Multi-dimensional QAM equivalent constellation using coherently coupled orbital angular momentum (OAM) modes in optical communication

Yuan Li; Kaitlyn Morgan; Wenzhe Li; J. Keith Miller; Richard Watkins; Eric G. Johnson

doi:10.1364/OE.26.030969

1. Introduction

In order to meet the exploding data transmission needs of recent years, advanced encoding schemes and multiplexing techniques have been developed and explored in both free-space and optical fiber based communication links. Apart from traditional modulation methods, multidimensional optical signals can be used to increase capacity of optical links. Here, more than one optical dimension such as wavelength, polarization, space, time, and quadrature are modulated in combination [1,2]. This provides a higher degree of freedom that can well exceed that of a traditional single-dimensional communications link [3]. One common example in which an optical signal can be mapped to a three-dimensional (3D) space is the mapping of non-orthogonal polarization states to the Poincaré-sphere using Stokes parameters. It is well known that light can carry angular momentum, not only due to circularly polarized light as spin angular momentum but also orbital angular momentum (OAM) due to a helical phase structure [4]. This helical phase structure can be described by exp(imψ), where m is the azimuthal mode index, or the charge number of the OAM beam and ψ represents the azimuthal spatial angular coordinates. Coherently coupled OAM (CCOAM) beams with charges ± m can be mapped to a 3D space similar to that of a high-order Poincaré sphere (HOPS) for OAM carrying beam and vector beam mapping [5–10].

Using optical OAM beams as data carrier for both free-space and underwater communications [11–14], PAM-4 modulation has been demonstrated in free space, for a transmission of 2 bits per symbol [15,16]; 64-QAM has been demonstrated in free-space communication links for transmission of 6 bits per symbol [17]. Recently we demonstrated a two-dimensional 32-QAM equivalent modulation free-space communication link using two CCOAM beams involving no coherent detection in the receiver end of the system [18]. These two-dimensional modulation scheme is limited in capacity by system noise which caps the maximum possible number of symbols of the system.

In this case, each spherical surface is populated with a discrete number of symbols and the concentric spheres represent the additional symbol dimension. This additional dimension to the symbol space is highlighted by 3D constellations with 64 and 128 symbols in a free space communication link using only two CCOAM beams with charge numbers of m = ± 2. Comparing with 2D mapping, the 3D mapping significantly reduces the BER by increasing the distance between the symbols in 3D space, therefore providing more symbols for an increase in spectral efficiency [19].

2. Encoding

As the basis of this work, the modulation concept of OAM beams is shown in Fig. 1(a) and the actual experimental setup is shown in Fig. 1(b). For two coherent coupled OAM beams with opposite charge number, or azimuthal mode index, the interference results in a petal-like beam with 2|m| distinct lobes with alternating phase. The relative phase delay between the two OAM beams will change the location of the lobes and therefore carrying phase modulation (PM) signals. Furthermore, by independently controlling the amplitude of the two interfering OAM beams, amplitude modulation (AM) can also be introduced into the picture. To describe the arbitrary amplitude and phase information of the CCOAM beams, a HOPS shown in Fig. 1(c) has been introduced [5,6]. For an equivalent HOPS the radius r, elevation angle θ, and phase angle φ determine the arbitrary location representing the state of the complex field of this CCOAM beam S(r, θ, φ). These components r, θ, and φ represent the spherical coordinates of the vector. S₁, S₂ and S₃ are the Stokes parameters defined in [10]. In this work, a discrete set of point locations on the equivalent HOPS are used as symbol locations in a 3D constellation map, with modulation information carried on amplitude and phase of the S(r, θ, φ) function.

Fig. 1 (a), the concept of the interference of OAM beams with opposite spiral phase with both amplitude and phase modulation; (b), the experimental setup of the OAM beams intereference; and (c), the OAM equivalent high order Poincaré sphere (HOPS).

