20
Comparison between Finite-Difference TimeDomain Method and Experimental Results for
Femtosecond Laser Pulse Propagation
Shinki Nakamura
Ibaraki University
Japan
1. Introduction
There has recently been significant interest in the generation of single-cycle optical pulses by
optical pulse compression of ultrabroad-band light produced in fibers. There have beensome experiments reported on ultrabroad-band pulse generation using a silica fiber(Nakamura et al, 2002a), (Karasawa et al, 2000) and an Ar-gas filled hollow fiber (Karasawa
et al, 2001) , and the optical pulse compression by nonlinear chirp compensation (Nakamura
et al, 2002a), (Karasawa et al, 2001). For these experiments on generating few-optical-cycle
pulses, characterizing the spectral phase of ultrabroad-band pulses analytically as well as
experimentally is highly important.
Conventionally, the slowly varying-envelope approximation (SVEA) in the beam propagation
method (BPM) has been used to describe the propagation of an optical pulse in a fiber(Agrawal, 1995). However, if the pulse duration approaches the optical cycle regime (<10 fs),
this approximation becomes invalid (Agrawal, 1995). It is necessary to use the finite-differencetime-domain (FDTD) method (Joseph & Taflove, 1997), (Kalosha & Herrmann, 2000) without
SVEA (Agrawal, 1995). Previous reports by Goorjian (Goorjian et al., 1992), (Joseph et al.,1993), Joseph (Joseph & Taflove, 1997), (Goorjian et al., 1992), (Joseph et al., 1993), Taflove(Joseph & Taflove, 1997), (Goorjian et al., 1992), (Joseph et al., 1993), (Taflove & Hagness., 2000)
and Hagness (Goorjian et al., 1992), (Taflove & Hagness., 2000) (JGTH) proposed an excellent
FDTD algorithm considering a combination of linear dispersion with one resonant frequency
and nonlinear terms with a Raman response function.
We performed an experiment of chirped 12 fs optical pulse propagation as described in
Section 3. In order to compare FDTD calculation results with the experimentally measured
ultrabroad-band spectra of such an ultrashort laser pulse, we extend the JGTH algorithm tothat considering all of the exact Sellmeier fitting values for ultrabroad-band spectra. Becauseof the broad spectrum of pulses propagating in a fiber, it becomes much more important totake the accurate linear dispersion into account. It is well known that at least two resonant
frequencies are required for the linear dispersion to fit accurately to the refractive index
data. A recent report by Kalosha and Herrmann considers the linear dispersion with two
resonant frequencies and the nonlinear terms without the Raman effect (Kalosha &Herrmann, 2000). For the single-cycle pulse generation experiment, we must use at least the
shortest pulse of 3.4 fs (Yamane et al., 2003) or sub-5 fs (Karasawa et al, 2001), (Cheng et al.,1998) or the commercially available 12 fs pulses. Such a time regime is comparable to thewww.intechopen.com

450

Coherence and Ultrashort Pulse Laser Emission

Raman characteristic time of 5 fs (Agrawal, 1995) in a silica fiber. Therefore, it is very
important to consider not only the accurate linear dispersion of silica but also the Raman
effect in a silica fiber in the few-optical-cycles regime. In addition, because of the highrepetition rate and pulse intensity stability, in particular, ultrabroad-band supercontinuum
light generation and few-optical-cycles pulse generation by nonlinear pulse propagation inphotonic crystal fibers (Ranka et al., 2000) and tapered fibers (Birks et al., 2000), which areboth made of silica, have attracted much attention. In this work, we have extended the
FDTD method with nonlinear polarization PNL involving the Raman response function(JGTH-algorithm) to 12 fs ultrabroadband pulse propagation in a silica fiber with theconsideration of linear polarization PL, including all exact Sellmeier-fitting values of silica
with three resonant frequencies, in order to compare the calculation results with ourexperimental results (Nakamura et al, 2002a), (Karasawa et al, 2000). We have alreadycompared the extended FDTD method (Nakamura et al., 2002b) with BPM by applying the
split-step Fourier (SSF) method which is the solution of a generalized nonlinear Schrödinger
equation (GNLSE) with SVEA(Agrawal, 1995), and the extended FDTD agreed better withthe experimental results than with BPM. However, we have not shown the details of thecalculation algorithm and temporal characteristics, and did not consider a chirp of the initial
incident pulse to a fiber. In this chapter, we show the details of the the extended FDTD
calculation algorithm (Nakamura et al., 2002c), (Nakamura et al., 2003) temporalcharacteristics of the pulse, and consider a chirp of the initial incident pulse to a fiber, andfinally, we demonstrate the group delay compensation which generates the compressed
pulse. Since 2004, the extended FDFD is called as the auxiliary differential Equation (ADE)FDTD (Fujii et al., 2004). Additionally, we compared between the extended FDTD
calculation and experimental result in dual wavelengths pulses propagation in a fiber.
Finally, we investigated the slowly varying envelope approximation breakdown bycomparing between BPM and the extended FDTD numerical results.

