Dynamically induced nonlinearity in a resonant-cavity interferometric intensity modulator

Janusz Murakowski; Garrett J. Schneider; Dennis W. Prather

doi:10.1364/OE.20.014683

1. Introduction

High-speed, high-dynamic-range modulators are essential for the performance of optical links both in analog and digital domains [1, 2], and have been the subject of intensive study in the quest for ever increasing communication bandwidth. A widely-employed class of modulators includes electro-optic modulators that rely on the Pockels effect in materials with no inversion symmetry such as, e.g., lithium niobate, to modulate the phase of a laser beam. In these devices, phase modulation needs to be converted to amplitude modulation for detection with a photodiode. The simplest configuration where such conversion is accomplished is a Mach-Zehnder interferometer, wherein the modulated beam is mixed with an un-modulated reference beam. While extremely successful thanks to their high speed and high optical power handling capability, Mach-Zehnder modulators suffer from inherent nonlinearity that limits their spur-free dynamic range (SFDR)—see [3, 4] and Appendix A.1 for the discussion of the SFDR concept. The nonlinearity is the result of a sine transfer function when a phase-modulated beam, in one arm of the Mach-Zehnder interferometer, is interfered with an unmodulated reference in quadrature. To overcome this nonlinearity, a new configuration has been proposed recently [5], which is referred to as a resonant cavity interferometric intensity (RCII) modulator, where the electro-optic phase modulator in one arm of the interferometer is replaced with a laser, and the modulation signal is applied to the cavity, Fig. 1 . In this case, the output of the master laser is injected into the modulated laser, and the system parameters, including the injection ratio and the quality factor Q of the injected laser cavity, are maintained such as to ensure locking conditions of the injected laser [6, 7]. For DC and low-frequency modulation, Adler’s equation, which governs the behavior of injected oscillators [4], predicts that the phase of the output of the injected laser is approximately an arcsine function of the applied signal. When combined with the sine transfer function of the Mach-Zehnder configuration operating at quadrature bias, this nominally produces a linear response.

Fig. 1 Resonant-cavity interferometric intensity (RCII) modulator.

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Clearly, such a configuration appears attractive as it potentially allows considerable improvement of linearity with only minimal increase of system complexity. However, the approximation leading to the arcsine phase dependence on the applied signal is fully justified only for static, i.e. DC, conditions. As the frequency of the modulating signal approaches half of the locking range of the injected laser, the approximation is expected to lose its validity, and the linearity of the modulator response to be compromised. In this paper, we present a quantitative analysis of the deviation from linearity as the frequency of the modulating signal is increased from DC.

The article is organized as follows. In Sec. 2, a method of solving Adler’s equation using Fourier series expansion is described and then applied to finding the frequency response of the RCII modulator to single-tone input at both the fundamental (Sec. 2.1) and third harmonic (Sec. 2.2) frequencies. Section 2.3 extends the method to calculate the third-order intermodulation spur amplitudes when two-tone input is used. Throughout Sec. 2, we use a Fourier series expansion of the sine of trigonometric series, which is proven in Appendix A.3. The results of Sec. 2 are used in Sec. 3 to compare the SFDR of the RCII modulator to the SFDR of the Mach-Zehnder modulator, which is calculated in Appendix A.2. Discussion of the results is presented in Sec. 4, which concludes the article.

2. Fourier-series solution to Adler’s equation

The starting point for the investigations is Adler’s equation in the form presented in [7]

\frac{d ϕ}{d t} = Δ ω_{0} - ω_{m} \sin ϕ,

where

ϕ

is the phase between the injected signal and the field oscillating in the cavity,

Δ ω_{0}

is the frequency difference between the injected signal and the free oscillations of the cavity, and

ω_{m}

is half of the locking range. At DC, i.e. time-invariant deviation between the free-running resonant frequency of the injected laser and the frequency of the injected signal, the time derivative of the phase on the left-hand side of Eq. (1) vanishes, and the phase becomes an arcsine function of the frequency offset

ϕ = arc \sin (\frac{Δ ω_{0}}{ω_{m}}),

which is the basis for the linearization scheme proposed in [5].

To study the behavior of the injected laser at modulation frequency other than zero, Eq. (1) is first simplified by redefining the frequency and time units so that $ω_{m} = 1.$ We divide Eq. (1) by to obtain

\begin{array}{l} \frac{1}{ω_{m}} \frac{d ϕ}{d t} = \frac{Δ ω_{0}}{ω_{m}} - \sin ϕ \\ \frac{d ϕ}{d \tilde{t}} = Δ {\tilde{ω}}_{0} - \sin ϕ, \end{array}

where the decorated time and frequency symbols are defined as

\tilde{t} = ω_{m} t

and

Δ {\tilde{ω}}_{0} = Δ ω_{0} / ω_{m},

respectively. To minimize clutter in the equations that follow, the tilde will be omitted with the understanding that the time and frequency variables are all rescaled such that

ω_{m} = 1.

Modulation of the effective cavity of the injected laser is equivalent to the modulation of the offset $Δ ω_{0}$ between the injected and the free-running frequencies. Thus, for a monochromatic, or single-tone, signal input the differential equation becomes

\frac{d ϕ}{d t} + \sin ϕ = Ω_{m} \sin (Ω t),

where

Ω

is the frequency of the modulating RF signal, and

Ω_{m}

is its amplitude expressed as the maximum deviation of the free-running oscillation frequency from that of the injected light.

Equation (3) has no known closed-form solutions. Below, it is solved approximately, where the approximation specifically targets the characterization of third-order nonlinearities. This means that the transient response of the system is of no interest here and therefore is omitted. In other words, it is assumed that $ϕ$ is a periodic function of $t$ with the period equal to the period of the driving signal, i.e.

ϕ (t + 2 π / Ω) = ϕ (t) .

