© Applied Optics [2016] Optical Society of America.]. One print or electronic copy may be made for personal use only. Systematic reproduction and1 distribution, duplication of any material in this paper for a fee or for commercial Vol. 55, No. 7 / March 2016 / Applied Optics 1786 Research purposes, Article or modifications of the content of this paper are prohibited.
Superluminal, luminal, and subluminal nondiffracting pulses applied to free-space optical systems: theoretical description ROGER L. GARAY-AVENDAÑO1,*
AND
MICHEL ZAMBONI-RACHED1,2
1
Department of Communications, School of Electrical and Computer Engineering, University of Campinas, Campinas, SP, Brazil e-mail:
[email protected] *Corresponding author:
[email protected] 2
Received 12 November 2015; revised 23 January 2016; accepted 25 January 2016; posted 29 January 2016 (Doc. ID 253846); published 1 March 2016
In this paper, we show theoretically nondiffracting pulses with arbitrary peak velocities that are suitable for data signal transmission without distortion over long distances using different techniques of signal modulation. Our results provide closed-form analytical solutions to the wave equation describing superluminal, luminal, and subluminal ideal nondiffracting pulses with frequency spectra commonly used in the field of optical communications. © 2016 Optical Society of America OCIS codes: (060.2605) Free-space optical communication; (060.4080) Modulation; (070.6110) Spatial filtering; (320.5540) Pulse shaping; (070.2615) Frequency filtering; (070.3185) Invariant optical fields. http://dx.doi.org/10.1364/AO.55.001786
1. INTRODUCTION In free-space optical (FSO) communication systems, the theoretical and experimental studies of light propagation in the form of beams or pulses are of obvious importance. In order to improve the efficiency of such systems, nondiffracting waves (also known as localized waves [LWs]) emerge as possible solutions against the undesirable effects caused mainly by environmental factors (rain, fog, pollution, etc.) [1–6]. Such waves, in the finite-energy case, propagate for long distances without distortion and, moreover, can rebuild their spatial form after being partially obstructed [7–10]. A review on the experimental generation of such waves (also known as pseudo-nondiffracting waves) is detailed in [11]. Ideal nondiffracting pulses (INPs) have spectra with specific spatiotemporal couplings and can travel with arbitrary peak velocities. It is well known that, independent of the peak velocity of a nondiffracting pulse, the transmission of energy/ information always occur with luminal or subluminal velocities, V , i.e., 0 < V < ∞ [10,12,13]. The so-called X-wave solutions (X-shaped waves) as well as the focus wave modes are well-known examples of superluminal (V > c) and luminal (V c) INPs, respectively, and both have been extensively studied [14–22]. The case of subluminal (V < c) INPs was studied, for instance, in [23–25]. In a very interesting work, Salem and Bağci [26] demonstrated, through numerical simulations, the possibility of transmitting information by using superluminal INPs with suitable 1559-128X/16/071786-09$15/0$15.00 © 2016 Optical Society of America
digital modulation formats. Such pulses were characterized as having temporal profiles similar to sinc-shaped pulses, known as ideal Nyquist pulses. That allowed for the propagation of several INPs without distortion and with zero intersymbol interference (ISI) within ideal conditions. Although we can use numerical simulations to describe INPs, the analytical description is always preferable in all aspects. Unfortunately, due to the complexity of the integrals involved in obtaining such pulses, these analytical solutions rarely can be found. So far, few methods exist which are capable of providing INPs as exact analytic solutions to wave or Maxwell equations [25,27–32], though none of them focused on the transmission of data. In this paper, we develop a method capable of providing simple closed-form expressions as approximate analytical solutions to the wave equation describing superluminal, luminal, and subluminal INPs (in this work, we only consider ideal nondiffracting pulses propagating in the vacuum) with frequency spectra commonly used in the field of optical communications. The method is based on the use of a new and unusual approximation of the ordinary zero-order Bessel function, which helps in the resolution of the integrals involved in obtaining INPs from suitable frequency spectra. The raised-cosine frequency spectrum is frequently used for pulse-shaping in digital modulation due to its ability to minimize ISI. This spectrum is characterized by a roll-off factor α which allows for a form modification (for example, α 0
