Mode locking theory of the Nyquist laser

Masataka Nakazawa; Toshihiko Hirooka

doi:10.1364/OE.24.004981

1. Introduction

Ultra-high capacity and high-speed optical transmission technology is being intensively studied to meet the demand for the construction of a high capacity information network, where increasing the spectral efficiency (SE) in a limited bandwidth is an important subject. A single-channel high-speed transmission with ultra-short pulses has already been reported using the optical time division multiplexing (OTDM) technique [1–4], but its drawback is a low SE caused by an inevitable increase in spectral width. To realize high-speed transmission with a high SE, we proposed the idea of Nyquist pulse OTDM transmission [5], where we interleaved the Nyquist pulse at every symbol period (the zero-crossing period of a sinc-like pulse) and employed demultiplexing with an optical sampling technique or an orthogonal property [6,7].

In the first Nyquist OTDM transmission experiment, the pulse was generated by the spectral filtering of a frequency comb from a CW beam or the spectral curving of mode-locked Gaussian/sech pulses. Since a Nyquist pulse with a high OSNR has advantages as regards Nyquist pulse transmission in terms of realizing higher multiplicities and longer transmission distances, we proposed a mode-locked Nyquist laser that can emit high OSNR Nyquist pulses directly [8]. The laser increased the optical signal to noise ratio (OSNR) by 10 dB compared with conventional methods. The Nyquist laser we demonstrated was a 40-GHz actively and regeneratively/harmonically mode-locked fiber laser with amplitude modulation, in which the manipulation of the spectral edges on both sides of the laser oscillation spectrum played an important role in the generation of Nyquist pulses with different roll-off factors. In [8], we also showed numerically that a stable Nyquist pulse can exist in a fiber cavity if we introduce such an edge-enhanced optical filter.

However, as yet there has been no report on an analytical expression for sinc pulse generation from a Nyquist laser. In this paper, based on Haus’s perturbative approach to mode-locking analysis [9–11], we derive a master equation for a Nyquist pulse laser and prove that there is a steady-state solution for the sinc function when the roll-off factor is zero.

2. Mode-locked Nyquist laser

In this section, we begin our analysis of the Nyquist laser by reviewing our experimental results [8]. Figure 1 shows a block diagram of a Nyquist laser, which consists of a fiber ring cavity with an erbium fiber amplifier, a special optical filter using a liquid crystal on silicon (LCoS) optical filter, an intensity modulator, an etalon, and an optical isolator to provide uni-traveling oscillation. All the devices were polarization-maintaining, and a regenerative and harmonic mode-locking scheme was adopted for pulse generation with a high repetition rate of, for example, 10~40 GHz. To suppress the mode-hopping of the longitudinal modes, we installed an optical etalon with a free spectral range (FSR) of 40 GHz. In addition the repetition rate was stabilized at 40 GHz by using a phase locked-loop (PLL) and controlling the cavity length with a PZT.

Fig. 1 Block diagram of a Nyquist laser. The laser is a regeneratively and harmonically mode-locked fiber laser at 40 GHz, where the thick and thin lines correspond to optical and electrical parts, respectively [8].

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Figure 2 shows typical output waveform characteristics when the roll-off factor was set at zero, where (a), (b), and (c) correspond to the output spectral profile, the spectral filter profile generated by the LCoS, and the obtained sinc Nyquist pulse, respectively. It is important to note that edge-enhanced filtering in the cavity (b) was used to generate a flat spectral profile (a), resulting in a sinc function output (c). To obtain a rectangular spectral profile, the edge of the optical filter was enhanced by approximately 5.3 dB, and we successfully obtained a full width at half maximum (FWHM) of 3 ps for the pulse duration and an output power of 15 mW for an EDFA pump power of 200 mW. We also successfully generated Nyquist pulses with different roll-off factors between 0 and 1.

Fig. 2 Output waveform characteristics when the roll-off factor was set at zero, where (a), (b), and (c) correspond to the output spectral profile, the spectral filter profile generated by the LCoS, and the obtained sinc Nyquist pulse, respectively [8].

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A numerical model of the Nyquist laser, which we described in [8], is shown in Fig. 3. The split-step Fourier method was used for Nyquist pulse propagation in the fiber cavity. Since the fiber cavity was 17 m-long with a large average dispersion of 28.9 ps/km/nm, fiber nonlinearity was not taken into account in the 3 ps pulse generation. Although there was a large dispersion in the fiber cavity it was compensated for linearly outside the laser. Therefore, we may simply ignore the GVD effect when describing the master mode-locking equation. We confirmed that the numerical results were identical even when we included the self phase modulation, which implies that the present laser operates in a linear regime.

Fig. 3 Numerical model for the Nyquist laser described in [8].

