JP7774804B2 - Optical Function Generator - Google Patents

Description

Article 30, Paragraph 2 of the Patent Act applies. Website address: https://doi.org/10.1109/JQE.2021.3114173 Date of website publication: September 20, 2021 (Reiwa 3).

The present invention relates to an optical function generator.

Ultrashort optical pulses have a wide range of applications, including ultrafast optical communications, optical measurement, and optical signal processing. Various pulse light sources have been developed, including Q-switched, gain-switched, and mode-locked lasers. Mode-locked lasers, among others, are widely used as picosecond to femtosecond ultrashort optical pulse sources with repetition rates in the MHz to GHz range. It is known that the shape of the optical pulses output from a mode-locked laser can generally be converted to Gaussian pulses by inserting an optical filter with Lorentzian transmission characteristics into the laser resonator (see, for example, Non-Patent Documents 1 and 2). Furthermore, by utilizing a nonlinear optical effect known as the soliton effect within the resonator, it is possible to generate sech pulses; such lasers are known as soliton lasers (see, for example, Non-Patent Documents 3 and 4). More recently, it has been demonstrated that parabolic pulses, which have a parabolic shape, can also be generated using non-linear optical effects (see, for example, Non-Patent Documents 5, 6, and 7). Such parabolic pulses are also known as similaritons because of their self-similar wave propagation characteristics (see, for example, Non-Patent Document 8).

To generate pulse waveforms other than the Gaussian, Sech, and parabolic waveforms mentioned above, a technique for spectrally shaping the waveform outside the mode-locked laser is used. Specifically, if the waveform and spectrum of the pulse directly output from the laser are u(t) and U(ω), respectively, and the waveform to be generated is a(t) and its frequency spectrum is A(ω), then by inserting an optical filter outside the laser whose transfer function is given by F(ω) = A(ω)/U(ω), an optical pulse with a waveform given by a(t) can be obtained.

One proposed method for realizing such an optical filter is to use a diffraction grating to separate ultrashort optical pulses and then use a spatial optical filter to impart the desired amplitude and phase changes to each frequency component. Furthermore, programmable optical filters using LCoS (Liquid Crystal on Silicon) have been put to practical use as such waveform-shaping optical circuits (see, for example, Non-Patent Document 9).

Another known technique is to use an optical frequency comb instead of a mode-locked laser. That is, when a CW laser is input into an optical comb generator, multiple longitudinal modes of equal intensity and spaced apart are generated from a single longitudinal mode. By passing the resulting flat optical comb through an optical filter with a transfer function given by A(ω), an optical pulse with a spectral shape given by A(ω) and a temporal waveform given by a(t) can be obtained.

D. Kuizenga and A. Siegman, “FM and AM mode locking of the homogeneous laser - Part I: Theory”, IEEE J. Quantum Electron., November 1970, vol. 6, no. 11, pp. 694-708 H. A. Haus, “A theory of forced mode locking”, IEEE J. Quantum Electron., July 1975, vol. QE-11, no. 7, pp. 323-330 L. F. Mollenauer and R. H. Stolen, “The soliton laser”, Opt. Lett., January 1984, vol. 9, no. 1, pp. 13-15 H. A. Haus and M. F. Islam, “Theory of the soliton laser”, IEEE J. Quantum Electron., August 1985, vol. QE-21, no. 8, pp. 1172-1188 D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave-breaking-free pulses in nonlinear-optical fibers”, J. Opt. Soc. Am. B, July 1993, vol. 10, no. 7, pp. 1185-1190 K. Tamura and M. Nakazawa, “Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers”, Opt. Lett., Jan. 1996, vol. 21, no. 1, pp. 68-70 T. Hirooka and M. Nakazawa, “Parabolic pulse generation by use of a dispersion-decreasing fiber with normal group-velocity dispersion”, Opt. Lett., March 2004, vol. 29, no. 5, pp. 498-500 M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers”, Phys. Rev. Lett., June 2000, vol. 84, no. 26, pp. 6010-6013 G. Baxter, S. Frisken, D. Abakoumov, H. Zhou, I. Clarke, A. Bartos, and S. Poole, “Highly programmable wavelength selective switch based on liquid crystal on silicon elements switching”, in OFC 2006, OTuF2 M. Nakazawa, E. Yoshida, and Y. Kimura, “Ultrastable harmonically and regeneratively mode-locked polarization-maintaining erbium fiber ring laser”, Electron. Lett., Sep. 1994, vol. 30, no. 19, pp. 1603-1604

However, with the waveform shaping methods described in Non-Patent Documents 1 to 9, the power of the original optical signal is reduced to one-third to one-fifth due to the insertion loss of the optical filter itself. Therefore, it is necessary to compensate for this optical loss using an optical amplifier after shaping. As a result, spontaneous emission noise from the optical amplifier is superimposed on the pulse, significantly degrading the optical signal-to-noise ratio (OSNR). In other words, conventional waveform shaping methods have the problem of making it difficult to generate optical pulses with high output and OSNR. In particular, with methods using an optical frequency comb, the power of the original CW light is dispersed into multiple longitudinal modes, resulting in low power per longitudinal mode during the optical comb signal generation stage, resulting in significant degradation of the OSNR. These waveform shaping methods still pose significant challenges for practical use.