Research Article corresponds to a rectangular spectrum form). Our methods are applied here to the cases based on this frequency spectrum, as well as for the Gaussian one, while ensuring that they satisfy the Nyquist ISI criterion. Additionally, we use two modulation techniques to verify the possibility of using the INPs to FSO systems. The first technique is characterized by utilizing an INP to encode each data bit using amplitude-shift keying (ASK) modulation through an on-off keying (OOK) scheme. Here, we use two signal encoding schemes: nonreturn to zero (NRZ) and alternative mark inversion (AMI). The second technique used is more complex because it involves using an INP to encoded multidata symbols per bit. For this, it uses both amplitude and phase modulations [quadrature amplitude modulation (QAM)]. Here, we use encoded data bits using 16-QAM, which is commonly used in digital telecommunication systems. 2. SUBLUMINAL, LUMINAL, AND SUPERLUMINAL IDEAL NONDIFFRACTING PULSES IN FREE SPACE Axially symmetric ideal nondiffracting pulses with peak velocity V are generally represented by a superposition of zero-order Bessel beams [5] with different angular frequencies ω and longitudinal wavenumbers β, both connected by a relation of the type ω V β b, which is a space–time coupling set by the pulse spectrum (with b being a constant). Due to this coupling, it is possible to write such a pulse solution as Z ωmax z dωSω Ψρ; z; t exp −ib V ωmin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ ω × J0 γ −2 ω2 2bω − b2 exp i ζ ; V V (1) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where γ 1∕ V ∕c − 1, ζ z − V t, V is the pulse-peak velocity with 0 ≤ V ≤ ∞, c is the light velocity in vacuum, b is a constant, and Sω is an arbitrary frequency spectrum. Of course, the INP will be subluminal when V < c, luminal when V c, and superluminal when V > c. The positive values of ωmax and ωmin depend on the value of V . Thus, • for subluminal (V < c) INPs: b > 0, ωmin cb∕ c V , and ωmax cb∕c − V ; • for luminal (V c) INPs: b > 0, ωmin b∕2, and ωmax ∞; • for superluminal (V > c) INPs: b ≥ 0, ωmin cb∕ c V , and ωmax ∞; or b < 0, ωmin cb∕c − V , and ωmax ∞. We note that solution Eq. (1) provides true INPs which can propagate without distortion indefinitely. Thus, it would be interesting to consider in Eq. (1) frequency spectra Sω commonly used in optical communications. In these cases, the finite-energy versions of the resulting INPs could be good candidates for data signal transmission over long distances without signal distortion. A pulse commonly used for transmitting digital signals is the Nyquist pulse, which possesses a particular frequency spectrum (frequency response) that allows the encoded data to be trans-
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mitted in a minimum bandwidth, thereby satisfying the Nyquist criterion of zero ISI. When such a frequency spectrum is used in Eq. (1), the resulting pulse cannot, in general, be obtained in analytical form and a numerical evaluation (much less satisfactory) should be considered. In this way, the search for methods capable of providing analytical solutions for Ψρ; z; t in Eq. (1) based on such a frequency spectrum becomes an important task. 3. METHOD: APPROXIMATE INTEGRAL SOLUTIONS TO THE INPs In this section, we are going to obtain approximate versions to the integral solution given by Eq. (1), which will be solved analytically in Section 4 for the three types of INPs (superluminal, luminal, and subluminal) with the frequency spectra used in optical communications. The method presented here depends on a new and unusual but very effective approximation of the ordinary zero-order Bessel function. A. New Approximation to the Zero-Order Bessel Function of the First Kind
Let us start by considering the following well-known approximation to the zero-order Bessel function of the first kind in the case of large arguments [33]: rffiffiffiffiffi 2 π cos r − ; for r ≫ 0: (2) J 0 r ≈ πr 4 For our purposes, however, we need an approximation valid for any value of the argument, and the expression above clearly fails for small values of r, especially for r 0. Of course, we could try to use the power series expansion to J 0 r, but it would be completely inefficient due the oscillatory character of the function. Here we propose the following new and unusual approximation to J 0 r that works very well for any value of r: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi J 0 r ≈ 2 cosr − π∕4∕ πr exp−π − 2r f1 − i expir 1 i exp−irg pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi × 1∕ 2 πr exp−π − 2r ; for r ≥ 0: (3) Figure 1 shows the graphics of the zero-order Bessel function and the approximations given by Eqs. (2) and (3). One can see
Fig. 1. Graphical representation of the zero-order Bessel function (red line) with the approximations given by Eqs. (2) (blue dashed line) and (3) (black dashed line).
