## Abstract

The novel concept of spatio-temporal modulation of Nyquist pulses is introduced, and the resulting wave-packets are termed Nyquist Localized Waves (LWs). Ideal Nyquist LWs belong to the generic family of LW solutions and can propagate indefinitely in unbounded media without attenuation or chromatic dispersion. The possibility of modulating Nyquist LWs for free-space optical (FSO) communication systems is demonstrated using two different modulation techniques. The first technique is on-off keying (OOK) with alternate mark inversion (AMI) coding for 1-bit per symbol transmission, and the second one is 16-ary quadrature amplitude modulation (16-QAM) for 4-bits per symbol transmission. Aspects related to the performance, detection and generation of the spatio-temporally coupled wave-packets are discussed and future research directions are outlined.

© 2012 Optical Society of America

## 1. Introduction

Free-space optical (FSO) communication is rapidly becoming an integral part of modern communication systems, between satellites, space platforms and airborne and other terrestrial platforms, and between mobile and stationary terminals within the atmosphere [1]. The stability and quality of terrestrial and in-atmosphere FSO links are highly dependent on atmospheric factors such as rain, fog, dust and heat [2, 3], whereas in outer space, the communication range of FSO communication is currently in the order of several thousand kilometers but has the potential to bridge interplanetary distances [4].

Even though the physical realizability of FSO communication systems is experimentally demonstrated, currently, the cost of constructing such systems is still too high for them to be widely used for commercial applications. As previously noted in [5], research and development in two different directions have the potential to dramatically reduce the cost for in-space and in-atmosphere FSO systems. The first research direction involves improving the physical-link communication efficiency by an order of magnitude using photon-counting receivers for incoherent (direct) detection in in-space channels. This will greatly reduce system complexity, weight, and power requirements for space systems since the additional components necessary for coherent detection, such as local oscillators and mixers, are eliminated. Recently, single photon detectors, which make use of superconducting nanowires, have been introduced. Experimental studies have demonstrated that they can achieve very high rates of detection (see [6] for a comprehensive review). The second research direction involves development of coherent-link systems for in-atmosphere channels; coherent processing can reduce interference in multiple access applications and significantly lower the overall system cost by increasing the number of subscribers accommodated by the channel. The design of the properly functioning front-end electronics in coherent receivers remains critical, especially at high data rates. Achieving near-quantum-limited performance in high data rate transfers with coherent detection is a challenging task because of the limitations of detection efficiency, maximum photocurrent, and trans-impedance gain. However, progress in designing coherent detection receivers aimed at alleviating these limitations has been recently reported [7–9].

The motivation of this work is to introduce propagation-invariant Localized Wave (LW) solutions as a viable asset for the commercialization of FSO communication systems. Localized Waves posses a number of characteristics, such as extended field depth, ultra-wideband, and spatio-temporal localization, among several others that position them as excellent candidates in various scientific and engineering applications [10–14]. These characteristics are most suitable for direct detection applications, since LWs can transfer the data signals without distortion for extended distances. However, the potential of LWs in telecommunication applications has not been fully demonstrated yet, partially due to the lack of electronic systems capable of launching or receiving LWs, and partially due to the strict condition of their spectral support needed to achieve propagation-invariance. In particular, conventional modulation of LWs with the data signals would inexorably violate the propagation-invariance condition. To this extent, the novel spatio-temporal modulation techniques proposed here are specifically designed to preserve the propagation-invariant nature of LWs and provide an attractive alternative for coherent detection in telecommunication systems.

In this work, to fill in this gap and demonstrate the possibility of using LWs in FSO communication links: *i)* We introduce the spectral structure of a novel type of LW solutions, which we give the name ‘Nyquist Localized Waves.’ Nyquist LWs are the extension of Nyquist pulses well-known in conventional communication channels into FSO communication, where invariant propagation is achieved and chromatic pulse dispersion is minimized or eliminated. *ii)* We demonstrate the possibility of modulating Nyquist LWs for use in wireless and optical telecommunication applications through two different examples of modulation techniques.