This periodicity allows the expansion of

ϕ

in a Fourier series

ϕ (t) = \sum_{n = 0}^{\infty} a_{n} \sin (n Ω t + φ_{n}) .

It is expected that the third-order nonlinearity will be dominant. Therefore, only terms up to $n = 3$ in the expansion (5) above need to be considered. However, the technique developed here is more general as it allows terminating the expansion after an arbitrary number of terms, which makes it applicable in other types of modulators.

With the help of Theorem (46) (found in the Appendix) and the Fourier expansion (5), Eq. (3) can be written as

Ω \sum_{n} n a_{n} \cos (n Ω t + φ_{n}) + \sum_{k} \prod_{n} J_{k_{n}} (a_{n}) \sin [\sum_{n} k_{n} (n Ω t + φ_{n})] = Ω_{m} \sin (Ω t),

where

J_{k_{n}} (a_{n})

is a Bessel function of order

k_{n}

with argument

a_{n}

(see Appendix for a complete explanation of the notation). Due to the linear independence of terms corresponding to different frequencies, Eq. (6) separates into a system of infinitely many equations, each containing terms corresponding to a single frequency, an integer multiple of

Ω

.

2.1 First-order approximation

Initially, only the fundamental response at is considered. In this case, only $n = 1$ and $k_{1} = \pm 1$ need to be considered since all other combinations involve higher order terms, higher frequencies, or constants. Therefore, $a_{n} = 0$ for $n > 1,$ and Eq. (6) becomes

Ω a_{1} \cos (Ω t + φ_{1}) + J_{1} (a_{1}) \sin (Ω t + φ_{1}) + J_{- 1} (a_{1}) \sin [- (Ω t + φ_{1})] = Ω_{m} \sin (Ω t) .

Since

J_{- 1} (z) = - J_{1} (z)

and

\sin (- z) = - \sin (z),

Eq. (7) can be simplified to

Ω a_{1} \cos (Ω t + φ_{1}) + 2 J_{1} (a_{1}) \sin (Ω t + φ_{1}) = Ω_{m} \sin (Ω t) .

Equation (8) must hold at all values of

t .

Inserting into Eq. (8) values of

t

such that

Ω t + φ_{1} = 0

and

Ω t + φ_{1} = π / 2

shows that the amplitude

a_{1}

and phase

φ_{1}

must satisfy the following system of equations

\begin{matrix} Ω a_{1} = - Ω_{m} \sin (φ_{1}) \\ 2 J_{1} (a_{1}) = Ω_{m} \cos (φ_{1}) . \end{matrix}

Equations (9) are a system of nonlinear equations that can be solved for parameters

a_{1}

and

φ_{1}

in terms of the frequency and amplitude of the modulating signal,

Ω

and

Ω_{m},

respectively. The result is the linear response of the phase

ϕ

to the modulation. In other words,

ϕ

is the following function of time:

ϕ (t) = a_{1} \sin (Ω t + φ_{1}),

with

a_{1}

and

φ_{1}

obtained from Eqs. (9) as functions of

Ω

and

Ω_{m} .

As mentioned in the Introduction, to detect the phase modulation $ϕ$ in a physical system, the modulated signal is interfered with a reference to produce amplitude modulation at a photo-detector. Mathematically, this procedure amounts to calculating the sine of phase $ϕ$ (assuming quatrature bias). Using Theorem (46) we find

\sin [ϕ (t)] = \sin [a_{1} \sin (Ω t + φ_{1})] = \sum_{k = - \infty}^{\infty} J_{k} (a_{1}) \sin [k (Ω t + φ_{1})] .

Since the approximation (10) only accounts for the first-order response, in the series of Eq. (11) only the terms with

| k | \leq 1

are meaningful:

\sin [ϕ (t)] ≃ \sum_{k = - 1}^{1} J_{k} (a_{1}) \sin [k (Ω t + φ_{1})] = 2 J_{1} (a_{1}) \sin (Ω t + φ_{1}) \equiv A_{1} \sin (Ω t + φ_{1}),

where

A_{1}

is defined in the last equality as the amplitude of the linear response of the modulator.

For small modulation amplitude, $Ω_{m} ≪ 1,$ the system of Eqs. (9) can be simplified. In this case, the first of Eqs. (9) leads to $a_{1} ≪ 1,$ which means that in the second of Eqs. (9) only the linear term in the Taylor expansion of the Bessel function, $J_{1} (a_{1}) = a_{1} / 2 - a_{1}^{3} / 16 + o (a_{1}^{3}),$ is significant. As a result, Eqs. (9) become

\begin{matrix} Ω a_{1} = - Ω_{m} \sin (φ_{1}) \\ a_{1} = Ω_{m} \cos (φ_{1}), \end{matrix}

which, by using trigonometric identity

\sin^{2} (φ_{1}) + \cos^{2} (φ_{1}) = 1,

can be solved to obtain the following expression for coefficient

a_{1}

as a function of modulation frequency

a_{1} = \frac{Ω_{m}}{\sqrt{1 + Ω^{2}}} .

The linear response amplitude

A_{1}

defined in Eq. (12) simplifies for small input amplitude

Ω_{m}

to

A_{1} = 2 J (a_{1}) ≃ a_{1} = \frac{Ω_{m}}{\sqrt{1 + Ω^{2}}} .

Expression (15) for the frequency dependence of the modulator response is identical to a low-pass filter. Figure 2 shows the frequency response of the modulator normalized to 0 dB at DC. The 3-dB drop occurs at the modulation frequency equal to half of the locking range, i.e. at

Ω = ω_{m} .

Fig. 2 Magnitude of the frequency response of the RCII modulator in the first-order approximation.