The present invention aims to solve these problems by providing an optical function generator that can generate high-power, high-quality optical pulses with any temporal waveform directly from an AM mode-locked laser without using waveform shaping technology external to the laser.

To achieve this objective, the optical function generator of the present invention has an AM mode-locked laser equipped with an optical intensity modulator, an optical amplifier, and an optical filter within a laser resonator, and the optical filter has variable amplitude and phase characteristics. By setting the amplitude and phase characteristics according to the shape of the optical pulse to be output, optical pulses of any shape can be generated.

In the optical function generator according to the present invention, the optical intensity modulator may be configured so as to be drivable by a sine wave having a repetition frequency of Ω _m , and the amplitude and phase characteristics of the optical filter may be configured by the modulation degree of the optical intensity modulator, the spectrum A(ω) of an optical pulse to be output, and functions A(ω-Ω _m ) and A(ω+Ω _m ) obtained by shifting A(ω) by Ω _m in both positive and negative directions.

In the optical function generator according to the present invention, the optical intensity modulator may superimpose a harmonic of Ω- _m on a driving sinusoidal modulation signal with a repetition rate of Ω- _m . In this case, the optical filter may have amplitude and phase characteristics consisting of functions A(ω-Ω- _m ) and A(ω+Ω-m) obtained by shifting the spectra A(ω) and A(ω) of the optical pulse to be output positively and negatively by the repetition rate of Ω _{- m} , and functions A(ω-NΩ- _m ) and A(ω+NΩ _-m ) obtained by shifting A(ω ₎ positively and negatively by a harmonic NΩ _{-m of Ω-m} . Furthermore, the optical filter may be configured to set a band limit on the amplitude and phase characteristics in accordance with the spectral width of the optical pulse to be output, thereby obtaining a transform-limited (TL) pulse.

Furthermore, in the optical function generator according to the present invention, the optical filter may be configured to be able to generate a Gaussian pulse by providing the amplitude characteristics in the form of a Gaussian function A(ω) and functions A(ω- _Ωm ) and A(ω+ _Ωm ) obtained by repeatedly shifting A(ω) positively and _negatively by an angular frequency _Ωm . Furthermore, the optical filter may be configured to be able to generate a Sech pulse by providing the amplitude characteristics in the form of a Sech function A(ω) and functions A(ω- _Ωm ) and A(ω+ _Ωm ) obtained by repeatedly shifting A(ω) positively and negatively by an angular frequency Ωm.

The optical filter may be configured to generate pulses having a bi-exponential shape by providing the amplitude characteristics as a Lorentz function A(ω) and functions A(ω- _Ωm ) and A(ω+ _Ωm ) obtained by repeatedly shifting A(ω) positively and negatively by an angular frequency _Ωm . The optical filter may be configured to generate pulses having a triangular shape by providing the amplitude characteristics as a squared sinc function A(ω) and functions A(ω- _Ωm ) and A(ω+ _Ωm ) obtained by repeatedly shifting A(ω) positively and negatively by an angular frequency _Ωm .

In the optical function generator according to the present invention, the optical intensity modulator may be configured to superimpose a sine wave of an angular frequency of 3 _{Ω m} on the modulation signal, and the optical filter may be configured to be able to generate a triangular pulse by providing the amplitude characteristic with a squared sinc function A(ω), and functions A(ω-Ω _m ), A(ω+Ω _m ), A(ω-3Ω _m ), and A(ω+3Ω _m ) obtained by repeatedly shifting A(ω ₎ positively and negatively by angular frequencies Ω m and 3 Ω m.

Furthermore, in the optical function generator according to the present invention, the optical filter may be configured to be able to generate a parabolic pulse by providing the amplitude characteristic with A(ω), which is given by the sum of functions sin ω/ ^ω3 and cos ω/ ^ω2 , and functions A(ω- _Ωm ) and A(ω+ _Ωm ) obtained by repeatedly shifting A(ω) positively and negatively by an angular frequency _Ωm . Furthermore, the optical filter may be configured to be able to generate a single-exponential pulse by providing the amplitude and phase characteristic with a Lorentz function A(ω), and functions A(ω- _Ωm ) and A(ω+ _Ωm ) obtained by repeatedly shifting A(ω) positively and negatively by an angular frequency _Ωm .

The optical function generator according to the present invention can generate a pulse train with any temporal waveform and a high OSNR, depending on the amplitude and phase characteristics of the optical filter inserted into the resonator of the mode-locked laser. Therefore, it is possible to provide a highly functional, high-quality optical function generator for optical measurement and optical signal processing with a simple configuration. Furthermore, the present invention can provide an optical function generator that can generate high-output, high-quality optical pulses with any temporal waveform directly from an AM mode-locked laser, without using waveform shaping technology external to the laser.