The modulation techniques used in this work maintain the unique spectral structure of LWs and hence their propagation-invariance is not violated. Ideally, this could lead to designing FSO communication systems that function over extensively long ranges with virtually no distortion in the received signal and minimal spatial interference with other optical signals. The first technique demonstrated, utilizes an on-off keying (OOK) scheme for streaming LWs in time along the propagation path. We show that modulating Nyquist LWs using a simple non-return to zero (NRZ) scheme would result in deterioration in the transverse localization. The application of a spatial filter to truncate the unwanted transversal extension would alter the spectral support of the LW and would deteriorate its propagation characteristics [15], thus we propose to use a simple alternate mark inversion (AMI) code, which is shown to maintain the intrinsic transverse localization of the wave-packet and thus minimize the likelihood of false positive detections off the optical axis. We suggest using direct detection with intensity-based threshold discriminator with this technique, since a positive detection of received power directly translates to a ‘1’ and the absence of power translates to a ‘0.’ The second technique introduces the concept of LW modulation for multi-bit per symbol transmission. Multi-bit per symbol modulation techniques are considered the building blocks of data packet transmission in network communication, because of their higher data throughput compared to OOK. Implementing a communication link with propagation-invariant LWs capable of transferring data packets could be a potential key element in deep space data communication as well as an essential advancement in airborne and terrestrial FSO communication. To this extent, we demonstrate encoding data bits using 16-ary quadrature amplitude modulation (16-QAM), which is one of the widely used modulation schemes in data communication networks. The encoding and modulation processes are designed in a way that does not disturb the spectral support of the LW; hence the resulting signal retains propagation-invariance, and diffraction- and dispersion-resistance features of unmodulated LWs.

In Section 2, we analyze the spectral structure of Nyquist LWs and their unique spectral support that establishes their propagation characteristics. Section 3 is devoted to the different LW modulation techniques, namely, OOK and 16-QAM. In Section 4 we briefly discuss the feasibility of the practical realization of the the proposed techniques. Finally, in Section 5, we summarize the study and point out further research directions related to LW modulation and their implementation in communication systems.

## 2. Spectral structure of Nyquist Localized Waves

In this section, we analyze the spectral structure of Nyquist LWs. To this extent, we start with the analysis of the spectral structure of “common” LW solutions in the cylindrical coordinate system. We identify the necessary condition to achieve propagation-invariance and its implication on the spectral support. Next, we derive the expressions for Nyquist LWs starting from the ideal Nyquist pulse and using the LW spectral support. Finally, we give the general representation for encoding *N*-bits using Nyquist LWs.

#### 2.1. “Common” Localized Waves

Since the early works [16–18] on the so-called non-diffracting waves (or LWs), many articles have been published on theoretical understanding of these waves spectral and spatial behavior as well as their practical generation. Initial studies focused on formulating LWs propagating in free-space; however, in recent years, researchers have started investigating characteristics of LWs propagating in dispersive, lossy, anisotropic, and/or non-linear media (see, e.g. Ch. 1 in [10]). All these studies have focused on understanding or improving the propagation characteristics of LWs, which makes them ideal candidates for carrying data for long distances without distortion. The concept of propagation-invariant wave-packets is based on establishing a linear relationship between the angular frequency, *ω*, and the wave-vector component in the direction of propagation. Such relationship ensures that all spectral components would interfere constructively, at least periodically, along the propagation direction. Thus it is only useful to apply this concept in coordinate systems, where the propagation direction is unbounded in order to benefit from the infinite invariant-propagation distance. However, one could correctly argue that a plane-wave packet with the aforementioned condition would also propagate invariantly for an infinite distance. Such wave-packets are indeed propagation-invariant, yet they cannot be considered LWs as their energy contents are spread throughout the whole transverse plane (i.e., plane normal to the propagation direction). In this regard, the cylindrical coordinate system (*ρ*, *ϕ*, *z*), became the first natural system to seek a propagation-invariant wave-packet solution that is localized in the transverse and longitudinal planes (i.e., plane of the propagation direction). It is already known that Bessel beams [19] are fundamental solutions in the cylindrical coordinate system and they are transversally localized, hence, a localized wave-packet could be constructed by superposition of Bessel beams. The linear (scalar) wave equation in free-space in cylindrical coordinate system is written as