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2.2 Third-order approximation

Next, the contributions to the third harmonic of the modulation signal are examined. In this case, in Eq. (6) we only need to consider $n \in {1, 3},$ $k_{1} \in {0, \pm 1, \pm 3},$ $k_{3} \in {0, \pm 1} .$ The following combinations of parameters contribute to the $3 Ω$ term:

\begin{array}{l} k_{1} = \pm 3, k_{3} = 0; and \\ k_{1} = 0, k_{3} = \pm 1. \end{array}

Note that cases

k_{1} = 2, k_{3} = - 1

and

k_{1} = - 2, k_{3} = 1

are omitted as they give rise to higher-order (5-th) amplitude terms. Since in this case

a_{1}

and

a_{3}

can take nonzero values, the equations for amplitude and phase at frequency

Ω

are modified from Eq. (9). The terms that contribute to frequency

Ω

correspond to

k_{1} = \pm 1, k_{3} = 0.

With the series truncated to the terms listed above, Eq. (6) becomes

\begin{matrix} Ω a_{1} \cos (Ω t + φ_{1}) + 3 Ω a_{3} \cos (3 Ω t + φ_{3}) + 2 J_{1} (a_{1}) J_{0} (a_{3}) \sin (Ω t + φ_{1}) + \\ 2 J_{3} (a_{1}) J_{0} (a_{3}) \sin (3 Ω t + 3 φ_{1}) + 2 J_{0} (a_{1}) J_{1} (a_{3}) \sin (3 Ω t + φ_{3}) = Ω_{m} \sin (Ω t), \end{matrix}

where the anti-symmetry of the odd-order Bessel functions has been used (compare Eqs. (7) and (8)). The linear independence of terms corresponding to different frequencies leads to the separation of Eq. (16) into two independent equations

\begin{matrix} Ω a_{1} \cos (Ω t + φ_{1}) + 2 J_{1} (a_{1}) J_{0} (a_{3}) \sin (Ω t + φ_{1}) = Ω_{m} \sin (Ω t) \\ 3 Ω a_{3} \cos (3 Ω t + φ_{3}) + 2 J_{3} (a_{1}) J_{0} (a_{3}) \sin (3 Ω t + 3 φ_{1}) + 2 J_{0} (a_{1}) J_{1} (a_{3}) \sin (3 Ω t + φ_{3}) . = 0 \end{matrix}

Note that the first of Eqs. (17) differs from Eq. (8) by the presence of

J_{0} (a_{3})

in the second term. This is because in the first-order analysis above,

a_{3}

was assumed to be zero, which is not the case here. Since Eqs. (17) have to hold at any time

t,

the following relations between the amplitudes and phases must be satisfied

\begin{matrix} a_{1} Ω = - Ω_{m} \sin (φ_{1}) \\ 2 J_{1} (a_{1}) J_{0} (a_{3}) = Ω_{m} \cos (φ_{1}) \\ 3 Ω a_{3} + 2 J_{3} (a_{1}) J_{0} (a_{3}) \sin (3 φ_{1} - φ_{3}) = 0 \\ 2 J_{3} (a_{1}) J_{0} (a_{3}) \cos (3 φ_{1} - φ_{3}) + 2 J_{0} (a_{1}) J_{1} (a_{3}) = 0. \end{matrix}

Equations (18) are a system of four nonlinear equations that allows the calculation of amplitudes

a_{1}

and

a_{3},

as well as phases

φ_{1}

and

φ_{3}

in terms of the modulation frequency

Ω

and amplitude

Ω_{m} .

Using this solution,

ϕ (t)

can be expressed up to the third order as

ϕ (t) = a_{1} \sin (Ω t + φ_{1}) + a_{3} \sin (3 Ω t + φ_{3}) .

To detect the phase modulation $ϕ$ in a physical system, the modulated signal is interfered with a reference to produce amplitude modulation at a photo-detector. Mathematically, this procedure amounts to calculating the sine of phase $ϕ$ (assuming quadrature bias). By rearranging Eq. (3), we find that

\sin ϕ = Ω_{m} \sin (Ω t) - \frac{d ϕ}{d t} .

Differentiating expression (19) and substituting the result into the right side of Eq. (20) yields

\sin ϕ = Ω_{m} \sin (Ω t) - Ω a_{1} \cos (Ω t + φ_{1}) - 3 Ω a_{3} \cos (3 Ω t + φ_{3}) .

Expression (21) shows that the spectral content of the signal detected at the output of the RCII modulator includes the fundamental frequency

Ω

of the input signal, and its third harmonic

3 Ω .

Such signal has a general form

\sin [ϕ (t)] \equiv A_{1} \sin (Ω t + ϕ_{1}) + A_{3} \sin (3 Ω t + ϕ_{3}),

where

A_{1}, ϕ_{1}, A_{3}, ϕ_{3}

are defined to be the amplitudes and phases of the modulator response at the fundamental and third harmonic, respectively. The amplitude

A_{3}

of the third harmonic is found immediately by inspection of Eq. (21), and is equal to (up to a sign)

A_{3} = 3 Ω a_{3},

whereas the amplitude of the fundamental tone is found by evaluating the first two terms in Eq. (21) at values of

t

such that

Ω t = 0

and

Ω t = π / 2

and combining the results in quadrature

\begin{matrix} A_{1} = {{(- Ω a_{1} \cos φ_{1})}^{2} + {[Ω_{m} - Ω a_{1} \cos (\frac{π}{2} + φ_{1})]}^{2}}^{1 / 2} \\ = {[{(Ω a_{1} \cos φ_{1})}^{2} + {(Ω_{m} + Ω a_{1} \sin φ_{1})}^{2}]}^{1 / 2} \\ = {[{(Ω a_{1})}^{2} + Ω_{m}^{2} + 2 Ω a_{1} Ω_{m} \sin φ_{1}]}^{1 / 2} \\ = {[- {(Ω a_{1})}^{2} + Ω_{m}^{2}]}^{1 / 2}, \end{matrix}

where the last equality is obtained by using the first of Eqs. (18). Formulas (24) and (23), where

a_{1}

and

a_{3}

are solutions to the system of Eqs. (18), provide a basis for calculating the frequency dependence of third-order spurs in the modulator.