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The numerical results, which correspond to the experimental results shown in Fig. 2, are given in Fig. 4, where (a)–(d) correspond to an edge-enhanced optical filter, the transient evolution of the Nyquist pulse from ASE noise, a pulse waveform in a steady-state, and a flat-top output spectral profile, respectively. Here, we obtained a stable sinc function waveform with a pulse width of 2.9 ps, which is almost the same pulse waveform as that obtained in the experiment. The output waveform was completely fitted with a sinc function, which indicates that the numerical simulation results agree well with the experimental results. Using these results as a basis, we start to derive a master equation for a mode-locked Nyquist laser.

Fig. 4 Numerical results, which correspond to the experimental results shown in Fig. 2. (a)–(d) relate to the edge-enhanced optical filter, the transient evolution of the Nyquist pulse from ASE noise, the pulse waveform in a steady-state, and the flat-top output spectral profile, respectively [8].

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3. Amplitude modulation of longitudinal modes and edge enhancement of optical filter for generation of flat-top spectral profile

As seen in section 2, handling of the spectral edge plays an important role in generating a flat-top spectrum. When we apply an amplitude modulation at Ω_m to a single longitudinal mode at ω = nΩ_m, it generates two new side bands on either side of the mode and the amplitude of the original mode is reduced by the modulation. For example, for the nth longitudinal mode v_n(t) = A_n exp(inΩ_mt), where the corresponding spectrum V_n(ω) is given by V_n(ω) = A_n δ(ω − nΩ_m), AM modulation −M(1 − cos (Ω_mt)) generates the spectrum V_n_,AM(ω) = − MA_n δ(ω − nΩ_m) + (M/2)A_n δ(ω − (n + 1)Ω_m) + (M/2)A_n δ(ω − (n − 1) Ω_m). When we have a finite number of longitudinal modes with the same amplitude as shown in Fig. 5(a), the spectrum can change to that in Fig. 5(b) due to the amplitude modulation. It is easily understood that all the longitudinal modes except for the edge modes and new modes are kept at the same amplitude as before the modulation because the side modes, which go out into the adjacent modes, return to the original modes due to the amplitude modulation applied to the adjacent longitudinal modes. The difference occurs at the edge mode, which can be reduced by a factor of 1 – M/2, where M is less than unity, and new high-frequency side bands generated at ω = ± (N + 1)Ω_m have an amplitude of M/2. To maintain a rectangular-like profile with a flat top, the edge mode has to be enhanced by a factor of 1/(1 – M/2) and the new high frequency side bands are suppressed with sharp edge filtering as shown in Fig. 5(c). Thus, we can obtain rectangular-like frequency combs as shown in Fig. 5(d) even after AM modulation. In our experiment, the edge enhancement was about 5.3 dB, and in our simulation it was set at 3 dB. This difference depends on the saturation intensity in the EDFA determined by the pump power, gain, loss, degree of modulation amplitude, and frequency dependence of each optical device.

Fig. 5 Change in the spectral profile caused by the AM modulation (a) a finite number of longitudinal modes with the same amplitude, (b) spectral changes after the amplitude modulation, where a difference occurs at the edge mode that can be reduced by a factor of 1 – M/2, (c) edge mode enhancement to maintain a rectangular-like profile with a flat-top, and (d) rectangular-like frequency comb generation in steady-state laser mode-locking.

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4. Perturbative analysis of laser components in the frequency and time domains and derivation of the master equation for the Nyquist laser

Based on Haus’s perturbative mode-locking approach [9–11], we define each small exponential perturbation that occurs in one circulation in the cavity as follows. Here A_n is the nth longitudinal mode. The perturbations, ΔA_n,gain and ΔA_n,loss, that are generated in the gain and loss sections are given by

Δ A_{n, g a i n} = e^{g} A_{n} - A_{n} ≅ g A_{n},

Δ A_{n, l o s s} = e^{- l} A_{n} - A_{n} ≅ - l A_{n}

and the optical loss caused by the AM modulation is given by

Δ A_{n, \mod} = e^{- M (1 - \cos Ω_{m} t)} A_{n} - A_{n} ≅ - M (1 - \cos Ω_{m} t) A_{n},

where the modulation factor is given by −M(1 − cos (Ω_mt)). Equation (3) can be rewritten in the following form on the assumption that there are many modes like a continuous spectrum

\frac{M}{2} (A_{n + 1} + A_{n - 1} - 2 A_{n}) ≅ \frac{M Ω_{m}^{2}}{2} \frac{\partial^{2} A}{\partial ω^{2}} for Ω_{m} < < 1.

Note here that M is the modulation amplitude corresponding to the degree of loss modulation, which is different from the ordinary degree of AM modulation given by (1 − M cos (Ω_mt)).

To generate the Nyquist pulse, we put the transfer function through an edge-enhanced optical filter to be H(ω). There are two models for handling H(ω). One is the direct use of H(ω), which obeys the following perturbation

Δ A_{n, f i l t e r} = H (ω) A_{n} - A_{n} = (H (ω) - 1) A_{n} .

