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Locking of microwave oscillators on the interharmonics of mode-locked laser signals

Meysam Bahmanian, Christian Kress, and J. Christoph Scheytt

Open Access Open Access

Abstract

In this paper, the theory of phase-locking of a microwave oscillator on the interharmonics, i.e. non-integer harmonics, of the repetition rate of the optical pulse train of a mode-locked laser (MLL) is developed. A balanced optical microwave phase detector (BOMPD) is implemented using a balanced Mach-Zehnder modulator and is employed to discriminate the phase difference between the envelope of the optical pulses and the microwave oscillator. It is shown mathematically that the inherent nonlinear properties of BOMPD with respect to the microwave excitation amplitude can be used for interharmonic locking. The characteristic functions of the phase detector for interharmonic locking are derived analytically and are compared with the measurement results. An opto-electronic phase-locked loop (OEPLL) is demonstrated whose output frequency locks on interharmonics of the MLL repetition rate when an appropriate modulator bias and sufficient RF amplitude are applied. Thus, for the first time theory and experiment of reliable locking on interharmonics of the repetition rate of a MLL are presented.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

Corrections

15 August 2022: A minor correction was made to Fig. #5.

1. Introduction

Femtosecond pulses from MLLs are popular for a wide range of applications from material processing to precision timing [1]. MLLs have also become compact and robust which makes them suitable for commercial applications. The envelope of the train of optical pulses from an MLL has also excellent noise and jitter properties which surpasses the noise and jitter performance of their electronic counterparts [2–5]. Generating a radio frequency (RF) signal using the optical pulses of an MLL has been implemented using different techniques [6–10]. A popular method with minimal introduction of excess noise is direct detection of the optical pulses using a photodiode. The signal to noise ratio (SNR) of this type of RF generator is limited by the thermal noise of output termination load and the maximum achievable RF power from the photodiode [11,12]. The output power of the photodiode is also limited at the exposure of high energy pulses mainly due to the nonlinear space-charge nonlinear effect. In order to mitigate the photodiode space-charge effect, uni-traveling carrier (UTC) photodiode and modified UTC (MUTC) photodiode use an undepleted p-layer to absorb light to inject the electrons into the drift region and reduce the transit time and the space-charge effect [13,14]. The illumination condition has also been optimized through beam shaping using a graded index (GRIN) lens coupling which increases the illumination cross section and reduces the peak magnitude of the optical field [15,16]. In addition to enhancement of photodiode linearity, pulse repetition rate multiplication based on mode filtering or pulse interleaving are popular techniques to reduce the peak power of the optical beam [9,11,17,18]. However, this limits the output frequency of the photodiode only to the harmonics of the multiplied repetition rate. Another approach for MLL-based RF generation is phase-locking of a microwave oscillator on the envelope of the MLL pulse trains [10,19,20]. This type of RF generator, shown in Fig. 2(a), requires a phase detector which operates in a mixed electro-optical domain, the so-called balanced optical microwave phase detector (BOMPD). This method provides an OEPLL which can potentially lock on any harmonic of the reference repetition rate while having excellent noise properties [2,21–23]. Such a wideband OEPLL is especially interesting for low noise and wideband frequency synthesizers with sub-Hz frequency resolution. This is illustrated graphically in Fig. 1(a) where the output frequencies of the OEPLL are shown as arrows and the frequency gaps are the hatched regions. These frequency gaps can be filled by a simple mixing scheme which multiplies the OEPLL output by a sub-Hz frequency resolution phase-locked loop and filtering the image signal as shown in Fig. 1(b) where the coarse tuning loop (CTL) is the harmonically locked OEPLL and the fine tuning loop (FTL) could be a direct digital synthesizer (DDS) or a fractional-N phase locked loop [24]. The alternative approach instead of direct mixing of CTL and FTL frequencies could be employing an offset phase-locked loop architecture which does not require the image reject filter (IRF) and is more suitable for wideband frequency synthesis [25].

 figure: Fig. 1.

Fig. 1. (a) Output frequencies of a harmonically locked OEPLL (shown as solid arrows) and frequency gaps between these frequencies (shown as hashed regions) where $\Delta f_{_\textrm {FTL}}$ is the frequency range of the FTL. (b) Simple mixing scheme to fill the frequency gaps of an OEPLL; CTL, coarse tuning loop; FTL, fine tuning loop; MX, frequency mixer; IRF, tunable image reject filter.