2. Extended FDTD algorithm
For simplicity, the electric and magnetic fields are expressed by Ey and Hx and onedimensional propagation along the z direction is considered. The optical fiber is assumed to
be isotropic and nonmagnetic. If a linear configuration is assumed, Maxwell’s equations are
as follows:

∂H x
1 ∂Ey
=
,
∂t
μ0 ∂z
∂Dy ∂H x
=
,
∂t
∂z

(1)

where µ0 is the permeability in a vacuum and Dy is the dielectric flux density. By means of
Yee’s central difference method, Eq. (1) can be expressed by the following, in which the timeand spatial steps are shifted by 1/2 step:H x i + 1/2 = H x i + 1/2 +n + 1/2

Dywww.intechopen.com

n+1
i

n − 1/2(n⎞Δt ⎛ n⎜ Ey i + 1 − Ey i ⎟ ,μ0 Δz ⎝⎠)

Δtn + 1/2
n + 1/2= Dy +
H x i + 1/2 − H x i − 1/2 ,
i
Δz
n

(2)

Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation

451

where n is the time step number and i is the spatial step number. They are t = n Δt and z =iΔz, respectively.
As the third step, let us use

to derive a new Eyn+ 1i

D (ω ) = ε 0ε r (ω ) E (ω )(3)

. Then, the iteration algorithm of the FDTD is complete. The authors

have introduced the linear polarization PL and the nonlinear polarization PNL, corresponding
to the Sellmeier fitting equation into the third step above, at the same time. First, the total
polarization is expressed as(

P = ε 0 χ ( )E + χ ( )E2 + χ ( ) E 3 + A
1

2

3= PL + PNL .

)(4)For simplicity, only the linear polarization is considered at first. The linear polarization isexpressed as
PL = ∑ Pi ,
3

(5)

i =1

Pi = ε 0 ∫ χ i( ) ( t − τ ) E (τ ) dτ ,
t

1

0

(6)

where χ i( ) has a Lorenz form with respect to frequency and is described as follows
1(Karasawa et al, 2001), (Agrawal, 1995):χ i( ) (ω ) =1

Giωi2 ( ε s − 1 )ωi2 + 2 jωδ i2 − ω 2, (i = 1, 2, 3).

(7)Here, j is the imaginary unit, εs is the permittivity in an electrostatic field, ωi is the resonant
angular frequency, δi is the attenuation constant with respect to resonant absorption, and∑ Gi = 1 . If the above is inversely Fourier-transformed by assuming that ωi is constant, the3

i =1

following differential equation is obtained:⎡ χ ( 1) ⎤ ″+ 2δ ⎡ χ ( 1) ⎤ ′+ ω 2 χ ( 1) = 0i⎢ i ⎥i i⎣⎢ i ⎦⎥
⎣
⎦

(8)

1
1
where χ i( ) ( t = 0 ) = 0 and ⎢⎡ χ i( ) ⎥⎤ ′ ( t = 0 ) = Giωi2 ( ε s − 1 ) . If the optical fiber is sufficiently⎣⎦
short that the attenuation term δi due to absorption can be neglected, then δi = 0, so that Eq.
(7) can be described as follows:χ i( ) (ω ) =1

www.intechopen.com

Gi ( ε s − 1 ) ⋅ ωi2

ωi2 − ω 2

, (i = 1, 2, 3).

(9)

452

Coherence and Ultrashort Pulse Laser Emission

Then, ε r (ω ) = 1 + ∑ χ i( ) (ω ) and the Sellmeier equation (Nakamura et al, 2002) is
3

i =1

1

ε r ( ω ) = n (ω ) = 1 + ∑bi ⋅ ωi2.
22
i = 1 ωi − ω
32Therefore,

Gi ( ε s − 1 ) ⋅ ωi2

χ i( ) (ω ) =ωi2 − ω 2

1

=

(10)

bi ⋅ ωi2
ωi2 − ω 2

(11)

where bi = Gi ( ε s − 1 ) . Also, since ∑bi = ∑ Gi ( ε s − 1 ) = ( ε s − 1 ) ∑ Gi = ( ε s − 1 ) , we have
3

3

3

i =1

i =1

i =1

ε s = 1 + ∑ bi .
3

(12)

i =1

The derivation above is summarized as follows:

D (ω ) = ε 0 E (ω ) + P ( ω )3⎛b ⋅ω2 ⎞
= ε 0 ⎜ 1 + ∑ 2i i 2 ⎟ E (ω )⎜⎟i = 1 ωi − ω ⎠⎝⎛ 3 b ⋅ω2 ⎞= ε 0E (ω ) + ε 0 ⎜ ∑ 2i i 2 ⎟ E (ω ) .
⎜ ω −ω ⎟⎝ i =1 i⎠(13)

If the nonlinear polarization is taken into account by returning to Eq. (4), then Eq. (13) can betreated as follows:
E=
where

PNL = ε 0 χ ( )E ∫
3

∞

3
⎞
1⎛
⎜ D − ∑ Pi − PNL ⎟
ε0 ⎝
i =1
⎠

(14)

⎡αδ ( t − τ ) + ( 1 − α ) g R ( t − τ ) ⎦⎤ ⎣⎡E (τ ) ⎦⎤ dτ .