The other is the same treatment given by the exponential expression as seen in Eqs. (1) and (2), and is given by

Δ A_{n, f i l t e r} = e^{h (ω)} A_{n} - A_{n} ≅ h (ω) A_{n},

where H(ω) = e^h^(ω). These differences correspond to differences in the heights of the potentials of quantum wells in quantum mechanics, that is, Eqs. (5) and (6) correspond to finite and infinite potential well heights, respectively. These two models are shown in Figs. 6(a) and 6(b), respectively, in which we define the relationship between each longitudinal mode and a frequency width of 2ε as the edge-enhanced bandwidth of the optical filter. Here we assume that ε is smaller than Ω_m, which enables us to extract and enhance only one longitudinal mode at the spectral edge. In our analysis, we use Eq. (6) to deal with all perturbations in an exponential manner.

Fig. 6 Two models for the optical filter transfer function. (a) and (b) correspond to finite and infinite heights of the potential well, respectively. We define the relationship between each longitudinal mode and a frequency width of 2ε as the edge-enhanced band width of the optical filter.

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From Eqs. (1)-(6), the sum of all exponential perturbations, ΔA, after one circulation in the laser cavity can be written as

Δ A = (g - l) A + h (ω) A + \frac{M Ω_{m}^{2}}{2} \frac{\partial^{2} A}{\partial ω^{2}} .

Since a steady-state condition is given by ΔA = 0, we obtain the following equation in the spectral region as a general expression

- \frac{M Ω_{m}^{2}}{2} \frac{\partial^{2} A}{\partial ω^{2}} - h (ω) A = (g - l) A .

This Schrödinger equation type analysis in the spectral region will be described in Section 8. By inverse-Fourier transforming Eq. (8) into the time domain, we obtain the following fundamental mode-locking equation

\frac{M Ω_{m}^{2}}{2} t^{2} a (t) - \frac{1}{2 π} \int_{- \infty}^{\infty} h (ω) A (ω) e^{i ω t} d ω = (g - l) a (t),

where

\begin{matrix} A (ω) = \int_{- \infty}^{\infty} a (t) e^{- i ω t} d t \\ a (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} A (ω) e^{i ω t} d ω \end{matrix}}

In Eq. (9), the second term on the left hand side plays an important role in Nyquist pulse generation. From Fig. 6(b), the filter integration becomes as follows

\begin{array}{l} I = \int_{- \infty}^{\infty} h (ω) A (ω) e^{i ω t} d ω \\ = \int_{- ω_{N} - ε}^{- ω_{N} + ε} h_{b} A (ω) e^{i ω t} d ω + \int_{- ω_{N} + ε}^{ω_{N} - ε} h_{a} A (ω) e^{i ω t} d ω + \int_{ω_{N} - ε}^{ω_{N} + ε} h_{b} A (ω) e^{i ω t} d ω . \end{array}

Here, the first term in Eq. (11) is rewritten in the time domain as follows

\begin{array}{l} \int_{- ω_{N} - ε}^{- ω_{N} + ε} h_{b} A (ω) e^{i ω t} d ω \\ = h_{b} \int_{- ω_{N} - ε}^{- ω_{N} + ε} \int_{- \infty}^{\infty} a (t^{'}) e^{i ω (t - t^{'})} d t^{'} d ω \\ = h_{b} \int_{- \infty}^{\infty} a (t^{'}) d t^{'} \int_{- ω_{N} - ε}^{- ω_{N} + ε} e^{i ω (t - t^{'})} d ω \\ = 2 h_{b} \int_{- \infty}^{\infty} a (t^{'}) e^{- i ω_{N} (t - t^{'})} \frac{\sin ε (t - t^{'})}{t - t^{'}} d t^{'} . \end{array}

Similarly, we obtain

\int_{- ω_{N} + ε}^{ω_{N} - ε} h_{a} A (ω) e^{i ω t} d ω = 2 h_{a} \int_{- \infty}^{\infty} a (t^{'}) \frac{\sin (ω_{N} - ε) (t - t^{'})}{t - t^{'}} d t^{'},

and

\int_{ω_{N} - ε}^{ω_{N} + ε} h_{b} A (ω) e^{i ω t} d ω = 2 h_{b} \int_{- \infty}^{\infty} a (t^{'}) e^{i ω_{N} (t - t^{'})} \frac{\sin ε (t - t^{'})}{t - t^{'}} d t^{'} .

