Parasitic elements of the equivalent circuit

## Abstract

Radio-over-fiber technology, used in the transport of radio signals over optical fiber by means of an optical carrier between a remote site and a central node of a cellular network, is an attractive solution for backhauling of a large number of remote antennas, enabling the shifting of the hardware complexity from base stations to a central station.

### Keywords

- RoF
- VCSEL
- OFDM
- SC-FDMA

## 1. Introduction

Radio-over-fiber (RoF) technology is used in transporting radio signals over optical fiber by means of an optical carrier between a remote site and a central node of a cellular network. RoF techniques are increasingly seen as a promising solution to facilitate the backhauling of a large number of remote antennas, enabling the shifting of the hardware complexity from base stations to the central station [1]. RoF allows for the combination of the advantages of optical systems such as their high bandwidth and low power consumption, with the advantages of wireless systems, namely the flexibility to use multiple standards such as Long-Term Evolution (LTE) or other upcoming standards.

The number of LTE users has been increasing due to advantages such as the ability to reach a peak throughput of 300 Mbps on the downlink and 75 Mbps on the uplink [2]. To achieve high radio spectral efficiency as well as enable efficient scheduling in time and frequency domains, a multicarrier approach for multiple access was chosen by the 3rd Generation Partnership Project. Orthogonal frequency division multiple access (OFDMA) and single carrier frequency division multiple access (SC-FDMA) were selected for the downlink and uplink, respectively, the latter also known as discrete Fourier transform–spread OFDMA.

## 2. Motivation

When OFDMA or SC-FDMA type of signals are transmitted through directly modulated vertical-cavity surface emitting lasers (VCSELs), they suffer from intermodulation distortion, due to the large number of electrical subcarrier combination in the laser cavity which degrades system performance in addition to relative intensity noise (RIN), clipping noise at the transmitter as well as shot noise and thermal noise at the receiver.

This phenomenon is well known in the literature [3], and is a result of the interaction between the electrons and photons in the active region, which is generally well described by the rate equations [4]. As a motivation for this problem, Figure 1 and Figure 2 show the spectrum of an OFDM electrical signal at the laser output (after conversion by means of a photodiode), for two values of the laser bias current, 3 and 6 mA, respectively. The original OFDM signal is centred at 1 GHz, and directly modulates the VCSEL. The high nonlinear distortion of the signal (with centre at 1 GHz) can be clearly seen, which is caused by intermodulation distortion. Third order intermodulation products (IMPs) of the type,

Figure 3 shows the total number of third-order IMPs as a function of channel number, for 128 subcarriers. For a large number of subcarriers

## 3. Vertical-cavity surface-emitting laser

The VCSEL has emerged as an important class of semiconductor lasers in recent years. Its main characteristics, associated with the vertical-cavity geometry, are light emission perpendicular to the surface of the wafer and single longitudinal mode due to its short cavity length. The VCSEL is a microcavity laser consisting of a thin active region (< 1 µm) sandwiched between epitaxially grown distributed Bragg reflectors (DBRs). Since the first demonstration of the VCSEL laser in Tokyo Institute of Technology in 1979 and after three decades of research, many potential applications have emerged. As a result several VCSEL manufacturers turned up in the market.

Due to their geometry, VCSELs offer a number of significant advantages over edge-emitting lasers listed below:

Low threshold currents: the volume of the active region is relatively small, hence they possess low threshold currents and therefore consume less power than edge-emitting lasers.

Circularly shaped beam: the symmetry in the wafer plane means the laser output is a narrow, low divergence circular beam, permitting high coupling efficiency to optical fibers with relaxed alignment tolerances.

Single-mode operation: due to the microscopic cavity length, VCSELs inherently operate in a single-longitudinal mode which makes them suitable for high-bit-rate fiber optic communications.

Low-cost-wafer fabrication: VCSELs allow for a high packing density in the form of two-dimensional arrays and cost-effective fabrication and testing at the wafer level.

High modulation speed: experimental studies [3] indicate that the VCSELs have very fast intrinsic dynamic properties with relaxation oscillation frequencies as high as 71 GHz.