The calculation of the spur-free dynamic range relies on the extrapolation of small-signal system response. In this case, both $a_{1}$ and $a_{3}$ are small, and it is sufficient to keep only the lowest-order terms in these parameters in Eqs. (18). Upon Taylor expansion of the Bessel functions, keeping only terms up to third order in $a_{1}$ and $a_{3},$ Eqs. (18) become

\begin{matrix} a_{1} Ω = - Ω_{m} \sin (φ_{1}) \\ a_{1} = Ω_{m} \cos (φ_{1}) \\ 3 Ω a_{3} + \frac{a_{1}^{3}}{24} \sin (3 φ_{1} - φ_{3}) = 0 \\ \frac{a_{1}^{3}}{24} \cos (3 φ_{1} - φ_{3}) + a_{3} = 0. \end{matrix}

By applying the trigonometric identity

\sin^{2} α + \cos^{2} α = 1

on the first and second pairs of Eqs. (25), the phases are eliminated, and the following relations between amplitudes

a_{1}

and

a_{3}

are obtained

\begin{matrix} a_{1} = \frac{Ω_{m}}{\sqrt{1 + Ω^{2}}} \\ a_{3} = \frac{a_{1}^{3}}{24 \sqrt{1 + {(3 Ω)}^{2}}} . \end{matrix}

Combining Eqs. (24), (23), and (26) yields the small signal approximation for the fundamental and third-harmonic response amplitudes

\begin{matrix} A_{1} = \frac{Ω_{m}}{\sqrt{1 + Ω^{2}}}, \\ A_{3} = \frac{Ω A_{1}^{3}}{8 \sqrt{1 + {(3 Ω)}^{2}}} = \frac{Ω Ω_{m}^{3}}{8 {(1 + Ω^{2})}^{3 / 2} \sqrt{1 + {(3 Ω)}^{2}}} . \end{matrix}

Expressions (27) show the fundamental and third-harmonic response of the modulator to a single-tone modulation at frequency

Ω

and amplitude

Ω_{m} .

The fundamental-frequency response of Eq. (27) is identical to that calculated when only linear terms were considered in the analysis, Eq. (15), as expected. The third-harmonic content is frequency dependent, and goes to zero when the modulation frequency

Ω

goes to zero, i.e. the system becomes linear, up to third order calculated here, in the DC limit, as anticipated from the DC limit of the Adler’s equation discussed in the introduction.

In the next section, we extend the methods developed here to calculate intermodulation when two signals at different frequencies constitute the input of the modulator. The two-tone response forms a basis for the calculation of the improvement of the SFDR in the RCII modulator over the Mach-Zehnder modulator as a function of modulation frequency $Ω .$

2.3 Third-order intermodulation

In the previous section, we have shown that at finite frequency, the RCII modulator exhibits third-order nonlinearity, which manifests itself in the presence of the third harmonic of the input signal, Eq. (27). Furthermore, the third-harmonic content is frequency dependent and disappears at DC. Here, we extend the analysis to calculate the frequency dependence of intermodulation when two different tones with frequencies $Ω_{1}$ and $Ω_{2}$ are fed into the modulator. The third-order intermodulation products occur on both sides of the input frequencies at $2 Ω_{1} - Ω_{2}$ and $2 Ω_{2} - Ω_{1} .$ In the RCII modulator, the two-tone modulation is accounted for by replacing the right-hand side of Eq. (3) with a superposition of two sine-waves

\frac{d ϕ}{d t} + \sin ϕ = Ω_{m} \sin (Ω_{1} t) + Ω_{m} \sin (Ω_{2} t) .

For simplicity, and consistency with commonly used approach, the modulation amplitude

Ω_{m}

is the same for the two tones. Since, in general, frequencies

Ω_{1}

and

Ω_{2}

may be incommensurable, the solution

ϕ

to the Eq. (28) is not necessarily periodic, and is sought in the form of a dual Fourier series

ϕ (t) = \sum_{n_{1}, n_{2} = - \infty}^{\infty} a_{(n_{1}, n_{2})} \sin [(n_{1} Ω_{1} + n_{2} Ω_{2}) t + φ_{(n_{1}, n_{2})}] = \sum_{n} a_{n} \sin (n \cdot Ω t + φ_{n}),

where compact notation has been introduced

n = (n_{1}, n_{2}), Ω = (Ω_{1}, Ω_{2}) .

To ensure uniqueness of the amplitudes

a_{n}

and phases

φ_{n},

the sum in the series (29) is carried over

n_{1}

and

n_{2}

such that

n_{1} + n_{2} \geq 0.

Substituting the series expansion (29) into Adler’s Eq. (28) yields, with the help of Theorem (46), the following equation

\begin{array}{r} \sum_{n} n \cdot Ω a_{n} \cos (n \cdot Ω t + φ_{n}) + \sum_{k} \prod_{n} J_{k_{n}} (a_{n}) \sin [\sum_{n} k_{n} (n \cdot Ω t + φ_{n})] = \\ Ω_{m} [\sin (Ω_{1} t) + \sin (Ω_{2} t)], \end{array}

which separates into a system of infinitely many equations, each containing terms corresponding to a single frequency—due to the linear independence of terms corresponding to different frequencies.