*c*is the speed of light in free-space. The generalized solution to Eq. (1) could be written in terms of Bessel beams as

*k*and

_{ρ}*k*are the transverse and longitudinal components of the wave-vector

_{z}**k**,

*n*is an integer,

*J*(

_{n}*x*) is the ordinary Bessel function of the first kind and order

*n*, and Ψ̃

*(*

_{n}*k*,

_{ρ}*k*,

_{z}*ω*) is the spectrum of the wave. The propagation-invariance condition could thus be fulfilled by satisfying the following relation [20] with

*V*the velocity of the LW centroid (plane with maximum localization) and

*α*an arbitrary parameter. Propagating the solution in Eq. (2) a distance Δ

_{m}*z*=

*z*

_{0}and time Δ

*t*=

*z*

_{0}/

*V*employing the invariant-propagation condition given by Eq. (3) and using the translation property of the Fourier transform FT[

*f*(

*x*+

*β*)] = exp(

*ikβ*)FT[

*f*(

*x*)], gives

*α*= 0, or it reconstructs its shape periodically at distances Δ

_{m}*z*

_{0}= 2

*mπV*/

*α*, where

_{m}*m*is an integer. It is important here to stress on the fact that though

*ω*and

*k*are coupled, the wave dispersion relation must hold true as well, i.e., This immediately puts a constraint on the choice of the transverse wave-vector components of LWs. Accordingly, the specification of any of spectral components (temporal or spatial), would determine the whole spatio-temporal spectrum [20].

_{z}The wave dispersion relation given by Eq. (4) and the invariant-propagation condition given by Eq. (3) could both be enforced simultaneously by means of introducing Dirac-delta functions into Ψ̃* _{n}*(

*k*,

_{ρ}*k*,

_{z}*ω*), such as

*S*(

_{nm}*ω*) is the temporal spectrum of the wave-packet,

*δ*(

*x*) is the Dirac-delta function, and the two Dirac-delta functions are to be understood in their generalized sense. Inserting Eq. (5) into Eq. (2) yields the general integral representation of ideal LWs

*ζ*=

*z*−

*Vt*, and the limits of integration ${\omega}_{m}^{-}$ and ${\omega}_{m}^{+}$ depend on the centroid velocity

*V*, so that the square root in the argument of the Bessel function yields real numbers. Conventionally,

*V*is chosen such that

*V*>

*c*in order to generate large bandwidth wave-packets ( ${\omega}_{m}^{+}\to \infty $) [10].

We should note here that LW solutions with *α _{m}* = 0 are named “X-Wave” solutions, whereas those with

*α*≠ 0 are named “Focus Wave Modes.” In what follows, we limit the analysis to rotationally-symmetric (

_{m}*n*= 0) LWs of the X-Wave family (

*α*= 0). This restriction is only for the sake of clarity of analysis, since each summand term in Eq. (6) constructs a valid LW solution. Moreover, since no angular modulation is introduced in this work, the variation of the wave amplitude in the transverse plane might result in false detection at the receiver end. In the following section, we will derive the propagation-invariant equivalent of the ideal Nyquist pulse using the principles discussed in this section.

