Maximum RF Input Signal on the Electro-Optic Modulator Transmission Boundary Area for Harmonic Distortion Compensation in Analog Radio Over Fiber

In this study, we perform an analytical investigation of the electro-optic (EO), Mach-Zehnder modulator (MZM) transmission boundary area with a nonlinear model. We propose a nonlinear model that represents the allowable RF input amplitude within the MZM transmission boundary area. We investigate the maximum RF input amplitude and confine its harmonics using nonlinear model theory, RoF simulation, and MZM characteristic experiments. The maximum RF input amplitude in the MZM transmission boundary area is a convenient technique that can confine harmonics without an optical amplifier, optical filter, or other optical devices in RoF links. We select several RF input amplitudes within the MZM boundary area and account for harmonic distortion in the investigation. The RF input amplitude is 0.8 times of the EO modulator switching voltages. The boundary areas show that near the switching voltage value, the harmonics-to-main RF ratio (HMR) is less than 10%. Furthermore, the maximum RF inputs are still appropriate for QPSK modulation. We evaluate the nonlinear model of these modulation signals in terms of the error vector magnitude and constellation diagram. After considering the optical fiber impairment, thermal noise, and harmonics distortion in the nonlinear model with the modulation signal of QPSK, the ACLR and EVMs satisfy the 3GPP specification. The nonlinear models are appropriate for a small or medium sized of radio-over-fiber links.

