1. Introduction

In accordance with the dramatic success and explosive demand for wireless communication systems based on microwave technology, there have been rapid developments in high-speed, radio frequency engineering over the past few decades. This continuously rising growth in the information and communication market is requiring even more rapid development of high-speed microwave circuits and high-frequency antennas to implement sophisticated transmitter/receiver systems that will be needed to meet the global demands for wireless markets. The functional information bandwidths of the wireless systems have been extended to the millimeter-wave regime to utilize the technically challenging higher frequency bands and to accommodate communication channels that are growing wider.

For the higher frequency applications, microwave and millimeter wave circuits and antennas (as well as arrays of their antennas) are becoming more compact and complicated, and thus accurate measurement methods have become increasingly important. Near-field electrical measurement and diagnostic techniques based on electro-optic (EO) sensing have served as powerful tools in the design and analysis of high frequency antennas or circuits, where complete field-pattern and polarization characterization are beneficial. Near-field information measured via these EO methods in proximity to guided-wave or radiating devices has been demonstrated to provide highly detailed electromagnetic field distributions [1].

To sense radio frequency (RF) fields with shorter wavelengths from the device under test (DUT), down-mixing techniques have been widely used [1–3]. As the signal frequencies increase, the local oscillator (LO) frequency ideally remains comparable to the RF in order to yield a reasonably low (typically KHz-MHz) intermediate frequency (IF). One of the main challenges in high-frequency EO sensing with the photonic-heterodyne technique is the creation of the high-frequency LO modulation sideband of the optical-carrier probe beam. Although millimeter-wave light modulation with a commercial external electro-optic modulator (EOM) is available, the system cost for the required bandwidth, including the need for millimeter-wave sources with which to drive the EOM, is still arguably large.

To address this issue, we utilized a method to enhance the LO bandwidth by generating the higher even-order-harmonic sidebands created within a series configuration of EOMs within an EO-sensing system. The efficiency of the higher-order harmonic components is demonstrated to be significantly increased by cascading two EOMs with a suitable optical gating method. The evolution of higher-even-order sidebands are investigated by analytically exploring the nonlinearity of each harmonic component. The practical threshold driving power level and rising slope of each nonlinear harmonic for EO sensing applications are presented. Near-electric field mapping results of a K-band antenna, using the fourth- and sixth- order harmonic sidebands of the optical-sideband LO, are also presented.

2. Principle and experiment

The generation of nonlinear, higher-order harmonics is a generally well-known effect [4–8]. Although such nonlinearities in an EOM are typically considered to be detrimental in optical communications, where the modulated light carries information, the nonlinear harmonics could, on the other hand, be beneficial for increasing the bandwidth of LO light modulation [4–8]. The even-order harmonics can be readily achieved by setting the DC bias of an EOM at its null transmission point so that the envelope of the transmitted light (i.e., the light-amplitude modulation) from the EOM (such as at point b in Fig. 1) is expressed as in Eq. (1).

where b₀=0.5(1-J₀(a)), b_2k=2J_2k(a) (k=natural number), and J(a) is the Bessel function of the first order. The relative strength of a sinusoidal drive-voltage input at a frequency of LO/4 (=V) divided by the half-wavelength voltage (V_π) of the modulator yields the quantity a (≡πV/V_π), which determines the values of the even-order harmonics [6].

 

Fig. 1. Cascaded EOMs for creating fourth-order harmonic (4×LO/4) modulations. (PC: polarization controller, a, b and c are reference points to be investigated)

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Compared to the phase of the sinusoidal input that drives EOM1, there is an additional phase delay, φ_b, mainly due to the pig-tailed optical-fiber path to point b. The efficient second-order harmonic generation (i.e., a strong b₂) and its applications to EO sensing have been reported recently to double the bandwidth [7,8] of the photonic-heterodyne EO sensing. Ideally, the harmonic sidebands (i.e., b_2k) could be enhanced further by mainly increasing the LO driving levels to boost the nonlinearity. Such nonlinear harmonic coefficients for higher driving voltages are presented in Fig. 2.

 

Fig. 2. Relative strength of even-order-harmonic coefficients at point b in Fig. 1 (i.e., b_2k). The dotted, solid, dashed, and dot-dashed lines for k=0, 1, 2, and 3, respectively.

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However, even driving the LO level up to the damage threshold of an EOM does not provide efficient fourth-order harmonics (i.e., b₄≪b₂). Such a limited nonlinearity for fourth-order harmonic generation in a single EOM can be overcome by cascading a second EOM [4,5], which imposes another identical amplitude-modulation envelope onto the output of the first EOM. Superimposing the two envelopes makes the second-order harmonic evolve to an efficient fourth-order harmonic that significantly increases the sensing bandwidth of the EOS system, even while employing microwave hardware having a bandwidth that is four times lower than if a single EOM was driven at the necessary fundamental frequency.