_{m}#### 2.2. Nyquist Localized Waves

Ideal Nyquist pulses are Sinc-shaped and spread into adjacent time slots, yet their spectra are perfectly rectangular in shape. Such pulses, by definition, require only the minimum Nyquist channel bandwidth. Though Nyquist pulses are well known in communication theory, they are relatively new in optical communication applications. Nyquist modulation format is the time-domain analogue to optical orthogonal frequency division multiplexing (OFDM), where Sinc-shaped sub-spectra extend into adjacent frequency slots, and symbols in time are rectangular in shape. Hence, Nyquist modulation is considered a form of orthogonal time division multiplexing techniques. We introduce the expression of the Nyquist pulse at the *q*-th time slot *y _{q}*(

*t*) and its Fourier transform

*Y*(

_{q}*ω*) as

*x*) = sin(

*πx*)/(

*πx*), rect(

*x*) =

*u*(

*x*+ 1/2) −

*u*(

*x*− 1/2),

*u*(

*x*) is the Heaviside step function,

*T*=

_{s}*t*

_{q}_{+1}−

*t*is the temporal spacing of symbols, and

_{q}*W*= 2

_{s}*π*/

*T*is the spectral symbol length. Figure 1 depicts three Nyquist pulses in time- and frequency-domains. The time-domain pulses are centered at three consecutive time instances

_{s}*t*

_{q−1},

*t*and

_{q}*t*

_{q+1}, whereas the sum of all three frequency-domain spectra represent the spectral content of a single specific Nyquist symbol centered at

*ω*= 0. The Nyquist pulse expression as given by Eq. (7) is a baseband pulse and generally cannot be directly transmitted over an optical channel. Conventionally, the baseband pulses are used to modulate a single-frequency carrier signal before transmission. On the receiver end, the baseband pulses must be first extracted from the modulated signal (demodulation) then passed on for detection. The modulation process essentially creates a signal that has a wider spectrum compared to the carrier, which is conventionally monochromatic. Here, we would like to emphasize that this spectrum change would necessarily result in distortion, in forms of spread and delays, in the transmitted signal, even if the carrier signal is ideal and propagation-invariant wave, such as Bessel beams. The reason for this distortion can be found in the dispersion relation given by Eq. (4), where the

*k*of the carrier is a single value (a Bessel beam with a certain transverse shape), while

_{ρ}*ω*acquires a wider spectrum. Hence,

*k*must vary with

_{z}*ω*in a non-linear fashion in order to satisfy Eq. (4), resulting in dispersion and delay since all spectral components travel with different phase velocities. While this distortion might not be very evident for narrow band modulation, their effects cannot be neglected for wide band modulation. On the other hand, wide band modulation is more favorable since it generates temporally compact signals, hence higher throughput and transmission rates could be achieved. By introducing the concept of invariant-propagation into Nyquit pulses, a distortion-free Nyquist wave-packet could be generated. First, the Nyquist pulse spectrum is shifted to a higher frequency

*ω*which is analogous to the carrier frequency in conventional modulation schemes. Next, the shifted Nyquist pulse spectrum is introduced in the LW integral representation in Eq. (6). As previously noted, only rotationally-symmetric X-Waves are considered, hence the Nyquist LW integral of the

_{c}*q*-th symbol reads

*A*is the encoding complex amplitude. The Nyquist X-Wave described by Eq. (8) is visualized in Fig. 2 for

_{jq}*ω*= 3 × 10

_{c}^{15}

*rad*,

*W*/

_{s}*ω*= 0.1 and

_{c}*V*= 1.2

*c*. The figure shows that the resulting wave-packet has the expected shape of a Nyquist pulse along the propagation axis. It also shows the transverse spatial localization of the wave-packet and that its spot-size is smaller than 1.5

*T*/

_{s}*V*. Note here that the horizontal axis is

*ζ*-axis, which tracks the centroid of the wave-packet, and emphasizes the propagation-invariant nature of the Nyquist X-Wave. A general representation of an encoded

*N*-long bit stream of Nyquist LWs would take the form

The concept of Nyquist LWs is not limited to the ideal Nyquist pulse. The choice of the Nyquist pulse in this work is for demonstration; however, in practical applications, other types of pulse shapes that satisfy Nyquist inter-symbol interference (ISI) could be employed, such as the raised-cosine or the family of parametric improved Nyquist pulses [21].