These advantages make the VCSEL device a suitable candidate for the applications targeted in ROF scenarios. However, the VCSEL as any semiconductor laser exhibits a nonlinear dynamic behavior that needs to be modeled accurately for the assessment of the impact of distortion on system performance. In the next section, the model of the VCSEL based on a set of rate equations for the carrier and photon density is presented. This model will be the basis for the investigation of the laser nonlinear distortion.

### 3.1. VCSEL modeling

The basic structure of the VCSEL is shown in Figure 4. The laser output is taken vertically through one of the mirrors, in contrast to conventional edge-emitting laser which emits light in the plane of the wafer surface. The conventional structure employs an active region consisting of multiple quantum wells between a n-type and a p-type DBR mirrors. Because of the short cavity length (≈ 1 µm) and thickness of the active region, the mirror reflectivities _{x}Ga_{(1-x)}As.

The operation of a VCSEL can be understood by accounting for the rate of recombination of carriers in the active region and the rate of generation and loss of photons. For laser emission to occur, stimulated emission should be the dominant recombination mechanism. The threshold gain is defined as the gain required to sustain the optical field after travelling one round trip in the cavity. Assuming the optical gain is constant over the whole length of the laser, this leads to the condition

where

The first term governs the rate at which the carriers are injected into the active layer due to external pumping;

A suitable form for the carrier recombination rate

where the terms with the coefficients

A corresponding rate equation for the photon density can be obtained from the Maxwell's equations using a classical approach [4, 6]. By a simple bookkeeping of the supply, annihilation, and creation of carriers and photons inside the laser cavity, we get

In which the photon lifetime is defined by

Equation (6) states that the rate of increase in photon density is equal to the photon generation by stimulated emission

The net stimulated rate which tells us how many photons are generated per unit of time per existing photon, yields a generation rate of new photons

The resulting stimulated gain coefficient relates to the stimulated emission coefficient,

where

where

where

Thus, we can rewrite the carrier and photon rate equations as

The evolution of the signal transmitted over the optical fiber requires knowledge of the phase of the electric field. To account for dispersion effects, these equations may be complemented by an additional equation for the phase:

where

The first-order transfer function of the intrinsic laser can be obtained by linearization of the previous equations and is given by the following equation

where for the ease of numerical calculation, the physical quantities have been normalized according to [9] and

### 3.2. Package and chip parasitic elements

When dealing with high-frequency electronics, the frequency limits are usually established by the parasitic elements. It is then required to know whether the laser modulation characteristics are due to the laser alone or due to the parasitic elements. To this aim, one must treat the laser as an electrical element and establish an equivalent circuit that includes the parasitic elements. Characterization of an electrical network at high frequencies is usually done using the scattering parameters.

The elements of the laser-equivalent circuit are derived from the rate equations augmented by the heterojunction voltage-current and space-charge characteristics. The resulting equivalent circuit is a parallel RLC resonant circuit [9, 10]. The carrier density and quasi-Fermi levels are clamped above threshold which manifests in the equivalent circuit as an “ac” short and no voltage can develop. The magnitude of the impedance of the entire circuit

Chip parasitic elements vary widely among different laser structures. In practice, they take the form of a resistance in series with the intrinsic device combined with a shunt capacitance. An equivalent circuit model of the package and chip parasitic elements is shown in Figure 5; it includes a series inductor representing the wirebond, a shunting capacitor representing the contact capacitance, and a series resistor representing the contact resistance and the Bragg mirror stacks.

We now define the following impedances:

The transfer function of the laser parasitic elements corresponding to ratio of the current flowing through the intrinsic laser,

The full laser transfer function is then the product of the intrinsic laser [equation (14)] and parasitic transfer functions (equation (26)).

### 3.3. Laser characterization

In modeling the VCSEL for simulation purposes, it is important to obtain a model as faithful as possible to the real device. This is achieved by extracting the laser parameters from experimental data. The

### 3.4. Extraction of laser parasitic elements

It is possible to determine the input impedance

The measured *P*-*I*) characteristic curve shown in Figure 8.

The laser parasitic elements are then obtained by means of an optimization process that fits the magnitude and phase of the input impedance of the equivalent circuit of the parasitics to the corresponding experimental result after the de-embedding procedure. From the results represented in Figure 9 and Figure 10, it is possible to verify the good approximation between the theoretical model (modeled) obtained using equation (23) and the experimental measurement (de-embedded), up to 7 GHz. Moreover, it is clear the importance of the de-embedding operation to obtain a good estimate of the laser parasitic elements.