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To implement the above mentioned concept into an optical communication system, a more general expression can be derived for an arbitrary symbol pulse representing the complex field of two co-polarized coherent OAM beams, with a total power, $r_{K} = P_{+ m} + P_{- m} = | U_{1} |^{2} + | U_{2} |^{2}$ , power ratio between one beam and the total power, $α_{N} = \sqrt{P_{+ m} / r_{K}}$ , and a relative phase difference between the two beams, φ_L, expressed as

\begin{matrix} S_{K N L} (r_{K}, α_{N}, φ_{L}, t) = U_{1} (r_{K}, α_{N}, φ_{L}, t) + U_{2} (r_{K}, α_{N}, φ_{L}, t) \\ = \sqrt{r_{K} g (t) α_{N}^{2}} \exp (i m ψ) \exp (- \frac{i φ_{L}}{2} g (t)) \\ + \sqrt{r_{K} g (t) (1 - α_{N}^{2})} \exp (- i m ψ) \exp (\frac{i φ_{L}}{2} g (t)) . \end{matrix}

In this case, g(t) is Gaussian pulse shaping function for the amplitude and phase. Using this expression for a symbol in time, a pulse train with amplitude and phase modulation can be generated by encoding a symbol pulse every T seconds resulting in the following,

p (t) = \sum_{n} S_{K N L} (r_{K}, α_{N}, φ_{L}, t) * δ (t - n T) .

$S_{K N L} (r_{K}, α_{N}, φ_{L}, t)$ is the complex field of the CCOAM optical signals which carry the multidimensional amplitude and phase modulation information encoded by the phase and amplitude modulators using the discrete power levels, discrete power ratios and the discrete phase levels mapped as 3D symbol locations denoted by combination of index K, N and L.

When the complementary power ratio of α_N and $\sqrt{1 - α_{N}^{2}}$ is satisfied such that the total power is constant with r_K, the symbols lie on the Kth sphere. N denotes the number of symbol latitude levels on Kth sphere, or the number of latitude positions of the symbols, changing α_N moves the Nth symbol latitude between pole to pole; and α_N can be decided by the elevation angle θ_N of the symbols as $α_{N} = \cos (θ_{N} / 2)$ and $\sqrt{1 - α_{N}^{2}} = \sin (θ_{N} / 2)$ , where $θ_{N} \in [0, π]$ . θ_N, or α_N is chosen so that the mapped latitudes have equal spacings and not too close to the poles. Experimentally, changing α_N is accomplished by picking the proper working voltage in amplitude modulator AM1; (shown in Fig. 4) and the working voltage on the other amplitude modulator AM2 needs to be chosen properly so that the amplitude ratio relation is satisfied. On the Nth latitude, symbols are discretized by phase location φ_L with equal phase spacing of 2π/L where L is the number of discrete phase positions of the symbols on each latitude. The total phase $θ_{L} \in [0, 2 π]$ is equally split between the two CCOAM states. In experiment, changing φ_L is accomplished by splitting the phase into both phase modulators PM1 and PM2 and choosing the proper working voltages. The function $g (t) = e x p (- t^{2} / 2 τ^{2})$ is the Gaussian shaping function with 30% pulse width τ and pulse period of T for both amplitude and phase modulation signals. In this work, the modulation rate f_s = 1/T is 1.0 GBd.

Thus, through the amplitude and phase control of both U₁ and U₂ through r_K, α_N and φ_L, symbols are mapped into 3D spherical space with K-sphere, N-latitude and L-phase, a total M_KNL-QAM equivalent 3D constellation scheme, where M = KNL. Data comprised of log₂(M) bits can be mapped into M symbols with different spherical radius, latitude and phase positions. Figure 2 shows a simple illustration of a 16-QAM 3D-QAM constellation with K = 2, N = 2 and L = 4. 8 symbols represented by blue color lie on the inner HOPS and the other 8 symbols represented by orange color lie on the outer HOPS with symbol number labled.

Fig. 2 Example 16-QAM constellation with 2-sphere (K = 2), 2-latitude (N = 2) and 4-phase(L = 4) and the corresponding symbol map. The two spheres are separated by radius r, or the total power.