−∞ ⎣

2

(15)

Here, δ(t-τ) is the delta function and α is the ratio of the intensities of the Kerr effect and theRaman effect. Also, g R ( t ) = ⎡ τ 12 + τ 22 τ 1τ 22 ⎤ e −t τ 2 sin ( t τ 1 ) . When Eqs. (11) and (12) are⎣⎦taken into consideration in Eqs. (14) and (15), we obtain

(

)

3
⎛
⎞
Pi″ + ωi2 Pi = ωi2 bi ⎜ D − ∑ Pi − PNL ⎟ .
i =1
⎝
⎠

Further, if the nonlinear polarization PNL in Eq. (15) is described aswww.intechopen.com(16)Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation

453

2
3
PNL = χ ( )E ( t ) ⎡ε 0α E ( t ) + ( 1 − α ) G ⎤ , then⎣⎦G = ε0 ∫g R ( t − τ ) ⎣⎡E (τ ) ⎦⎤ dτ .

∞

2

−∞

(17)

Also, if Eq. (16) is replaced by G, we obtain
3
⎛
⎞
3
3
Pi″ + ωi2 Pi = ωi2 bi ⎜ D − ∑ Pi − χ ( ) ( 1 − α ) E ⋅ G − ε 0 χ ( )α E3 ⎟
i =1
⎝
⎠

leading to the form

(18)

P1″ + ω12 ( 1 + b1 ) P1 + ω12 b1 P2 + ω12 b1 P3

3+ω12 b1 χ ( ) ⋅ ⎣⎡( 1 − α ) EG + ε 0α E3 ⎦⎤ = ω12 b1D

P2″ + ω22 ( 1 + b2 ) P2 + ω22 b2 P1 + ω22 b2 P3+ω22 b2 χ ( ) ⋅ ⎡⎣( 1 − α ) EG + ε 0α E3 ⎤⎦ = ω22 b2 D

(19)

3

P3″ + ω32 ( 1 + b3 ) P3 + ω32 b3 P1 + ω32 b3 P2

3+ω32 b3 χ ( ) ⋅ ⎣⎡( 1 − α ) EG + ε 0α E3 ⎦⎤ = ω32 b3D.

Further, applying a central difference scheme centered at time step n, these equations yield
a1 P1

n+1

+ c1 P2(

n+1= c1 D+ c1 P3

n+1

n+1

+D

n−1

) − c (P1

+ 4 P1 + g1 P1

n−1

nc 2 P1

n+1+ a2 P2(

n+1= c2 D+ c 2 P3

n+1

+D

n+ 1
n−1

) − c (P1

+ 4 P2 + g2 P2

n−1

n+1

+ c 3 P2(

n+1

= c3 D+ a3 P3

n+ 1

+D

n+1
n−1

3

1

+ 4 P3 + g3 P3

n−1

n

Where

www.intechopen.com) − c (P

+ P3

n−1

+ P3

n−1

+ P2

n−1

)

( )

n
n
n 3⎤
3 ⎡
− ω12 b1 χ ( ) ⋅ ⎢( 1 − α ) E G + ε 0α E
⎥
⎣
⎦

n−1

2

n

c 3 P1

n−1

2

)

( )

n
n
n 3⎤
3 ⎡
− ω22 b2 χ ( ) ⋅ ⎢( 1 − α ) E G + ε 0α E
⎥
⎣
⎦

n−1

)

( )

n
n
n 3⎤
3 ⎡
− ω32 b3 χ ( ) ⋅ ⎢( 1 − α ) E G + ε 0α E
⎥,
⎣
⎦

(20)

454

Coherence and Ultrashort Pulse Laser Emissionai = 2 + ωi2 Δt 2 ( 1 + bi )ci = ωi2 Δt 2 bi

(21)

gi = −2 − ωi2 Δt 2 ( 1 + bi ) .
nnHere, E and G have already been updated in a previous time step. From the above, the
n+1
n+1
n+1
unknown P1, P2
and P3can be derived. Here, by using
F = PL = ∑ Pi
3

i =1
3

= ∑ ε 0 ∫ χi
t

i =1

0

n+1

= P1

( 1)

( t − τ ) E (τ ) dτ ,(22)

the linear polarization PL is replaced by PL = F . Then,
F

n+1

= PL

n+1

+ P2

n+1

+ P3

n+1

(23)