The output after the second EOM in Fig. 1 is expressed in Eq. (2).

where c_2k is the even-order coefficient, which can be re-expressed in terms of multiple b_2k coefficients, and φ_c is an additional phase delay at point c. To obtain higher coefficients c_2k, the two identical modulation envelopes of the EOMs need to be overlapped in-phase or quadrature-phase. Such condition (i.e., φ_b=0° or ±90°) can be realized by adjusting the fiber length between the EOMs or the electrical cable length connected to EOM2 [4,5]. As opposed to φ_b, φ_c does not impact the amplitude modulation or the EO sensing quality throughout the system.

The full expression of c_2k, in terms of b_2k, would be quite complex. However, since the sixth-order harmonic component of a single EOM is typically negligible (i.e., b₆≈0) as observed in Fig. 2, the c_2k up to the k=4 (=eighth) order can be expanded as c₀=b²₀+b²₂/2+b²₄/2, c₂=2b₀b₂+b₀b₄, c₄=2b₀b₄+b²₂/2, c₆=b₀b₄, and c₈=b²₄/2, respectively.

 

Fig. 3. Relative 5 strength of even-order-harmonic coefficient at point c in Fig. 1 (i.e., c_2k). The dotted, solid, dashed, dot-dash, and double dot-dash lines are for k=0, 1, 2, 3, and 4, respectively.

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Typical fiber-pigtailed EOMs have 3–4 dB insertion loss and different modulation responses versus driving frequency. Here, we assumed ideal cascaded EOMs (i.e., no insertion loss and uniform frequency response). The relative scales of b_2k and c_2k are compared in Fig. 4 on a log scale. The EO-modulation harmonics are plotted over 40 dB, which is the typical EO dynamic range in our system. The signal threshold points, notated with b^thr_2k. and c^thr_2k., and their relative harmonic slopes are computed as shown in Fig. 4. These relative threshold and slope comparisons among the harmonic components indicate the required driving power levels and harmonic generation efficiencies, respectively.

 

Fig. 4. Normalized strength of even-order-harmonic coefficients versus log scale input driving power. The style and color of the plots in Figs. 2 and 3 are reused accordingly.

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Here, one of the most valuable operating points for the work of this paper would be b^thr.₂≪c^thr.₂<c^thr.₄≪b^thr.₄. This means, for generating the second-order harmonic, the use of a single EOM is more effective (i.e.,b₂≪c₂), while adding the second EOM makes the fourth-order harmonics much more effective (i.e., c₄≪b₄). The feasibility of higher-order harmonics is determined by the damage threshold level of an EOM. The level is ~27 dBm for our modulator (a JDS Uniphase OC-192 device), and the position of the vertical threshold line was extrapolated from the experimental b₂ plot in our previous publication [8].

One interesting result is that c₂ and c₄ have thresholds and rising slopes that are closely spaced, which is very different from the single EOM case. The contrast between c₂ and c₄ is less than 6 dB. Thus, the modulation output of the second EOM contains a strong combination of the first two even-order harmonics. The experimental survey of this condition can be observed through the spectra in Fig. 5. The relative strength between the carrier and its two sidebands regulates the second-order harmonic, while the beating between the sidebands controls the fourth-order harmonic component. The overall spectral shape is maintained in general with the LO/4 drive-power levels because of the flat ratio of c₂ and c₄.

 

Fig. 5. Attenuated modulation spectra at point c in Fig. 1 (LO/4=4.633 GHz, driving power=9, 12, 15, 18, 21, and 24 dBm, starting from the bottom spectrum, respectively).

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Fig. 6. Experimental schematic of the entirely fiber-coupled, photonic down-conversion EO sensing system using modulated cw light [3]. (DFB LD: distributed feedback, continuous-wave laser diode; PC: polarization controller; PD: photo detector; OSA: optical spectrum analyzer). The gray and black lines are optical fibers and electrical connections, respectively.

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To demonstrate the utility of the fourth-order-harmonic carrier sideband in photonic-heterodyne EO sensing, we employed the same methodology for near-field mapping of a K-band antenna as reported previously [8]. The single EOM in the previous system could yield up to the third-order harmonic of the drive signal, and the corresponding evolution of the field mapping data was demonstrated. Since the second-order coefficient can be significantly increased from b₂ to c₂ by the use of the second EOM, the creation of LO harmonics with the cascaded EOMs can even result in the fourth-order-harmonic generation.

 

Fig. 7. Evolution of the transverse EO near-field distribution (vertical polarization) from the K-band patch antenna for different modulator-drive powers (listed along the bottom).

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The transverse electric field parallel to the vertical edge of the K-band patch antenna in Fig. 7, also considered in our work of Ref. 8 when using the second-order harmonic of the EOM driving frequency, is measured here with the fourth-order harmonics of the cascaded EOMs. The evolution of the transverse near-electric field imaging scans from the antenna is presented in Fig. 7, where the modulator-drive input power to the cascaded EOMs has been increased in 3 dB increments. The EO measurement of the antenna used 18.532 GHz for the LO-modulation optical-probe-beam sidebands so that mixing could take place in the EO sensor to create an easily resolved low-frequency replica of the K-band output. This frequency was obtained using an EOM-drive frequency of 4.633 GHz (=LO/4). Fig. 8 shows the rapid evolution of the strength of the field-mapping EO signal with respect to the LO/4 and LO/6 drive power to the EOMs. The threshold LO/4 drive power required to observe an EO signal was ~8 dBm. After this, the EO signal increased rapidly at 3.57 dB per 1 dB LO/4 until it reached the 1 dB suppression point. This saturation actually indicated power transfer to the next higher even-order (i.e., sixth) mode, which grew more rapidly than the fourth order after exhibiting a much higher threshold level. Adding a third EOM could make these threshold levels lower.

