Optical Signals Using Superposition of Optical Receiver Modes

Lee
Jae Seung

doi:10.3807/COPP.2017.1.4.308

OA학술지
Current Optics and Photonics

Optical Signals Using Superposition of Optical Receiver Modes

DOI : 10.3807/COPP.2017.1.4.308
Author: Lee Jae Seung
Publish: Current Optics and Photonics Volume 1, Issue4, p308~314, 25 Aug 2017

ABSTRACT

A particular optical receiver has its own optical receiver modes (ORMs) determined by its optical and electrical filters. Superposing the ORM waveforms at the transmitter, we can generate a new type of optical signals, called ORM signals. After optical detection, they produce pre-specified voltage waveforms accurately, which is advantageous for digital signal processing. Assuming a Gaussian optical receiver, where the optical and electrical filters are Gaussian, we illustrate various phase-shift keying ORM signals using two ORMs by changing their relative phase. We also illustrate multi-level ORM signal patterns using two or more ORMs.

KEYWORD

Optical receiver modes , Optical receiver , Fiber optics and optical communications

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I. INTRODUCTION

Optical communication is one of the most important solutions for our information oriented society [1, 2]. Its transmission capacity has increased steadily due to many technical advances. Recently, various modulation formats and detection schemes have become increasingly practical [3-9]. One of their driving forces is the digital signal processing (DSP) technique [10, 11]. As the DSP technique flourishes, sending many different optical signals over a given optical channel becomes plausible. For this purpose, it is important to make their voltage waveforms distinguishable from each other after optical detection. However, the optical receiver has a photodetector which is a nonlinear device. So it is not easy to synthesize the received voltage waveform precisely.

In this paper, we use optical receiver modes (ORMs) to address this problem. A typical optical receiver can be regarded as an optical filter, a photodetector, an electrical filter, and a DSP circuit in series. The optical receiver has its own ORMs determined by its optical and electrical filters [12-16]. The optical filter acts as a demultiplexer for wavelength-division multiplexing (WDM) optical channels. The electrical filter suppresses beat noises after optical detection caused by the amplified spontaneous emission (ASE) from optical amplifiers [17]. ORMs were used to analyze direct-detection optical receivers to account for the effects of the ASE [12-16]. Similar receiver modes were also used in earlier times for the analysis of radio receivers [18, 19]. If we transmit a superposition of ORMs, called ORM signal, we can very easily control the voltage waveform before the DSP circuit by changing the amplitude and phase of each ORM. Assuming a Gaussian optical receiver, we illustrate various phase-shift keying (PSK) ORM signals using two ORMs by changing their relative phase. We also illustrate multi-level ORM signal patterns using two or more ORMs.

II. TIME DOMAIN ORMS

The m-^th ORM satisfies the following homogeneous Fred-holm integral equation having a Hermitian kernel K(ω,ω ') [12]:

where ϕ_m(ω) is the m-^th ORM’s mode function in the optical frequency domain having a real eigenvalue λ_m and mode index m (= 0, 1, 2, ...). The Hermitian kernel K(ω,ω') is a product of filter transfer functions such that

where H_o(ω) and H_e(ω) are the transfer functions for the optical and electrical filters, respectively.

The corresponding time-domain mode function is the inverse Fourier transform of ϕ_m(ω)

We normalize the time-domain mode functions as

where δ_mn is the Kronecker delta function. The mode functions in the optical frequency domain satisfy

This choice is slightly different from that in [15].

We find the integral equation for ψ_m(t) from Eq. (1). The transfer functions are Fourier transforms of impulse responses,

where h_o(t) and h_e(t) are the impulse responses of the optical and electrical filters, respectively. Using Eqs. (6) and (7), and the formula for the Dirac delta function,

we obtain the homogeneous Fredholm integral equation in the time domain

where the time-domain kernel K(t, t ') is given as

Since this kernel is real and Hermitian, both ψ_m(t) and its complex conjugate satisfy Eq. (9). Thus we will choose ψ_m(t) to be a real function.

III. TIME-DEPENDENT MODE FUNCTION COEFFICIENT AND CORRELATION FUNCTIONS

Denoting the complex electric field amplitude of the optical channel in the optical frequency domain as ε_in(ω) at the optical filter’s input, the voltage output from the electrical filter is given by [15]

where k is a constant. The mode functions of Eq. (1) form a complete set of orthonormal basis functions. Thus we expand the ε_in(ω) exp(jωt) factor in Eq. (11) as

where V_m(t) is the time-dependent mode function coefficient of the m-^th ORM [16]. Then y(t) is a linear sum of |V_m(t)|²,

From Eq. (12), we can remove the phase shift factor, exp( jωt), by setting t = 0 to obtain

Thus t = 0 can be considered the origin of the ORM signal. Taking the inverse Fourier transform of both sides of Eq. (14), we find the complex electric field amplitude in the time domain, E_in(t), to be