It is worth to mention that the flexibility in LW structure facilitates the design of OFDM pulses using the same approach used with Nyquist pulses, except that Sinc-shaped sub-spectra are employed in place of the rectangular spectra in Eq. (8).

## 3. Modulation of Localized Waves

In this section, we present two different techniques for modulating Nyquist LWs for telecommunication applications. The first approach utilizes AMI OOK using an individual LW to encode each data bit, while the second approach introduces a more comprehensive encoding scheme, namely 16-QAM, to transmit 4-bits per symbol. The presented basic modulation techniques are mainly meant to demonstrate the potential of using LWs in practical telecommunication applications; nevertheless, it should be noted that LW modulation techniques are not limited to the ones presented here and more complex techniques could be employed in a similar fashion.

#### 3.1. Alternate mark inversion scheme

Alternate mark inversion is a type of bipolar encoding, which is commonly used on T-carriers. In this coding scheme, a binary ‘1’ is referred to as a “mark,” while a binary ‘0’ is called a ”space.” Conventionally, marks are transmitted as pulses with a voltage that changes its polarity with each mark transmission. Spaces, on the other hand, are transmitted as zero volts. This prevents a significant build-up of DC, as the positive and negative pulses average to zero volts. The alternation of the polarity of the marks also serves to establish synchronization between the transmitter and the receiver, where the extraction of the transmitter clock from the bit stream is relatively easier and cheaper. However, with AMI scheme, long sequences of zeroes could result in no transitions and a loss of synchronization. Data channels thus use pulse-stuffing techniques, where, e.g., the coder inserts an additional mark after every seven consecutive spaces, in order to maintain synchronization. On the decoder side, this extra mark is removed. The pulse-stuffing results in a transmitted bit stream that is nominally longer than the original data by less than 1% on average [22].

In terms of localized optical power transmission, Nyquist LWs, as discussed in the previous section, could be streamed as bits of data. Applying the AMI encoding, for each binary ‘1’ a mark is transmitted and for each binary ‘0’ a space. The marks are encoded as Nyquist LWs with their amplitude polarity prescribed by the polarity of the marks, while the spaces designate no transmission of any optical power. Such modulation scheme is expressed using the generic Nyquist X-Wave bit stream given by Eq. (9), by letting *A _{jq}* = 0∀

*q*≠ 0,

*A*

_{j0}= 0 for a space, and

*A*

_{j0}= ±

*A*for a mark where the sign alters for every consecutive mark and

*A*is a constant. Additionally, at the end of every sequence of seven spaces, an auxiliary mark is inserted to maintain synchronization as discussed in the previous paragraph.

Figure 3 shows a sample 8-bit stream for OOK modulated bits, encoded in unipolar NRZ and in AMI, and the intensities of their corresponding modulated streams of Nyquist LWs. The figure shows that AMI is favorable for maintaining better transverse localization of the transmitted bit stream. Additionally, the AMI encoded bit stream allows for easier synchronization, since the wave amplitude, and consequently the intensity, go down to zero at the end of every bit. A comparison between the two encoding schemes for the modulated LW stream along the optical axis is shown in Fig. 4. The figure shows that both encoding schemes satisfy the Nyquist ISI criterion; however, consecutive ‘1’ bits in the NRZ scheme result in higher peak average power ratio (PAPR), whereas this effect is subdued in AMI and at no additional cost in terms of necessary bandwidth.

#### 3.2. Quadrature amplitude modulation scheme

Quadrature amplitude modulation is a scheme used in analog and in digital modulation alike. In digital QAM, two bit streams are simultaneously transmitted by modulating the amplitudes of two carrier signals, using the amplitude-shift keying (ASK) modulation scheme. The carrier signals are set out of phase with each other by *π*/2 radians, i.e., in quadrature, hence the name of the scheme. The modulated signals are then summed, resulting in a waveform that is a combination of phase-shift keying (PSK) and ASK. QAM schemes are used extensively in digital telecommunication systems, specifically in optical fiber systems as bit rates increase. In QAM, the constellation points are usually arranged in a square grid with equal vertical and horizontal spacing and a power of two number of constellation points. A QAM constellation with 2* ^{M}* constellation points, with