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that our approximation, Eq. (3), works very well for any value of r and it even furnishes a correct value to the derivative at r 0. Thus, this approximation does not cause any problem in the integrals in Eq. (1). In the next subsection, we use our approximation to write the integral solution given by Eq. (1) in an approximate (new) form, which will allow the derivation, in Section 4, of highly precise analytical solutions to the wave equation describing superluminal (V > c), luminal (V c), and subluminal (V > c) INPs, with the frequency spectra used in optical communications. B. New Approximate Integral Solution to the Superluminal INPs
Besides using the new approximation p forffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the zero-order Bessel function defined in Eq. (3) with r ρ γ −2 ω2 2bω − b2 ∕V in Eq. (1), we also use the fact that we are only going to consider narrow frequency spectra centeredpon ωc , which allows us ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2 to approximate the function f ω γ ω2 2bω − b2 by its second-order Taylor series around ωc , that is, f ω ≈ f 2 ω c 1 ω2 c 2 ω c 3 with c 1 T 2 f−b2 ∕2f 3c − b2 γ −2 ∕2f 3c g; c 2 b∕f c ωc γ −2 ∕f c T 2 fωc b2 ∕f 3c b2 γ −2 ωc ∕f 3c g; c 3 ≡ f c − b ωc γ −2 ωc ∕f c − T 2 fb2 γ −2 1ω2c ∕2f 3c g; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f c f ωc γ −2 ω2c 2bωc − b2 ; (4) we should note that if T 2 0, then the where T 2 1. Here pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi function f ω γ −2 ω2 2bω − b2 is approximated by its first-order Taylor expansion around ωc , that is, f ω ≈ f 1 ω c 2 ω c 3 . Now, we can get a simpler approximate integral solution by replacing the Bessel function appearing in Eq. (1) with the approximation defined in Eq. (3) with r ρc 1 ω2 c 2 ω c 3 ∕V ρf 2 ∕V . Also, due to the narrow frequency spectrum, we extend the lower and upper limits of the integral in Eq. (1) to −∞ and ∞, respectively; furthermore, we consider
1 exp−ibz∕V Ψρ;z;t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 πρf 2c ∕V exp−π − 2ρf 2c ∕V Z ∞ ρ ω Sωexp i f 2 exp i ζ dω × 1 − i V V −∞ Z ∞ ρ ω Sωexp −i f 2 exp i ζ dω ; 1 i V V −∞ (5) where we recall that f 2 c 1 ω2 c 2 ω c 3 with c 1 , c 2 , and c 3 are given by Eq. (4). In the next section we are going to show examples of analytical superluminal, luminal, and subluminal INPs obtained from some frequency spectra commonly used in telecommunications.
4. APPLICATION TO SOME IMPORTANT SPECTRA: RAISED-COSINE AND GAUSSIAN In order to propose an alternative way for transmitting information in free space using superluminal, luminal, and subluminal nondiffracting pulses, the spectra Sω have to be chosen so that the corresponding time pulse signals can be transmitted satisfying the Nyquist criterion of zero ISI. In [26], this kind of analysis was made numerically in the case of superluminal nondiffracting pulses only. In the following, we are going consider this for the three types of nondiffracting pulses by using a full analytical approach. A. Raised-Cosine Spectrum
A possible representation of a Nyquist pulse defined in the qth time slot is given by yq t
cosπαt − t q ∕T s t − tq sinc ; Ts 1 − 2αt − t q ∕T s 2
(6)
where T s t q1 − t q is the pulse duration between zero crossing, i.e., it is the time interval elapsing between pulses t q and t q1 (see Fig. 2).