for either the optical link or the process after the photodiode. 90 Intermodulation will appear if there are two or more tones 91 at the transmitter [52], [53], [54]. Nonetheless, this condition sensitivity or active area [39], [41] is one solution to decrease 95 harmonic distortion and intermodulation. 96 Adjusting the bias point voltage in the MZM transmission 97 to increase the RF signal or reduce the noise figure is dis-98 cussed in [55]. However, this mechanism requires additional 99 devices to adjust the bias point. The analytical method of 100 the MZM and EO silicon transfer function using the Taylor 101 series [33] aims to explain a nonlinear model in harmonics 102 and the change in its refractive index. The presence of the 103 EDFA optical amplifier in the RoF link still gives less power 104 to the main RF than the second harmonic, even at the initial 105 RF input power [33]. However, this method still uses a phase 106 shifter in one of the MZM arms and gives rise to intermodu-107 lation that needs to be addressed. Due to the dual-wavelength 108 of a light source and the adjustment of the bias point near 109 the minimum transmission point [41], this technique brings 110 more power to the second harmonic than to the main RF at an 111 RF input power of 8 dBm. Another method using linearity 112 enhancement is MZM design [52]; however, the resulting 113 MZM characteristic curve is quite different from the ideal 114 curve, which is a cosine curve. Modeling an MZM distortion 115 using a transfer function with phase change imbalances and 116 fabricating a modulator using the Y branch to control the 117 extinction ratio and chirp is discussed in [56]. This method 118 requires four direct current (DC) biases to linearize the modu-119 lator. Expanding the Taylor series as a transfer function model 120 is discussed in [57]; while this mechanism can minimize the 121 second intermodulation (IMD2), and does not include the 122 fiber optic dispersion. 123 Previous techniques [33], [41], [52], [56], [57] have shown 124 that exploring or designing MZM transmission using the 125 transfer function model can decrease harmonics distortions. 126 Nevertheless, these techniques still need additional devices. 127 Furthermore, there are restrictions on the scope of investiga-128 tion of the MZM transmission area to confine the harmonics 129 without adding more devices to the link. Specifically, to the 130 best of our knowledge, no study has explored the allowable 131 RF input amplitude through the MZM transmission boundary 132 area, which is defined by the analytical and experimental 133 investigation of MZM characteristics. This research is pro-134 posed to confine harmonics by maximizing the RF input 135 power in the MZM transmission boundary area. An analytical 136 investigation using a nonlinear model was proposed, investi-137 gated, and tested. 138 In detail, this study aims to find the allowable maximum 139 RF input signal in the MZM transmission boundary area. 140 The MZM transmission boundary area mechanism is con-141 venient and can be used to confine harmonics in the RoF 142 link without an optical amplifier, optical filter, or other opti-143 cal devices. Therefore, we only need to explore the char-144 acteristics of MZM. The techniques described in this study 145 involve minimal additional devices in the RoF link, a RoF 146 link with simple structure, a lower optical power input, 147 an optimized RF input amplitude to confine second harmonic, 148 and no intermodulation. There is no intermodulation in this 149 experiment because the maximum RF input amplitude in 150 where V m is RF input amplitude, and f m is the RF frequency  The MZM electrical field e, E MZM in (2) [53] is found from 181 the optical input power of the MZM, P in .
where V π is either the MZM half-wave voltage or MZM 184 switching voltage, m and ∅ are the modulation index and 185 the depth-of-intensity modulation respectively, and the phase 186 shift constant can be written in (3) and (4). . This modulator has two electrodes in one arm and no electrode in the others. The modulator has three input signals: the optical carrier input, the RF input, and the DC bias voltage. The optical carrier input will pass through two MZM arms [29] and the output is a modulated signal.
the existence of this RF harmonic signal with (5), which is 198 expanded by the Jacoby-Anger expansion as follows.
We substitute (6) and (7) into (5) where n is the multiple of 202 the harmonics that appear periodically in each multiple of 203 the RF frequency therefore, we can model the electrical field 204 equation output of the MZM in (8).
where J 2n+1 (π m ) and J 2n (π m ) are the n th first kind of 212 the Bessel function. Therefore, if we expand (8) again using 213 (6) to obtain (9), as shown at the bottom of the next page, the 214 electrical field output of the MZM consists of odd harmonics, 215 as shown in (10), as shown at the bottom of the next page, 216 and even harmonic as seen in (11), as shown at the bottom 217 of the next page,. Equation (9) explains the optical spectrum 218 of the harmonics. The distribution of these harmonics is a 219 function of the intensity modulation index as reflected by the 220 parameter m . Therefore, we model the MZM electrical field 221 signal as simplified in (10) and (11), where n = 0 in E even , 222 and we obtain (12) as follows:  signal, which can be expressed in the form of optical power 235 normalization [59]. 236 We determine the MZM transfer function based on the non- If QBP is precisely in the middle of the switch-262 ing voltage, it lies on − V π 2 , 0.5 or V π 2 , 0.5 . The MZM 263 transmission boundary area is explored in (16) to obtain V m 264 as in (16).
The value of V m is the RF input amplitude and has a 267 maximum value of V π . If m is half of the peak-to-peak voltage 268 V pp and is determined by the boundary area, then 0 < 269 m < 0.5. This study uses three values of m; 0.2, 0.3, and 270 0.4. Nevertheless, V m should be less than V π , to minimize the 271 nonlinear distortion. Therefore, we assume that V m is the RF 272 input amplitude and that the boundary areas for V m are 0.7V π , 273 0.8V π and 0.9V π . These three values of m are investigated 274 for the optimized RF input amplitude to limit harmonics.

275
For these RF input amplitudes, we assume there is an RF 276 optimized signal with tolerable harmonics and the absence 277 of an optical filter, or optical amplifier in the RoF link.
In the beginning, we assume that this investigation in the where α loss is the field attenuation coefficient ( dB km ), j β| ω 0 is 321 the propagation constant that represents the group delay per 322 unit length, and ∂β We select three RF input amplitudes and subsequently per-330 form RoF simulation. We use several experimental data 331 parameters in the simulation. The purpose of this simulation 332 is to determine the value of V m in the transmission boundary 333 area; that results in an acceptable number of harmonics after 334 the photodiode output and an RF output power that fits the 335 ITU-T standard. Furthermore, to determine the MZM output 336 power, P MZM and the photodetector output, P RoF , we add 337 noise disturbances that predominantly appear in the RoF link, 338 such as relative intensity noise (RIN) in light sources and 339 thermal noise in photodiodes. RIN can be shown by (19) as 340 discussed [61] where k RIN is 150 dB/Hz. For this simulation, 341 3 dB RIN noise is used and for the thermal noise photodetec-342 tor in (19), the value of 10 −022 W/Hz [62] is selected. Fig. 4 shows the RoF link configuration in the simulation, 346 where the signal generator generates the RF signal. The 347 optical input signal is a continuous wavelength (CW) laser 348 with RIN noise. The end of the RoF link is a PIN photodiode 349 output, which is measured by an electric power meter and an 350 RF spectrum analyzer to explore the RF output with its har-351 monics. Meanwhile, we choose a PIN for the simulation and 352 the experiment that justifies the MZM transmission boundary 353 area condition.

