Differential Modulator: Principles and Applications

• Differential modulators are devices that encode information by utilizing the differences between successive symbols, eliminating the need for instantaneous CSI.
• Integrated photonic implementations, such as dual-output Mach–Zehnder modulators on thin-film LiNbO₃, achieve high extinction ratios and low half-wave voltages for efficient broadband operation.
• In wireless and URLLC scenarios, differential modulation supports high mobility and low latency by enabling noncoherent detection and leveraging diversity techniques.

Differential modulators are physical and algorithmic devices or protocols that implement or enable differential modulation—a technique wherein digital information is encoded not in absolute signal states but in differences between consecutive symbol intervals. This approach obviates the requirement for instantaneous channel state information (CSI) at the receiver, making it especially valuable in high-mobility, bandwidth-constrained, or integrated photonics contexts. Differential modulation can be realized in diverse domains: in photonic integrated circuits via dual-output Mach–Zehnder modulators (MZMs), in wireless downlink MIMO settings with unitary space–time codes, and in cooperative protocols using noncoherent detection. Differential modulators play central roles in URLLC, RF photonics, linearized optical links, and cooperative relay networks.

1. Principles of Differential Modulation and Modulator Architectures

In differential modulation, the transmitted symbol $s_k$ is recursively constructed from the previous symbol and the current data symbol, usually as $s_k = s_{k-1} \cdot x_k$ for $x_k$ drawn from a conjugate-symmetric unitary constellation such as M-PSK (Bhatnagar, 2012, Zheng et al., 2023). The receiver recovers $x_k$ based on the relative change in phase or state between $s_{k-1}$ and $s_k$, inherently enabling noncoherent detection insensitive to channel phase shifts or certain hardware drift.

Optically, state-of-the-art differential modulators utilize dual-output MZMs on thin-film lithium niobate (LiNbO₃) integrated with silicon nitride (SiN). Such devices split a continuous-wave input field into two arms, impart a differential phase shift via the Pockels effect, and recombine at a symmetric 2×2 multimode-interference (MMI) coupler. The resultant outputs $I_1$ and $I_2$ correspond to $½ I_0[1 \pm \cos(\Delta\phi(t))]$, directly encoding the differential information (Nelan et al., 2022). Dual parallel architectures (DPD-MZM) can further employ differential photodetection and feed asymmetry to linearize RF-to-optical transfer characteristics, vastly improving intermodulation suppression (Perez et al., 2014).

2. Integrated Photonic Differential Modulators: Principles and Performance

Recent advances in thin-film LiNbO₃ photonics have produced dual-output folded electro-optic MZMs with ultra-high extinction ratios (>45 dB per output) (Nelan et al., 2022). The phase shift is accomplished via traveling-wave electrodes aligned with the LiNbO₃ Pockels tensor maximum ($r_{33}\approx 30$ pm/V), with the transfer function

$I_{1,2}(t) = ½ I_0 [1 \pm \cos(\pi V(t)/V_\pi)],$

where $V_\pi$ is the half-wave voltage ($<3\,\mathrm{V}$ DC, $V_\pi L \approx 3.3\,\mathrm{V}\cdot\mathrm{cm}$). A symmetric 2×2 MMI coupler ensures that when the arms are $\pi$ out-of-phase, all optical power is routed to the complementary output, suppressing leakage to below $-45$ dB without needing active control or cascaded couplers. The folded device layout achieves compactness (7.8 mm), matched arms, single fiber-array coupling, and maintains RF polarity to avoid cancellation.

Table 1: Representative Metrics—Dual-Output Thin-Film LiNbO₃ Modulator (Nelan et al., 2022)

Metric	Value	Note/Significance
Half-wave voltage $V_\pi$	$<$3.0 V (DC)	Enables low-voltage drive
Modulation efficiency	$V_\pi L \approx 3.3\,\mathrm{V}\cdot\mathrm{cm}$	Compact, energy efficient
Extinction ratio (ER)	$>$45 dB	Enables robust differential decoding
Bandwidth	$\sim$30 GHz	Suitable for broadband operation
Insertion loss	$\sim$12 dB	Typical for integrated photonic circuits

This architecture enables direct differential detection, which is essential for advanced modulation formats such as differential phase-shift keying (DPSK) and differential quadrature phase-shift keying (DQPSK), maximizing noise rejection and improving shot-noise-limited SNR by up to 6 dB in balanced photodiode receiver setups.

3. Differential Modulation in RF Photonics and Linearized Modulator Schemes

The dual-parallel differential Mach–Zehnder modulator (DPD-MZM) achieves broadband third-order distortion suppression of up to 20 dB (HD₃ improvement) relative to conventional single MZMs at 5 GHz (Perez et al., 2014). This is accomplished by:

• Splitting both the optical and RF drive powers asymmetrically between two MZMs biased at opposite quadrature points ($\pm V_\pi/2$).
• Utilizing differential photodetection (summed/subtracted currents from two photodiodes).
• Tuning the RF and optical split ratios to cause destructive interference of the third-order intermodulation products.

Key amplitudes for the fundamental and third-order intermodulation tones are governed by their respective Bessel products, with configuration parameters $\{a, B\}$ (optical and RF ratios) chosen to minimize $i_1(2\omega_1-\omega_2)$. This methodology avoids narrowband filtering and is intrinsically broadband.