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3. Decoding

To recover the modulation information, the position of the symbols on 3D spherical space must be projected onto the three orthogonal axis set, which are equivalent to the Stokes parameters, S₁, S₂ and S₃ [10]. To be more compatible with the communication concept, the orthogonal parameters used are re-named as I, Q and Z. I and Q channels are similar to the traditional QAM constellation, while Z is another axis introduced for the third dimension. The recovery of both modulation signals utilizes a similar optical correlation setup to our previous work [18]. Optical correlators are commonly used in many mode detection applications [13,14,20] where a phase-matched element is designed to match and cancel the phase profile of the incident beam to form a correlation peak in the detection plane by exploiting the Fourier-transformation property of lenses. In order to recover the 3D information, four detections are necessary: two for amplitude modulation detection and two for phase modulation detection. The on-axis correlation spot power in the Fourier-plane indicates how closely the phase of the incident mode matches that of the phase-matched element. As we have shown in previous work, the phase difference between the CCOAM beams, φ, is recoverable through correlations with two azimuthally offset cosine functions, Ф_I and Ф_Q.

Φ_{I} (ψ) = \cos (m ψ),

Φ_{Q} (ψ) = \cos [m (ψ - π / 4 m)] .

For the detection of the amplitude modulation θ, spiral phase plates are used as the matched filter, given by Ф_-m and Ф_+m.

Φ_{- m} (ψ) = \exp (- i m ψ),

Φ_{+ m} (ψ) = \exp (+ i m ψ) .

The optical correlation integrals of these four phase expressions Eqs. (3)-(6) with the input signal given by Eq. (2) produce four farfield on-axis correlation spots with varying intensity, respectively, given as,

P_{I} (t) = \sum_{n} P_{I} \cdot g (t - n T) = \sum_{n} P_{R X} [1 + \sin (θ) \cos (φ)] \cdot g (t - n T),

P_{Q} (t) = \sum_{n} P_{Q} \cdot g (t - n T) = \sum_{n} P_{R X} [1 + \sin (θ) \sin (φ)] \cdot g (t - n T),

P_{+ m} (t) = \sum_{n} P_{+ m} \cdot g (t - n T) = \sum_{n} \frac{1}{2} P_{R X} (1 + \cos (θ)) \cdot g (t - n T),

P_{- m} (t) = \sum_{n} P_{- m} \cdot g (t - n T) = \sum_{n} \frac{1}{2} P_{R X} (1 - \cos (θ)) \cdot g (t - n T) .

Here, P_RX is the power in the on-axis correlation spot, which is determined by the diffraction limit of the system and is proportional to the radius of the Kth sphere r_K, or the total power in Kth sphere. The optical correlation powers in Eqs. (7)-(10) are detected and sampled at the peak to obtain the intensity information P_I, P_Q, P_-m and P_+m. AC coupling the signals represented by Eqs. (7) and (8) centers the constellation map and produces the I and Q components of the spherical constellation map,

I = \sum_{n} P_{R X} \sin (θ) \cos (φ) \cdot g (t - n T),

Q = \sum_{n} P_{R X} \sin (θ) \sin (φ) \cdot g (t - n T) .

Taking the difference of Eqs. (9) and (10) produces the conversion from spherical coordinates to the Z-component,

Z = \sum_{n} P_{R X} \cos (θ) \cdot g (t - n T) .

Therefore, the modulation signals $S_{K N L} (r_{K}, θ_{N}, φ_{L})$ is mapped from spherical coordinates to the projection on I, Q and Z axis by using four detections. Figure 3 shows a block- diagram of the receiver overview to help better understanding the 3-channel correlation detection.

Fig. 3 Block diagram of receiver setup sampled at t = T_b .