*M*an integer, is capable of transmitting

*M*-bits per symbol; however, the most common schemes are 16-QAM, 64-QAM and 256-QAM. Higher-order constellation schemes are capable of transmitting more bits per symbol at the expense of transmitted power or the susceptibility to noise. Demodulation of QAM signals needs to take into account both the amplitude and the phase of the received signal for correct detection [22]. Figure 5 depicts a Gray-coded (all adjacent symbols differ in only one bit) 16-QAM constellation, which is the most common choice in communication systems. The in-phase vector amplitude is determined by the two most significant bits, while the quadrature vector amplitude is determined by the two least significant bits. The summation of both vectors determines the complex modulation amplitude of the symbol

*A*.

_{jq}Using the expression of the general expression of modulated Nyquist X-Waves given by Eq. (9), the 16-QAM is constructed by assigning every consecutive four bits to one symbol, then letting *A _{jq}* = 0∀

*q*≠ 0, and

*A*

_{j}_{0}determined by the sum of the in-phase and quadrature vectors of the Gary-coded constellation in Fig. 5. For illustration purpose, we will encode the bit stream ‘1101000111100010’ using the 16-QAM scheme. Since the 16-QAM scheme encodes 4-bits per symbol, the 16-bit stream is encoded in the four symbols

*A*

_{j}_{0}= {1 +

*i*, −3 +

*i*, 1 − 3

*i*, −3 − 3

*i*}, where real and imaginary values represent the in-phase and quadrature amplitudes, respectively. The in-phase amplitude, quadrature amplitude, and intensity of the resulting four Nyquist LWs are plotted in Fig. 6. The Nyquist LWs employed have the same spectral characteristics as those employed in the previous sub-section. The figure also shows that the transverse localization could be greatly affected by the content of the transmitted data, i.e., the modulated LW stream might suffer from intensity ‘pulsing’ in the transverse plane in addition to that along the propagation direction. The decoding of QAM modulated Nyquist LWs follows an analogous approach to the standard detection; however, the in-phase and the quadrature components are the real and the imaginary parts of the Nyquist LW since both components are orthogonal to each other [23, 24].

## 4. Discussion

In this section in the light of the aforementioned analysis, we shall discuss a number of issues related to the use of LWs in telecommunication systems. In particular, we will focus on the issues related to performance, reception and detection, and generation and transmission. To start with, in order to evaluate the performance of modulated Nyquist LWs; it is necessary to consider the spatial dimension. However, if only the wave field along the direction of propagation is considered, and particularly in the vicinity of the optical axis, then the performance analysis should not differ from their conventional modulation counterparts. For instance, statistical performance analyses and simplified empirical descriptions for Nyquist Sinc-based pulses in OOK and *M*-ary QAM are given in [25, 26]. The performance analysis for off the axis portion of the wave-packet requires more careful analysis. This is due to the essential coupling between the spectral components, which alters the temporal structure of the wave-packet away from the optical axis. Here, we should also note that such analysis might not be useful or even necessary, since the main purpose of using Nyquist LWs is to facilitate the transmission of highly localized bit streams while avoiding dispersion, delay and cross-talk with the neighboring communication channels.

Next, we look into the issue of reception and detection of modulated LWs. Whereas the reception of OOK Nyquist LWs is straightforward when using an intensity-based discriminator, this detection process is prone to errors due to noise interference; it is however, the highly localized nature of Nyquist LWs and their propagation-invariant nature, which ensures that all the energy of the wave-packet is transmitted along the propagation direction could compensate for the effect of ambient noise. Another advantage of direct optical intensity detection is to avoid the bandwidth limitation of the electronic circuitry components, which are a major limitation for high-speed signal digital-to-analog (DACs) and analog-to-digital converters (ADCs). The reception of a non-spatio-temporally coupled Nyquist pulse (*M*-ary QAM) signal is similar to the reception of a conventional unfiltered NRZ signal. The complex-modulated optical field is down-converted to the baseband using a coherent receiver, and then the electrical signal is sampled by ADCs before being digitally processed. The detection of spatio-temporally coupled Nyquist wave-packets requires additional optical elements to decouple the spectral component of the transverse direction and compensate for the change in the resulting signal spectrum. The decoupled spectrum could then undergo the same detection process with conventional QAM signals.