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Under certain conditions the microwave oscillator in the OEPLL can also lock on non-integer harmonics of the repetition rate of MLLs. This operating regime is called interharmonic locking and is often an undesired operational mode of the OEPLL. However, interharmonic locking can be intentionally used in wideband and low noise OEPLL-based frequency synthesizers with sub-Hz frequency resolution which is explained in the following. One of the main challenges of the circuit in Fig. 1(b) is to keep the phase noise of the FTL below that of the CTL and transferring the output signal of CTL to the synthesizer output with minimum additive phase noise from the FTL which is usually a fractional-N PLL. This can be achieved by using high clock division ratios to divide the frequency of – relatively – noisy FTL and thus reduce its phase noise. However, this technique also reduces the frequency range of the FTL and therefore it becomes necessary to reduce the frequency gaps of the CTL which is an OEPLL. A simple approach is using an MLL with lower repetition rate. One drawback is that with a constant average optical power, the pulse energy increases and exacerbates the nonlinear effects in the optical fiber and the BOMPD photodiodes. Decreasing the MLL average power mitigates these nonlinear effects but has adverse effects on the phase noise of BOMPD [2]. Besides, in an optical clock distribution network where a master MLL is used to synchronise multiple devices, solutions which do not require the modification of the master MLL are preferred. Therefore, a CTL-OEPLL which can reliably generate interharmonics of the repetition rate of the MLL is highly desired for OEPLL-based frequency synthesizers with sub-Hz resolution.

In this paper, interharmonic locking of OEPLL using a BOMPD is analyzed thoroughly. It is shown that operating a BOMPD in this regime requires proper biasing of the balanced intensity modulator (BIM) and sufficient RF excitation amplitude. The characteristic curves of the BOMPD (its output current as a function of the phase difference between the RF signal and the target optical interharmonic) are derived and the results are compared with the measured ones. Finally, an OEPLL is demonstrated which locks on the $40$’th, $40\frac 12$’th, $40\frac 13$’th and $40\frac 14$’th multiples of the reference repetition rate.

2. Theory of interharmonic locking

Figure 2(a) shows the block diagram of the OEPLL. The tunable oscillator output voltage is sampled with the MLL pulses at the BIM. The BIM can be a Mach-Zehnder modulator or a fiber-Sagnac loop. The output optical pulses of the BIM are converted to an electrical current via a pair of photodiodes. This current is then integrated via the loop filter and fed back to the tunable oscillator, closing the loop. When the OEPLL is phase locked, the upper and lower BOMPD photodiodes in Fig. 2(a) have photocurrent pulses with equal amplitudes. If the tunable oscillator has a phase difference with respect to the optical reference, the amplitudes of photocurrent pulses of the upper and lower photodiodes change and a non-zero current is generated, as illustrated in Fig. 2(b). This non-zero current is then integrated via the loop filter and generates a dc voltage applied to the tunable oscillator which realigns the phase of tunable oscillator with the phase of optical reference. The output current of BOMPD can be written as [2]