The linear response is obtained by keeping the lowest-order terms in Eq. (30), which separates then into two equations—one for each of the input frequencies

\begin{matrix} Ω_{1} a_{(1, 0)} \cos (Ω_{1} t + φ_{(1, 0)}) + 2 J_{1} (a_{(1, 0)}) \sin (Ω_{1} t + φ_{(1, 0)}) = Ω_{m} \sin (Ω_{1} t), \\ Ω_{2} a_{(0, 1)} \cos (Ω_{2} t + φ_{(0, 1)}) + 2 J_{1} (a_{(0, 1)}) \sin (Ω_{2} t + φ_{(0, 1)}) = Ω_{m} \sin (Ω_{2} t) . \end{matrix}

Each of Eqs. (31) is identical with Eq. (8), and yields the same linear-response solution as given in Eq. (14)

\begin{array}{l} a_{(1, 0)} = \frac{Ω_{m}}{\sqrt{1 + Ω_{1}^{2}}}, \\ a_{(0, 1)} = \frac{Ω_{m}}{\sqrt{1 + Ω_{2}^{2}}} . \end{array}

To calculate the amplitudes of third-order intermodulation spurs, the terms contributing to frequencies

2 Ω_{1} - Ω_{2}

and

2 Ω_{2} - Ω_{1}

in the series Eq. (30) must be identified. It can be shown that at frequency

2 Ω_{1} - Ω_{2},

corresponding to one of the third-order spurs, Eq. (30) yields

\begin{matrix} (2 Ω_{1} - Ω_{2}) a_{(2, - 1)} \cos [(2 Ω_{1} - Ω_{2}) t + φ_{(2, - 1)}] + 2 J_{1} (a_{(2, - 1)}) \sin [(2 Ω_{1} - Ω_{2}) t + φ_{(2, - 1)}] - \\ 2 J_{2} (a_{(1, 0)}) J_{1} (a_{(0, 1)}) \sin [(2 Ω_{1} - Ω_{2}) t + 2 φ_{(1, 0)} - φ_{(0, 1)}] = 0. \end{matrix}

The first term in Eq. (33) corresponds to the only term in the first series of Eq. (30) with frequency

2 Ω_{1} - Ω_{2},

whereas the last two terms of Eq. (33) are the lowest-order terms in the second series of Eq. (30) with the same frequency. Note the explicit absence of the input amplitude

Ω_{m}

in Eq. (33). The dependence of the intermodulation amplitude

a_{(2, - 1)}

on the input amplitude

Ω_{m}

is indirect—via linear-response amplitudes

a_{(1, 0)}

and

a_{(0, 1)}

found in Eqs. (32). By using techniques similar to those used in Sec. 2.2 when calculating the amplitude of the third harmonic, Eq. (33) can be solved for coefficient

a_{(2, - 1)}

in the dual Fourier series expansion (29)

a_{(2, - 1)} = \frac{a_{(1, 0)}^{2} a_{(0, 1)}}{8 \sqrt{1 + {(2 Ω_{1} - Ω_{2})}^{2}}},

and the amplitude of the intermodulation term with frequency

2 Ω_{1} - Ω_{2}

at the output of the RCII modulator obtained from

\sin ϕ

is found to be

A_{(2, - 1)} = \frac{(2 Ω_{1} - Ω_{2}) a_{(1, 0)}^{2} a_{(0, 1)}}{8 \sqrt{1 + {(2 Ω_{1} - Ω_{2})}^{2}}} = \frac{Ω_{m}^{3}}{8 (1 + Ω_{1}^{2}) \sqrt{1 + Ω_{2}^{2}} \sqrt{1 + {(2 Ω_{1} - Ω_{2})}^{- 2}}} .

Similarly, the output amplitude at intermodulation frequency

2 Ω_{2} - Ω_{1}

is found

A_{(- 1, 2)} = \frac{(2 Ω_{2} - Ω_{1}) a_{(0, 1)}^{2} a_{(1, 0)}}{8 \sqrt{1 + {(2 Ω_{2} - Ω_{1})}^{2}}} = \frac{Ω_{m}^{3}}{8 (1 + Ω_{2}^{2}) \sqrt{1 + Ω_{1}^{2}} \sqrt{1 + {(2 Ω_{2} - Ω_{1})}^{- 2}}} .

For frequencies

Ω_{1}

and

Ω_{2}

such that the difference between them is much smaller than either of these frequencies,

| Ω_{1} - Ω_{2} | ≪ Ω_{1}

and

| Ω_{1} - Ω_{2} | ≪ Ω_{2},

i.e.

Ω_{1} ≃ Ω_{2} = Ω,

expressions for the spur amplitudes (35) and (36) simplify to

A_{spur} = \frac{Ω a_{1}^{3}}{8 \sqrt{1 + Ω^{2}}} = \frac{Ω Ω_{m}^{3}}{8 {(1 + Ω^{2})}^{2}},

where

a_{1}

is the linear coefficient of the Fourier-expansion solution, Eqs. (15), (26), or (32), and at the same time is equal to the (linear) output amplitude of the modulator at the fundamental frequency, Eqs. (15) or (27). Note that the expression for the intermodulation amplitude Eq. (37) is similar in structure to the third-harmonic amplitude Eq. (27). This should come as no surprise since both are the result of the same underlying nonlinear phenomena of the RCII modulator.

The explicit dependence of the spur amplitude on the input-signal frequency given in Eq. (37) is the basis for calculating the SFDR of the RCII modulator, and its comparison with the SFDR of the Mach-Zehnder modulator undertaken in the next section.

3. Spur-free dynamic range of RCII modulator

In the previous section, the frequency response of the RCII modulator has been found, including the linear response, the third-order harmonic content, and the third-order intermodulation spur amplitude. Here, we will use these results to compare the performance of the RCII modulator, in terms of its SFDR, to the conventional Mach-Zehnder modulator. Normally, when SFDR is estimated, the amplitude of the spur is compared to the noise floor of the investigated system. Yet, in the analysis presented above, no noise was assumed. To make the results as general as possible, noise will not be considered, and instead the performance of the modulator under investigation will be compared to a conventional Mach-Zehnder modulator operating with the same noise constraints. This way, the effect of the anticipated linearity improvement afforded by the new modulator is isolated, and at the same time a baseline comparison is made.