Therefore, Eq. (11) becomes

\begin{array}{l} I = 2 h_{b} \int_{- \infty}^{\infty} a (t^{'}) [e^{i ω_{N} (t - t^{'})} + e^{- i ω_{N} (t - t^{'})}] \frac{\sin ε (t - t^{'})}{t - t^{'}} d t^{'} + 2 h_{a} \int_{- \infty}^{\infty} a (t^{'}) \frac{\sin (ω_{N} - ε) (t - t^{'})}{t - t^{'}} d t^{'} \\ = 2 h_{b} \int_{- \infty}^{\infty} a (t^{'}) \frac{\sin (ω_{N} + ε) (t - t^{'})}{t - t^{'}} d t^{'} + 2 (h_{a} - h_{b}) \int_{- \infty}^{\infty} a (t^{'}) \frac{\sin (ω_{N} - ε) (t - t^{'})}{t - t^{'}} d t^{'}, \end{array}

where we used the relationship

\cos ω_{N} (t - t^{'}) \sin ε (t - t^{'}) =

(1 / 2) [\sin (ω_{N} + ε) (t - t^{'}) - \sin (ω_{N} - ε) (t - t^{'})] .

Thus, we can derive a master equation for a Nyquist laser in the time domain as follows

\begin{array}{l} \frac{M Ω_{m}^{2}}{2} t^{2} a (t) - \frac{h_{b}}{π} \int_{- \infty}^{\infty} a (t^{'}) \frac{\sin (ω_{N} + ε) (t - t^{'})}{t - t^{'}} d t^{'} \\ - \frac{(h_{a} - h_{b})}{π} \int_{- \infty}^{\infty} a (t^{'}) \frac{\sin (ω_{N} - ε) (t - t^{'})}{t - t^{'}} d t^{'} = (g - l) a (t) . \end{array}

In the perturbed amplitude modulation given in Eqs. (3) and (9), cos Ω_mt is parabolically approximated as 1−(1/2)(Ω_mt)². Since we have the relationship ε < Ω_m, cos ε(t−t′) and sin ε(t−t′) can also be approximated as 1−(1/2)ε²(t−t′)² and ε(t−t′), respectively. In other words, functions

a_{n} \frac{\sin ω_{N} (t - t^{'})}{t - t^{'}}

and

a_{n} \frac{\cos ω_{N} (t - t^{'})}{t - t^{'}}

change much faster than cos ε(t−t′) and sin ε(t−t′), and therefore, a parabolic approximation can be applied to slow modulation terms given by cos ε(t−t′) and sin ε(t−t′). Thus, we have the following approximation

\frac{\sin (ω_{N} \pm ε) (t - t^{'})}{t - t^{'}} ≅ \frac{\sin ω_{N} (t - t^{'})}{t - t^{'}} \pm ε \cos ω_{N} (t - t^{'}) - \frac{1}{2} ε^{2} (t - t^{'}) \sin ω_{N} (t - t^{'})

and therefore, Eq. (16) can be modified as follows

\begin{array}{l} \frac{M Ω_{m}^{2}}{2} t^{2} a (t) - \frac{h_{a}}{π} \int_{- \infty}^{\infty} a (t^{'}) \frac{\sin ω_{N} (t - t^{'})}{t - t^{'}} d t^{'} + \frac{ε (h_{a} - 2 h_{b})}{π} \int_{- \infty}^{\infty} a (t^{'}) \cos ω_{N} (t - t^{'}) d t^{'} \\ + \frac{ε^{2} h_{a}}{2 π} \int_{- \infty}^{\infty} a (t^{'}) (t - t^{'}) \sin ω_{N} (t - t^{'}) d t^{'} = (g - l) a (t) . \end{array}

With respect to laser mode locking we can generally assume that the spectral amplitude of the longitudinal mode at ω = ω_N is equal to that at ω = −ω_N, that is A(ω_N) = A(−ω_N) = A_N > 0. This symmetry occurs in a Gaussian or sech solution with respect to a center frequency. Thus, the following relationships can be obtained.

\int_{- \infty}^{\infty} a (t^{'}) \cos ω_{N} (t - t^{'}) d t^{'} = \frac{A_{N}}{2} (e^{i ω_{N} t} + e^{- i ω_{N} t}) = A_{N} \cos ω_{N} t,

and, for the 4th term on the left hand side of Eq. (18), it can be rewritten in the form

\int_{- \infty}^{\infty} a (t^{'}) (t - t^{'}) \sin ω_{N} (t - t^{'}) d t^{'} = - {A^{'}}_{N} \cos ω_{N} t + A_{N} t \sin ω_{N} t .