The stability of the phase delay is a critical issue in optical communication applications where long distances and durations are of concern. Whereas, for EO sensing applications, the LO modulation associated with the phase delay between the EOMs was stable enough due to the system compactness. The overall fiber-path length (gray lines in Fig. 6) is less than 3 meters, with the last 1 meter being the EO sensor part of the assembly. Each field-mapping scan in Fig. 7 was obtained in approximately one hour in order to achieve fine spatial resolution, and the phase was observed to be very stable over that time.

 

Fig. 8 Evolution of the peak EO signal levels in Fig. 7 versus driving powers.

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The fourth-order harmonic-based EO signal measured when using a single EOM is poor due both to its high threshold and low signal levels, as well as to its high noise floor. The latter is mainly due to the laser intensity noise which is proportional to the laser power. The noise level decreases after the second EOM because of its insertion loss. In other words, the result in Fig. 8 indicates that the second EOM significantly enhances the higher-order modulation per unit beam power. The optical carrier component f_c degrades the optical modulation depth and should be suppressed for higher even order modulation. Based on the spectra in Fig. 5, the optical power portion of f_c ranges from about one third to one quarter of the spectra, and this contributes to the second harmonic rather than the fourth. This carrier component can be further suppressed using a Bragg grating filter or by adjusting optical or electrical delays to the second EOM. Thus, the EO or LO modulation depth per unit optical power could be enhanced if desired.

Figure 9 shows the modulated power level at point c and the ratio of this power to that of the input beam, both versus the EOM drive power. Operating an EOM at a null DC bias basically creates a frequency-doubled temporal envelope over the cw input beam with an insertion loss. Then, cascading the second EOM sharpens the envelope of the amplitude-modulated beam by superimposing the same envelope over the first EOM output with a proper timing delay. The distributed-feedback Bragg laser diode delivers 50 mW of power to the EOMs, while the output of the EOMs is less than 4% of the input due to the cascading insertion losses. Although the output power is small, it has strong higher-even-order harmonics for its power level. The cascaded insertion loss and relatively low conversion efficiency for higher harmonic orders could be readily compensated through an erbium doped fiber amplifier. Furthermore, common mode laser noise can be suppressed using a balanced detection scheme. Thus, while an enhanced signal-to-noise ratio (SNR) could be realized, if desired [9], the SNR achieved in our experimental system was more than adequate to produce reasonable electric-field maps.

 

Fig. 9. Evolution of the modulated power with increasing EOM input power.

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As the nonlinearity of cw light modulation driven with the fundamental modulation frequency f_m is further increased, and more energy is dedicated to its higher order harmonics, the modulation eventually evolves to that of short-pulsed laser operation with a pulse-repetition rate f_m. The pulse separation (=1/f_m) and width determine the fundamental harmonic and number of effective higher harmonic orders. An extreme case is a near-infrared ultrafast laser with a microwave repetition rate. Typical ~100 fs pulsed lasers with repetition rate 50–100 MHz possess thousands of higher-order-harmonics reaching into the terahertz regime. For this reason, ultrafast lasers have been an excellent but expensive solution for EO sensing at greater than 100 GHz [10].

As an alternative to this ultrafast-modulation method, the use of high-even-order-harmonic generation for the creation of a local oscillator is a powerful technique and a reasonable solution for significantly enhancing EO sensing bandwidth in high-frequency applications. This bandwidth enhancement is potentially very useful for millimeter-wave or even terahertz applications where the feasibility of LO light modulation is a substantial challenge when using devices approaching their electronic limits. Such cost and complexity of high frequency light modulation for photonic-down-conversion sensing can be overcome by utilizing nonlinear high-order-harmonic generation with cascaded EOMs.

3. Conclusions

We have demonstrated a technique to enhance the measurement range of an electro-optic sensing system by cascading multiple EOMs in order to create high-frequency sidebands using low drive frequencies. Adding an EOM boosts the nonlinearity for generating higher-order harmonics that can be used as a local-oscillator light source in heterodyne-down-mixed EO sensing. The analytical solution for higher-even-order harmonics of both single and cascaded EOMs has provided clear guidelines for utilizing sideband harmonics to increase the bandwidth of EO-sensing without requiring high-frequency microwave instrumentation. The near electric field of a K-band antenna was successfully measured utilizing the fourth-and sixth-order harmonics of two cascaded EOMs as the LO in the EO-sensing technique. This methodology could extend the sensing bandwidth of photonic heterodyne electro-optic sensing to greater than 100 GHz using only K-band (18-26.5 GHz) microwave instruments and components.