8 T s exp−iωt q ; > > < n h i h io π π S 0 ω T2s exp−iωt q − T4is exp i T2αs jωj − 2α − exp −i T2αs jωj − 2α exp−iωt q ; > > : 0;
jωj ≤ 1 − απ∕T s ; 1 − απ∕T s ≤ jωj ≤ 1 απ∕T s ; jωj > 1 απ∕T s ; (7)
the denominator of Eq. (3) with a constant value of ω, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi evaluated at ωc , i.e., πρf 2 ∕V exp−π − 2ρf 2 ∕V ≈ ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi πρf 2c ∕V exp−π − 2ρf 2c ∕V , where f 2c f 2 ωc c 1 ω2c c 2 ωc c 3 . With all this, we can write Eq. (1) approximately as
where 0 ≤ α ≤ 1 is a parameter called the roll-off factor, indicating the excess of bandwidth over the ideal situation (when α 0), and W s 2π1 α∕T s is the bandwidth. The normalized frequency spectrum S 0 ω is shown in Fig. 3(a) for three values of α, namely, 0.0, 0.5, and 1.0. The corresponding temporal functions yt are shown in
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where
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ρ I 1 exp i c 3 fexpia1 W 1 − expia1 W 3 expia1 W 2 V a − expia1 W 4 g 1 fexpq 1 W 2 im1 2q1 − expq 1 W 1 im1 g a − 1 fexpq2 W 2 − im2 − expq2 W 1 − im2 g 2q2 a 1 fexpq 2 W 4 − im3 − expq2 W 3 − im3 g 2q2 a1 fexpq1 W 4 im4 − expq1 W 3 im4 g; − 2q1
Fig. 2. Three time-domain sinc-shaped pulses centered at t q−1 , t q , and t q1 and corresponding frequency spectra (real part).
Fig. 3(b). The frequency spectrum given by Eq. (7) is known as low-frequency or baseband signal because it is centered at ω 0. It cannot be used for transmission over optical channels without first passing through a modulation process, which shifts the baseband signal spectrum to a higher carrier frequency ωc, known as higher-frequency or bandpass signal. Due to this, we make ω → ω − ωc in Eq. (7) and use Sω S 0 ω − ωc in Eq. (5). The result of that will be an expression with hard integrals, which will be resolved considering T 2 0 in Eq. (4), so c 1 0, f 2 f 1 c 2 ω c 3 , and f 2c f 1c (5). This last fact simply means that the funcc 2 ωc c 3 in Eq. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi −2 2 2 tion f ω −γ ω 2bω − b used in Eq. (1) now will be approximate by its first-order Taylor series around ωc , that is, f ω ≈ f 1 ω c 2 ω c 3 , which remains valid for our analysis. Considering this, Eq. (5) can be written as Ψρ; z; t
exp−ibz∕V expiωc t q iT s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 πρf 1c ∕V exp−π − 2ρf 1c ∕V I I × 1 − i 1 1 i 2 ; a1 a2
and
ρ I 2 exp −i c 3 fexpia2 W 1 − expia2 W 3 expia2 W 2 V a − expia2 W 4 g 2 fexpq3 W 2 im2 2q3 − expq 3 W 1 im2 g a − 2 fexpq4 W 2 − im1 − expq4 W 1 − im1 g 2q4 a 2 fexpq4 W 4 − im4 − expq 4 W 3 − im4 g 2q4 a2 fexpq3 W 4 im3 − expq 3 W 3 im3 g; − 2q3
m3 a1 q1 q3
Fig. 3. (a) Raised-cosine spectra and corresponding time-domain pulses (b) for α 0.0, 0.5, and 1.0.