1 is a block diagram showing the configuration of an optical function generator according to an embodiment of the present invention. 1 is a block diagram for explaining the operating principle of an optical function generator according to an embodiment of the present invention. (a) Transfer function of the optical filter, (b) Shape of the filter with a band limit of 200 GHz (±100 GHz), (c) Waveform of a stationary pulse (Gaussian pulse) obtained by computer analysis, and (d) Its spectrum of the optical function generator according to the first embodiment of the present invention. (a) Transfer function of the optical filter, (b) shape of the filter with a band limit of 200 GHz (±100 GHz), (c) waveform of a stationary pulse (sech pulse) obtained by computer analysis, and (d) its spectrum of the optical function generator according to the second embodiment of the present invention. (a) Transfer function of the optical filter, (b) Shape of the filter that imposes a band limit of 640 GHz (±320 GHz), (c) Waveform of a stationary pulse (biexponential pulse) obtained by computer analysis, and (d) Its spectrum of the optical function generator of the third embodiment of the present invention. (a) Transfer function of the optical filter, (b) Shape of the filter that imposes a band limit of 640 GHz (±320 GHz), (c) Waveform of a stationary pulse (triangular pulse) obtained by computer analysis, and (d) Its spectrum of the optical function generator of the fourth embodiment of the present invention. (a) modulation function, (b) transfer function of the optical filter, (c) shape of the filter that imposes a band limit of 640 GHz (±320 GHz), (d) waveform of a stationary pulse (triangular pulse) obtained by computer analysis, and (e) its spectrum of the optical function generator according to the fifth embodiment of the present invention. (a) Transfer function of the optical filter, (b) Shape of the filter that imposes a band limit of 640 GHz (±320 GHz), (c) Waveform of a stationary pulse (parabolic function pulse) obtained by computer analysis, and (d) Its spectrum of the optical function generator of the sixth embodiment of the present invention. (a) Real part and (b) imaginary part of the transfer function of the optical filter of the optical function generator of the seventh embodiment of the present invention, (c) real part and (d) imaginary part of the filter that imparts a band limit of 640 GHz (±320 GHz), (e) waveform of a stationary pulse (single-exponential pulse) obtained by computer analysis, and (f) its spectrum. FIG. 1 is a block diagram showing the configuration of a 1.55 μm wavelength band harmonic AM mode-locked erbium fiber laser used in an experiment of an optical function generator according to an embodiment of the present invention. 1A shows an optical filter implemented with an LCoS element, and FIG. 1B shows an enlarged waveform of the central portion of the optical function generator according to the first embodiment of the present invention. 11A and 11B show (a) an intensity waveform, (b) a waveform converted into amplitude by taking the square root of (a), and (c) an optical spectrum of a Gaussian pulse obtained using the optical filter of FIG. 11. 10A shows an optical filter implemented with an LCoS element, and FIG. 10B shows an enlarged waveform of the central portion of the optical function generator according to the second embodiment of the present invention. (a) Intensity waveform of the SECH pulse obtained using the optical filter of FIG. 13, (b) waveform converted into amplitude by taking the square root of (a), and (c) optical spectrum. 10A shows an optical filter implemented with an LCoS element, and FIG. 10B shows an enlarged waveform of the central portion of the optical function generator according to the third embodiment of the present invention. 16 shows (a) the intensity waveform, (b) the waveform converted into amplitude by taking the square root of (a), and (c) the optical spectrum of a biexponential pulse obtained using the optical filter of FIG. 15. 10A shows an optical filter implemented with an LCoS element, and FIG. 10B shows an enlarged waveform of the central portion of the optical function generator according to the fourth embodiment of the present invention. 17A and 17B show the intensity waveform of a triangular pulse obtained using the optical filter, (b) the waveform obtained by taking the square root of (a) and converting it into amplitude, and (c) the optical spectrum. 13A shows the absolute value of an optical filter implemented with an LCoS element, and (b) its phase, in the optical function generator according to the sixth embodiment of the present invention. 19A and 19B show (a) an intensity waveform, (b) a waveform converted into amplitude by taking the square root of (a), and (c) an optical spectrum of a parabolic pulse obtained using the optical filter of FIG. 19A. 13A shows the amplitude characteristics and the phase characteristics of an optical filter implemented with an LCoS element in the optical function generator according to the seventh embodiment of the present invention. 22 shows (a) the intensity waveform of a single exponential pulse obtained using the optical filter of FIG. 21, (b) the waveform obtained by taking the square root of (a) and converting it into amplitude, and (c) the optical spectrum.

Hereinafter, an embodiment of the present invention will be described with reference to the drawings.
Mode-locked lasers are generally classified into two types: passively mode-locked lasers and actively (forced) mode-locked lasers. Passively mode-locked lasers can easily generate ultrashort pulses in the femtosecond range by inserting a saturable absorber into the resonator or by generating nonlinear polarization rotation. However, their repetition rates are generally slow, ranging from several tens to several hundred MHz. On the other hand, actively (forced) mode-locked lasers can generate pulses synchronized with an externally supplied modulation signal by inserting an optical modulator into the resonator. In this case, the pulse width is picosecond to subpicosecond, and the repetition rate can be increased to several tens of GHz. Actively (forced) mode-locked lasers can be further classified into two types depending on the modulation method: AM (amplitude modulation) mode-locked lasers and FM (frequency modulation) mode-locked lasers. In this invention, an AM mode-locked fiber laser is used.