Given the definition of SFDR provided in Appendix A.1, to make the comparison of SFDR in the RCII modulator with the same in Mach-Zehnder modulator, only the output coordinates of the third-order intercept points $({OIP}_{3})$ are required for the two modulator configurations. The ${OIP}_{3}$ for the Mach-Zehnder is calculated in the Appendix, Eq. (45). In this section, the ${OIP}_{3}$ for the RCII is obtained. It is easily found by noting that—by the definition of intercept point—when the modulation amplitude $Ω_{m}$ is equal to the input coordinate of the intercept point $({IIP}_{3}^{(RCII)}),$ the output amplitude of the linear response $A_{1}$ given, e.g., in Eq. (27), is equal to the amplitude of the spur $A_{spur}$ found in Eq. (37)

A_{1} = \frac{Ω_{m}}{\sqrt{1 + Ω^{2}}} = \frac{Ω Ω_{m}^{3}}{8 {(1 + Ω^{2})}^{2}} = A_{spur} .

Solving Eq. (38) for

Ω_{m}

yields the input coordinate of the intercept point

{IIP}_{3}^{(RCII)} = \sqrt{\frac{8 {(1 + Ω^{2})}^{3 / 2}}{Ω}} .

The output coordinate of the intercept point

{OIP}_{3}^{(RCII)}

is obtained from

{IIP}_{3}^{(RCII)}

by multiplying the latter with the (linear) gain of the system, which is simply the ratio of the output amplitude at the fundamental frequency over the amplitude of the modulating signal

A_{1} / Ω_{m} = {(1 + Ω^{2})}^{- 1 / 2} .

The result is

{OIP}_{3}^{(RCII)} = 2 \sqrt{2} {(1 + Ω^{- 2})}^{1 / 4} .

The ratio of this value to the output intercept coordinate corresponding to the Mach-Zehnder configuration found in Eq. (45) is the amount of linearity improvement of RCII modulator over the Mach-Zehnder modulator. Converted to dB, the ratio becomes a difference

{SFDR}^{(RCII)} - {SFDR}^{(MZ)} = \frac{2}{3} 20 \log_{10} {(1 + Ω^{- 2})}^{1 / 4} [{dB-Hz}^{2 / 3}],

where the factor of

2 / 3

accounts for the same factor present when calculating SFDR, see Fig. 4 in Appendix A.1. Note that calculating this difference renders the exact position of the noise floor irrelevant to the comparison of the SFDR-s of two modulators. Expression (41) for the SFDR improvement in RCII modulator over MZ modulator is plotted in Fig. 3 .

Fig. 3 Improvement of the SFDR in the RCII modulator over a Mach-Zehnder (MZ) modulator as a function of modulation frequency.

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4. Discussion

Based on Eqs. (15) and (41), and Figs. 2 and 3, the following observations can be made regarding the performance of the resonant-cavity interferometric intensity modulator (RCII) as compared to the conventional Mach-Zehnder (MZ) configuration.

1. Modulation efficiency of the RCII modulator falls as the modulation frequency is increased; the 3-dB point is reached at $Ω = ω_{m},$ i.e. when the modulation frequency is equal to half of the locking range.
2. SFDR of the RCII modulator approaches that of the MZ modulator when the modulation frequency approaches half of the locking range, i.e. when $Ω \to ω_{m} .$
3. With every octave of modulation frequency increase, the SFDR drops by ~2 dB as long as the modulator is operated at frequencies below about 25% of the locking range. For higher frequencies, the SFDR of the RCII modulator asymptotically converges to the SFDR of the MZ modulator.
4. If we consider a practical example where the injected laser is set to operate with 10-GHz locking range, then operating the modulator at frequencies up to 150 MHz (3% of half locking range) would produce a relatively modest 10 dB improvement of SFDR over the Mach-Zehnder modulator. This result seems to stand in direct contradiction with the conclusions of [5] where hundreds of dB in SFDR improvement in the RCII over the Mach-Zehnder have been hypothesized. In fact, no such contradiction exists since the results of [5] are strictly valid only for the modulation frequency equal to zero, whereas the analysis presented here extends that range to arbitrary modulation frequencies. Using Eq. (41), it can be calculated that in order to obtain a 100-dB improvement of SFDR in the RCII modulator over a Mach-Zehnder, the modulation frequency must not exceed $1.0 \times 10^{- 15} ω_{m},$ which in the example considered here corresponds to 5 μHz.

Intuitively, the frequency limitation of the RCII modulator can be thought of as follows. As the modulation frequency approaches the edge of the locking range, it is expected that the oscillator will be hard-pressed to follow the modulating signal. Therefore, the phase-modulation response will be diminished—hence the low-pass-filter behavior and 3-dB drop seen in Fig. 2 at $Ω = ω_{m}$ . At the same time, since the locking phenomenon plays a diminished role at these modulation frequencies, the improvement in linearity over a conventional MZ modulator is expected to be erased as well. At low modulating frequencies, the 2-dB/octave improvement in SFDR as the frequency is reduced is the result of the $2 / 3$ factor that accompanies the calculation of the SFDR.