(10)
with Ts π ρ ω − 2α c 2α V T π ρ s ωc − 2α V 2α ζ ρ c2 − t q ; V V T i a1 − s ; 2α T i a2 − s ; 2α
m1 (8)
(9)
Ts π ρ ω − − c ; 2α c 2α V 3 T π ρ c3; m4 s ω c c 3 ; 2α V 2α ζ ρ a2 − c 2 − t q ; V V T q 2 i a1 s ; 2α T q 4 i a2 s ; (11) 2α c3;
m2
and where the integration limits are represented by W 1 ωc − 1 απ∕T s , W 2 ωc − 1 − απ∕T s , W 3 ωc 1 − απ∕T s , and W 4 ωc 1 απ∕T s . Recall that c 2 and c 3 are given by Eq. (4) for T 2 0. The expression Eq. (8), with the Eqs. (9) and (10), represents a closed-form analytic solution to the wave equation and describes superluminal (V > c), luminal (V c), and subluminal (V < c) INPs obtained from the raised-cosine frequency spectrum defined in Eq. (7) centered on ωc . One can use Eq. (8) to get analytical INPs with the above raised-cosine frequency spectrum given by Eq. (7) for different values of α, for example, α 0, 0.5, and 1. To this end, let us choose an angular carrier frequency ωc 1.21 × 1015 rad∕s (λc 1.550 × 10−6 m) and W s 0.001 ωc 1.216 × 1012 rad∕s (193.5 GHz). For the superluminal case, we choose V 1.00000001c and b 0; for the
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Fig. 5. Four time-domain Gaussian-shaped pulses centered at t q , T s , 2T s , and 3T s and the corresponding frequency spectrum (real part) for pulses centered at t q .
(see Fig. 5). Thus, we have defined a Gaussian-shaped signal, for a qth time slot, given by
4 y q t exp − 2 t − t q 2 : Ts
(12)
The Fourier transform S 0 ω of Eq. (12) is Fig. 4. 3D field intensities of (a), (b) superluminal, (c) luminal, and (d) subluminal analytic INPs obtained from Eq. (8) with the raisedcosine spectrum defined in Eq. (7) at ωc 1.21 × 1015 rad∕s with their respective numerical results (right).
luminal case, we choose V c and b 1 × 107 rad∕s; and for the subluminal case, we choose V 0.99999999c and b 2 × 107 rad∕s. The resulting superluminal, luminal, and subluminal INPs are shown in Fig. 4. Note that the superluminal, luminal, and subluminal INPs defined in Eq. (8) using the frequency spectrum Eq. (7) can also be obtained in numerical form from Eq. (1). Generally, however, this results in a loss of the physical meaning of the results or leads to time-consuming simulations. In Fig. 4, we also show the respective numerical results for better comparison. We can see that the longitudinal and transverse behavior agrees with the provisions made above, i.e., these pulses behave as Nyquist pulses in the longitudinal direction and they are propagation-invariant. Moreover, we can see that these pulses are localized in the transverse direction, possessing a spot radius (let us define the radius of the central “spot” of a Bessel beam as the distance, along the transverse direction ρ, at which the first zero occurs for the Bessel function characterizing its transverse shape) of approximately 4.21 mm, 4.63 mm, and 5.21 mm for the superluminal, luminal, and subluminal cases, respectively.