1 to 22 show an optical function generator according to an embodiment of the present invention.
FIG. 1 shows an example of the configuration of an optical function generator according to an embodiment of the present invention. Optical fiber 1 constitutes a laser ring resonator. The resonator includes an optical amplifier 2 used as a gain medium, an optical intensity modulator 3 used as a mode locker, and an optical filter 4 whose amplitude and phase characteristics can be arbitrarily set. The optical amplifier 2 may be an erbium-doped fiber amplifier (EDFA), a semiconductor optical amplifier (SOA), or a solid-state laser element. The optical intensity modulator 3 may be a Mach-Zehnder modulator using an LN ( _LiNbO3 ) crystal, and is externally driven by a sine wave M _AM cos(Ω _m t) with a frequency f _m (angular frequency Ω _m = 2πf _m ) and a modulation depth M _AM . As described below, harmonics of f _m may be superimposed on the modulation signal. The optical filter 4 may be a programmable optical filter made of LCoS or the like. As will be described later, it is assumed that a pulse waveform having an asymmetric shape is generated, and it is desirable that the optical filter 4 be able to control not only the amplitude (transmission) characteristics but also the phase characteristics.

The operating characteristics of this optical function generator (AM mode-locked fiber laser) and the design method for the optical filter 4 will be explained using the block diagram in Figure 2. In Figure 1, the loss of the resonator is L, the gain is G, and the transfer function of the optical filter 4 is F _A (ω). In this case, if the electric field amplitude of the light circulating in the resonator is a(t) and its spectrum is A(ω), then in the steady state A(ω) satisfies the following equation (1):

Here, Φ[a(t)] is the Fourier transform of a(t), and Φ[(1+ M _AM cosΩ _m t)a(t)] represents the spectrum of the signal (1+ M _AM cosΩ _m t)a(t) after the signal a(t) has been intensity-modulated by the optical intensity modulator 3. Φ[(1+ M _AM cosΩ _m t)a(t)] is expressed by equation (2) using the spectrum A(ω) of a(t).

Therefore, equation (1) can be written as equation (3). Solving this for F _A (ω) gives equation (4). Since the gain G is chosen to balance with the loss L, we can set GL = 1, and as a result, F _A (ω) from equation (4) is given by equation (5).

Therefore, when it is desired to generate a desired pulse waveform a(t) using an AM mode-locked laser, the spectrum A(ω) can be used to design the transfer function F _A (ω) of the optical filter 4 as shown in equation (5). As will be described later, F _A (ω) is generally a complex function, and the optical filter has not only amplitude transmission characteristics but also phase characteristics. Specifically, when the waveform is symmetric, such as Gaussian or Sech, F _A (ω) has only a real part and does not have phase characteristics. However, when the waveform is asymmetric, such as a single exponential function described in the seventh embodiment, F _A (ω) has not only a real part but also an imaginary part, and the optical filter has phase characteristics.

[First embodiment]
The first embodiment of the present invention can generate a Gaussian pulse. The waveform a(t) of the Gaussian pulse is given by equation (6). T is a parameter representing the pulse width, and the full width at half maximum (FWHM) _Wp of ^a2 (t) and T have the relationship _Wp = 2(ln2) ^1/2 T. From the Fourier transform of equation (6), the spectrum A(ω) of the Gaussian pulse is given by equation (7). The shape of the optical filter 4 for generating this Gaussian pulse is given by equation (8) by substituting equation (7) into equation (5). Gaussian pulses can be generated by inserting this optical filter into the resonator instead of the Lorentz-type optical filter normally used in mode-locked lasers.

FIG. 3(a) shows an example of the shape of the transfer function F _A (ω) of the optical filter 4 used in this embodiment. Here, the pulse width T = 6 ps (W _p = 10 ps), the modulation frequency Ω _m = 2π×10 GHz, and the modulation index M _AM = 1. The black dots in the figure represent the frequencies at which longitudinal modes (10 GHz intervals) of the optical spectrum exist. Although the base of _{F A} (ω) gradually decays, band-limiting may be applied to the extent that the waveform is not distorted. FIG. 3(b) shows the shape of the filter in FIG. 3(a) with a band-limit of 200 GHz (±100 GHz). The results of computer analysis of the steady-state solution of the laser using this filter are shown in FIGS. 3(c) and (d). FIG. 3(c) shows the waveform of a steady-state pulse, and FIG. 3(d) shows its spectrum and the ideal spectral shape (A(ω) in Equation (7)) as black lines. It can be seen that this laser can output a Gaussian pulse.

Second Embodiment
In the second embodiment of the present invention, a Sech pulse can be generated. The waveform a(t) of the Sech pulse is given by equation (9). The full width at half maximum (FWHM) _Wp of ^a2 (t) and T have the relationship _Wp = [ln(3+2×2 ^1/2 )]T. From the Fourier transform of equation (9), the spectrum A(ω) of the Sech pulse is given by equation (10). The shape of the optical filter 4 for generating this Sech pulse is given by equation (11) by substituting equation (10) into equation (5).