To achieve at least 10 dB of improvement in SFDR over Mach-Zehnder, the RCII modulator would have to operate at frequencies below 3% of the half-locking range. If we set the operating frequency at 10 GHz, a reasonable bandwidth for practical applications, it means the half-locking range needs to be of the order of 300 GHz. Since the half-locking range is determined by the locked oscillator Q and the ratio of the injected power $I_{1}$ to the output power $I_{0}$ according to [7],

ω_{m} = \frac{ω_{0}}{Q} \sqrt{\frac{I_{1}}{I_{0}}},

such wide locking range can be achieved by either reducing the Q of the laser cavity or by increasing the injection power relative to the output power of the injected laser. The injection ratio is limited by the emergence of nonlinearities and instabilities, as analyzed in Ref [8]. On the other hand, reducing the Q of the laser cavity requires higher pump currents to achieve lasing, but may be a preferable avenue. In this case, reducing the cavity size would be beneficial. Therefore, it is expected that VCSELs would be preferable over DFB lasers in this regard. Even better, photonic-crystal lasers or plasmonic lasers [9] occupying a fraction of

λ^{3}

could outperform other types of lasers in RCII modulators.

Appendix

A.1 Spur-free dynamic range (SFDR)

SFDR is defined as a difference (on a dB scale) between the minimum detectable input signal level, and the input level which will produce distortion products (spurs) equal to the minimum detectable signal referred to the input of the system [3, 4]. For a system limited by third-order nonlinearity, such as the modulator analyzed in this article, the situation is illustrated in Fig. 4 . The linear response of the system is shown as a green line with a slope of 1 (on dB scale). The minimum detectable signal is determined by the noise floor (for a pre-defined finite signal bandwidth), and the system gain that controls the position (but not the slope) of the green line on the graph. The input coordinate of the intersection of the linear response (green line) with the noise floor (horizontal blue line) is the minimum detectable signal.

Fig. 4 Obtaining SFDR from the coordinates of the third-order spur intercept and the noise floor. SFDR is equal to 2/3 of the separation of the OIP₃ from the noise floor.

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The red line represents the third-order spurs. On the dB scale, the spur line has a slope of 3. The input coordinate of the intersection of the spur (red) line with the noise floor is the (input) signal level that produces spurs equal to the minimum detectable signal. According to the definition provided above, the SFDR is the difference between the input coordinates of the intersection of the red line with the blue line, and the input coordinate of the intersection of the green line with the blue line, as indicated in Fig. 4 by the horizontal interval labeled SFDR. Note that since the slope of the linear-response line is 1 in this graph, the SFDR can also be defined with respect to output signal levels as indicated by the vertical interval labeled SFDR.

Because the linear response line has a slope 1, and the spur line has a slope 3, the two lines intersect. Furthermore, if any useful signal is to be recovered, the intersection has to occur above the noise floor. Define ${IIP}_{3}$ as the input coordinate of the intercept point between the linear-response line and the spur line. Similarly, define ${OIP}_{3}$ as the output coordinate of the intercept point between the linear-response line and the spur line. Since the slope of the green line is 1 and the slope of the red line is 3, it is easy to ascertain using elementary geometry that the SFDR, measured using the vertical interval, is $2 / 3$ of the difference between the ${OIP}_{3}$ and the noise floor. This is the definition of the SFDR that is used here.

A.2 Third-order spurs in Mach-Zehnder modulator

As an illustration of the method of finding the intercept with the third-order spur discussed at the beginning of Sec. 3, consider a Mach-Zehnder (MZ) modulator. If the modulator is biased in quadrature, the transfer function is

f (x) = \sin (x) .

To obtain intermodulation, a superposition of two sine-waves with the same amplitude, but different frequencies is used for the argument of the transfer function

f (m \sin (Ω_{1} t) + m \sin (Ω_{2} t)) = \sin [m \sin (Ω_{1} t) + m \sin (Ω_{2} t)] .

The series expansion of the sine function of a trigonometric series is carried out according to the Theorem of Appendix A.3, Eq. (46). The series is then truncated at lowest-order terms in the modulation amplitude m

\begin{matrix} f (x) = \sin [m \sin (Ω_{1} t) + m \sin (Ω_{2} t)] \\ = \sum_{k_{1}, k_{2} = - \infty}^{\infty} J_{k_{1}} (m) J_{k_{2}} (m) \sin [(k_{1} Ω_{1} + k_{2} Ω_{2}) t] \\ ≃ 2 J_{1} (m) J_{0} (m) \sin (Ω_{1} t) + 2 J_{0} (m) J_{1} (m) \sin (Ω_{2} t) - \\ 2 J_{2} (m) J_{1} (m) \sin [(2 Ω_{1} - Ω_{2}) t] - 2 J_{1} (m) J_{2} (m) \sin [(2 Ω_{2} - Ω_{1}) t] \\ ≃ 2 J_{1} (m) J_{0} (m) [\sin (Ω_{1} t) + \sin (Ω_{2} t)] - \\ 2 J_{2} (m) J_{1} (m) {\sin [(2 Ω_{1} - Ω_{2}) t] + \sin [(2 Ω_{2} - Ω_{1}) t]} \\ ≃ (m - \frac{3}{8} m^{3}) [\sin (Ω_{1} t) + \sin (Ω_{2} t)] - \frac{1}{8} m^{3} {\sin [(2 Ω_{1} - Ω_{2}) t] + \sin [(2 Ω_{2} - Ω_{1}) t]}, \end{matrix}

where the last equality is obtained by Taylor expansion of the Bessel functions and retaining terms up to third order in the modulation amplitude m. Equating the coefficients of the linear response and the third-order intermodulation amplitude in the last of Eqs. (44),

m = \frac{1}{8} m^{3},

immediately yields the third-order spur output intercept coordinate

{OIP}_{3}^{(MZ)} = 2 \sqrt{2},

which is identical to the input coordinate of the third-order spur intercept point

{IIP}_{3}^{(MZ)}

since the (linear) gain of the system is one, per the last of Eqs. (44).

Note that the frequency response of an ideal MZ modulator considered here is flat, i.e. the output is frequency independent, see Eq. (43). As a result, the output intercept coordinate given in Eq. (45) that characterizes the third-order spurs does not depend on the frequency of the input signal.