pffiffiffi 1 ω2 π exp − S 0 ω T s exp−iωt q ; 2 W s ∕42 2
(13)
pffiffiffi where W s 8 2∕T s . The frequency spectrum defined in Eq. (13) will be shifted to carrier frequency ωc and will be defined as Sω S 0 ω − ωc for all cases: superluminal (though it is worth noting that superluminal pulses [X-type waves] with Gaussian frequency spectra have been studied by Saari and his collaborators [34–36] through a different approach), luminal, and subluminal. Therefore, using this new spectrum Sω in Eq. (5), we obtain pffiffiffi Ts π exp−ibz∕V pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 πρf 2c ∕V exp−π − 2ρf 2c ∕V
Z ∞ 1 ω − ωc 2 ρ × 1 − i exp − exp if 2 2 W s ∕42 V −∞ ω × exp i ζ dω V
Z ∞ 1 ω − ωc 2 ρ 1 i exp − exp −if 2 2 W s ∕42 V −∞ ω × exp i ζ dω : (14) V
Ψρ; z; t
B. Gaussian Spectrum
Although the Gaussian spectrum does not satisfy the Nyquist criterion when this is defined from −∞ to ∞, we can have approximately zero ISI defining this spectrum in a finite interval
The difficult integrals in Eq. (14) may be evaluated by changing variables, so letting ω W s x ωc , with d ω W s d x and using [33], we get
Research Article 2T exp−ibz∕V expiωc t q expiωc ζ∕V Ψρ; z; t pffiffiffis pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π πρf 2c ∕V exp−π − 2ρf 2c ∕V 2 rffiffiffiffiffi q˘ π × 1 − i expip0 exp 1 p˘ 1 4˘p1 2 rffiffiffiffiffi q˘ π 1 i exp−ip0 exp 2 ; (15) p˘ 2 4˘p2 where p˘ 1 8 − ic 1 ρW 2s ∕V , q˘ 1 iW s ζ 2c 1 ωc ρ c 2 ρ − V t q ∕V , p˘ 2 8 ic 1 ρW 2s ∕V , q˘ 2 iW s ζ − 2c 1 ωc ρ − c 2 ρ − V t q ∕V , and p0 ρf 2c ∕V . Recall that the c 1 , c 2 , and c 3 constants were defined in Eq. (4) with T 2 1. The expression in Eq. (15) represents a closed-form analytic solution to the wave equation and describes superluminal (V > c), luminal (V c), and subluminal (V < c) INPs obtained from the Gaussian frequency spectrum defined in Eq. (13). One can use Eq. (15) to get analytical INPs with the above Gaussian frequency spectrum given by Eq. (13). To this end, let us choose an angular carrier frequency ωc 1.21 × 1015 rad∕s (λc 1.550 × 10−6 m) and W s 0.001ωc 1.216 × 1012 rad∕s (193.5 GHz). For the superluminal case, we choose V 1.00000001c and b 0; for the luminal case, we choose V c and b 1 × 107 rad∕s; and for the subluminal case, we choose V 0.99999999c and b 2 × 107 rad∕s. The resulting superluminal, luminal, and subluminal INPs are shown in Fig. 6 with their respective numerical results for a better comparison. The longitudinal and transverse behaviors illustrated by these plots agree with the provisions made above. The radius of the central spot sizes of the INPs are approximatively 4.21 mm, 4.63 mm, and 5.21 mm for the superluminal, luminal, and subluminal cases, respectively.
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Before moving on, it is worth making some comments about the field depth of the spatially truncated versions of the ideal pulses considered here. Because we are considering narrow spectra (W s ∕ωc ≪ 1), we can say approximately that the field depth of the truncated versions of our pulses will be very similar to that obtained by a truncated Bessel beam withpangular frequency ωc and k z ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ωc ∕V − b∕V (where kρ ω2c ∕c 2 − k2z ). A Bessel beam truncated by a finite aperture of radius R has a field depth, Z , given by Z R∕k ρ ∕k z Rkz Δρ0 ∕2.4, where Δρ0 is the beam spot radius, which be considerably smaller than R, i.e., Δρ0 2.4∕kρ ≪ R. Also, because of that we are working in the paraxial regime, and we can approximate kz to k to get Z RkΔρ0 ∕2.4 R∕2.4Δρ0 z diff , where z diff kΔρ20 approximately represents the diffraction length of a Gaussian beam with the same spot size. Therefore, the pulses here considered can present a field depth R∕2.4Δρ0 times greater than that of a Gaussian pulse with the same spectrum and spot size. In our examples, where λc 1.550 × 10−6 m (k 4.05 × 6 10 m−1 ), if we choose a finite aperture of radius R 3.5 cm, then the superluminal pulse (V 1.00000001c, b 0, and Δρ0 0.42 cm) will have a field depth Z RkΔρ0 ∕2.4 248.29 m, the luminal pulse (V c, b 1 × 107 rad∕s, and Δρ0 0.46 cm) will have a field depth Z 271.93 m, and the subluminal pulse (V 0.99999999c, b 2 × 107 rad∕s, and Δρ0 0.52 cm) will have a field depth Z 307.40 m. Here, it is important to note that for efficient data transmission of our signal, the analytic INPs were obtained for a high carrier frequency ωc. This process is called modulation and the high-frequency carrier signal—for digital modulation—can be modulated through three basic methods: amplitude-shift keying, frequency-shift keying, and phase shift keying. 5. MODULATION OF IDEAL NONDIFFRACTING PULSES
Fig. 6. 3D intensity patterns of (a) superluminal, (b) luminal, and (c) subluminal analytic INPs obtained from Eq. (15) from the Gaussian spectrum defined in Eq. (13) at ωc 1.21 × 1015 rad∕s, with their respective numerical results (right).