FIG. 4(a) shows an example of the shape of the transfer function F _A (ω) of the optical filter 4 used in this embodiment. Here, the pulse width T = 6 ps (W _p = 10.6 ps), the modulation frequency Ω _m = 2π×10 GHz, and the modulation index M _AM = 1. The black dots in the figure represent the frequencies at which longitudinal modes (10 GHz intervals) of the optical spectrum exist. Because F _A (ω) has a flat base, band limiting is necessary. FIG. 4(b) shows the shape of the filter in FIG. 4(a) with a band limit of 200 GHz (±100 GHz). The results of computer analysis of the steady-state solution of the laser using this filter are shown in FIGS. 4(c) and (d). FIG. 4(c) shows the waveform of a steady-state pulse, and FIG. 4(d) shows its spectrum and the ideal spectral shape (A(ω) in Equation (10)) as black lines. It can be seen that this laser can output a sech pulse. Although optical nonlinearity is essential for outputting sech pulses from a normal mode-locked laser, the present invention is characterized in that it does not require optical nonlinearity.

[Third embodiment]
In the third embodiment of the present invention, a bi-exponential pulse can be generated. The waveform a(t) of the bi-exponential pulse is given by equation (12). The full width at half maximum (FWHM) _Wp of ^a2 (t) and T have the relationship _Wp = (ln2)T. From the Fourier transform of equation (12), the spectrum A(ω) of the bi-exponential pulse is given by equation (13). The shape of the optical filter 4 for generating this bi-exponential pulse is given by equation (14) by substituting equation (13) into equation (5).

FIG. 5(a) shows an example of the shape of the transfer function F _A (ω) of the optical filter 4 used in this embodiment. Here, the pulse width T = 6.25 ps (W _p = 4.33 ps), the modulation frequency Ω _m = 2π×10 GHz, and the modulation index M _AM = 1. The black dots in the figure represent the frequencies at which longitudinal modes (10 GHz intervals) of the optical spectrum exist. Because F _A (ω) has a flat base, band limiting is necessary. FIG. 5(b) shows the shape of the filter in FIG. 4(a) with a band limit of 640 GHz (±320 GHz). The steady-state solution of the laser obtained by computer analysis using this filter is shown in FIGS. 5(c) and 5(d). FIG. 5(c) shows the waveform of a steady-state pulse, and FIG. 5(d) shows its spectrum and the ideal spectral shape (A(ω) in Equation (13)) as black lines. It can be seen that this laser can output both exponential pulses.

[Fourth embodiment]
In the fourth embodiment of the present invention, a triangular pulse can be generated. The waveform a(t) of the triangular pulse is given by equation (15). The full width at half maximum (FWHM) _Wp of ^a2 (t) and T have the relationship _Wp = (2-2 ^1/2 )T. From the Fourier transform of equation (15), the spectrum A(ω) of the triangular pulse is given by equation (16). The shape of the optical filter 4 for generating this triangular pulse is given by equation (17) by substituting equation (16) into equation (5).

FIG. 6(a) shows an example of the shape of the transfer function F _A (ω) of the optical filter 4 used in this embodiment. Here, the pulse width T = 37.5 ps (W _p = 22.0 ps), the modulation frequency Ω _m = 2π×10 GHz, and the modulation index M _AM = 1. The black dots in the figure represent the frequencies at which longitudinal modes (10 GHz intervals) of the optical spectrum exist. Because _{F A} (ω) does not attenuate and maintains a constant magnitude even with increasing frequency, a band limit is necessary. FIG. 6(b) shows the shape of the filter in FIG. 6(a) with a band limit of 640 GHz (±320 GHz). The results of a steady-state solution for a laser obtained by computer analysis using this filter are shown in FIGS. 6(c) and (d). FIG. 6(c) shows the waveform of a steady-state pulse, and FIG. 6(d) shows its spectrum and the ideal spectral shape (A(ω) in Equation (16)) as black lines. It can be seen that a triangular pulse can be output from this laser.

Fifth Embodiment
The fifth embodiment of the present invention can generate triangular pulses with higher accuracy than the fourth embodiment. In this embodiment, a 3 Ω _{m harmonic is superimposed on a sine wave with an angular frequency of Ω m as} the modulation signal driving the optical intensity modulator 3. The modulation function is given by equation (18). Here, the coefficient 1/9 is due to the coefficient of the 3 Ω _m component when the triangular wave is expanded into a Fourier series. This modulation function is shown in Figure 7(a). It can be seen that the shape of the modulated signal is closer to a triangle than the normal sine wave cosΩ _m t shown by the black line. When the modulation function of equation (18) is used, the transfer function F _A (ω) of the optical filter 4 for generating the triangular pulse is given by equation (19) instead of equation (17).

The shape of this F _A (ω) is shown in Figure 7(b), and the shape with a 640 GHz (±320 GHz) band limit is shown in Figure 7(c). The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figures 7(d) and (e). Figure 7(d) shows the waveform of the steady-state pulse, and Figure 7(e) shows its spectrum and the ideal spectral shape (A(ω) in equation (16)) as black lines. It can be seen that by superimposing a 3 Ω- _m harmonic on the modulation signal, a triangular pulse can be generated with higher accuracy than in Figure 6(c).