Inserting Eq. (14) into Eq. (12) and using the orthogonal relation Eq. (5), we find the mode function coefficient at any time from their values at t = 0 as follows [16]:

where C_nm(t) is a real correlation function between the n-^th and m-^th modes [16],

IV. OPTICAL SIGNAL GENERATION USING THE ORMS

Using Eq. (16), we expand y(t) as

Denoting

we express y(t) as a linear sum of terms

At the transmitter, for the synthesis of y(t), we set V_m(0) = κ_m exp(jθ_m), where κ_m = |V_m(0)|. Then, we have

where θ_mn = θ_m − θ_n . Changing the mode function coefficients, we obtain various waveforms of y(t) from Eq. (24). We can assign each waveform a unique digital datum whose number of digits increases as the available number of waveforms increases.

V. GAUSSIAN OPTICAL RECEIVER

We present some examples using a Gaussian optical receiver, where the optical and electrical filters are Gaussian [15]. The Gaussian optical receiver has Hermite functions for its mode functions. Thus the ORM signals can be evaluated very efficiently. The optical filter has a Gaussian impulse response

where t_o is the time delay of the optical filter. The electrical filter also has a Gaussian impulse response

where likewise t_e is the filter’s time delay. We assume that these time delays are sufficiently long not to violate the causality condition. The mode functions are as follows:

where H_m(⋅) is the m-^th Hermite polynomial and t_d = t_o + t_e. The parameters a, α, and β are related by α² = a²(1+ q) / (1− q) and β² = a² (1− q² ) / 2q, where q is a positive quantity less than 1 given by

r (= 2α / β) is the 3-dB bandwidth ratio of the optical and electrical filters. Here we have made ψ_m(t) real in contrast to [15]. The recursive relation for the eigenvalues is (λ_m+1)^-1 = q / λ_m, where the inverse of λ₀ is given as

Some expressions of y_mn(t) for the first three modes are given as follows:

where y_c = k / 4πλ₀. From Eq. (19), y(0) is given as

In our analysis, the 3-dB bandwidth of the optical filter is chosen to be 10 GHz. The 3-dB bandwidth of the electrical filter is chosen to be 7 GHz. In Figs. 1(a) and 1(b), we plot ψ_m(t) and |ϕ_m(ω)|² , respectively. The center wavelength of the received optical channel is 1550 nm. The ORMs have broader bandwidths in both time and spectral domains, as m increases.

[FIG. 1.] (a) ψm(t) for m = 0, 1, and 2. (b) | ？m(ω) |2 for m = 0, 1, and 2. The Gaussian optical filter has a 3-dB bandwidth of 10 GHz centered at 1550 nm. The Gaussian electrical filter has a 3-dB bandwidth of 7 GHz.

VI. SINGLE ORM DETECTION

When E_in(t) has only one ORM, E_in(t)= V_m(0) ψ_m(t) , we have from Eq. (24). In Fig. 2, we plot the normalized output voltage, y(t) / y_c, with κ_m = 1, for m = 0, 1, and 2. As the mode index m increases, y(t) spreads out with smaller peak values. y(0) / y_c is q^m which becomes small as m increases. In our case, the q value is 0.175.

[FIG. 2.] Normalized output voltage, y(t) / yc, when Ein(t) = Vm(0) ψm(t) for m = 1, 2, and 3.

VII. IN-PHASE PSK WITH TWO ORMS

When E_in(t) has two optical receiver modes, E_in(t) = V_m(0)ψ_m(t) +V_n(0)ψ_n(t) (m ≠ n), the phase difference between the mode function coefficients, V_m(0) and V_n(0), also becomes important. From Eq. (24), we obtain

The phase term appears as cos θ_mn. Thus in-phase PSK is possible with two ORMs. The y(0) value is related as

which is independent of cos θ_mn.

In Fig. 3, we show some possible PSK signals with κ_m = κ_n =1. For each curve, the corresponding value of cos θ_mn is shown in the figure. Five cos θ_mn values are used in Figs. 3(a), 3(d), and 3(f). For clear sketches, three cos θ_mn values are used in Figs. 3(b), 3(c), and 3(e). If we perform amplitude-shift-keying (ASK) or use higher-order ORMs, we can have more ORM signals.

[FIG. 3.] Normalized output voltage, y(t) / yc, for the in-phase PSK signals using the m-th and n-th order modes with κm = κn =1. (a) m = 1, n = 0. (b) m = 2, n = 0. (c) m = 3, n = 0. (d) m = 2, n = 1. (e) m = 3, n = 1. (f) m = 3, n = 2. For each curve, the corresponding value of cos θmn is shown in the figure.

