginary) data stream representing in-phase (quadrature) signal. After digital-to-analog conversion (DAC), the corresponding in-phase and quadrature signals are used as inputs to the I/Q modulator. The band selection within the band group is performed by complex multiplication with the exp(j2π

*f*)-term (

_{n}kT*T*is the sampling interval), as shown in Fig. 16(b), where

*f*is the center frequency of the

_{n}*n*-th band in band-group. Such obtained signals are initially spectrally-multiplexed to create the spectral band group. The spectral multiplexing can be achieved by the complex multiplication (to be performed in the electrical domain as shown in Fig. 16(b)) of corresponding 2

*M*-dimensional signals by exp[j2π(

*f*+

_{c}*f*)

_{n}*kT*], where

*f*is the central frequency of the

_{c}*c*-th spectral band group, and by combining them by a power coupler. Alternatively, the all-optical OFDM approach can be used for both spatial bands and spatial band groups multiplexing. The corresponding spectral band-group signals are then coupled into the orthogonal-division multiplexer, as shown in Fig. 16(a). The basic parameters of DPSS-FBG based ODM, corresponding to symbol rate of 31.25 GS/s are provided in Table 2. With LDPC code rate of 0.8, the information symbol rate would be 25 GS/s. To facilitate the demodulation process, both the central frequencies of bands within the band-group and frequencies among the band-groups are properly chosen so that principle of orthogonality is satisfied.

On the receiver side, see Fig. 16(c), after mode-demultiplexing, followed by orthogonal-division demultiplexing, every mode/ODM projection is forwarded to the conventional polarization-diversity receiver, which provides the projections along the basis functions in both polarizations (and in-phase/quadrature channels). Each projection (in-phase/quadrature in either polarization) represents *M*-dimensional electrical signal. Two *M*-dimensional projections (corresponding to x-/y-polarizations) are passed through analog-to-digital conversion (ADC) blocks and complex multiplier by exp[-j2π(*f _{c}* +

*f*)

_{n}*kT*], and used as inputs to corresponding matched filters with impulse responses

*h*(

_{m}*n*) =

*Φ*(-

_{m}*nT*). For configuration of 2

*M*-dimensional demodulator please refer to [18]. Finally, the re-sampled outputs represent projections along the corresponding basis functions, and these projections are used as inputs to the multidimensional

*a posteriori*probability (APP) demapper, which calculates symbol log-likelihood ratios (LLRs). We iterate the extrinsic information between LDPC decoders and APP demapper until convergence is achieved, or until pre-determined number of iterations has been reached. To compensate for the mode-coupling, optical MIMO detection principles described in [1] are used.

## 4. Concluding remarks

As the response to never ending high rate demands, 100 GbE standard has been adopted recently (IEEE 802.3ba), and 400 GbE/1 TbE and beyond Ethernet technologies have been intensively studied. It has become evident that terabit optical Ethernet technologies will be affected not only by limited bandwidth of information-infrastructure, but also by its energy consumption, heterogeneity of network segments, and security issues.

As a solution to all problems, in this paper, we have advocated the use of both electrical basis functions and optical basis functions, implemented as FBGs with orthogonal impulse response in addition to spatial modes. We have designed the Bragg gratings with orthogonal impulse responses by means of the discrete layer peeling algorithm. The target impulse responses have been derived from the class of discrete prolate spheroidal sequences, which are mutually orthogonal regardless of the sequence order. We have designed the corresponding encoders/decoders and studied their application in all-optical encryption, OCDMA, optical steganography, and orthogonal-division multiplexing. Finally, we have proposed the spectral multiplexing-ODM-spatial multiplexing scheme enabling beyond 10 Pb/s serial optical transport.

The future research topics include the fabrication of DPSS-based FBGs, their experimental evaluation, the demonstration of tunability, and proof-of-concept experimental demonstration of various applications that have been proposed in this paper. However, these topics are out of scope of this paper, and will be addressed in a follow-up publication.

## Acknowledgments

This work was supported by TU Darmstadt Fellowship supporting Dr. Djordjevic sabbatical (during 2013). This work was also supported in part by the NSF under Grant CCF-0952711.

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