Table 2: Third-Order Distortion Suppression—DPD-MZM at 5 GHz (Perez et al., 2014)

RF Attenuation B (dB)	Optical Attenuation a (dB)	HD₃ Suppression (dB)
3.0	18.5	37.6
6.0	22.5	43.7

Differential modulator architectures of this type are optimal for analog fiber-optic links demanding large spurious-free dynamic range (SFDR).

4. Algorithmic Differential Modulation: Multiuser MIMO, Cooperative, and Relay Networks

In wireless multiuser MIMO systems, differential modulation is employed with unitary space–time codes (e.g., differential Alamouti) and block diagonalization precoding to eliminate inter-user interference (Alsifiany et al., 2017). The core recursion is

$S_k = S_{k-1} D_k,$

with $S_k$ the differentially encoded block and $D_k$ drawn from a unitary codebook. Differential decoding metrics (maximizing $\Re\{\mathrm{tr}(D R_{k-1}^H R_k)\}$) obviate the need for instantaneous CSI.

Block Diagonalization (BD) precoding uses SVD to construct $P_i$ such that $H_j P_i = 0$ for all $j \neq i$, shifting complexity to the transmitter. Combining BD with differential Alamouti coding preserves full spatial diversity ($M N_r$), at a typical $\sim$3 dB SNR penalty compared to coherent schemes.

In cooperative and relay networks, differential modulation enables decode-and-forward (DF) and amplify-and-forward (AF) strategies without requiring channel knowledge (Bhatnagar, 2012, Avendi, 2014). ML and piecewise-linear decoders have been derived for unitary and nonunitary constellations, with the PL variant achieving near-optimal performance at $O(M)$ computational complexity.

Table 3: Diversity Orders in Differential Relaying (Unitary Constellations) (Bhatnagar, 2012, Avendi, 2014)

Network Type	Diversity Order	SNR Scaling of Error Probability
Single Relay (DF/AF)	2	$O(1/\gamma^2)$
$N$ Relays (DF)	$N+1$	$O(1/\gamma^{N+1})$

For time-varying channels, AR(1) models quantify performance degradation due to Doppler, and multiple-symbol detection can partially recover diversity lost in fast-fading.

5. Differential Modulation for Low-Latency and Ultra-Reliable Communications

Differential modulation protocols are particularly adapted to ultra-reliable low-latency communication (URLLC) scenarios with short packet transmissions, where pilot overhead for channel estimation is prohibitive (Zheng et al., 2023). The principle $s_n = s_{n-1} e^{j\Delta\phi_n}$ enables noncoherent detection and full utilization of transmission blocks for payload. In high-Doppler environments (e.g., normalized autocorrelation $\alpha < 0.9$), differential PSK outperforms pilot-based schemes by $1$–$3$ dB, especially as packet size $n$ decreases.

Selection combining (SC) and maximal-ratio combining (MRC) across $K$ branches enhance diversity gain—effective SNRs scale as

$$\rho_{\text{dif}}^{\text{SC}} = \frac{\alpha^2 \rho \max_k \theta_k}{(1-\alpha^2)\rho + (1+\alpha^2)}, \quad \rho_{\text{dif}}^{\text{MRC}} = \frac{\alpha^2 \rho \sum_k \theta_k}{(1-\alpha^2)\rho + (1+\alpha^2)},$$

with $\theta_k = |h_k|^2$ per branch.

Payload gains exceeding $50\%$ are typical for short DP-MSK packets at target BLER $10^{-5}$ due to pilot-free operation. Differential modulation supports bandwidth and processing savings and is preferred whenever Doppler or latency constraints dominate.

6. Implementation and Fabrication Considerations

In integrated photonics, thin-film LiNbO₃ dual-output modulators are fabricated atop SiN strip waveguides deposited over Si substrates by CIS bonding. Electrodes are formed using 1 μm-thick Au coplanar waveguide structures, with 450 nm SiO₂ buffer to mitigate RF and optical loss (Nelan et al., 2022). Folded layouts allow for compact footprints and matched delay between arms, facilitating scalable packaging. DPD-MZM architectures leverage commercial LiNbO₃ chips, optical/RF attenuators, and precise quadrature biasing. Delay-lines equalize path differences to maintain phase alignment, crucial for broadband linearization.

7. Applications, Advantages, and Trade-Offs

Differential modulators are foundational to:

• Coherent and RF photonic links requiring high SFDR, ultra-low noise, and broad bandwidth (Nelan et al., 2022, Perez et al., 2014).
• Multiuser MIMO and cooperative relay networks needing CSI-free operation, spatial diversity, and complexity shifted to the transmitter (Alsifiany et al., 2017, Bhatnagar, 2012, Avendi, 2014).
• URLLC and high-Doppler regimes in wireless communications, where pilot overhead and estimator latency would otherwise be prohibitive (Zheng et al., 2023).

The principal trade-off is a systematic 3 dB SNR loss in slow-fading environments vs. coherent modulation, with differential architectures regaining this loss through diversity combining, multi-connectivity, and advanced circuit techniques.

A plausible implication is that future differential modulators will continue to combine hardware innovations (e.g., photonic integration, precise electrode patterning) and algorithmic advances (e.g., higher-order differential STBC, multi-symbol noncoherent detection) to meet demands for bandwidth, latency, and reliability across diverse scientific and engineering domains.