VIII. MULTI-LEVEL PATTERN GENERATION USING ORMS

As another form of ORM signals, there are multi-level ORM signal patterns, (multi-level patterns, to be brief). For example, when E_in(t) = V₀(0)ψ₀(t) + V₁(0) ψ₁(t), we have three parameters at our disposal, κ₀, κ₁, and cos θ₁₀. Thus we can generate y(t) with pre-specified values at three different times, which will be called time points. Let us assume that y(t) has equally-spaced voltage levels at three time points, t = 0 and ±t_s. In Fig. 4(a), we plot the multilevel patterns having four equally-spaced voltage levels with t_s = 45 ps. A total of 22 multi-level patterns are obtained, including the all-zero one. In Fig. 4(b), the number of multi-level patterns is plotted as a function of t_s. The curve has its maximum at t_s = 45 ps, which corresponds to Fig. 4(a). From now on, we will denote t_M as the t_s value where the pattern number is maximum. In Fig. 5, we show similar results when the level number is increased to eight. Fig. 5(a) shows 236 patterns for t_M = 50 ps, as indicated in Fig. 5(b).

[FIG. 4.] (a) Normalized output voltage, y(t) / yc, for 4-level patterns with three time points. (b) Pattern number versus ts.

[FIG. 5.] (a) Normalized output voltage, y(t) / yc, for 8-level patterns with three time points. (b) Pattern number versus ts.

When E_in(t) = V₀(0)ψ₀(t) + V₁(0)ψ₁(t) + V₂(0)ψ₂(t), we have two more parameters at our disposal, the amplitude and phase of V₂(0). Thus the number of time points can be increased to five and so on. In Figs. 6 and 7, we use the first three and four ORMs, respectively, to obtain four-level patterns. In Fig. 6, the time points are at t = 0, ±t_s, and ±2t_s, where the 79 patterns at t_M = 35 ps are shown in Fig. 6(a). In Fig. 7, the time points are at t = 0, ±t_s, ±2t_s, and ±3t_s, where the 178 patterns at t_M = 32 ps are shown in Fig. 7(a). In our examples, the maximum number of patterns increases as the number of ORMs increases because we can build the patterns more accurately using more ORMs. We have increased the number of time points as the number of ORMs increases. So the t_M value decreases. Also, the width of the pattern number versus t_s distribution decreases as the number of ORMs increases. The number of time points can even be unchanged when the ORM number is increased.

[FIG. 6.] (a) Normalized output voltage, y(t) / yc, for 4-level patterns with five time points. (b) Pattern number versus ts.

[FIG. 7.] (a) Normalized output voltage, y(t) / yc, for 4-level patterns with seven time points. (b) Pattern number versus ts.

IX. DISCUSSION

We have shown that we can synthesize the received voltage waveform using ORM signals. So far, we have neglected the existence of electrical amplifiers within the optical receiver but there may be saturation effects from any electrical amplifiers. Our analysis holds when the electrical amplifiers operate in their linear regimes. Their frequency responses can be absorbed into the frequency response of the electrical filter.

When we use ORM signals, there can be crosstalks between adjacent optical WDM channels, or between adjacent ORM signals of the same optical channel. With our Gaussian optical receiver, the optical channel spacing needs to be around 20 GHz or greater. The time interval between ORM signals needs to be around 200 ps or longer. If we assume the optical channel spacing to be 20 GHz and the time interval of each ORM signal to be 200 ps, the spectral efficiency becomes 1 bit s^-1 Hz^-1 with 2⁴ = 16 different ORM signals. To obtain this value, we may use the 24 PSK signals presented in Fig. 3. A spectral efficiency of 2 bits s^-1 Hz^-1 requires 16 × 16 = 256 different ORM signals. In this case, we may use the multi-level patterns presented in Figs. 4, 6, and 7. If we take into account the crosstalk during the evaluation of mode function coefficients, the time interval between adjacent multi-level patterns can be reduced or removed. In addition, we may use different time point numbers or positions including unequally spaced cases.

The patterns have no zero voltage crossings in our Gaussian optical receiver. This is evident from Eq. (13) and from the fact that all eigenvalues are positive. If we allow finite extinction values for y(t), the number of patterns can be increased. For example, if we use {0.3, 1, 2, 3} levels instead of {0, 1, 2, 3} levels for y(t) / y_c at the given time points, we obtain 232 patterns for the case of Fig. 7(a). The pattern number still has its maximum at t_s = 32 ps.

X. CONCLUSION

A new method to generate optical signals has been introduced using the superposition of ORMs. Changing the amplitude and phase of each ORM, we can synthesize the optical receiver’s voltage waveform before its DSP circuit. Assuming a Gaussian optical receiver, we have shown the optical receiver’s responses to some in-phase PSK signals using two ORMs, where the PSK is performed for the phase difference between the two ORMs. If we do the ASK further, we can increase the number of optical signals. We have also shown multi-level patterns using two or more ORMs. As the ORM number increases, we can assign more time points where the optical receiver’s voltage has pre-specified values.