Although LWs posses many characteristics that pose them as good candidates for carrying data in telecommunication systems, there are several difficulties that should be resolved before they are considered a tangible and competitive solution. One of these difficulties is the lack of LW generating devices. This problem has two aspects; the first one is that ideal LW solutions have infinite energy content and thus are not physically realizable. Nevertheless, one method to overcome this restriction is to truncate the wave field by means of an aperture. In the literature, there exist rather simple analytic expressions to describe the on-axis evolution of some rotationally-symmetric LWs truncated by finite apertures [27]. These truncated wave fields are thus physically realizable; however, they are not capable of propagating undistorted indefinitely compared to their infinite energy analogues. Even though the truncated LWs exhibit deterioration in their propagation characteristics, they still outperform other truncated pulsed wave fields [15].

The second aspect is the difficulty in maintaining the invariant-propagation condition given by Eq. (3) precisely while launching the wave fields. To this end, a few attempts were made on generating LWs in the optical regime with good agreement with theoretical expectations (see Ch. 7 in [10]). However, these experiments required optical elements, such as axicons and prisms, which might not be suitable for integrated manufacturing of optical LW launchers and receivers.

One strategy to tackle the previously mentioned difficulties in generating modulated LWs is to separate the temporal modulation step from the spatial one. A schematic demonstrating a practical LW modulator for FSO applications is shown in Fig. 7. In the shown setup, a digital signal-processing unit generates an electrical digitized Nyquist pulse; to convert the digitized Nyquist pulse to an analog signal, it is fed into a digital-to-analog converter. The analog Nyquist pulse is then used in an electro-optical modulator to modulate an optical signal to generate a Nyquist optical pulse. Finally, the temporally modulated Nyquist optical pulse is inputted into a LW generation setup and spatially modulated to generate a Nyquist LW. It should be noted here that this setup could easily be extended to generate QAM modulated LWs (see [25] for the QAM setup).

Finally, we would like to emphasize that the modulation of spatio-temporally coupled wave-packets is not limited to FSO communication applications. The same techniques outlined in this paper could be carried over to be applied in optical fiber, infrared, radio frequency (RF), and line-of-sight microwave communication systems.

## 5. Conclusion

In this work, we have introduced the concept of propagation-invariant LW modulation for telecommunication applications, specifically for FSO communication systems. These modulation techniques take advantage of the complete spatio-temporal coupling of LWs, thus are sought to outperform current conventional pulse modulation techniques in optical telecommunication systems in general. We have derived expressions for Nyquist LWs, which are the spatio-temporally coupled equivalent of the temporal Nyquist pulses. We have also demonstrated a simplified OOK modulation scheme with AMI coding to transmit a bit stream using Nyquist LWs. We have also introduced a technique for multi-bit per symbol modulation, namely 16-QAM, which is an essential modulation technique for high throughput FSO data networking applications.

Furthermore, we have discussed the various aspects of extending this work for other modulation schemes and their implementation for other optical communication networks such as optical fibers. In addition, we have discussed the limitations of the proposed implementations, such as the infinite energy content and the LW generation techniques, and have given suggestions of how to overcome these limitations.

To the best of our knowledge, no previous attempts were done on modulating spatio-temporally coupled wave-packets; hence the work presented in this paper is preliminary and subject to further development and improvement. Future work in this area could possibly include further investigation of other modulation techniques, such as polarization division multiplexing (PDM), the analysis of the truncation effects on modulated LWs and how to enhance their performance and propagation characteristics, and investigation and implementation of the devices capable of launching and detecting such wave-packets.

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