The parasitic elements obtained before the optimization process are represented in Table 1:

Element | Value | Element | Value |

The frequency response of the circuit, as defined in equation (26), is shown in Figure 11, where the lowpass characteristic is observed.

### 3.5. Extraction of laser intrinsic parameters

The method employed for the extraction of the laser intrinsic parameters follows the frequency subtraction method described in reference [14]. To that purpose the laser transfer functions at different bias currents are obtained, using the

The experimental results, for different bias currents (above the threshold current), are represented as dashed lines in Figure 12. These results were used to extract the laser intrinsic parameters

It is then possible to fit the corresponding theoretical response to the measured data through an optimization process of the laser parameters. That procedure was applied to the 1550 nm VCSEL, RC33xxx1-F from *RayCan*, using the Optimization Toolbox of MATLAB; the final result is shown in Table 2.

Parameter | Value |

The frequency response of the VCSEL is represented in Figure 12, for different bias currents, including the simulated and experimental results. A good approximation is obtained between the simulated and experimental results.

## 4. OFDM and SC-FDMA over fiber applications

The VCSEL can be directly modulated using signals such as OFDM or SC-FDMA to encode digital data on multiple subcarrier frequencies. OFDM and SC-FDMA are used in applications such as wireless networks and LTE mobile communications. In order to avoid the high peak-to-average-power ratio (PAPR) inherent to OFDM modulation, the LTE standard employs SC-FDMA [16-18], an alternative modulation technique for the uplink with a similar low-complexity. Additionally, OFDM requires highly linear power amplifiers operating with a large backoff from their peak power, which results in low power efficiency [19]. In this context, it becomes pertinent to study the impact of employing SC-FDMA modulation within a RoF system based on directly modulated VCSELs.

### 4.1. Method and setup

The OFDM and SC-FDMA signal generation and demodulation is carried out in MATLAB environment. The experimental RoF setup illustrated in Figure 13 includes the Vector Signal Generator for the generation of the radio frequency (RF) signal that directly modulates the VCSEL, the optical fiber, the optical attenuator, and the optical receiver. The received signal is then sampled with a digital sampling oscilloscope at 20 Gsamples/s for offline demodulation in MATLAB. The validation of the transmitter and receiver blocks was performed in back-to-back configuration, by comparison with the theoretical PAPR results from the literature.

### 4.2. Transmitter

Both OFDM and SC-FDMA signals were modulated with a bit sequence, using three different possible modulation formats: QPSK (Quadrature Phase Shift Keying), 16-QAM (Quadrature Amplitude Modulation) and 64-QAM. Pilot subcarriers were added to the resulting symbols to estimate the effect of channel propagation. A diagram of both transmitters is shown in Figure 14.

In the case of OFDM, the first symbol is used to facilitate time synchronization (finding the first symbol of the signal) at the receiver. This symbol is obtained using the inverse fast Fourier transform (IFFT) of a sequence formed by the real part of a pseudorandom noise sequence at even frequencies and by zeros at odd frequencies, as proposed by Park [20]. After applying zero padding (adding zeros on both sides of the signal spectrum) in order to ease the filtering operation, a N-point IFFT is applied to convert the signal to the time domain. Before the digital-to-analog converter (DAC) where the signal is upsampled, the cyclic prefix (CP) (copy of the last part of the signal) is added to prevent multipath delay. Finally, the signal is upconverted to a RF carrier.

In the case of SC-FDMA, a Zadoff-Chu sequence [21] is used in the LTE standard, which functions as the first SC-FDMA symbol for synchronization purposes at the receiver. After this, a N-point fast Fourier transform (FFT) is performed and the resulting N subcarriers are mapped into M subcarriers using one of two different mapping methods: the interleaved mapping also known as interleaved frequency division multiple access, where the subcarriers are equidistantly distributed over the entire spectrum; and the localized mapping, also called localized frequency division multiple access, where the subcarriers are confined to a fraction of the spectrum. Thereafter, the zero padding and M-point IFFT are applied. Lastly, the CP is added and, as in the case of OFDM, the upsampling and the upconversion operations are performed.