참고문헌

1. Keiser G. 2000 Optical fiber communications
2. Kaminow I. P., Li T., Willner A. E. 2008 Optical fiber telecommunications V-B, Systems and Networks
3. Armstrong J. 2009 OFDM for optical communications [J. Lightw. Technol.] Vol.27 P.189-204
4. Winzer P. J. 2012 High-spectral-efficiency optical modulation formats [J. Lightw. Technol.] Vol.30 P.3824-3835
5. Ke J. H., Gao Y., Cartledge J. C. 2013 400 Gbit/s single-carrier and 1 Tbit/s three-carrier superchannel signals using dual polarization 16-QAM with look-up table correction and optical pulse shaping [Opt. Express] Vol.22 P.71-83
6. Lyubomirsky I., Ling W. A. 2014 Digital QAM modulation and equalization for high performance 400 GbE data center modules [OFC/NFOEC]
7. Rezania A., Cartledge J. C. 2015 Transmission performance of 448 Gb/s single-carrier and 1.2 Tb/s three-carrier superchannel using dual-polarization 16-QAM with fixed LUT based MAP detection [J. Lightw. Technol.] Vol.33 P.4738-4745
8. Kikuchi K. 2016 Fundamentals of coherent optical fiber communications [J. Lightw. Technol.] Vol.34 P.157-179
9. Zhu C., Song B., Corcoran B., Zhuang L., Lowery A. J. 2015 Improved polarization dependent loss tolerance for polarization multiplexed coherent optical systems by polarization pairwise coding [Opt. Express] Vol.23 P.27434-27447
10. Ip E., Kahn J. M 2007 Digital equalization of chromatic dispersion and polarization mode dispersion [J. Lightwave Technol.] Vol.25 P.2033-2043
11. Roberts K., Beckett D., Boertjes D., Berthold J., Laperle C. 2010 100G and beyond with digital coherent signal processing [IEEE Commun. Mag.] Vol.48 P.62-69
12. Lee J. S., Shim C. S. 1994 Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain [J. Lightw. Technol.] Vol.12 P.1224-1229
13. Forestieri E. 2000 Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and post-detection filtering [J. Lightw. Technol.] Vol.18 P.1493-1503
14. Holzlohner R., Grigoryan V. S., Menyuk C. R., Kath W. L. 2002 Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization [J. Lightw. Technol.] Vol.20 P.389-400
15. Lee J. S., Willner A. E. 2013 Analysis of Gaussian optical receivers [J. Lightwave Technol.] Vol.31 P.2987-2993
16. Seo K. H., Lee J. S., Willner A. E. 2013 Time-dependent analysis of optical receivers using receiver eigenmodes [J. Opt. Soc. Korea] Vol.17 P.305-311
17. Olson N. A. 1989 Lightwave systems with optical amplifiers [J. Lightw. Technol.] Vol.7 P.1071-1082
18. Kac M., Siegert A. J. F. 1947 On the theory of noise in radio receivers with square law detectors [J. Appl. Phys.] Vol.18 P.383-397
19. Emerson R. C. 1953 First probability densities for receivers with square law detectors [J. Appl. Phys.] Vol.24 P.1168-1176

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[ FIG. 1. ] (a) ψm(t) for m = 0, 1, and 2. (b) | ？m(ω) |2 for m = 0, 1, and 2. The Gaussian optical filter has a 3-dB bandwidth of 10 GHz centered at 1550 nm. The Gaussian electrical filter has a 3-dB bandwidth of 7 GHz.
[ FIG. 2. ] Normalized output voltage, y(t) / yc, when Ein(t) = Vm(0) ψm(t) for m = 1, 2, and 3.
[ ]
[ ]
[ FIG. 3. ] Normalized output voltage, y(t) / yc, for the in-phase PSK signals using the m-th and n-th order modes with κm = κn =1. (a) m = 1, n = 0. (b) m = 2, n = 0. (c) m = 3, n = 0. (d) m = 2, n = 1. (e) m = 3, n = 1. (f) m = 3, n = 2. For each curve, the corresponding value of cos θmn is shown in the figure.
[ FIG. 4. ] (a) Normalized output voltage, y(t) / yc, for 4-level patterns with three time points. (b) Pattern number versus ts.
[ FIG. 5. ] (a) Normalized output voltage, y(t) / yc, for 8-level patterns with three time points. (b) Pattern number versus ts.
[ FIG. 6. ] (a) Normalized output voltage, y(t) / yc, for 4-level patterns with five time points. (b) Pattern number versus ts.
[ FIG. 7. ] (a) Normalized output voltage, y(t) / yc, for 4-level patterns with seven time points. (b) Pattern number versus ts.