$$i = RI(t)\sin\left(\frac{v_\textrm{RF}\pi}{V_{\pi ,\textrm{RF}}} + \psi_\textrm{dc}\right),$$
where $i$ is the output current of the BOMPD, $R$ is the photodiode responsivity, $I(t)$ is the optical cycle averaged intensity of the optical input of the BOMPD, $\psi _\textrm {dc}$ is the optical phase shift introduced by the dc electrode, $V_{\pi,\textrm {RF}}$ is the $\pi$-voltage of RF electrode and $v_\textrm {RF}$ is the sinusoidal RF modulation voltage with an amplitude of $V_\textrm {RF}$, an angular frequency of $\omega _\textrm {RF}$ and an offset phase of $\phi$
$$v_\textrm{RF} = V_\textrm{RF}\sin(\omega_\textrm{RF}t+\phi).$$
The sine term in Eq. (1) shows that the RF voltage undergoes a nonlinear transformation which generates harmonics of a single tone RF excitation. The BOMPD can therefore be modeled as a nonlinear block and a balanced frequency mixer as illustrated in Fig. 3(a). The nonlinear behavior of the BOMPD can be used to lock a harmonic of the microwave oscillator, rather than its fundamental tone, to an integer multiple of the optical reference repetition rate. The frequency content of different nodes of the BOMPD are shown in Fig. 3(a) where $M$ is the harmonic index of the reference repetition rate and $N$ is the harmonic index of RF voltage. The OEPLL locks when the frequency of the inputs of the mixer match
$$f_\textrm{RF} = \frac{M}{N}f_\textrm{ref},$$
where $M$ and $N$ are positive integers with greatest common divisor (gcd) of 1. We call the operating regime of an OEPLL with $N=1$ as harmonic locking and with $N>1$ as the $N$’th order interharmonic locking. Figure 3(b) illustrates the tunable oscillator waveform and the MLL pulses for $\omega _\textrm {RF}/\omega _\textrm {ref}=2,2\frac 12$, and $2\frac 14$ corresponding to $N=1,2$, and $4$, respectively. Assuming the optical pulses of the MLL are much shorter than the RF signal period, each pulse can be approximated as a Dirac delta function
$$I(t) = I_0 T_\textrm{ref} \sum_{m=-\infty}^{+\infty} \delta(t-\frac{m}{f_\textrm{ref}}),$$
where $I_0$ is the average intensity of the optical input and $T_\textrm {ref}=1/f_\textrm {ref}$. Equation (1) can therefore be written as
$$i = RI_0T_\textrm{ref} \sum_{m=-\infty}^{+\infty} \delta(t-\frac{m}{f_\textrm{ref}}) \sin\left[\alpha\sin(\omega_\textrm{RF}t+\phi) + \psi_\textrm{dc}\right],$$
where we introduced $\alpha =\pi V_\textrm {RF}/V_{\pi,\textrm {RF}}$. The characteristic function of the phase detector is defined as its average output current with respect to the phase difference between its inputs. For the $N$’th order interharmonic phase detection, the output current in Eq. (5) has a period of $NT_\textrm {ref}$. Therefore, the $N$’th order characteristic function of BOMPD, $H_N(\phi )$, can be written as
$$H_N(\phi) = \frac{1}{NT_\textrm{ref}}\int_0^{NT_\textrm{ref}} \textrm{d}t\hspace{.1em} i = \frac{RI_0}{N} \sum_{m=0}^{N-1} \sin\left[\alpha\sin(\frac{2\pi m}{N}+\phi) + \psi_\textrm{dc}\right].$$

 figure: Fig. 2.

Fig. 2. (a) Simplified block diagram of OEPLL, (b) corresponding waveforms: (black) voltage waveform at the RF port of BIM, (red) upper photodiode current and (blue) lower photodiode current when (solid) the phase of RF signal and the optical intensity are aligned, (dashed) the optical intensity has a phase lead and (dotted) the optical intensity has a phase lag; BIM, balanced intensity modulator; BOMPD, balanced optical microwave phase detector; LPF, low-pass filter.

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 figure: Fig. 3.

Fig. 3. (a) OEPLL with nonlinear model of BOMPD, and (b) corresponding waveforms: (black) RF voltage waveform, (red) MLL pulses when $\omega _\textrm {RF}/\omega _\textrm {ref}=2$, (blue) MLL pulses when $\omega _\textrm {RF}/\omega _\textrm {ref}=2\frac 12$, (violet) MLL pulses when $\omega _\textrm {RF}/\omega _\textrm {ref}=2\frac 14$. The time axis is normalized to the RF signal period $T_\textrm {RF}=1/f_\textrm {RF}$; BIM, balanced intensity modulator; BOMPD, balanced optical microwave phase detector; LPF, low-pass filter.

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The index $M$ in Eq. (6) was suppressed since we assumed gcd$(M,N)=1$. The first, 2nd and 4th order characteristic functions of the BOMPD (corresponding to harmonic locking, 2nd order interharmonic locking and 4th order interharmonic locking, respectively) can consequently be found using Eq. (6) as