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4. Experiment and results

A schematic diagram of the CCOAM communication system is shown in Fig. 4 and includes two main parts: the transmitter, which generates and controls the dynamic CCOAM modes with signal encoding; and the receiver, which detects the transmitted CCOAM modes and decodes the amplitude and phase modulation information. The transmitter consists of a 1550 nm fiber-to-free-space Mach-Zehnder interferometer (MZI), a polarized single-mode, single-frequency seed laser source (Agilent 8164A) is amplified by an Erbium doped fiber amplifier (EDFA, Nufern NUA-1550-PB-0010-B3) and then split 50:50 by a fiber splitter. Each leg of the MZI includes a fiber coupled amplitude modulator (AM1 and AM2, Thorlabs LN81S-FC) and a fiber coupled phase modulator (PM1 and PM2, Thorlabs LN65S-FC). All fiber components are polarization maintained. The two fiber outputs are collimated and passed through spiral phase plates with OAM charge m = ± 2 and then combined with a polarization independent 50:50 cube beam splitter. Electrical modulation signals at 1.0 GBd are generated by an arbitrary waveform generator (Tektronix, AWG5208) and amplified through an amplifier (Tektronix PSPL5865) before sent to AMs and PMs.

Fig. 4 The experiment setup of the CCOAM communication system and the actual receiver picture.

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The output of the transmitter, which is a quad-lobed interference pattern of ± 2 CCOAM carrying vortices, is propagated in free–space 1.1 m to the receiver. The received beam is split into four measurements of equal power, with each detection leg consisting of a corresponding matched filter, an anti-reflection (AR)-coated Fourier lens with focal length of 150 mm, a 100 μm pinhole and a high-speed photo-detector (Thorlabs DET08CL). The detected signals pass through a DC block and a preamplifier (Fairview Microwave, SLNA-030-32-30-SMA) and then are collected by a real-time oscilloscope (Tektronix TDS7404B) at 5Gs/s sampling rate. For each detection channel, the matched filter is fabricated in house in our cleanroom facility and each phase profile corresponds to Eqs. (3)-(6).

In order to demonstrate the feasibility of this communication link, multidimensional 64- and 128-star QAM maps are used to transmit symbols at 1.0 GBd. MATLAB is used to create a uniform pseudorandom bit sequence (PRBS) using a Mersenne Twister pseudorandom number generator. The experimental results for both 64- and 128-QAM equivalent star constellations are shown in Fig. 5 at the corresponding highest measured SNR level, 16.8 and 17.3 dB, respectively. The symbols of different latitudes are represented by alternating dark and light colors. Sphere radii are represented with blue, orange, or purple colors. The 64-QAM constellation represents K = 2, N = 4, L = 8 and M = KNL = 64. The 128-symbol constellation should ideally represent K = 2, N = 8 and L = 8. However, due to the limited optical power of the transmitter, an alternative with K = 3, N₁ = 2, N₂ = 8, N₃ = 6, L = 8 yielding M = (N₁ + N₂ + N₃)L = 128 symbols is shown. The separation on both Z-axis and azimuthal direction are distinguishable and therefore enables decoding of each symbols. In order to visualize the 3 spheres of the 128-QAM constellation, symbols on each sphere are separated and plotted individually.

Fig. 5 The measured double sphere 64-QAM and triple-sphere 128-QAM 3D star constellations plotted in a normalized optical power scale. For 128-QAM constellation, individual spheres are also shown in different scales.

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The measured symbol error ratio (SER) is converted to a bit error ratio (BER) by assuming gray coding and is plotted against the signal-to-noise ratio (SNR) in Fig. 6. SNR is measured using the modulation error ratio (MER) method,

Fig. 6 The measured BER plotted against measured SNR for 64- and 128- QAM constellations with simulated results.