In order to demonstrate the possibility of using the analytical INPs obtained in Eqs. (8) and (15) from raised-cosine and Gaussian spectra applied to FSO systems, we will show two different signal modulation techniques that can be used on INPs. The first technique is characterized by utilizing an INP to encode each data bit using ASK modulation. Here, we will use two signal encoding schemes: nonreturn to zero and alternative mark inversion. The second technique is more complex as it uses an INP to encode 4 data bits using both amplitude and phase modulations [quadrature amplitude modulation (QAM)]. Let us begin by assuming that we want to have a transmission of M bits per symbol. Thus, the variable q or t q used above, which indicates the position of the symbolic spectrum, will be zero. This is because we do not wish to transmit several spectra per bit but only one. Then, a representation of an N-long encoded bit stream of INPs defined above could be yt
N X j0
Aj Ψρ; ζ − jV T s ;
(16)
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Fig. 7. Sample 8 bit stream for OOK-modulated bits, encoded in (a) unipolar NRZ and (b) in AMI.
where Aj is a constant which will be defined below. [This expression is a simple version of the one shown in [26], which also takes into account the possibility of having several spectral symbols per bit (t q ≠ 0).] A. Nonreturn to Zero Encoding and Alternative Mark Inversion Schemes
NRZ and AMI encoding refer to a form of digital data transmission in which the high and low binary states, represented
Research Article by the numerals 0 and 1, are transmitted by specific constant direct-current voltages. This scheme type is called unipolar encoding or OOK, which denotes the simplest form of ASK. The constant voltage value is referred to as a “mark” for a binary “1,” while the zero voltage value is referred to as a “space” for a binary “0.” Conventionally, such marks are transmitted as pulses. Moreover, in the AMI scheme we have alternating polarization of the marks, so this scheme type is called bipolar encoding. Applying the AMI and NRZ schemes, the marks are encoded as INPs whose amplitudes are now dependent on Aj through Eq. (16), while the spaces designate no transmission of power. Thus, in Eq. (16), we set Aj A as a “mark” and Aj 0 as a “space,” where A is a positive constant. For the AMI scheme, the sign of A changes for each consecutive mark, i.e., Aj A. As an example, we consider a sample 8 bit stream for OOKmodulated bits, encoded in unipolar NRZ and in AMI (see Fig. 7). Using these scheme types, we show in Fig. 8 the intensity distributions of the superluminal (V 1.0000001) INP stream with their respective projections on the plane (ρ, ζ), obtained from the raised-cosine spectrum with α 0 [Figs. 8(a) and 8(b)] and α 1 [Figs. 8(c) and 8(d)]; and the Gaussian spectrum [Figs. 8(e) and 8(f)] with the following parameters: ωc 1.21 × 1015 rad∕s (λc 1.550 × 10−6 m) and W s ∕ωc 0.001. The figures show that AMI encoding is favorable for maintaining the transverse localization of the transmitted signal. Here, we can see that both schemes satisfy the Nyquist ISI criterion; however, consecutive “1 s” in the NRZ scheme leads to an accumulation of power (higher peak average power ratio), whereas this effect is not present in the AMI scheme. B. Quadrature Amplitude Modulation Scheme
Commonly used in digital telecommunication systems that require high transfer rates of information (such as digital TV),
Fig. 8. Intensity distributions of a superluminal (V > c) analytic INP 8 bit stream encoded in NRZ (left) and AMI (right), which are shown with their projections on the plane (ρ, ζ). Parts (a)–(d) are obtained from the raised-cosine frequency spectrum and parts (e) and (f) from the Gaussian spectrum.