$$H_1(\phi) = RI_0 \sin(\alpha\sin(\phi)+\psi_\textrm{dc}),$$
$$H_2(\phi) = RI_0 \sin(\psi_\textrm{dc})\cos[\alpha\sin(\phi)],$$
and
$$H_4(\phi) = \frac 12 RI_0 \sin(\psi_\textrm{dc})\left[\cos(\alpha\sin(\phi))+\cos(\alpha\cos(\phi))\right].$$
Equations (8) and (9) show that the 2nd and 4th order characteristic functions linearly scale with $\sin (\psi _\textrm {dc})$ and become zero when $\psi _\textrm {dc}=0$. This is expected since the nonlinear characteristic function of the BOMPD in Eq. (1) has even symmetry with respect to $v_\textrm {RF}$ at $\psi _\textrm {dc}=\pm \pi /2$ and odd symmetry at $\psi _\textrm {dc}=0$. With a single tone excitation, only odd harmonics at $\psi _\textrm {dc}=0$ and only even harmonics at $\psi _\textrm {dc}=\pm \pi /2$ are generated. The average of $H_N(\phi )$ can be found using Jacobi-Anger expansion of Eq. (6) as
$$<H_N(\phi)> = RI_0 \sin(\psi_\textrm{dc}) J_0(\alpha),$$
where $<.>$ denotes the average value and $J_0$ denotes the Bessel function of the first kind of order 0. A zero-average $H_N(\phi )$ guarantees a zero-crossing in the transfer characteristic which is necessary for operating the BOMPD in an OEPLL [2]. Proper operation of the OEPLL requires coarse tuning of the tunable oscillator around the desired frequency which can be very close to another interharmonic frequency of the same or different order. Therefore, although a zero-crossing in the characteristic function can be generated by having sufficient RF amplitude, locking the OEPLL on very high order interharmonics can be practically difficult.

3. Measurement results

The BOMPD is characterized using the measurement setup shown in Fig. 4(a). A commercial MLL with a center wavelength of 1560 nm, a pulse width below 200 fs and a repetition rate of 250 MHz is used as the optical reference [26]. The MZM is a balanced Lithium-Niobate (LiNbO3) modulator with two complementary outputs and two separate electrodes for RF excitation and dc bias. The dc electrode is biased via a low noise digital to analog converter (DAC) which is controlled with a microcontroller. A pair of InGaAs photodiodes are used to convert the output optical pulses of the MZM to an electrical current. An arbitrary waveform generator (AWG) and an amplifier are used to feed the RF port of the modulator. The output of the amplifier is filtered to suppress the harmonics of the signal fed to the RF electrode of the MZM and guarantee single tone excitation of the RF port according to Eq. (2). The AWG is synchronized with the MLL and the phase of the RF signal is swept by introducing a 100 kHz offset frequency between the RF signal and the target frequency of Eq. (3).

 figure: Fig. 4.

Fig. 4. (a) Test setup for characterization of BOMPD. MCU, microcontroller unit; DAC, digital to analog converter; MLL, mode-locked laser; MZM, Mach-Zehnder modulator; PC, polarization controller; AWG, arbitrary waveform generator; OSC, oscilloscope; AMP, amplifier; LPF, low-pass filter. BOMPD characteristic curves according to (dashed) theory and (solid) measurement for (b) harmonic locking at $\psi _\textrm {dc}=0$, (c) 2nd order interharmonic locking at $\psi _\textrm {dc}=\pi /2$, (d) 3rd order interharmonic locking at $\psi _\textrm {dc}=0$ and (e) 4th order interharmonic locking at $\psi _\textrm {dc}=\pi /2$ for different RF amplitudes: (black) $\alpha =0.8$, (red) $\alpha =1.2$ and (blue) $\alpha =2.3$.

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Figures 4(b), 4(c), 4(d) and 4(e) compare the measured phase detector characteristic functions with the theoretically derived ones in Eq. (6) for $N=1$ to $4$ which correspond to harmonic locking and 2nd to 4th order interharmonic locking, respectively. The measured characteristic curves have a good matching with the theory. The negative values of $H_1(\phi )$ exceeds -1 which is due to mismatch in the responsivity of the BOMPD photodiodes. The dc value of the measured curves is also slightly different from what is expected from theory which is the result of limited extinction ratio of the MZM and measurement errors due to temperature variations of the temperature-sensitive Lithium-Niobate MZM.

Figure 5 shows the block diagram of the implemented OEPLL. An Yttrium Iron Garnet (YIG) oscillator is used as a tunable oscillator. The Main coil of the YIG oscillator is used for coarse tuning of the frequency and is biased for a 10 GHz center frequency via a low noise DAC and a current driver. Another low noise DAC is used to bias the dc electrode of the MZM. The output current of the BOMPD is integrated with a low-pass filter and is fed back to the FM coil of the YIG oscillator via a low noise current driver. For the OEPLL setup, the amplifier is driven into deep saturation region to have a high RF voltage for excitation of the MZM. This ensures the characteristic function has a zero crossing and the locking conditions are satisfied.

 figure: Fig. 5.