is explicitly obtained. Hence, an unknown linear polarization can be found. The reason why
only one resonant frequency can be taken into account for the linear polarization in the
JGTH algorithm (Joseph & Taflove, 1997), (Goorjian et al., 1992), (Joseph et al., 1993) (Tafloven& Hagness., 2000) is as follows. Since we take G as G in applying the central difference toEq. (19) to obtain Eq. (20), we can obtain the linear polarization explicitly. However
n+1
n −1
(G
+G
)/2 was used in the previous JGTH algorithm to improve the finite-differencen+1
is necessary, and therefore the successive solution
accuracy. In this case, the unknown Gof the nonlinear polarization after the linear polarization is derived is not possible
algorithmically. Therefore, in the JGTH algorithm, the linear polarization is restricted to one
so that symmetry is provided to the differential equations for linear polarization andnonlinear polarization. This is followed by the solution of the simultaneous finite differencenequations to obtain two unknowns simultaneously. Since only the known G is used byour method in the finite difference equation for the linear polarization, it is possible toderive the nonlinear polarization after the linear polarization is found. The accuracy of thefinite difference method is sufficient in this procedure. As described in detail below, the useof only one resonant frequency for the linear polarization when the results are comparedwith experimental results is a serious problem due to its more significant approximation ofdispersion, as follows.
Let us now describe the approximation of the dispersion in the conventional FDTD method.
The equation used in Refs. (Joseph & Taflove, 1997)-(Taflove & Hagness., 2000), in a series of
conventional FDTD methods, called the JGTH algorithm, is the case of i = 1 in Eq. (7) andεr(ω) = n(ω)2 = ε∞ +b1ω1/(ω12-ω2) in Eq. (10). Here, ε∞ is the relative permittivity when thefrequency is infinite. It is not possible in principle to approximate the refractive index dataof fused silica in the entire frequency range in this equation. However, it is possible within alimited frequency range. For example, if an approximation is introduced such that the errorof the refractive index is 0.1% at 250-517 THz, then the parameters can be determined as ε∞ =www.intechopen.com

Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation

455

0.9560, b1 = 1.130 and ω1 = 1.1639 × 1016 rad/s. Using these values, the values of β1, β2 and β3
are found to be 4.901 × 10-9 s/m, 7.287 × 10-26 s2/m and 3.469 × 10-41 s3/m, respectively.
β n (n = 1, 2, 3,A ) is the n-th order derivative of propagation constant β 0 at an incident(

center angular frequency ω0 defined as β n = d n β dω n)ω ω (Agrawal, 1995) . In contrast,= 0

the Sellmeier equation with three resonant frequencies used in the extended FDTD methodby the authors accurately approximates the refractive index in the entire frequency range
where measurement is possible (Malitson., 1961). If this equation is used, β1 = 4.894 ×10-9s/m, β2 = 3.616 × 10-26 s2/m and β3 = 2.750 × 10-41 s3/m. Since the conventional method andthe extended FDTD method approximately agree with regard to β1, the results for the groupvelocity vg = (β1)-1 can be relied upon to a certain extent in the conventional method (Joseph& Taflove, 1997), (Taflove & Hagness., 2000). This implies that the conventional method canbe used for the calculation of the propagation of a pulse of nanosecond order, for which thespectral width is sufficiently narrow and the effect of the group velocity dispersion is
negligible. However, since the group velocity dispersion β2 = ∂( v g )−1 ∂ω differs from thevalue obtained from the refractive index data by a factor of 2, the comparison of theconventional method with the propagation experiment of a pulse of sub-nanosecond orderor shorter is not possible. Therefore, in the conventional FDTD method of the JGTH
algorithm (Joseph & Taflove, 1997) (Goorjian et al., 1992), (Taflove & Hagness., 2000),
calculation of the propagation of femtosecond pulses can be carried out only for ahypothetical medium after the dispersion value is set to an easily calculated value and χ(3) is
set to an unrealistic value that is 1020 times the usual value. The comparison between simple
JGTH algorithm with parameters of ε∞ = 0.9560, b1 = 1.130 and ω1 = 1.1639 × 1016 rad/s andthe extended FDTD method is shown as temporal and spectral characteristics in section IV.
For the above reason, the nonlinear polarization is derived next from the linear polarization
n+1
F
derived from Eq. (23). The definition of the nonlinear polarization has already beenpresented in Eq. (15). When Eq. (15) is Fourier-transformed, we obtain{}

P#NL (ω ) = ε 0 χ 0(3)E# (ω ) × α E# 2 (ω ) + ( 1 − α ) g# RE# 2 (ω )g# R (ω ) =ω02,2
ω0 − jωδ − ω 2

(24)

(25)

where δ = 1 τ 2 and ω02 = ( 1 τ 1 ) + ( 1 τ 2 ) . Next, if Eq. (17), the definition of G, is Fourier2

transformed, we obtain

2

G# (ω ) = ε 0 g# R (ω ) E# 2 (ω ) .

(ω − jωδ − ω ) G# (ω ) = ε ω E# (ω ) .

(26)

When this is used, Eq. (25) becomes
2
0

2

2 2
0 0

(27)

Hence, the equation to be solved is expressed as follows by taking the inverse Fouriertransform of Eq. (27):www.intechopen.com

456

Coherence and Ultrashort Pulse Laser Emission

1 ∂ 2G δ ∂G+
+ G − ε 0 E2 = 0.
ω02 ∂t 2 ω02 ∂t(28)

From the relationship between the dielectric flux density and the electric field,# = ε E# + P#D0
= ε E# + P# + P#

= ε 0 E# + ε 0 χ (1)E# + ε 0 χ (3)E# ⎡⎣α E# 2 + ( 1 − α ) g# RE# 2 ⎤⎦
0

(

L

NL

)

# #.
= ε 0 1 + αχ (3)E# 2 E# + F# + ( 1 − α ) χ (3) EG(29)

Eq. (28) can be modified to the following:1 ∂ 2G δ ∂G+
ω02 ∂t 2 ω02 ∂t

( )
( )