Here, we put derivatives of the spectral profile with respect to ω as

{\frac{d A}{d ω} |}_{ω_{N}} = {- \frac{d A}{d ω} |}_{- ω_{N}} = {A^{'}}_{N} < 0

because the mode-locked laser spectrum is symmetric and has a convex profile with respect to the laser center frequency. The derivation of Eq. (20) is given in Appendix 1. Substituting Eqs. (19) and (20) into Eq. (18), we obtain the following simplified master equation

\begin{array}{l} \frac{M Ω_{m}^{2}}{2} t^{2} a (t) - \frac{h_{a}}{π} \int_{- \infty}^{\infty} a (t^{'}) \frac{\sin ω_{N} (t - t^{'})}{t - t^{'}} d t^{'} + \frac{ε}{π} {(h_{a} - 2 h_{b}) A_{N} - \frac{1}{2} ε h_{a} {A^{'}}_{N}} \cos ω_{N} t \\ + \frac{ε^{2}}{2 π} h_{a} A_{N} t \sin ω_{N} t = (g - l) a (t) . \end{array}

5. Equation that satisfies a sinc function

A spherical wave equation from a polar Maxwell equation, E(r) is given by

\frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial E}{\partial r}) = \frac{\partial^{2} E}{\partial r^{2}} + \frac{2}{r} \frac{\partial E}{\partial r} = \frac{1}{υ^{2}} \frac{\partial^{2} E}{\partial t^{2}},

where r indicates the radial direction, and the phase velocity υ is equal to ω/k, where k is the wave number. Assuming E(r, t) = R(r) T(t), we obtain the following two equations.

\frac{d^{2} R}{d r^{2}} + \frac{2}{r} \frac{d R}{d r} + {(\frac{ω}{υ})}^{2} R = 0,

\frac{d^{2} T}{d t^{2}} + ω^{2} T = 0.

Noting here that Eq. (23-1) is written as

\frac{d^{2} (r R)}{d r^{2}} + {(\frac{ω}{υ})}^{2} (r R) = 0

, we obtain

E (r, t) = \frac{A}{r} e^{i (ω t + \frac{ω}{υ} r)} + \frac{B}{r} e^{i (ω t - \frac{ω}{υ} r)},

where the first term on the right shows a backward spherical wave and the second term shows a forward spherical wave. Here it is important to note that a spherical wave has a

\frac{1}{r} e^{- i \frac{ω}{υ} r}

dependence, which we call the Green function

G (r, r^{'}) = \frac{1}{4 π} \frac{e^{- i k | r - r^{'} |}}{| r - r^{'} |}

, expressing the electromagnetic wave radiation from a point source. This spherical wave has a solution of

\frac{\sin (\frac{ω}{υ} r)}{r}

, which is a sinc function with a variable r. Thus we have a time domain equation that satisfies the sinc function

a (t) = \frac{\sin ω_{N} t}{t}

as follows

(\frac{d^{2}}{d t^{2}} + \frac{2}{t} \frac{d}{d t} + ω_{N}^{2}) a (t) = 0.

In other words, Eq. (23-1) is the lowest order expression of the spherical Bessel equation, which has a sinc function solution.

6. Derivation of sinc function solution from the master equation

To obtain the solution a(t) = sin ω_Nt/t in the master equation, Eq. (21) has to satisfy Eq. (25). To prove this, we newly prepare an operator O given by

O = (\frac{d^{2}}{d t^{2}} + \frac{2}{t} \frac{d}{d t} + ω_{N}^{2})

and operate it on both sides of Eq. (21). Here Eq. (21) can be further simplified by applying Fourier transformation and inverse Fourier transformation. Noting that

ℱ {\frac{\sin ω_{N} (t - t^{'})}{t - t^{'}}} = {\begin{matrix} π, & | ω | \leq ω_{N} \\ 0, & | ω | > ω_{N} \end{matrix},

we can rewrite Eq. (21) in the form

\frac{M Ω_{m}^{2}}{2} t^{2} a (t) - h_{a} a (t) + α \cos ω_{N} t + β t \sin ω_{N} t = (g - l) a (t),

where

α = \frac{ε}{π} {(h_{a} - 2 h_{b}) A_{N} - \frac{ε}{2} h_{a} {A^{'}}_{N}},

β = \frac{ε^{2}}{2 π} h_{a} A_{N} .

Since we have

O {t^{2} a (t)} = t^{2} \frac{d^{2} a (t)}{d t^{2}} + 6 t \frac{d a (t)}{d t} + 6 a (t) + ω_{N}^{2} t^{2} a (t)

and

O {α \cos ω_{N} t + β t \sin ω_{N} t} = - 2 α ω_{N} \frac{\sin ω_{N} t}{t} + 4 β ω_{N} \cos ω_{N} t + 2 β \frac{\sin ω_{N} t}{t},

Equation (28) can be rewritten in the form

\begin{array}{l} \frac{M Ω_{m}^{2}}{2} t^{2} (\frac{d^{2} a (t)}{d t^{2}} + \frac{2}{t} \frac{d a (t)}{d t} + ω_{N}^{2} a (t)) + \frac{M Ω_{m}^{2}}{2} (4 t \frac{d a (t)}{d t} + 6 a (t)) + 4 β ω_{N} \cos ω_{N} t \\ + (2 β - 2 α ω_{N}) \frac{\sin ω_{N} t}{t} = (g - l) (\frac{d^{2} a (t)}{d t^{2}} + \frac{2}{t} \frac{d a (t)}{d t} + ω_{N}^{2} a (t)) . \end{array}

Noting O{a(t)} is equal to zero for a sinc solution, we obtain the following equation from Eq. (33)

\frac{d a (t)}{d t} + \frac{3}{2 t} a (t) = \frac{(α ω_{N} - β)}{M Ω_{m}^{2}} \frac{\sin ω_{N} t}{t^{2}} - \frac{2 β ω_{N}}{M Ω_{m}^{2}} \frac{\cos ω_{N} t}{t} .