Sixth Embodiment
In the sixth embodiment of the present invention, a parabolic pulse can be generated. The waveform a(t) of the parabolic pulse is given by equation (20). The full width at half maximum (FWHM) _Wp of ^a2 (t) and T have the relationship _Wp = 2(1-1/2 ^1/2 )T. From the Fourier transform of equation (20), the spectrum A(ω) of the parabolic pulse is given by equation (21). The shape of the optical filter 4 for generating this parabolic pulse is given by equation (22) by substituting equation (21) into equation (5).

FIG. 8(a) shows an example of the shape of the transfer function F _A (ω) of the optical filter 4 used in this embodiment. Here, the pulse width T = 25.0 ps (W _p = 27.1 ps), the modulation frequency Ω _m = 2π×10 GHz, and the modulation index M _AM = 1. The black dots in the figure represent the frequencies at which longitudinal modes (10 GHz intervals) of the optical spectrum exist. Because F _A (ω) does not attenuate and maintains a constant magnitude even with increasing frequency, a band limit is necessary. FIG. 8(b) shows the shape of the filter in FIG. 8(a) with a band limit of 640 GHz (±320 GHz). The results of a steady-state solution for a laser obtained by computer analysis using this filter are shown in FIGS. 8(c) and (d). FIG. 8(c) shows the waveform of a steady-state pulse, and FIG. 8(d) shows its spectrum and the ideal spectral shape (A(ω) in Equation (21)) as black lines. It can be seen that this laser can output a parabolic pulse.

Seventh Embodiment
While the above-described embodiments have focused on pulses with time-symmetric waveforms, the seventh embodiment of the present invention can generate a semi-exponential pulse as an example of a pulse with an asymmetric waveform. The waveform a(t) of a semi-exponential pulse is given by Equation (23). The full width at half maximum (FWHM) _Wp of ^a2 (t) and T have the relationship _Wp = (ln2)T/2. By Fourier transforming Equation (23), the spectrum A(ω) of the semi-exponential pulse is given by Equation (24). Unlike the symmetric pulses (first to sixth embodiments), the spectrum of an asymmetric pulse has not only a real part but also an imaginary part. The shape of the optical filter 4 for generating this semi-exponential pulse is given by Equation (25) by substituting Equation (24) into Equation (5). Since A(ω) has an imaginary part, the optical filter also has an imaginary part. Therefore, the optical filter 4 has not only amplitude but also phase characteristics.

Figures 9(a) and 9(b) show an example of the shape of the transfer function F _A (ω) of the optical filter 4 used in this embodiment. Figure 9(a) shows the real part of F _A (ω), and Figure 9(b) shows the imaginary part. Here, the pulse width T = 6.25 ps (W _p = 2.17 ps), the modulation frequency Ω _m = 2π×10 GHz, and the modulation index M _AM = 1. The black dots in the figure represent the frequencies at which longitudinal modes (10 GHz intervals) of the optical spectrum exist. Since _{F A} (ω) does not attenuate and maintains a constant magnitude even with increasing frequency, a band limit is necessary. The shape of the filter in Figures 9(a) and 9(b) with a band limit of 640 GHz (±320 GHz) is shown in Figures 9(c) (real part) and 9(d) (imaginary part). The results of a steady-state solution of the laser obtained by computer analysis using this filter are shown in Figures 9(e) and 9(f). Figure 9(e) shows the waveform of a steady pulse, and Figure 9(f) shows its spectrum and the ideal spectral shape (A(ω) in equation (24)) in black lines. It can be seen that this laser can output a single-exponential pulse.

A specific experimental example is shown below. In the experiment, as shown in FIG. 10, a harmonic AM mode-locked erbium fiber laser (see Non-Patent Document 10) with a repetition rate of 10 GHz and oscillating at a wavelength of 1.55 μm was used, and an LCoS element was inserted into the resonator (resonator length 15.6 m). The filter function F _A (ω) shown in the first to seventh embodiments was implemented in the LCoS element using software. The frequency resolution of the LCoS element was 1 GHz. In the following experimental example, the modulation index M _AM = 1.

The shape of the optical filter used to generate Gaussian pulses in the first embodiment of the present invention is shown in Figure 11. Figure 11(b) is an enlarged view of Figure 11(a). The solid line represents the filter (Figure 3(b)) in which F _A (ω) in equation (8) is band-limited at 200 GHz, and the dashed line represents the filter shape implemented in an LCoS element after approximating this filter with a step function every 10 GHz. The LCoS element used in the experiment originally has a frequency resolution of 1 GHz, but considering fluctuations in the longitudinal mode frequency of the laser, F _A (ω) was approximated in a step-like manner every 10 GHz, which is the longitudinal mode spacing, and this was implemented in the LCoS element.

The waveform and optical spectrum of the Gaussian pulse generated using this filter are shown in Figure 12. Figure 12 (a) shows the intensity waveform ^a2 (t) observed with an optical sampling oscilloscope, (b) shows the amplitude a(t) obtained by taking its square root, and (c) shows the optical spectrum A(ω) observed with an optical spectrum analyzer, displayed in dB (20 log|A(ω)|). The black line shows the experimental results, and the solid line shows the computer analysis results shown in Figures 3(c) and (d). The two lines are in good agreement, demonstrating that the Gaussian pulse was generated as designed using the optical filter in Figure 11. Furthermore, the time-bandwidth product is 0.44, demonstrating that a TL Gaussian pulse was obtained directly from the laser.