354
The parameters for the CW laser, SD MZM, and signal 355 generator are summarized in Table 1. The photodetector 356 VOLUME 10, 2022  parameters are listed in Table 2. The photodetector output, 357 P RF is observed using a power meter and a dedicated RF spec-  Table 3. 383 We measure the MZM transmission boundary area using a  is P LS , as a result, P LS = P in = |E in | 2 equals the square of 395 the electric field input, with the optical signal electric field 396 input represented as E in = E 0 .E j2πf c t .

397
The other MZM input, DC bias voltage, V b is chosen 398 because the MZM is a transverse modulator and it is measured 399 in volts. In the first experiment, the MZM output power 400 signal, P MZM is measured with an optical power meter (OPM) 401 and the output power, P MZM = P o , is the value measured in 402 the OPM.

403
The second experiment used a signal generator as the 404 RF input and was observed at the PIN output. The results 405 are investigated on the RF spectrum analyzer. The second 406 experiment uses a continuous wave DFB laser that is operated 407 at an output power of −5.8 dBm and wavelength of 1550 nm. 408 The RF input is a sine wave with a frequency of 2 GHz. The 409 MZM is biased at the quadrature point. in (21) when there is no RF signal. Consequently, we need to evaluate whether harmonic dis-435 tortion will occur in the RF output signal. Fig. 7 shows 436 the RF output power in the time domain. The RF output 437 power will remain strong until the RF input amplitude is 438 0.8V π ; afterward, the RF output power will be distorted.

439
If the RF input amplitude is more than 0.9V π , the RF 440 output signal will become more distorted. Moreover, after 441 reaching an RF input amplitude of 1.8V π , the RF output 442 signal changes in terms of its minimum and maximum peaks.

443
The minimum signal, which should be the lowest value, 444 becomes a distorted signal. However, the maximum sig-445 nal shape is distorted by decreasing its peak at 0.001 ns.

446
Considering these conditions, based on

452
We need to evaluate the domain effect of the main RF and 453 harmonics in the simulation based on the RF input amplitude. 454 Optimizing the RF input amplitude will increase the RF 455 output power, nonetheless, when the total harmonics power 456 is larger than the main RF output, the RF output power will 457 decrease. We can see the domain effect of the main RF and 458 harmonics in Fig. 8. The red gradient area within the RF 459 input amplitude of 1.6V π is the domain effect of the main RF. 460 Meanwhile, harmonics become dominant after an RF input 461 amplitude of 1.6V π .

462
For this simulation, the RF output power is observed to 463 obtain the power difference ratio that results from applying 464 unfiltered and filtered signals after the photodiodes in the 465 simulation. We simulate the unfiltered RF output power, P RF , 466 and compare the results to the filtered signal in Fig. 8. The 467 solid red curve is the RF output power before the filter, which 468 is larger than the filtered output signal, especially after the RF 469 input amplitude reaches 2V π volts. The solid blue curve with 470 blue square markers is the RF output power after the LPF.