Since

(\frac{d}{d t} + \frac{3}{2 t}) (\frac{\sin ω_{N} t}{t}) = \frac{ω_{N} \cos ω_{N} t}{t} + \frac{1}{2} \frac{\sin ω_{N} t}{t^{2}}

, Eq. (34) becomes

(\frac{d}{d t} + \frac{3}{2 t}) (a (t) - \frac{\sin ω_{N} t}{t}) = 0,

when

\frac{2 β}{M Ω_{m}^{2}} = - 1,

\frac{α ω_{N} - β}{M Ω_{m}^{2}} = \frac{1}{2} .

Thus, Eq. (33) finally becomes

{\frac{M Ω_{m}^{2}}{2} t^{2} - (g - l)} (\frac{d^{2}}{d t^{2}} + \frac{2}{t} \frac{d}{d t} + ω_{N}^{2}) a (t) + 2 M Ω_{m}^{2} t (\frac{d}{d t} + \frac{3}{2} \frac{1}{t}) (a (t) - \frac{\sin ω_{N} t}{t}) = 0.

To satisfy Eq. (37) at all times, we have a sinc function solution a(t) = sin ω_Nt/t from the second term and the first term simultaneously satisfies Eq. (25). Thus, we could prove that the master equation given in Eq. (16) has a sinc function solution when ε(t−t′) << 1.

7. Derivation of constraints for steady-state mode locking

From Eqs. (36-1) and (36-2), we obtain

α = 0.

Since we have the following relationship (A( ± (ω_N + ε)) = 0 for a rectangular spectrum up to ± ω_N),

{A^{'}}_{N} = \frac{A (ω_{N} + ε) - A (ω_{N})}{ε} = - \frac{A (ω_{N})}{ε},

α can be simplified as

α = \frac{ε}{π} {(h_{a} - 2 h_{b}) A_{N} - \frac{ε}{2} h_{a} {A^{'}}_{N}} = \frac{ε A_{N}}{π} (\frac{3}{2} h_{a} - 2 h_{b}) .

Thus, we obtain

3 h_{a} = 4 h_{b} .

Since we defined h_a = ln H_a and h_b = ln H_b, we have

H_{b} = H_{a}^{3 / 4} .

The filter amplitudes H_a and H_b must be less than unity and therefore,

H_{a} < H_{b} for H_{b} < 1.

This result indicates that there is edge enhancement of the optical filter on both spectral edges. From Eqs. (30) and (36-1),

M Ω_{m}^{2} = - \frac{ε^{2}}{π} h_{a} A_{N}

. As shown in Eq. (27), noting that the spectral amplitude A_N is equal to π, we obtain

h_{a} = - \frac{M Ω_{m}^{2}}{ε^{2}},

and therefore,

H_{a} = e^{h_{a}} = e^{- \frac{M Ω_{m}^{2}}{ε^{2}}} (< 1),

H_{b} = H_{a}^{3 / 4} = e^{- \frac{3}{4} \frac{M Ω_{m}^{2}}{ε^{2}}} (> H_{a}) .

For the relationship between filter amplitude h_a and net gain g – l, from Eq. (28) when a(t) = sin ω_Nt/t, α = 0, and

β = - \frac{1}{2} M Ω_{m}^{2}

we obtain

- h_{a} = g - l > 0.

This indicates that the net gain g – l is used to compensate for the Nyquist filter loss –h_a (> 0).

The sinc function solution can be confirmed quite simply by substituting a(t) = sin ω_Nt/t into Eq. (21), where we use

\int_{- \infty}^{\infty} \frac{\sin ω_{N} t^{'}}{t^{'}} \frac{\sin ω_{N} (t - t^{'})}{t - t^{'}} d t^{'} = π \frac{\sin ω_{N} t}{t} .

