Thesis organization

This thesis is consisted of four chapters. Chapter 1 is an overview of the GVD measurement methods and our motivation for doing this research. Chapter 2 is the theory of the new GVD measurement method derived form the ASM laser dynamics. Chapter 3 presents the experimental setup and the measurement results to demonstrate the feasibility of the new dispersion measurement method.

Finally Chapter 4 gives a summary about the obtained results and discuss possible future work.

Reference

[1] L. G. Cohen, “Comparison of Single-Mode Fiber Dispersion Measurement Techniques,” IEEE J. Lightw. Technol, vol. LT-3, pp. 958–966, 1985.

[2] J. M. Wiesenfeld and J. Stone, “Measurement of Dispersion Using Short Lengths of an Optical Fiber and Picosecond Pulses from Semiconductor Film Lasers,” IEEE J. Lightw. Technol, vol. LT-2, pp. 464–468, 1984.

[3] C. Palavicini, Y. Jaouën, and G. Debarge, “Phase-sensitive optical

low-coherence reflectometry technique applied to the characterization of photonic crystal fiber properties,” Opt. Lett. vol. 30, pp. 361 – 363, 1984.

[4] P. Merritt, R. P. Tatam, and D. A. Jackson, “Interferometric Chromatic Dispersion Measurements on Short Lengths of Monomode Optical Fiber,”

Opt. Lett., vol. 7, pp. 703-716, 1989.

[5] T. N. Nguyen, T. Chartier, M. Thual, P. Besnard, L. Provino, A. Monteville and N. Traynor, “Simultaneous measurement of anomalous group-velocity dispersion and nonlinear coefficient in optical fibers using soliton-effect compression,” Opt. Commun., vol. 278, pp. 60-65, 2007.

Chapter 2

Theories of the new GVD measurement method

2.1 Theory of ASM modelocking

For many applications of high speed fiber communication[1] and the ultrafast optical processing[2-3], a modelocking fiber laser that can directly output ultrashort (< ps) pulses at a very high repetition rate (>= 10GHz) [4-6] is very advantageous but challenging to achieve. Compared to the traditional modelocking techniques, asynchronous soliton modelocking (ASM) has the benefit to provide a good solution for related issues due to the employed hybrid modelocking effects. In an asynchronous soliton modelocking laser, the deviation frequency, (i.e., the difference between the modulation frequency and the cavity harmonic frequency), is not zero but is about several kHz to tens kHz.

The soliton pulses can react against the frequency shift caused be the modulation frequency deviation and still remain stable mode-locking with a short pulsewidth (<1ps). Moreover, by asynchronous operation the laser noises will also be reduced through the equivalent sliding-filter effect at each roundtrip and the supermode suppression ratio (SMSR) of the laser can be smaller than -70dBm[7]. Most interestingly, the ASM laser exhibits slow periodic variation of the output pulses parameters as will be explained in more details below.

2.1.1 Master equation of ASM modelocking

Under the small roundtrip change assumption, an asynchronously modelocked fiber soliton laser can be described by the master equation as follows: [8] [9]

2 2

represents the effect of the optical filtering, is group velocity dispersion, represents the effect of equivalent fast saturable absorption caused by the polarization additive pulse modelocking (P-APM), is the self phase modulation coefficient, angular modulation frequency, is the number of the cavity round trip, is the time axis measured in the moving frame propagating at a specific group velocity along with the pulse, and is the linear timing walk-off per roundtrip due to asynchronous phase modulation, which can be expressed by

T t

Here δ f is the deviation frequency between the N-th cavity harmonic frequency Nf and the modulation frequency_R f . In the following analyses, the _m sinusoidal modulation curve of the phase modulator will be expanded by the Taylor’s series at the center of the pulse t T₀( ) to the second order:

Such an approximation should be quite accurate since the laser pulsewidth is much shorter than the modulation time period in modelocked fiber lasers.

2.1.2 Variational analysis of ASM

The master equation (2.1) describing ASM can be reformulated as a variational problem and then solved approximately by assuming a reasonable pulse solution ansatz [10]. In the variational approach, the Lagrangian corresponding to the master equation (2.1) is

2 2 *

+

∫

is the saturated gain and the modulation curve has been expanded by (2.3)-(2.5). The master equation (2.1) can de derived from the Lagrange by taking the variation of the functional with respective to and :

which is equivalent to the following equation:

* * functions so that they do not take part in the variational procedure in (2.7).