Research Article
Fig. 9. Constellation for gray-coded 16-QAM.
QAM consists of two carriers which are processed simultaneously through two types of modulations: ASK (described above) and phase shift keying (PSK) modulations. The PSK modulation produces signals that are out of phase, π∕2 rad, i.e., in quadrature. Before both signals are transmitted, they are summed, forming a single wave with complex amplitude, in our case, Aj . In Fig. 9, we see a common gray-coded QAM constellation with 24 constellation points capable of transmitting 4 bits per symbol, where the two most significant bits determine in-phase vector amplitude, and the two least significant bits determine quadrature vector amplitude. This is usually used in communication systems and is called a 16-QAM constellation. The demodulation for this case is detailed in [37]. To build a 16-QAM, in Eq. (16), 4 bits are encoded in the amplitude and in the phase difference between consecutive
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symbols (M 4), so Aj will be determined by the sum of the in-phase and quadrature vectors of the gray-coded constellation shown in Fig. 9. As a way to illustrate the application of the method, we are going to encode the bit stream “1101000111100010” using a 16-QAM scheme. Thus, using the gray-coded constellation we get 1101 1 i for j 0, 0001 −3 i for j 1, 1110 1 − 3i for j 2, and 0010 −3 − 3i for j 3, so Aj0;1;2;3 f1 i; −3 i; 1 − 3i; −3 − 3ig, where real and imaginary values represent the in-phase and quadrature amplitudes, respectively. This modulation type was applied to subluminal (V 0.9999999c) [Figs. 10(a)–10(c)], luminal (V c) [Figs. 10(d)–10(f )], and superluminal (V 1.0000001c) [Figs. 10(g)–10(i)] INPs using the Gaussian and raised-cosine (α 0 and 1) spectra, respectively. In Fig. 10, we can see again that the spectra used satisfy the Nyquist ISI criterion; however, consecutive “1” bits in these three cases result in a higher peak average power ratio. In addition, we can see that the in-phase amplitude of superluminal INPs with the raised-cosine (α 1) spectrum is better compared with the others because it possesses more longitudinal spatial localization [see Figs. 10(b), 10(e), and 10(h)]. 6. CONCLUSIONS We have demonstrated analytically that superluminal, luminal, and subluminal INPs characterized by spatiotemporal frequency coupling propagate longitudinally like a Nyquist pulse; therefore, their finite-energy form could be modulated through different modulation schemes over long distances without distortion. Analytic solutions to the wave equation describing superluminal, luminal, and subluminal INPs were obtained from
Fig. 10. Normalized 3D intensity patterns along with their respective orthogonal projections on the (ρ, ζ) plane for the 16-QAM-modulated bit stream “1101000111100010,” using four analytic INPs for each of the following cases: (a) subluminal with Gaussian spectrum, (d) luminal with raised-cosine spectrum (α 0), and (g) superluminal with raised-cosine spectrum (α 1), respectively. Plots (b), (e), (h) are their in-phase amplitudes and (c), (f ), (i) their quadrature amplitudes, respectively.
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frequency spectra commonly used in the field of optics communications. The mathematical method used in this article reduces complex integrals to simple ones providing simple analytical expressions for the description of these pulses that are analytically expressible in a closed form. Moreover, this approach could be used as a new tool for finding new approximate analytical solutions to the wave equation for describing other kinds of nondiffracting waves. In this work, we only considered ideal nondiffracting pulses propagating in vacuum, but future work will be done focusing on the study of nondiffracting pulses with finite energy, capable of resisting diffraction and attenuation effects. Funding. Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) (2014/23854-8). Acknowledgment. The authors thank Mohamed A. Salem for his valuable discussions and kind collaboration.
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