2
⎡
( 1 − α ) χ (3) Ey ⎤⎥+ ⎢1 +
⋅G
2 ⎥
⎢
(3)
E
+
αχ
1
y
⎣⎢
⎦⎥

⎡
⎤
Ey
⎢
⎥⋅F
+⎢
2⎥
(3)
αχ
E
+
1
y
⎣⎢
⎦⎥

(30)

( )

⎡
⎤
Ey
⎥ ⋅D .
= ⎢⎢
y
2⎥
(3)
αχ
1
E
+
y
⎢⎣
⎥⎦

( )

(

)

(

)

Further, applying a central difference scheme centered at time step n, these equations yield
⎛ 1 ⎞
⎛ δ ⎞
n+ 1
n
n−1
n+1
n−1
+ ⎜⎜ 2 ⎟⎟ ⋅ G i − G i⎜⎜ 2 2 ⎟⎟ ⋅ G i − 2G i + G i
t
t
ω
ω
Δ
Δ
⎝ 0
⎠
⎝ 0 ⎠
2⎤
⎡
n
( 1 − α ) χ (3) ⎜⎛ Ey i ⎟⎞ ⎥ G ni + 1 + G ni −1
⎢
⎝
⎠ ⎥⋅
⎢
+ 1+
n ⎞2 ⎥⎢2(3) ⎛1 + αχ ⎜ Ey ⎟ ⎥
⎢
i
⎝
⎠ ⎦
⎣
⎡
⎤
n
n+1
n−1
⎢
⎥ Fi +Fi
Ey
i
⎢
⎥
+
⋅
n ⎞2 ⎥
⎢
2
(3) ⎛
E
αχ
1
+
⎜ yi⎟ ⎥
⎢
⎝
⎠ ⎦
⎣
⎤ ⎛
⎡
n+1
n−1 ⎞
n
⎥ ⎜ Dy i + Dy i ⎟
⎢
Ey
⎝
⎠
i
⎥⋅
=⎢
n ⎞2 ⎥⎢2.
(3) ⎛⎢ 1 + αχ ⎜ Ey i ⎟ ⎥
⎝
⎠ ⎦
⎣

(

(

www.intechopen.com

)

)

(31)

Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation

When F
F

n−1

n+1
n

457

derived from Eq. (23) is substituted into Eq. (31) together with the known

，G ，G

n −1

n

and E , the unknown G

n+ 1

can be derived. Here, an explicit form of

updating G is presented as
n+ 1

Gi

= [d + e + f ]

−1

(

)

⎧
⎛ D n+ 1 + D n−1 ⎞ ⎫
n+1
n−1
⎜ yi
⎟⎪
Fi +Fiy⎪⎪f ⎞ n−1in ⎛⎠⎪× ⎨2 d ⋅ G i + ⎜ e − d + ⎟ ⋅ G i − g ⋅
+ g⋅⎝
⎬
2
2
2
⎝
⎠
⎪
⎪
⎩⎪
⎭⎪.

(32)

where

When the obtained G

n+1

⎛ 1 ⎞
d=⎜ 2 2 ⎟⎜ ω Δt ⎟
⎝ 0
⎠
⎛ δ ⎞
e=⎜ 2 ⎟⎜ ω Δt ⎟
⎝ 0 ⎠
⎡
n 2⎤
( 1 − α ) χ (3) ⎛⎜ Ey i ⎞⎟ ⎥⎢⎝⎠ ⎥f = ⎢1 +n ⎞2 ⎥⎢(3) ⎛1 + αχ ⎜ Ey ⎟ ⎥⎢⎝ i⎠ ⎦
⎣
⎡
⎤
n
⎢
⎥
Ey
i
⎥⋅
g=⎢
2
n⎞ ⎥
⎢
(3) ⎛
⎢ 1 + αχ ⎜ Ey i ⎟ ⎥
⎝
⎠ ⎦
⎣

and F

n+1

(33)

are substituted into the following, the unknown E

n+ 1

can be derived and the iteration algorithm for the FDTD is terminated. Hence,n+1=
Eyp + 1

Dy

n+1

−F

p = 0,1, 2,ANote that the initial condition is Ey0

n+1

n+1

− ( 1 − α ) χ 0( ) Eyp

( )⎦
3

2
3
ε 0 ⎡⎢1 + αχ 0( ) Eyp ⎤⎥

⎣

n+1

G

n+1

.

(34)

= Ey . Here, Eq. (34) is derived from Eq. (29).
n

3. Femtosecond single pulse propagation
3.1 Experimental Setup
Figure 1 shows the setup used for the authors’ experiments (Nakamura et al, 2002a),(Karasawa et al, 2000). The 12 fs and 10 nJ pulses (sech2) at the center wavelength of 800 nm
were generated from a mode-locked Ti:sapphire laser (Femtolaser GmbH, Femtosource M-

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458

Coherence and Ultrashort Pulse Laser Emission1). The incident pulse width was measured by a fringe-resolved autocorrelator (FRAC). The
12 fs pulses were coupled into a 2.5 mm silica fiber by a 36x reflective objective (Ealing). The
advantage of this kind of reflective objective is that no additional group-delay dispersion(GDD) or third-order dispersion (TOD) is introduced to the pulses. However, the 12 fs
pulses are slightly chirped because the pulse width of the full width at half maximum(FWHM) resulting from the Fourier transformation of the fiber input spectrum is 8.5 fs.
FRAC orSpectrometer

M

Silica fiber

Modelocked laser

Spectrometer
F
R1

R2

Fig. 1. Layout of the experimental setup (Nakamura et al., 2002a). M, silvercoated plane
mirrors; F, flipper beam splitter, R1, R2, reflective objectives (36x); FRAC, fringe resolved
autocorrelator.