Thus we obtain

\begin{array}{l} \frac{M Ω_{m}^{2}}{2} t \sin ω_{N} t - h_{a} \frac{\sin ω_{N} t}{t} - \frac{h_{a} ε^{2}}{2 π} {A^{'}}_{N} \cos ω_{N} t \\ + \frac{h_{a} ε^{2}}{2 π} A_{N} t \sin ω_{N} t - A_{N} \frac{ε}{π} (2 h_{b} - h_{a}) \cos ω_{N} t = (g - l) \frac{\sin ω_{N} t}{t} . \end{array}

To satisfy Eq. (48) at all times, we have the following equalities

\begin{matrix} Coefficient of t \sin ω_{N} t : & \frac{M Ω_{m}^{2}}{2} + \frac{h_{a} ε^{2}}{2 π} A_{N} = 0 \\ Coefficient of \frac{\sin ω_{N} t}{t} : & - h_{a} = g - l \\ Coefficient of \cos ω_{N} t : & - \frac{h_{a} ε^{2}}{2 π} {A^{'}}_{N} - A_{N} \frac{ε}{π} (2 h_{b} - h_{a}) = 0 \end{matrix}} .

Since A_N = π and

{A^{'}}_{N} = - \frac{A_{N}}{ε}

, we obtain and confirm

h_{a} = - \frac{M Ω_{m}^{2}}{ε^{2}}

and

h_{b} = \frac{3}{4} h_{a}

, respectively, as have already been obtained in Eqs. (44) and (41).

Let us evaluate how H_a, H_b and M are related each other in a steady-state condition. Putting ε = δΩ_m (δ < 1) H_a and H_b become

H_{a} = e^{- \frac{M}{δ^{2}}} and H_{b} = e^{- \frac{3}{4} \frac{M}{δ^{2}}} .

For example, when M = 0.8 (0 < M ≤ 1) and δ = 0.5, we obtain H_a = 0.041 and H_b = 0.091, that is, H_b = 2.21H_a. Here the edge enhancement factor, which is defined by 10 log(H_b/H_a)², is approximately 6.9 dB. In our experiment, the edge enhancement factor was 5.3 dB.

8. Derivation of sinc-like spectral profile from a time-independent Schrödinger equation model

The master equation in the spectral domain given in Eq. (8) has the same form as the time-independent Schrödinger equation in quantum mechanics

- \frac{ℏ^{2}}{2 m} \frac{\partial^{2} u}{\partial x^{2}} + V (x) u = E u .

Here, a wave function u(x), an energy eigen value E, and potential well V(x) correspond respectively to a mode-locked spectral profile A(ω), net gain g – l, and a filter function –h(ω) in Eq. (8). As derived by Haus [9–11], the potential well becomes parabolic when the filter function is given by a Lorenztian or Gaussian profile, and in such a case the Schrödinger equation becomes a harmonic oscillator, which results in Hermite-Gaussian solutions.

In the present case, the situation becomes very simple since the potential is given by constant values. Noting here that V(x) = –h(ω) = – lnH(ω), a potential well V(x) is depicted as shown in Fig. 7, where the potential well has an infinite height so that the wave function is completely confined within V(x) and there is no tunnel effect. Here, we put H_b = 1, H_a = e^h⁰, ω_o = ω_N – ε = x_o, and ω₁ = ω_N + ε = x₁. Therefore, solving the spectral profile A(ω) in Eq. (8) is equivalent to looking for the wave function in a dual potential well with V(x) = V₀ for |x| < x₀ denoting region A and V(x) = 0 for x₀ < |x| < x₁ denoting region B. The Schrödinger equation has boundary conditions such that the wave function u(x) and its first derivative $\frac{\partial u}{\partial x}$ are continuous at the potential boundaries. To obtain a flat-top profile for A(ω), the Schrödinger equation should have no oscillations in region A (|x| < x₀) so that V_o is equal to E. In this case, we have $\frac{\partial^{2} u}{\partial x^{2}} = 0$ , and the solution is u₁(x) = C₁x + C₂. Since the potential well given in Fig. 7 is symmetric, we have a condition of u₁(x₀) = u₁(–x₀) corresponding to A(ω) = A(–ω). Thus, we obtain u₁(x) = C₂. In region B, V₀ = 0 and the solution of the Schrödinger equation is given by

u_{2} (x) = A \sin k x + B \cos k x, k = \sqrt{\frac{2 m E}{ℏ^{2}}} .

Since the potential well has an infinite height, the wave function should be zero at x = x₁, that is u₂(|x₁|) = 0. In addition, the conditions u₂(|x₀|) = C₂ and

{\frac{\partial u_{2}}{\partial x} |}_{x = | x_{0} |} = 0

must be satisfied from the continuity of the Schrödinger equation. Thus, we obtain the wave function in the form

u_{2} (x) = C_{2} \cos k (x - x_{0}),

where

k (x_{1} - x_{0}) = (n + \frac{1}{2}) π (n = 0, 1, 2, \dots) .