However, they should be replaced by and respectively after performing the variation. This is the standard technique to deal with loss terms under the variational formulation, since all the non-conserved terms can not be directly handled in the usual Lagrangian formulation. For the ASM fiber lasers, the reasonable pulse solution ansatz [9] is given by

u0 u^*₀ θ( )T is the phase. One can obtain the evolution equations of all the pulse parameters from the reduced Lagrange <L> :

0 ,

and L_ansaltz represents the Lagrange L in which the ansatz (2.9) has been substituted for the function and . u u^*

The evolution equations of the pulse parameters are derived from the corresponding Lagrange equations:

,

where represents each pulse parameter in the ansatz (2.9). The final derived equations for the pulse center frequency, the timing, the chirp, the amplitude, and the pulsewidth are given below:

xi

For simplicity the evolution equation for the phase has been omitted because of irrelevance. Please also note that the linear timing drift effects caused by the asynchronous phase modulation are included in the expressions for and

in (2.4) and (2.5), which are periodic functions of T and .

m1

m2 t T₀( )

2.1.3 Periodic variation of the pulse parameters in ASM

The laser dynamics of ASM lasers can be investigated in terms of the evolution of the pulse parameters described by (2.13)-(2.17). To the lowest-order approximation, only the evolution of the pulse timing and the pulse center frequency are needed to be considered and the variation of all the other pulse parameters can be ignored. Equations (2.13) and (2.14) for chirpless pulse

(β =0) can be further simplified to the following two equations under the assumption that the oscillating pulse timing is much less than the linear timing drift

0( ) t T RT due to asynchronous modulation:

2

These two simplified coupled equations can also be derived from the soliton perturbation theory [8]. The solutions of (2.18) and (2.19) have the following forms: the half peak-to-peak displacements of the sinusoidal variation in the pulse center frequency and the pulse timing. The two solutions (2.20) and (2.21) indicate that to the lowest-order approximation the variation of the pulse timing and the pulse center frequency of ASM laser is simply sinusoidal at the deviation frequency fδ . In addition, the phase difference between the pulse timing and the pulse center frequency is exactly π / 2. This knowledge will be

utilized when we try to experimentally determine the pulse center frequency variation.

The accurate simulation results of the full set of coupled equations (2.13)-(2.17) are shown in Fig. 2.1. The simulation parameters used are given as follows: d_i =0.2, d_r =0.05, k_i =0.4, k_r =0.1, g₀ =4, l₀ =0.8, E₀ =0.47, M =0.8, f = 8 MHz, _R f =1250_H f , and _R δ f =25 kHz. These parameters are estimated by the following procedure. The units for time t is chosen to be 0.5 ps, which is of the same order with the laser pulsewidth. The value of is then determined from the known filter bandwidth (13.5 nm) and the value of is determined from the estimated cavity average dispersion (-4.1 ) as well as the cavity length (25 m). The values of and are from the roughly estimated loss and gain of the cavity. The values of and are estimated from the values of and under the assumption that the pulse is a chirpless fundamental soliton with roughly the unit normalized pulsewidth and the unit normalized amplitude. To be more specific, we have simply set

and required estimated based on the above normalization assumption. The other parameters can be directly estimated from the actual experimental conditions. In this way, a reasonable set of parameters that correspond to the studied fiber laser can be obtained for illustrative studies. These numbers should not be very far from the actual operating conditions of the studied laser.

E0

From the obtained plots, one can clearly observe the slow periodic variation at the deviation frequency as shown in Fig. 2.1 (a)–(d). The half peak-to-peak

displacement of the pulse center frequency variation is found to be ~125 GHz in Fig. 2.1(a), corresponding to the variation of the pulse center wavelength of 1 nm around central wavelength 1550 nm, and the half peak-to-peak displacement of the pulse timing is ~3 ps in Fig. 2.1 (b). Besides these two parameters, other pulse parameters are also found to exhibit smaller but more complicated slow periodic variation. In particular the evolution of the pulse amplitude and the pulse energy are shown in Fig. 2.1 (c) and Fig. 2.1(d) respectively, which clearly indicates that the oscillation is not purely sinusoidal and the components of the higher-order harmonics of the deviation frequency appear. Direct numerical simulation has also been performed to verify the obtained results and the excellent agreement between the variational approach and the direct numerical simulation has been found.

Fig. 2.1 Slow periodic evolution of the pulse parameters: (a) pulse center frequency; (b) pulse timing; (c) pulse amplitude; (d) pulse energy.