A.3 Fourier expansion of the sine of trigonometric series

Theorem:

\sin (\sum_{n} a_{n} \sin x_{n}) = \sum_{k} \prod_{n} J_{k_{n}} (a_{n}) \sin (k \cdot x),

where $k = (k_{1}, k_{2}, \dots, k_{n}, \dots),$ each $k_{n}$ spans all integers ( $k_{n} \in ℤ, \forall n$ ), and $k \cdot x = \sum_{n} k_{n} x_{n} .$

Proof:

Begin by rewriting the outer sine function in exponential form, and replacing the exponential of a sum with the product of the exponentials of each term,

\begin{matrix} \sin (\sum_{n} a_{n} \sin x_{n}) = \frac{1}{2 i} [\exp (i \sum_{n} a_{n} \sin x_{n}) - \exp (- i \sum_{n} a_{n} \sin x_{n})] \\ = \frac{1}{2 i} [\prod_{n} \exp (i a_{n} \sin x_{n}) - \prod_{n} \exp (- i a_{n} \sin x_{n})] . \end{matrix}

Using Fourier expansion of the exponential function

\exp (i z \sin ϕ) = \sum_{k = - \infty}^{\infty} J_{k} (z) e^{i k ϕ},

where coefficients

J_{k} (z)

are Bessel functions of the first kind, with

z = a_{n}

and

ϕ = \pm x_{n},

the last expression can be written as

\sin (\sum_{n} a_{n} \sin x_{n}) = \frac{1}{2 i} [\prod_{n} \sum_{k_{n} = - \infty}^{\infty} J_{k_{n}} (a_{n}) \exp (i k_{n} x_{n}) - \prod_{n} \sum_{k_{n} = - \infty}^{\infty} J_{k_{n}} (a_{n}) \exp (- i k_{n} x_{n})] .

The distributive property of multiplication over addition, and commutative and associative properties of multiplication allow the following:

\begin{matrix} \sin (\sum_{n} a_{n} \sin x_{n}) = \frac{1}{2 i} [\sum_{k_{1} = - \infty}^{\infty} \sum_{k_{2} = - \infty}^{\infty} \dots \sum_{k_{n} = - \infty}^{\infty} \dots \prod_{n} J_{k_{n}} (a_{n}) \exp (i k_{n} x_{n}) - \\ \sum_{k_{1} = - \infty}^{\infty} \sum_{k_{2} = - \infty}^{\infty} \dots \sum_{k_{n} = - \infty}^{\infty} \dots \prod_{n} J_{k_{n}} (a_{n}) \exp (- i k_{n} x_{n})] \\ = \frac{1}{2 i} [\sum_{k} \prod_{n} J_{k_{n}} (a_{n}) \exp (i k_{n} x_{n}) - \sum_{k} \prod_{n} J_{k_{n}} (a_{n}) \exp (- i k_{n} x_{n})] \\ = \frac{1}{2 i} {\sum_{k} [\prod_{n} J_{k_{n}} (a_{n})] [\prod_{n} \exp (i k_{n} x_{n})] - \sum_{k} [\prod_{n} J_{k_{n}} (a_{n})] [\prod_{n} \exp (- i k_{n} x_{n})]} \\ = \frac{1}{2 i} {\sum_{k} [\prod_{n} J_{k_{n}} (a_{n})] \exp (i \sum_{n} k_{n} x_{n}) - \sum_{k} [\prod_{n} J_{k_{n}} (a_{n})] \exp (- i \sum_{n} k_{n} x_{n})} \\ = \sum_{k} [\prod_{n} J_{k_{n}} (a_{n})] \frac{1}{2 i} (e^{i k \cdot x} - e^{- i k \cdot x}) \\ = \sum_{k} \prod_{n} J_{k_{n}} (a_{n}) \sin (k \cdot x), \end{matrix}

which completes the proof.

References and links

1. C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006). [CrossRef]

2. D. J. F. Barros and J. M. Kahn, “Optical modulator optimization for orthogonal frequency-division multiplexing,” J. Lightwave Technol. 27(13), 2370–2378 (2009). [CrossRef]

3. “Spurious-free dynamic range,” Wikipedia, The Free Encyclopedia,http://en.wikipedia.org/wiki/Spurious-free_dynamic_range, (accessed April 9, 2012).

4. W. F. Egan, Practical RF system design (IEEE Press; Wiley-Interscience, 2003), pp. xxv, 386 p.

5. N. Hoghooghi, I. Ozdur, M. Akbulut, J. Davila-Rodriguez, and P. J. Delfyett, “Resonant cavity linear interferometric intensity modulator,” Opt. Lett. 35(8), 1218–1220 (2010). [CrossRef] [PubMed]

6. R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946).

7. A. E. Siegman, Lasers (University Science Books, 1986), pp. xxii, 1283 p.

8. N. Hoghooghi and P. J. Delfyett, “Theoretical and experimental study of a semiconductor resonant cavity linear interferometric intensity modulator,” J. Lightwave Technol. 29(22), 3421–3427 (2011). [CrossRef]

9. R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009). [CrossRef] [PubMed]

Dynamically induced nonlinearity in a resonant-cavity interferometric intensity modulator

Abstract

1. Introduction

2. Fourier-series solution to Adler’s equation

2.1 First-order approximation

2.2 Third-order approximation

2.3 Third-order intermodulation

3. Spur-free dynamic range of RCII modulator

4. Discussion

Appendix

A.1 Spur-free dynamic range (SFDR)

A.2 Third-order spurs in Mach-Zehnder modulator

A.3 Fourier expansion of the sine of trigonometric series

Theorem:

Proof:

References and links

Cited By

Figures (4)

Equations (55)

Optics Express