Intensity (a. u. )The input pulse is obtained by calculating the inverse Fourier transform of the spectrum of
the input pulse. However, the transform-limited pulse width (8.5 fs) is smaller than theexperimentally measured pulse width (12 fs) obtained by a fringe-resolved autocorrelator.
To use the pulse with the correct spectrum and the pulse width in the calculation, thetemporal duration is adjusted by adding the quadratic spectral phase in the formϕ (ω ) = c(ω − ω0 )2 to the experimental spectrum, to render the inverse Fourier transform
pulse width the same as that of the measured one. This gave the values of c = -9.043 fs2/radand ∂ 2ϕ (ω ) / ∂ω 2 |ω =ω0 = -113.6 fs2. The peak power of an input pulse was 175 kW. A singlemode fused silica fiber (Newport F-SPV) with a core diameter of 2.64 μm was used. The
output from the 2.5 mm fiber was collimated by another reflective objective and measured
by a spectrometer. The obtained input and output spectra of the fiber are shown in Fig. 2.
The dashed line in Figure 2 shows the input spectrum and the solid line shows the output
spectrum. The dashed line and short-dashed line in Fig. 3 show the assumption of an input
spectrum whose form is a sech2 shape, using this as field excitations in the extended FDTD
calculation (dashed line) and the simpler JGTH-FDTD calculation (short-dashed line). Thereis no difference between the spectra of the extended FDTD and JGTH-FDTD. The solid line
in Fig. 3 shows the experimentally obtained input pulse spectrum. Note that the intensity at
1

0

Input
Output600

800
Wavelength (nm)1000

Fig. 2. Input and output spectra measured in the 12 fs laser propagation experiment in asilica fiber (Karasawa et al, 2000). Dashed line shows the input spectrum and solid line
shows the output spectrum.

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Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation
1459This result

Intensity (a. u. )

JGTH
Exp.

0
600

800

1000

Wavelength (nm)Fig. 3. Spectra of 12 fs incident laser pulse into a 2.5-mm silica fiber, under the assumption of
having slight chirp, using this as field excitations in the extended FDTD (dashed line)(Nakamura et al., 2005a) and the simpler JGTH-FDTD (short-dashed line) (Nakamura et al.,2005a) , and our previously reported experimental result (Karasawa et al, 2000) (solid line).
the 650-750 nm of the assumed spectrum is zero for the solid line, however, the intensity inthe wavelength region is represented by a dashed line, and disagreement could be expected
at 650-750 nm in the calculated results. The inverse Fourier transformation of the assumed
input spectrum (dashed line in Fig. 3) indicates a pulse width of 8.5 fs (FWHM), which is thesame as that of the experimental spectrum (solid line in Fig. 3). In addition, the output pulse
was characterized by our second-harmonic generation frequency-resolved optical gating(SHG-FROG) apparatus set up in place of the spectrometer. Figures obtained by the SHGFROG are shown in section 3.2-A.

3.2 Numerical results
A. Comparison between Experimental and Numerical Results

In our calculation for a fused silica fiber, the parameters in Eq. (10) are set as b1 = 0.6961663, b2= 0.4079426, b3 = 0.8974794, λ1 = 0.0684043 μm, λ2 = 0.1162414 μm, and λ3 = 9.896161 μm(Agrawal, 1995) , where λi = 2πc/ωi and c is the velocity of light in vacuum. We used the valueof the nonlinear refractive coefficient n2 = 2.48×10-20 m2/W from Ref. (Taylor, 1996), and thethird-order susceptibility χ(3) was found to be χ(3) = 1.85×10-22 m2/V2 at 800 nm, as given by χ(3)= (4/3)ε0cn(ω0)2n2, where ω0 is the center angular frequency of the optical pulse. The
parameters α, τ1, and τ2 in Eq. (15) are set to be α = 0.7, τ1 = 12.2 fs and τ2 = 32 fs (Agrawal,
1995) . A single time step of the finite difference is set as Δt = 4.4475215×10-17 s at the
wavelength of 800 nm. The time step of Δt at 800 nm is defined as the optical cycle at 800 nm of
2.6666667 fs divided by 60. In the extended FDTD calculation, we set all parameters to be thesame as in our experiment (Nakamura et al, 2002a), (Karasawa et al, 2000).
We compare the FDTD calculation with the experimental result for the pulse peak power of
175 kW in order to generate an ultrabroad spectrum which can finally be compressed to 7.1
fs (Nakamura et al, 2002) or even shorter. The total fiber length of L = 2.5 mm corresponds

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1

(a) Input

0

–1
0

50

Time (fs)100Electric Field Amplitude (a.u.)Electric Field Amplitude (a. u. )Coherence and Ultrashort Pulse Laser Emission(b) Output
0.2