2.2 Sinusoidal swept-wavelength pulse light source

Different wavelength components will induce different time delays when the light propagates through the dispersive medium. In the proposed method we choose a swept-wavelength pulse light source to be the light source for the dispersion measurement. The asynchronously modelocked Er-fiber soliton laser exhibits the slow periodic swept-wavelength characteristics and thus is an ideal candidate. With the reasonable assumption that the pulse timing variation is mainly a simple sinusoidal function at the deviation frequency fδ , the photocurrent from the fast photodiode detecting the pulse train can be expressed by is the half peak-to-peak displacement of the sinusoidal pulse timing variation,

t0

Δ

δ( )⋅ is the Dirac’s delta function, and ⊗ stands for the operation of convolution. The periodic pulse timing variation effect can be illustrated as in Fig. 2.2.

Fig. 2.2 The sinusoidal timing position variation

We have assumed that the variation of other pulse-shape parameters can be ignored at least to the first order approximation. This should be a reasonable assumption given with the theoretical results, where the variation of the pulse energy is much smaller in percentage. The Fourier transform of the photocurrent

can be expressed as: the fast photodiode and the pulse intensity distribution respectively, and J_n( )⋅ is the Bessel function of the first kind of order n. The periodic Dirac’s delta functions with the sinusoidal timing variation in the time domain gives rise to the comb-like sub-components in the frequency domain as shown in Fig. 2.3.

That is, the pure sinusoidal timing variation will produce the frequency

components with the amplitudes of J_n(ωΔ , which are spaced equally by the t₀) deviation frequency fδ around the pulse train harmonics mf . When the _H pulsewidth is sub-ps short and the response speed of the photodiode is also fast enough compared to the slow modulation frequency, then the intensity of the n-th sub-component of ( )I ω in (2.23) is simply proportional to |J_n(ωΔt₀) |². Therefore by comparing the peak intensity at the main pulse train harmonic (n=0) to the peak intensity at the first sub-component (n=1) from the experimental data,

0 0 2

the half peak-to-peak displacement Δ of the sinusoidal pulse timing variation t₀ can be identified

10.0255 10.0256 10.0257 10.0258 10.0259 -90

20.0512 20.0513 20.0514 20.0515 20.0516 -90

30.0769 30.0770 30.0771 30.0772 30.0773 -90

40.1027 40.1028 40.1029 40.1030 40.1031 -90

Fig. 2.3 RF spectra of the laser output with 500 kHz span. Data are taken (a) near 10 GHz; (b) near 20 GHz; (c) near 30 GHz; (d) near 40 GHz.

2.3 Experimental determination of the pulse timing variation by analyzing the RF spectra of laser output

Experimentally the peak intensity ratio Δ in (2.24) can be obtained directly by analyzing the RF spectra of the laser output. The output pulse train from the 10 GHz asynchronously mode-locked soliton Er-fiber laser is detected by a fast photodiode and the amplified electric signals of the photodiode are connected to a RF spectrum analyzer. The RF spectra with a smaller span of 500 kHz near the 10 GHz, 20 GHz, 30 GHz, and 40 GHz of the main pulse train harmonics have been shown in Fig. 2.3 (a)-(d) respectively. The values of the RF peak intensities I at A mf and _H I_Bat mf_H +δ f for m=1 to m=4 are represented by the solid squares in Fig. 2.4. According to (2.24), the differences between I and _A are equal to the value of the half peak-to-peak displacement of the sinusoidal pulse timing variation . The value is found to be around 3.5 ps for all the four values of m. The consistency of the estimated values from different orders of pulse train harmonics indicates that the proposed method should be able to give consistent and reasonable results for the pulse timing variation.

t0

Δ

10 15 20 25 30 35 40 45 6

9 12 15 18 21

Intensity Difference (dB)

RF Frequency (GHz)

Fig. 2.4 Intensity difference between the 0th and 1st frequency subcomponents around the 10, 20, 30, 40 GHz pulse train harmonic frequencies.

2.4 Method to identify sinusoidal variation of pulse center frequency

The method developed above for determining the pulse timing variation can be further extended to identify the pulse center frequency (wavelength) variation as well. The pulse center wavelength variation will turn into extra pulse timing variation after the pulse train of ASM propagates through an external section of dispersive optical fiber. As indicated by (2.20) and (2.21), the phase difference between the two sinusoidal variations of the pulse timing and the pulse center frequency is π /2, i.e., sine and cosine respectively. Thus the variation of the pulse timing Δt T₁( ) of the ASM pulse train after propagating through a length L of the optical fiber with the dispersion parameter D will be given by:

0

where ΔλDLcos(RT) is the extra pulse timing variation introduced by the dispersion of the external optical fiber. Based on (2.26), the half peak-to-peak displacement of the pulse center wavelength variation Δ can be determined λ according to

with the additional experimental measurement for Δt₁ .