471
Based on the P RF results before and after passing through 472 the LPF, the black dashed-dotted curve shows the power ratio. 473 This result is shown in Table 4. The increase in RF input 474 amplitude will result in a greater power difference in RF 475 output. However, for the RF input amplitude in the trans-476 mission boundary area at V m ≤ 0.9V π , the power ratio is 477 approximately 0%. In addition, the RF input amplitudes range 478 from V π to 5V π , and the average power ratio is approximately 479 1.7%. When, V m ranges from 5V π to 10V π , the power ratio 480 difference is 8.5%. The most prominent effect of using a filter 481 results in a power ratio of approximately 12% and the corre-482 sponding RF input amplitude is over 10V π volts. Therefore, 483 we choose 0.8V π as the limiting RF input amplitude as it is 484 the value before the harmonics distort the RF main signal, and 485 it results in a power ratio 0%. 486 In Fig. 8, there is an RF input amplitude threshold, 487 V m threshold , which is the limiting value that can be used to 488 design an integrated antenna-EO modulator more than the 489 VOLUME 10, 2022  The minimum RF input threshold is larger than 0.003V π ; 496 therefore, the modulation process will occur. The device 497 design requires a narrow MZM transmission area by using a 498 low switching voltage. As discussed in [63], a low switching 499 voltage or drive voltage can stabilize the modulation process 500 using higher RF input power.  to the main RF power in the form of the total harmonics-527 power-to-main RF ratio (HMR) in percentage.

528
Based on (9), the value of n ranges from −10 to 10 for the 529 harmonics. Afterward, (9) will transform into (22), as shown 530 at the bottom of the next page. Therefore, the HMR can be 531 written as in (23) and (24), as shown at the bottom of the next 532 page. In contrast, we obtain (25), as shown at the bottom of 533 the next page, for the main RF-to-harmonics (MHR) ratio, 534 and we use dB for this parameter. Based on (23), the results are illustrated in Figure 11(a) 537 where the RF input amplitude determines the HMR value. 538 Fig. 10 (a) shows the HMR with RF input amplitude chosen in 539 the analytical investigation of the MZM transmission bound-540 ary area. The ratio reaches 100% at 2.2V π ; and reaches under 541 10% at RF input amplitude within V π as detailed in Table 5. 542 This comparison can be formulated as (25), the main-RF-to 543 harmonics ratio (MHR). Fig. 10 (b) shows that the weaker the 544 RF input power is, the greater the MHR at the initial RF input 545 amplitude.

546
The main RF is dominant until an RF input amplitude of 547 1.6V π ; afterward, the total harmonics power is greater than 548 the main RF. The possibility that the MHR can be infinite is 549 determined by the RF input amplitude. Initially, the harmon-550 ics are very low compared to the main RF. These low har-551 monics occur because the harmonics are approximately zero; 552 therefore, the MHR peaks at 96 dB. Afterward, the MHR 553 decreases with an increase in RF input amplitude because the 554 harmonics begin to increase.

555
According to Table 5, if we want to optimize the RF input 556 amplitude after the MHR peaks, the RF input amplitude 557 must be below V π ; if we use amplitudes from V π to 2V π , 558 the harmonics increase to until 1.6V π . The dominant effect 559 of harmonics is after 1.6V π and the main RF is no longer 560 dominant. We can choose the RF input amplitude to confine 561 the harmonics using the HMR. area, and they occur at RF input powers of V π and 2V π , 574 respectively.

575
3GPP Release 16 confines ACLR of the E-UTRA 576 (Enhanced UMTS terrestrial radio access) at 30 dB. 577 Fig. 11 shows that the nonlinear model can be used for 60 km 578 at an RF input of 0.7, and 55 km at 0.8. For a single-carrier RF 579 input amplitude in (1), without additional optical devices in 580 the RoF link and while, considering only harmonics distortion 581 and fiber attenuation, the nonlinear model matches 3 GPP 582 Release 16. The analog RoF performance is related to the signal-to-noise 586 ratio, SNR RoF . This SNR RoF is the photodiode output, which 587 was compared with the noise in the RoF system. In the last 588 step, we used the nonlinear model to estimate the signal-589 to-noise-harmonic ratio (SNHR) including thermal noise in 590 the photodiode receiver. The noise power can be written 591 as (27)  Based on HMR and MHR modeling in (25) and (26), respec-594 tively, we can estimate the SNHR by comparing the funda-595 mentals of the RF output divided by the total harmonics added 596 by quantum electronics noise in the receiver and, thermal 597 2P in e jω c t 1 + 2e j∅ · · · + −jJ −5 (π m ) e j−5ω RF t + J −4 (π m ) e j−4ω RF t + jJ −3 (π m ) e j−3ω RF t + J −2 (π m ) e j−2ω RF t + −jJ −1 (π m ) e j−ω RF t + J 0 (π m ) + jJ 1 (π m ) e jω RF t + J 2 (π m ) e j2ω RF t + −jJ 3 (π m ) e j3ω RF t + J 4 (π m ) e j4ω RF t + jJ 5 (π m ) e j5ω RF t + . . .