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S N R = 10 \log_{10} \frac{\sum_{j} ({\bar{I}}_{j}^{2} + {\bar{Q}}_{j}^{2} + {\bar{Z}}_{j}^{2})}{\sum_{j} [{(I_{j} - {\bar{I}}_{j})}^{2} + {(Q_{j} - {\bar{Q}}_{j})}^{2} + {(Z_{j} - {\bar{Z}}_{j})}^{2}]},

where I_j, Q_j and Z_j is the components of the 3D vector of the jth symbol; and ${\bar{I}}_{j}$ , ${\bar{Q}}_{j}$ and ${\bar{Z}}_{j}$ are the ideal location vector components of the jth symbol. It is worth mentioning that all the symbol clusters on the sphere with the same radius have similar SNRs, with a standard deviation of 0.5 dB. Major noise contributions come from the intensities of the individual CCOAM beams. The noise on the azimuthal phase φ direction is caused by a combination of electrical signal noise of the phase modulators and signal temporal alignment timing, fitted to be 21 dB for all the cases. By changing the incident optical power and evaluating MER and BER, the simulation is shown in Fig. 6 as well, which takes into account of noises from the amplitude and phase signals, but not sampling time errors and sampling jitter noises. Comparing the measurement and the simulation, the overall trends for each case fit well.

As show in Fig. 5, the experimental constellation plots do not have perfect symmetry and distributions; multiple parameters could change the symmetry and distort the sphere which could cause additional errors in decoding. One defect is a tilt of the sphere axis, subtle but present in all experimental measured constellations shown in Fig. 5. This is caused by crosstalk between the two detections, P_+m and P_-m, mainly due to pinhole mis-alignment before the photodetector yielding an imperfect optical correlation. In the Z-axis direction, errors will be caused by latitude level distortion, or uneven level spacing due to uneven power between the two OAM modes. As long as the uneven balance of optical powers is minimal, the symbol locations will remain separable in Z-axis and will therefore minimally affect the BER.

Another distortion is azimuthal phase plane ellipticity. As mentioned before, the rotation orientation of the two azimuthal phase-matched elements, Ф_I and Ф_Q, are oriented such that the detected powers form orthogonal I- and Q-channels. When there is a misalignment with respect to the rotation angles of these phase-matched filters, the recovered symbols with different phases will fall into an elliptical projection, visible from the top-down view. In our case the alignment error of the phase plate will be no more than ± 0.5°, which causes a negligible amount of ellipticity. All the above mentioned distortion factors cause recovered 3D constellation sphere distortion, but minimally change the separation between symbols given that the symbol boundaries are adjusted accordingly when decoding. In this work, all these factors are kept within a minimum range that the measured BER is minimally affected. Changing the OAM mode number will require a re-design of the matched phase filters, which reduces the flexibility of the system.

5. Discussion and conclusion

In conclusion, we demonstrated a M-ary 3D constellation modulation with 64- and 128-QAM equivalent modulation using only two CCOAM modes with opposite OAM mode numbers, which corresponds to a spectral efficiency of log₂(M) bit/s/Hz of 6 bit/s/Hz and 7 bit/s/Hz respectively. OAM charge numbers of +/−2 were chosen as a tradeoff of crosstalk between the amplitude detection and phase detection. However, the method can be implemented with different charge numbers of opposite polarity with properly designing the matched phase filters in the detection leg. Comparing with traditional I and Q detection for a 2-channel optical communication link, this method involves a similar number of modulators and detectors without the requirement of a local oscillator. However, the power ratio between the two modes, both on the encoding and decoding side, needs to be well balanced to create/recover a perfect 3D spherical space. This proposed modulation scheme is inspired by an OAM equivalent Poincaré/Bloch sphere and is utilizing 3D space, therefore increases spectral efficiency while not sacrificing BER comparing with our previously demonstrated 32-QAM 2D star constellation modulation scheme. This modulation method can be further implemented in parallel with other multiplexing schemes such as wavelength division multiplexing (WDM) and polarization division multiplexing (PDM) to furthermore increase the link capacity.

Funding

Office of Naval Research (ONR) (N00014-16-1-3090, N00014-17-1-2779)

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Multi-dimensional QAM equivalent constellation using coherently coupled orbital angular momentum (OAM) modes in optical communication

Abstract

1. Introduction

2. Encoding

3. Decoding

4. Experiment and results

5. Discussion and conclusion

Funding

References

Cited By

Figures (6)

Equations (14)

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