In order to allow the performance comparison between the two modulation formats, only one user is considered.

The In-phase (*I*) and Quadrature (*Q*) components of both modulations formats were loaded to the signal generator, represented in Figure 13, where the frequency and the power of the RF-transmitted signal was specified.

### 4.3. Receiver

As shown in Figure 15, the received OFDM signal is first baseband filtered, to eliminate the noise outside the band, and then downconverted at the same frequency specified on the generator in order to obtain the baseband signal, followed by the lowpass filtering to remove the harmonics generated. Then the downsampling and quantization operations are applied to the signal followed by a temporal synchronization using Park’s method. Then, the inverse of the operations carried out in the OFDM transmitter are performed at the receiver, namely the channel estimation and corresponding equalization in the frequency domain, followed by the removal of both the pilot subcarriers after the FFT and zero padding operations.

In the case of the SC-FDMA receiver, the operations are as follows: first baseband filtering, then temporal synchronization using the cross-correlation between the received signal and the reference signal (Zadoff-Chu sequence), and finally downconversion and downsampling. In order to recover the transmitted symbols, the inverse of the operations carried out at the SC-FDMA transmitter are performed at the receiver, including the channel estimation and the frequency domain equalization, similarly to the case of the OFDM receiver, after the IFFT and the subcarrier demapping.

## 5. System performance assessment

In this section, we present and discuss the relevant results using the signal-to-noise ratio (SNR) as a figure of merit to compare the OFDM and SC-FDMA signals with the same signal parameters.

### 5.1. Signals parameters

The signals generated are characterized by the following parameters: 16-QAM modulation, 10 Mbps, 1 user, 128 subcarriers, 8 pilot subcarriers for channel estimation, ^{st} symbol for synchronization and the others for data) and RF carrier located at 2.4 GHz. On the receiver side, the zero forcing equalization is used after the least squares channel estimation.

### 5.2. Analysis

We now assess the performance of a RoF system and compare the performance of OFDM and SC-FDMA wireless signals over fiber. Theoretically, the optical SNR is given by the following equation [22]:

where the four noise currents are:

The performance of the system is assessed on the basis of SNR or the error vector magnitude (EVM) figures of merit. The EVM expresses the quality of a digital modulated signal and is defined as the difference vector between the measured and the reference signals. Then the SNR can be calculated from the EVM. These figures of merit are calculated as follows:

where

In Figures 16 to 18, the SNR results for OFDM and SC-FDMA transmissions are represented for three different laser bias currents (

where

From the results presented, it is possible to conclude that there is a good matching between the simulation and the experimental results. It is clear from Figures 16 to 18 that for lower RF power, noise is dominant whereas for higher RF power, the intermodulation distortion, introduced by the VCSEL, becomes the limiting performance factor. The SC-FDMA signal is more sensitive to noise then the OFDM signal for lower RF power, while for higher RF power, the SC-FDMA signal is more robust to the intermodulation distortion. Despite this fact, the maximum SNR values attained are identical in both cases, albeit at a higher RF power for the SC-FDMA case, with no clear performance improvement of the SC-FDMA with respect to the OFDM.

## 6. Conclusion

We have assessed the performance of OFDM and SC-FDMA in the context of a RoF scenario, based on a directly modulated VCSEL operating at 1550 nm, and direct detection. To that purpose an accurate theoretical model of the laser was presented, for which device parameters were extracted based on fitting the model to experimental data of frequency response and input impedance. The simulation model fits well the experimental results, and we conclude that the SC-FDMA modulation presents a lower PAPR than the OFDM modulation as expected. Moreover, it is observed that the SC-FDMA is more susceptible to noise, yet it is more immune to intermodulation distortion than the OFDM modulation.

## Acknowledgments

We acknowledge support from Project "NORTE-07-0124-FEDER-000058" financed by the North Portugal Regional Operational Programme (ON.2 - *O Novo Norte*), under the National Strategic Reference Framework (NSRF), through the European Regional Development Fund (ERDF), and by national funds, through the Portuguese funding agency, *Fundação para a Ciência e a Tecnologia* (FCT). This work was carried out with the support of the TEC4SEA research infrastructure (www.tec4sea.com).