2.5 Binary frequency-shift-keying (FSK) pulse light source

Slow periodic frequency modulation produces slow periodic timing variation when the pulse train propagates in the dispersive test fiber. Through this effect the fiber GVD can be inferred accurately and sensitively. In principle, the frequency modulation format can be sinusoidal, binary-step-wise, or some other forms. Preliminary experimental demonstration has been achieved with an asynchronous modelocked (ASM) Er-fiber soliton laser and has been described in the previous sections [11-12], where the frequency modulation is sinusoidal.

The possibility by using a binary frequency-shift-keying (FSK) pulse light source instead will be explored in the present section.

The pulse train with a periodic binary frequency modulation will experience a periodic pulse timing position modulation with only two different values. The Fig. 2.5 illustrates the single time delay difference of FSK signals without/with dispersion.

T

Fig. 2.5 The FSK pulse signals (a) without (b) with dispersion

By using the binary frequency-shift-keying modulation format as the example, the final detected signal can be written as:

( ) ( ) ( ) [1 ( )] ( ) optical wavelength shift (Δ ) through the fiber GVD ( D ) with the fiber length λ ( ). Finally is the frequency of the binary frequency-shift-keying modulation. By carrying out the Fourier transform of

L Ω

tot( )

f t , it is not difficult to show that near the first harmonic component at ω_H =2 /π T_H, the power ratio of the 0th (the cavity harmonic frequency) and 1st spectral side-peaks with a frequency separation of Ω can be written as:

This ratio values can be determined accurately from the experimental RF

spectrum of the signals.

2.5.1 Principle of FSK

Frequency-shift keying (FSK) is a frequency modulation scheme in which digital information is transmitted through discrete frequency changes of a carrier wave. The simplest FSK is binary FSK (BFSK). BFSK literally implies using two discrete frequencies to transmit binary (0s and 1s) information. With this scheme, the "1" is called the mark frequency and the "0" is called the space frequency. The time domain of an FSK modulated carrier is illustrated in the Fig.

2.6.

Fig. 2.6 Frequency-shifting keying modulation

The FSK signal can be generated by a FSK modulator consisted of two Mach­

Zehnder Modulators (MZM) and a phase modulator. The Mach ­ Zehnder Modulator can be schematically illustrated in Fig.2.7.

Φ(V2) V₁

V2

Φ(V₁)

in out

Fig. 2.7 Mach­Zehnder Modulator

The modulator has two waveguide arms with a phase modulator respectively and the refraction index is changed by the applied electric field. Then, the refraction index modulation gives rise to phase modulation and the modulator as a whole operates like an interferometer. If the phase difference between Φ(V1) and Φ(V2) is 180°, the output port sees destructive interference with no output light. On the contrary, if the phase difference between Φ(V1) and Φ(V2) is 0°, the output port see constructive interference with highest output light.

2.5.2 Operation of FSK

V_π

V_π

2 V_π

±

Fig. 2.8 FSK modulator

The layout of a FSK modulator can be shown as in Fig. 2.8. The input CW light has a carrier frequency ω_c. The frequency spectrum of “a” point has two peaks at ω ω_c− _RF and ω ω_c + _RF due to the amplitude modulation of the first modulator stage, which is illustrate by Fig. 2.9. Therefore, the main optical spectrum components to be considered are at ω_c, ω ω_c − _RF and ω ω_c + _RF.

t

φ

cos((ω ω_c − _RF) )t cos((ω ω_c + _RF) )t

ωc

c RF

ω ω− ω ω_c + _RF Fig. 2.9 The double side band

When we inject the modulation signal with a 90° phase shit to the Mach­

Zehnder Modulator (MZM) in the bottom of Fig. 2.8, the phases of the spectral components will be different. The phenomenon is shown in Fig. 2.10.

cos[(ω_c −ω_RF) ]t cos[(ω ω_c+ _RF) ]t

Fig. 2.10 Effects of the 90° phase shift

At the final stage of the modulator, the signals from two arms are added with a phase difference determined by the applied voltage of the modulator. When the modulation signal is at

2 V_π

, the spectrum is illustrated in Fig. 2.11. The signal at

c RF

ω ω+ will be totally suppressed by the 180° phase difference.

cos[(ω ω_c− _RF) ]t cos[(ω ω_c+ _RF) ]t

− , the spectrum is illustrated in Fig. 2.12. This time the signal at ω ω_c− _RF will be totally suppressed by the 180° phase difference.

cos[(ω ω_c− _RF) ]t cos[(ω ω_c+ _RF) ]t

ωc

c RF

ω ω− ω ω_c+ _RF

cos[(ω ω_c− _RF)t−180 ]° cos[(ω ω_c+ _RF) ]t

ωc

c RF

ω ω− ω ω_c+ _RF

ωc

c RF

ω ω− ω ω_c+ _RF

Fig. 2.12 Sideband at ω ω_c− _RF is canceled

By the above operation method of FSK, we can get an output signal with double frequencies which can be switched in a controllable way [13].