The shape of the optical filter used to generate the sech pulse in the second embodiment of the present invention is shown in Figure 13. Figure 13(b) is an enlarged view of Figure 13(a). The solid line represents the filter (Figure 4(b)) in which F _A (ω) in equation (11) is band-limited to 200 GHz, and the dashed line represents the filter shape implemented on an LCoS element after approximating this with a step function every 10 GHz.

The waveform and optical spectrum of the SECH pulse generated using this filter are shown in Figure 14. Figure 14 (a) shows the intensity waveform ^a2 (t) observed with an optical sampling oscilloscope, (b) shows the amplitude a(t) obtained by taking its square root, and (c) shows the optical spectrum A(ω) observed with an optical spectrum analyzer, displayed in dB (20 log|A(ω)|). The black line shows the experimental results, and the solid line shows the computer analysis results shown in Figures 4(c) and (d). The two lines are in good agreement, demonstrating that the SECH pulse was generated as designed using the optical filter in Figure 13. Furthermore, the time-bandwidth product is 0.32, demonstrating that a TL SECH pulse was obtained directly from the laser.

The shape of the optical filter used to generate a biexponential pulse in the third embodiment of the present invention is shown in Fig. 15. Fig. 15(b) is an enlarged view of Fig. 15(a). The solid line represents the filter (Fig. 5(b)) in which F _A (ω) in equation (14) is band-limited to 640 GHz, and the dashed line represents the filter shape implemented on an LCoS element after approximating this filter with a step function every 10 GHz.

The waveform and optical spectrum of the bi-exponential pulse generated using this filter are shown in Figure 16. Figure (a) shows the intensity waveform ^a2 (t) observed with an optical sampling oscilloscope, (b) shows the square root of this waveform converted to amplitude a(t), and (c) shows the optical spectrum A(ω) observed with an optical spectrum analyzer, displayed in dB (20 log|A(ω)|). The black line shows the experimental results, and the solid line shows the computer analysis results shown in Figures 5(c) and (d). The two are in good agreement, demonstrating that the optical filter in Figure 15 can generate bi-exponential pulses as designed.

The shape of the optical filter used to generate triangular pulses in the fourth embodiment of the present invention is shown in Figure 17. Figure 17(b) is an enlarged view of Figure 17(a). The solid line represents the filter (Figure 6(b)) in which F _A (ω) in equation (17) is band-limited to 640 GHz, and the dashed line represents the filter shape implemented on an LCoS element after approximating this with a step function every 10 GHz.

The waveform and optical spectrum of a triangular pulse generated using this filter are shown in Figure 18. Figure (a) shows the intensity waveform ^a2 (t) observed with an optical sampling oscilloscope, (b) shows the square root of this waveform converted to amplitude a(t), and (c) shows the optical spectrum A(ω) observed with an optical spectrum analyzer, displayed in dB (20 log|A(ω)|). The black line shows the experimental results, and the solid line shows the computer analysis results shown in Figures 6(c) and (d). The two are in good agreement, demonstrating that the optical filter in Figure 17 can generate bi-exponential pulses as designed.

The shape of the optical filter used to generate parabolic pulses in the sixth embodiment of the present invention is shown in Figure 19. Figure 19(a) shows the absolute value, and Figure 19(b) shows the phase. The solid line shows the filter (Figure 8(b)) in which F _A (ω) in equation (22) is band-limited to 640 GHz, and the dashed line shows the filter shape implemented on an LCoS element after approximating this with a step function every 10 GHz.

The waveform and optical spectrum of the parabolic pulse generated using this filter are shown in Figure 20. Figure (a) shows the intensity waveform ^a2 (t) observed with an optical sampling oscilloscope, (b) shows the square root of this waveform converted to amplitude a(t), and (c) shows the optical spectrum A(ω) observed with an optical spectrum analyzer, displayed in dB (20 log|A(ω)|). The black line shows the experimental results, and the solid line shows the computer analysis results shown in Figures 8(c) and (d). The two are in good agreement, demonstrating that the optical filter in Figure 19 can generate bi-exponential pulses as designed.

The shape of the optical filter used to generate a single-exponential pulse in the seventh embodiment of the present invention is shown in Figure 21. Figure 21(a) shows the amplitude characteristics, and Figure 21(b) shows the phase characteristics. The solid line in Figure 21(a) shows the filter (Figure 9(c)) in which F _A (ω) in equation (25) is band-limited to 640 GHz, and the dashed line shows the filter shape implemented on an LCoS element after approximating this with a step function every 10 GHz.

The waveform and optical spectrum of a single-exponential pulse generated using this filter are shown in Figure 22. Figure (a) shows the intensity waveform ^a2 (t) observed with an optical sampling oscilloscope, (b) shows the amplitude a(t) obtained by taking the square root of the intensity waveform, and (c) shows the optical spectrum A(ω) observed with an optical spectrum analyzer, displayed in dB (20 log|A(ω)|). The black line shows the experimental results, and the solid line shows the computer analysis results shown in Figures 9(e) and (f). The two are in good agreement, demonstrating that a single-exponential pulse can be generated as designed using the optical filter in Figure 21.