646
In this study, for the nonlinear model, SD-MZM uses two 647 RF frequencies: 2 GHz and 3.5 GHz. The RF signal equation 648 is shown in (31).
The 3 rd IMD is shown in (33), as shown at the bottom of the 654 next page.
655 Fig. 15 shows the three curves, the blue solid curve is the 656 main RF output, the red dashed curve is 2 nd H, and the black 657 dashed-dotted curve is 3 rd IMD. The 2 nd H power is lower 658 than the main RF within 0.8V π of the RF input amplitude.

659
The 0.8V π is the limit of maximum RF input with tolerable 660 2 nd H and 3 rd IMD. For 0.9V π of RF input amplitude, the main 661 give 2 nd IMD suppression, but the 2 nd H remained constant.

674
The 3 rd IMD in this study gives a better fundamental-RF-to-675 third-intermodulation ratio (FIR) of 20 dB than [65].  optical power, as shown in Fig. 16 (a). The second order is 686 the optical signal with second harmonics (f c + 2f RF ) which 687 increases more than the third (f c + 3f RF ), fourth (f c + 4f RF ), 688 and fifth (f c + 5f RF ) harmonics. Fig. 16 (a) shows that the 689 dominant harmonics are second and third harmonics and the 690 SNHR RoF = MHR P harmonic P harmonic +P noptical (29)  the RF input amplitude is 0.8V π , as shown in Fig. 16 (b).

693
Meanwhile, the second harmonic increases by 89% in power 694 when the RF input voltage is 0.9V π , as shown in Fig. 16 (c).  From the RF input amplitude based on the analytical inves-712 tigation in the MZM transmission boundary area, we obtain 713 OCSRs of 2.91 dB for 0.7V π , 2.365 dB for 0.8V π , and 714 1.97 dB for 0.9V π . This OCSR is better than the OCSRs in 715 [19] and [20]; nevertheless, the best OCSR is 1 dB.

717
A previous technique [36] used an EDFA optical amplifier 718 in the RoF link; and gave the main RF power less than the 719 second harmonic at a wavelength of 1552.4 nm, even at the 720 initial RF input power. Moreover, the previous technique did 721 not include fiber dispersion in the link. The transmission 722 boundary area uses no optical amplifier, optical filter, or any 723 other optical devices except a CW laser, SD MZM, and PIN 724 photodiode.

725
Our technique gives better main RF power than the second 726 harmonic from up to 25 dBm of RF input amplitude at a 727 wavelength of 1550 nm. The main RF power and 2 nd H are 728 −83 dBm and −113 dBm, respectively; the main RF remains 729 better than the 2 nd H by approximately 30 dB, while [36], 730 the main RF only remained approximately 20 dB from the 731 2 nd H at a wavelength of 1552.1 nm. Our technique gives 732 better main RF output power than the second harmonic from 733 up to 12.46 dBm of RF input amplitude at a wavelength 734 of 1550 nm. In this study, the main RF power and 2 nd H 735 are −76 dBm and −78 dBm, the main RF remains better 736 than the 2 nd H by approximately 2 dB in the transmission 737 boundary area, while in [33], the main RF only remained 738 lower by approximately 20 dB from the 2 nd H at a wavelength 739 of 1552.4 nm. Moreover, in [41], this technique gave the 740 second harmonic more power than the main RF at an RF input 741 power of 10 dBm due to the increase in the 2 nd H. 743 We proposed an analytical investigation and experiment of 744 the MZM transmission boundary area using nonlinear mod-745 els. This technique is an alternative solution to improve the 746 RF input in an integrated antenna-EO modulator, by using 747 the maximum allowable RF input amplitude and by limiting 748 the harmonics. We experimentally measured the MZM trans-749 mission area and theoretically investigated the allowable RF 750 input on an analog radio over fiber link. We proposed a non-751 linear model to obtain the RF output signal and its harmonics 752