Reference

[1] M. Nakazawa, H. Kubota, K. Suzuki, E. Yamada, and A. Sahara, "Ultra-high speed and long-distance TDM and WDM soliton transmission technology,”

IEEE J. Sel. Top. Quantum Electron., vol. 6, pp. 363-396, 2000.

[2] J. P. Wang, B. S. Robinson, S. A. Hamilton, and E. P. Ippen, “Demonstration of 40-Gb/s Packet Routing Using All-Optical Header Processing,” IEEE Photon. Technol. Lett., vol. 18, pp. 2275-2277, 2006.

[3] Y. Miyoshi, K. Ikeda, H. Tobioka, T. Inoue, S. Namiki, and K. Kitayama,

“Ultrafast all-optical logic gate using a nonlinear optical loop mirror based multi-periodic transfer function,” Opt. Express, vol. 16, pp. 2570-2577, 2008.

[4] C. R. Doerr, H. A. Haus, and E. P. Ippen, “Asynchronous soliton mode locking,” Opt. Lett., vol. 19, pp. 1958-1960, 1994.

[5] H. A. Haus, D. J. Jones, E. P. Ippen, and W.S. Wong, “Theory of soliton stability in asynchronous modelocking,” IEEE J. Lightwave Technol., vol.

14, pp. 622-627, 1996.

[6] W.-W Hsiang, C.-Y Lin, M.-F Tien, and Y. Lai, “Direct generation of a 10 GHz 816 fs pulse train from an erbium-fiber soliton laser with asynchronous phase modulation,” Opt. Lett., vol. 30, pp. 2493-2495 , 2005.

[7] J. N. Kutz, B. C. Collings, K. Bergman, W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J.

Quantum Electron., vol. 34, pp.1749-1757 ,1998.

[8] H. A. Haus, D. J. Jones, E. P. Ippen, and W.S. Wong, “Theory of soliton stability in asynchronous modelocking,” IEEE J. Lightwave Technol., vol.

14, pp. 622-627, 1996.

[9] H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B, vol. 8, pp. 2068-2076 ,1991.

[10] D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A, vol. 27, pp. 3135 – 3145, 1983.

[11] W.-W Hsiang, C.-Y Lin, M.-F Tien, and Y. Lai, “Direct generation of a 10 GHz 816 fs pulse train from an erbium-fiber soliton laser with asynchronous phase modulation,” Opt. Lett., vol. 30, pp. 2493-2495, 2005.

[12] W.-W. Hsiang, C. Lin, N. Sooi, and Y. Lai, “Long-term stabilization of a 10 GHz 0.8 ps asynchronously mode-locked Er-fiber soliton laser by deviation-frequency locking,” Opt. Express, vol. 14, pp. 1822-1828, 2006.

[13] S. Shimotsu, S. Oikawa, T. Saitou, N. Mitsugi, K. Kubodera, T. Kawanishi, and M. Izutsu, “Single Side-Band Modulation Performance of a LiNbO3 Integrated Modulator Consisting of Four-Phase,” IEEE Photon Tech. Lett., vol. 13, pp. 364-366, 2001.

Chapter 3

Experimental setup and results

3.1 The dispersion measurement system

The theory of our new dispersion measurement method has been explained in the previous chapter and in this chapter we will experimentally demonstrate the feasibility of the method. The pulse train from a wavelength-swept pulse light source will experience dispersion-induced periodic timing variation, which can be easily characterized by a RF spectrum analyzer. In this way the fiber GVD can be inferred accurately. The way of wavelength-sweeping can be sinusoidal, bi-wavelength sweeping, or some other sweeping formats. In this thesis work, the ASM is used as the sinusoidal format and FSK is used as the binary-step-wise format. The schematic setup

The theory of our new dispersion measurement method has been explained in the previous chapter and in this chapter we will experimentally demonstrate the feasibility of the method. The pulse train from a wavelength-swept pulse light source will experience dispersion-induced periodic timing variation, which can be easily characterized by a RF spectrum analyzer. In this way the fiber GVD can be inferred accurately. The way of wavelength-sweeping can be sinusoidal, bi-wavelength sweeping, or some other sweeping formats. In this thesis work, the ASM is used as the sinusoidal format and FSK is used as the binary-step-wise format. The schematic setup

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