Fig. 5. Block diagram of the implemented OEPLL. MCU, microcontroller unit; DAC, digital to analog converter; MLL, mode-locked laser; MZM, Mach-Zehnder modulator; PC, polarization controller; LPF, low pass filter; YIG, Yttrium Iron Garnet; AMP, amplifier.

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Figure 6 shows the power spectrum of the OEPLL output signal and Fig. 7 shows the measured phase noise of the OEPLL output at 10 GHz, 10.125 GHz, $10.08\overline {3}$ GHz and 10.0625 GHz carrier frequencies which are the 40’th harmonic and $40\frac 12$’th, $40\frac 13$’th and $40\frac 14$’th interharmonics of the 250 MHz repetition rate and correspond to harmonic locking and 2nd, 3rd and 4th order interharmonic locking, respectively. The power spectrum was measured using MS2760A from Anritsu and the phase noise was measured using APPH20G phase noise analyzer from Anapico. The BOMPD has a lower gain for the 2nd, 3rd and 4th order interharmonic phase detection compared to harmonic phase detection. This leads to a lower loop bandwidth for the OEPLL locked on the reference interharmonics, which can be observed in Fig. 7 around the offset frequency where the phase noise starts to follow the YIG oscillator phase noise. The in-band phase noise of the BOMPD (at offset frequencies between 10 kHz and 1 MHz) for interharmonic locking is slightly higher than that for harmonic locking. For both harmonic-locking and interharmonic locking, the in-band phase noise of the signal at offset frequencies below 10 kHz is dominated by the phase noise of the reference MLL. In overall, the phase noise performance of the OEPLL at interharmonics is similar to the harmonic locking case. As a result, with proper biasing of the MZM and sufficient RF amplitude, frequency resolution of the OEPLL has been enhanced to lock on the interharmonics of the reference repetition rate.

 figure: Fig. 6.

Fig. 6. Power spectrum of the OEPLL output signal at (a) 10 GHz corresponding to harmonic locking, (b) 10.125 GHz corresponding to 2nd order interharmonic locking, (c) $10.08\overline {3}$ GHz corresponding to 3rd order interharmonic locking and (d) 10.0625 GHz corresponding to 4th order interharmonic locking.

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 figure: Fig. 7.

Fig. 7. Measured phase noise of the OEPLL output signal at (red,left) 10 GHz, (blue,left) 10.125 GHz, (orange, right) $10.08\overline {3}$ GHz and (cyan,right) 10.0625 GHz carrier frequencies corresponding to harmonic locking and 2nd, 3rd and 4th order interharmonic locking, respectively, (dashed black) phase noise of the MLL reference scaled to 10 GHz carrier frequency and (dashed violet) phase noise of the YIG oscillator.

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4. Conclusion

In this paper, an optoelectronic PLL was demonstrated which can lock on the interharmonics of the repetition rate of a MLL. The PLL uses a balanced optical microwave phase detector (BOMPD) to discriminate the phase difference between the optical reference and the RF signal. The characteristic functions of the BOMPD were derived analytically and the results were compared with the measured values. This is the first time the characteristic functions of BOMPD for interharmonic locking are derived and the first demonstration of a microwave oscillator locked on the interharmonics of the repetition rate of a MLL.

Funding

Deutsche Forschungsgemeinschaft (370491995, 403579441).

Disclosures

The authors declare that they have no relevant or material financial interests that relate to the research described in this paper.

Data availability

Simulation and measurement data underlying the results presented in this paper may be obtained from the authors upon reasonable request.

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Data availability

Simulation and measurement data underlying the results presented in this paper may be obtained from the authors upon reasonable request.

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Figures (7)
Fig. 1.
Fig. 1. (a) Output frequencies of a harmonically locked OEPLL (shown as solid arrows) and frequency gaps between these frequencies (shown as hashed regions) where $\Delta f_{_\textrm {FTL}}$ is the frequency range of the FTL. (b) Simple mixing scheme to fill the frequency gaps of an OEPLL; CTL, coarse tuning loop; FTL, fine tuning loop; MX, frequency mixer; IRF, tunable image reject filter.

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Fig. 2.
Fig. 2. (a) Simplified block diagram of OEPLL, (b) corresponding waveforms: (black) voltage waveform at the RF port of BIM, (red) upper photodiode current and (blue) lower photodiode current when (solid) the phase of RF signal and the optical intensity are aligned, (dashed) the optical intensity has a phase lead and (dotted) the optical intensity has a phase lag; BIM, balanced intensity modulator; BOMPD, balanced optical microwave phase detector; LPF, low-pass filter.