0

–0.2

12400

12600

12800

Time (fs)Fig. 4. Input (a) and output (b) temporal electric fields of a 2.5 mm silica fiber calculated by
the extended FDTD Maxwell equation method considering all orders of dispersions and theRaman response [α = 0.7 in Eq. (15)]. (Nakamura et al., 2005a)

to 136,500 spatial steps, which means that L = 136,500×Δz, where Δz is a unit spatial step inthe z direction. We need 293,000 time steps to measure the electric field until the pulse tailpasses completely. The peak power of an input pulse is set to be 175 kW (soliton number N= 2.09). The effective core area Aeff is set to be 5.47 μm2. The iteration number in Eq. (34) is
three. An iteration number of two is sufficient, as shown in Refs. (Joseph & Taflove, 1997)(Goorjian et al., 1992), (Taflove & Hagness., 2000). Figure 4 shows the input (a) and output(b) temporal electric fields of the fiber. Since the chirp is very small ( ∂ 2ϕ (ω ) / ∂ω 2 |ω =ω0 = 113.6 fs2), it is seen that the pulse has no chirp in Fig.4(a), while Fig. 4(b) seems to have large
chirp because of the linear and nonlinear effects in fiber propagation. Figure 5 shows the
temporal intensity profile of the pulse after propagation of the fiber, where the solid line
indicates measurement by FROG in the previous experiment, and the dashed line and theshort-dashed line indicate the results calculated by the extended FDTD Maxwell equation
method and the simpler JGTH-FDTD method, respectively.
The solid line shows a ringing behavior but generally agrees with the calculated result
except for the ringing. The envelope of the ringing appears to be the dashed line. The featureof a steep increase (from -100 fs to 0 fs) and a slow decrease (from 0 fs to 200 fs) of thedashed line is also seen in the solid line. The ringing of the experimental result occurs
because of the interference between core-mode light and cladding-mode light. Because oursimulation is performed by 1D calculation, the cladding mode is not considered. Hence,
there is no ringing in the numerical result. The experimentally measured pulse width(FHWM) is 92 fs. Numerically, the pulse width of FWHM is obtained as 108 fs. Therefore,
the pulse widths are in good agreement with each other. However, the pulse width(FWHM) obtained by the simpler JGTH-FDTD method is 146 fs, which is 1.5 times theexperimental result. The pulse width obtained by JGTH-FDTD does not agree with theexperimental result. This disagreement is caused by the approximation of dispersion in
JGTH-FDTD as described in section 3.2. Hence, the extended FDTD is superior to thesimpler JGTH-FDTD method.

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Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation

Normalized Intensity I(t)1

461

ExperimentThis result
JGTH

92 fs

0.5

0
–100

0

100

200

300

Time (fs)Intensity (a. u. )Fig. 5. Temporal intensity profile obtained experimentally (Nakamura et al., 2002) (solidline) and numerically by the extended FDTD (dashed line) and the simpler JGTH-FDTD(short-dashed line) as a function of time for 12 fs laser transmission through a 2.5-mm silica
fiber. (Nakamura et al., 2005a)1This result
JGTH
Exp.

0
600

800

Wavelength (nm)1000

Fig. 6. Spectra after propagation of a slightly chirped 12 fs laser pulse through a 2.5-mm
silica fiber, calculated by the extended FDTD (dashed line) (Nakamura et al., 2005a) and thesimpler JGTH-FDTD (short-dashed line) (Nakamura et al., 2005a), and our previously
reported experimental result (Karasawa et al., 2000) (solid line).
Figure 6 shows the results calculated by the extended FDTD Maxwell equation method(dashed line) and the simpler JGTH-FDTD method (short-dashed line), and our previously
reported experimental result (solid line, experiment [Exp.]) (Karasawa et al, 2000). The
spectra obtained by the extended FDTD and the simpler JGTH-FDTD show similar results.
The trapezium form of the FDTD results is roughly similar to that of the experimental result.
Several peaks in the experimental result appear due to interference between core-mode lightand cladding-mode light. In BPM solving GNLSE by SSF with SVEA, the second derivativeof the electric field with respect to z, ∂ 2 Ey / ∂z 2 , is neglected, which corresponds towww.intechopen.com

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Coherence and Ultrashort Pulse Laser Emission

Phase φ ( ω ) (rad)neglecting the backward propagation wave. On the other hand, our extended FDTD
Maxwell equation method accurately includes the delayed Raman response and all orders ofthe dispersion in silica using Sellmeier’s equation, and does not use SVEA. The shortest
wavelengths (intensity of 1%) of the spectra obtained by the extended FDTD, the simpler
JGTH-FDTD, and the experimental result (Exp.) are 653 nm, 660 nm and 626 nm,
respectively. The longest wavelengths of the extended FDTD, the simpler JGTH-FDTD, andthe experimental result (Exp.) are 1046 nm, 1027 nm, and 972 nm (maximum measured
wavelength of the spectrometer), respectively. The spectral bandwidths Δλ (1% of peakintensity) of the extended FDTD, the simpler JGTH-FDTD, and the experimental result(Exp.) are 393 nm, 367 nm, and 346 nm, respectively. Thus, the spectrum bandwidth of theexperimental result (Exp.) is narrower than those of the extended FDTD and the simpler
JGTH-FDTD. Note that the intensity at 650-750 nm of the assumed spectrum is lower thanthe experimental result, and also that the intensity at 650-750 nm of the assumed spectrumof the input pulse is zero; however, the intensity of the region of the wavelengths is not zeroin the experimental result of the input pulse. This affects the weak intensity at 650-750 nm in
the fiber output spectrum obtained by the extended FDTD and the simpler JGTH-FDTD. In
considering wavelengths longer than 800 nm in Fig. 6, the intensity of the experimental data
is lower than those of the extended FDTD and the simpler JGTH-FDTD results. We assume
that the sensitivity of the spectrometer detector is low at wavelengths longer than 800 nm
because the detector is made of silicon. We believe our extended FDTD result is correct. We
suggest that it is better to use an IR detector made of a material such as InGaAs for the
spectrometer.