Noting here that E = g – l = – h₀, we have

k (x_{1} - x_{0}) = \sqrt{\frac{- 2 h_{0}}{M Ω_{m}^{2}}} \cdot 2 ε = \frac{π}{2},

and hence

h_{0} = \frac{- M Ω_{m}^{2}}{ε^{2}} \cdot \frac{1}{2} {(\frac{π}{4})}^{2},

under the normalization of h_b = 0, or

H_{a} = e^{- h_{0}} and H_{b} = 1.

In Eqs. (45-1) and (45-2), normalizing H_b at unity, we have

H_{a} = e^{- \frac{M Ω_{m}^{2}}{ε^{2}} \cdot \frac{1}{4}},

which is almost the same as Eq. (54) assuming π²/8 ~1. Thus, we obtain a rectangular-like solution in the spectral region as follows.

A (ω) = {\begin{matrix} C_{2}, & | ω | < ω_{0} \\ C_{2} \cos k (ω - ω_{0}), & ω_{0} < | ω | < ω_{1}, \\ 0, & | ω | > ω_{1} \end{matrix} k = \sqrt{\frac{- 2 h_{0}}{M Ω_{m}^{2}}} .

Figure 8 shows the relationship between the spectral profile thus obtained and the edge-enhanced optical filter as the dual flat potential well.

Fig. 7 A dual flat potential well for solving a time-independent Schrödinger equation. The potential well is equivalent to the edge-enhanced optical filter.

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Fig. 8 The relationship between the spectral profile A(ω) obtained from a Schrödinger equation and the spectral filter used as a dual flat potential well. (a) is the spectral profile and (b) is the optical filter profile.

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9. Conclusion

A master equation for a Nyquist laser was derived by using a perturbation theory. Since a sinusoidal AM modulation is less effective at the edge of the longitudinal modes, an edge enhanced optical filter plays an important role in keeping the flat-top spectral profile in the Nyquist laser. By adopting a second-order differential equation that satisfies a sinc function as an operator, we proved directly that the master equation has the sinc solution. Based on the fact that the master equation in the spectral domain is equivalent to the one-dimensional time-independent Schrödinger equation, we showed that there is a sinc-like solution as the wave function in a dual flat potential well. The edge enhancement factor was almost the same as that obtained directly from the master equation in the time domain.

Appendix 1

\begin{array}{l} \int_{- \infty}^{\infty} a (t^{'}) (t - t^{'}) \sin (t - t^{'}) d t^{'} \\ = \frac{1}{2 i} t e^{i ω_{N} t} \int_{- \infty}^{\infty} a (t^{'}) e^{- i ω_{N} t^{'}} d t^{'} - \frac{1}{2 i} e^{i ω_{N} t} \int_{- \infty}^{\infty} t^{'} a (t^{'}) e^{- i ω_{N} t^{'}} d t^{'} \\ - \frac{1}{2 i} t e^{- i ω_{N} t} \int_{- \infty}^{\infty} a (t^{'}) e^{i ω_{N} t^{'}} d t^{'} + \frac{1}{2 i} e^{- i ω_{N} t} \int_{- \infty}^{\infty} t^{'} a (t^{'}) e^{i ω_{N} t^{'}} d t^{'} \end{array}

Since we have the following relationship

\int t a (t) e^{- i ω t} d t = i \frac{d A}{d ω},

the right hand side of Eq. (58) can be rewritten in the form

- \frac{1}{2} ({\frac{d A}{d ω} |}_{ω_{N}} e^{i ω_{N} t} - {\frac{d A}{d ω} |}_{- ω_{N}} e^{- i ω_{N} t}) + \frac{t}{2 i} (A (ω_{N}) e^{i ω_{N} t} - A (- ω_{N}) e^{- i ω_{N} t}) .

Acknowledgment

This work was supported by the JSPS Grant-in-Aid for Specially Promoted Research (26000009). We especially thank to Prof. Erich Ippen of MIT for fruitful discussions and valuable comments regarding our research of the Nyquist laser and Nyquist pulse transmission.

References and links

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7. D. O. Otuya, K. Harako, K. Kasai, T. Hirooka, and M. Nakazawa, “Single-channel 1.92 Tbit/s, 64 QAM coherent orthogonal TDM transmission of 160 Gbaud optical Nyquist pulses with 10.6 bit/s/Hz spectral efficiency,” in OFC (2015), paper M3G.2.

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Mode locking theory of the Nyquist laser

Abstract

1. Introduction

2. Mode-locked Nyquist laser

3. Amplitude modulation of longitudinal modes and edge enhancement of optical filter for generation of flat-top spectral profile

4. Perturbative analysis of laser components in the frequency and time domains and derivation of the master equation for the Nyquist laser

5. Equation that satisfies a sinc function

6. Derivation of sinc function solution from the master equation

7. Derivation of constraints for steady-state mode locking

8. Derivation of sinc-like spectral profile from a time-independent Schrödinger equation model

9. Conclusion

Appendix 1

Acknowledgment

References and links

Cited By

Figures (8)

Equations (65)

Optics Express