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Fig. 3.
Fig. 3. (a) OEPLL with nonlinear model of BOMPD, and (b) corresponding waveforms: (black) RF voltage waveform, (red) MLL pulses when $\omega _\textrm {RF}/\omega _\textrm {ref}=2$ , (blue) MLL pulses when $\omega _\textrm {RF}/\omega _\textrm {ref}=2\frac 12$ , (violet) MLL pulses when $\omega _\textrm {RF}/\omega _\textrm {ref}=2\frac 14$ . The time axis is normalized to the RF signal period $T_\textrm {RF}=1/f_\textrm {RF}$ ; BIM, balanced intensity modulator; BOMPD, balanced optical microwave phase detector; LPF, low-pass filter.

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Fig. 4.
Fig. 4. (a) Test setup for characterization of BOMPD. MCU, microcontroller unit; DAC, digital to analog converter; MLL, mode-locked laser; MZM, Mach-Zehnder modulator; PC, polarization controller; AWG, arbitrary waveform generator; OSC, oscilloscope; AMP, amplifier; LPF, low-pass filter. BOMPD characteristic curves according to (dashed) theory and (solid) measurement for (b) harmonic locking at $\psi _\textrm {dc}=0$ , (c) 2nd order interharmonic locking at $\psi _\textrm {dc}=\pi /2$ , (d) 3rd order interharmonic locking at $\psi _\textrm {dc}=0$ and (e) 4th order interharmonic locking at $\psi _\textrm {dc}=\pi /2$ for different RF amplitudes: (black) $\alpha =0.8$ , (red) $\alpha =1.2$ and (blue) $\alpha =2.3$ .

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Fig. 5.
Fig. 5. Block diagram of the implemented OEPLL. MCU, microcontroller unit; DAC, digital to analog converter; MLL, mode-locked laser; MZM, Mach-Zehnder modulator; PC, polarization controller; LPF, low pass filter; YIG, Yttrium Iron Garnet; AMP, amplifier.

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Fig. 6.
Fig. 6. Power spectrum of the OEPLL output signal at (a) 10 GHz corresponding to harmonic locking, (b) 10.125 GHz corresponding to 2nd order interharmonic locking, (c) $10.08\overline {3}$  GHz corresponding to 3rd order interharmonic locking and (d) 10.0625 GHz corresponding to 4th order interharmonic locking.

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Fig. 7.
Fig. 7. Measured phase noise of the OEPLL output signal at (red,left) 10 GHz, (blue,left) 10.125 GHz, (orange, right) $10.08\overline {3}$  GHz and (cyan,right) 10.0625 GHz carrier frequencies corresponding to harmonic locking and 2nd, 3rd and 4th order interharmonic locking, respectively, (dashed black) phase noise of the MLL reference scaled to 10 GHz carrier frequency and (dashed violet) phase noise of the YIG oscillator.

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Equations (10)

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i = R I ( t ) sin ⁡ ( v RF π V π , RF + ψ dc ) ,
v RF = V RF sin ⁡ ( ω RF t + ϕ ) .
f RF = M N f ref ,
I ( t ) = I 0 T ref ∑ m = − ∞ + ∞ δ ( t − m f ref ) ,
i = R I 0 T ref ∑ m = − ∞ + ∞ δ ( t − m f ref ) sin ⁡ [ α sin ⁡ ( ω RF t + ϕ ) + ψ dc ] ,
H N ( ϕ ) = 1 N T ref ∫ 0 N T ref d t i = R I 0 N ∑ m = 0 N − 1 sin ⁡ [ α sin ⁡ ( 2 π m N + ϕ ) + ψ dc ] .
H 1 ( ϕ ) = R I 0 sin ⁡ ( α sin ⁡ ( ϕ ) + ψ dc ) ,
H 2 ( ϕ ) = R I 0 sin ⁡ ( ψ dc ) cos ⁡ [ α sin ⁡ ( ϕ ) ] ,
H 4 ( ϕ ) = 1 2 R I 0 sin ⁡ ( ψ dc ) [ cos ⁡ ( α sin ⁡ ( ϕ ) ) + cos ⁡ ( α cos ⁡ ( ϕ ) ) ] .
< H N ( ϕ ) >= R I 0 sin ⁡ ( ψ dc ) J 0 ( α ) ,
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