As explained in detail above, the present invention makes it possible to easily generate pulses with any time waveform by appropriately designing the amplitude and phase characteristics of an optical filter inserted into a laser resonator. Pulse trains with time waveforms such as exponential, triangular, and parabolic waveforms obtained by this invention can be used in a wide range of applications, including signal pulses for ultra-high-speed time-division multiplexed optical communications and sampling pulses for ultra-high-speed measurement.

1 Optical fiber 2 Optical amplifier 3 Optical intensity modulator 4 Optical filter 5 Polarization-maintaining erbium fiber 6 Pumping LD
7 WDM coupler 8 Intensity modulator 9 Etalon 10 PZT element 11 Coupler 12 Isolator 13 LCoS element 14 Amplifier 15 Phase shifter 16 10 GHz clock extraction circuit

Claims (10)

an AM mode-locked laser including an optical intensity modulator, an optical amplifier, and an optical filter in a laser resonator;
the optical intensity modulator is provided so as to be drivable by a sine wave having a repetitive angular frequency of Ω _m ;
The optical filter has variable amplitude and phase characteristics, and the amplitude and phase characteristics are given by a transfer function F A (ω) expressed by the following equation (1) using the modulation depth M _AM of the optical intensity modulator, the spectrum A(ω) of the optical pulse to be output, and functions A(ω-Ω _m ) and A(ω+Ω _m ) obtained by shifting A ₍ ω) by Ω _m in both positive and negative directions:
An optical function generator that generates an optical pulse of any shape by setting the amplitude and phase characteristics according to the shape of the optical pulse to be output. an AM mode-locked laser including an optical intensity modulator, an optical amplifier, and an optical filter in a laser resonator;
The optical intensity modulator is driven at a repetition rate of Ω _m A sinusoidal modulated signal of Ω _m The harmonic components are superimposed on the
The optical filter has variable amplitude and phase characteristics, and the amplitude and phase characteristics are such that the spectrum A(ω) and A(ω) of the optical pulse to be output are repeatedly changed at an angular frequency Ω _m A function A(ω-Ω) shifted positively or negatively by _m ), A(ω+Ω _m ), and A(ω) as Ω _m Harmonics of NΩ _m A function A(ω-NΩ) shifted positively or negatively by _m ), A(ω+NΩ _m )
By setting the amplitude and phase characteristics according to the shape of the optical pulse to be output, it is possible to generate an optical pulse having any shape.
An optical function generator characterized by: An optical function generator as described in claim 1 or 2, characterized in that the optical filter is configured to set a band limit on the amplitude and phase characteristics according to the spectral width of the optical pulse to be output, thereby obtaining a TL (Transform-limited) pulse. The optical function generator according to claim 1, characterized in that the optical filter is capable of generating a Gaussian pulse by giving the amplitude characteristic as a Gaussian function A(ω) and functions A(ω-Ω _m ) and A(ω+Ω _m ) obtained by repeatedly shifting A(ω) positively and negatively by an angular frequency Ω _m . The optical function generator according to claim 1, characterized in that the optical filter is configured to be able to generate a sech pulse by giving the amplitude characteristic as a sech function A(ω) and functions A(ω-Ω _m ) and A(ω+Ω _m ) obtained by repeatedly shifting A(ω) positively and negatively by an angular frequency Ω _m . The optical function generator according to claim 1, characterized in that the optical filter is configured to be able to generate pulses having the shapes of both exponential functions by giving the amplitude characteristics as a Lorentz function A(ω) and functions A(ω-Ω _m ) and A(ω+Ω _m ) obtained by repeatedly shifting A( _ω) positively and negatively by an angular frequency Ω m. The optical function generator according to claim _{1, characterized in that the optical filter is capable of generating a triangular pulse by providing the amplitude characteristics as a function A(ω) of the square of a sinc function, and functions A(ω-Ω m )} and A(ω+Ω _m ) obtained by repeatedly shifting A(ω) positively and negatively by an angular frequency Ω m. the optical intensity modulator is provided so as to superimpose a sine wave having an angular frequency of 3 Ω- _m on the modulation signal;
The optical function generator of claim 2, wherein the optical filter is configured to be able to generate a triangular _pulse by providing the amplitude characteristics as a squared sinc function A( _ω ), and functions A(ω- _Ωm ), A(ω+ _Ωm ), A(ω- _3Ωm ), and A(ω+ _3Ωm ) obtained by repeatedly shifting A(ω) positively and negatively by angular frequencies Ωm and 3Ωm. The optical function generator according to claim 1, characterized in that the optical filter is configured to be able to generate a pulse having a parabolic shape by giving the amplitude characteristic A(ω) given by the sum of functions sin ω/ ^ω3 and cos ^ω / _ω2 , and functions A(ω- _Ωm ) and A(ω+ _Ωm ) obtained by repeatedly shifting A(ω) positively and negatively by an angular frequency Ωm. The optical function generator according to claim 1, characterized in that the optical filter is capable of generating a single-exponential pulse by providing the amplitude and phase characteristics as a Lorentz function A(ω) and functions A(ω-Ω _m ) and A(ω+Ω _m ) obtained by repeatedly shifting A(ω ₎ positively and negatively by an angular frequency Ω m.