90
80
70
60
50
40
30
20
10
0

This result
JGTH
Experiment

Angular Frequency ω (rad/fs)
2

2.5

Fig. 7. Spectral phase as a function of angular frequency after a chirped 12 fs pulse
propagation through a 2.5-mm silica fiber previously obtained experimentally (solidline)(Nakamura et al., 2002) and numerically by the extended FDTD (dashed line)(Nakamura et al., 2005a) and the simpler JGTH-FDTD (short-dashed line) (Nakamura et al.,2005a)
Figure 7 shows the spectral phase obtained experimentally by the FROG (solid line) andnumerically by the extended FDTD (dashed line) and the simpler JGTH-FDTD (shortdashed line) as a function of angular frequency for 12 fs laser transmission through a 2.5mm silica fiber. From our previous experimental result corresponding to the solid line, we

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Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation

463

determine the group-delay dispersion (GDD) and the third-order dispersion (TOD) values
to be 342 fs2 and 187 fs3, respectively. From our calculation corresponding to the dashed line,
we determine the GDD and TOD values to be 518 fs2 and 333 fs3, respectively. The
calculated GDD and TOD values do not agree with the experimental results. We have notyet clarified the reason for this disagreement. We must consider assuming the sech2
temporal shape of the input pulse, but not the complex waveform obtained by the inverse
Fourier transform of the experimentally observed spectrum. However, the calculated result
by the extended FDTD (dashed line) from the angular frequency of 1.8 rad/fs to 2.4 rad/fs(lower frequency) appears to be in good agreement with the experimental result. On theother hand, the simpler JGTH-FDTD (short-dashed line) shows a large difference in not only
the higher angular frequency but also the lower angular frequency. The phase values at thelowest angular frequency of the extended FDTD, the simpler JGTH-FDTD, and theexperimental result are 47.4 rad, 74.2 rad, and 38.9 rad, respectively. In this comparison, thephase value of the simpler JGTH-FDTD is twice that of the experimental result. This marked
difference between the simpler JGTH-FDTD and the experimental result is caused by thedipersion problem of the simpler JGTH-FDTD, which is described in section II and above inthis section. The spectral phase is very important for single-cycle pulse generation via phasecompensation of the nonlinear-chirped supercontinuum generated in a silica fiber. Hence,the extended FDTD is superior to the simpler JGTH-FDTD in predicting the spectral phasefor the phase compensation.
B. Compensation of group delay by using SLM

We assume the model in which the output beam from the fiber is collimated by a sphericalmirror and is directed to a 4-f system by the SLM. The 4-f system consisted of two sphericalmirrors and two reflective gratings (Nakamura et al, 2002a), (Karasawa et al, 2001), (Yamane
et al., 2003). The optical path lengths from the gratings to the spherical mirrors with a focal
length f and those from the spherical mirrors to the SLM are all set to be f. The liquid crystal
SLM (Citizen Company) (Karasawa et al, 2001) consisted of 648 97−μm-wide pixels, with a 5
μm gap between adjacent pixels.
The SLM is used as a phase modulator, which can impose a phase variation of 2π on light
whose wavelength is less than 1400 nm with a resolution of 192 (Karasawa et al, 2001). The
phase φSLM(x) from the SLM at position x is given by a polynomial of the form

φSLM(x) =

α

2

β

γ

6

24

[ω( x ) − ω0 ]2 + [ω( x ) − ω0 ]3 +[ω(x ) − ω0 ]4 ,(35)

where ω0 is the center angular frequency for a Taylor expansion, ω (x) is the angularfrequency at position x and is given by ω (x) = 2 πc/{d sin[tan-1(x/f ) + sin-1(λc/d)]}, c is the
speed of light, and λc = 800 nm is the wavelength at the central position of the SLM, where x= 0. λc does not have to be equal to the center wavelength for the Taylor expansion λ0 ,where λ0 = 2 πc/ω0 . α, β, and γ are GDD, TOD, and the fourth-order dispersion (FOD),
respectively, at ω0 . The group delay td,SLM(x) imposed by the SLM is∂ϕSLM (ω )td,SLM(x) =∂ω= α [ω ( x ) − ω0 ] +www.intechopen.com

β

2

γ

[ω( x ) − ω0 ]2 + [ω( x ) − ω0 ]3 .
6

(36)

464

Coherence and Ultrashort Pulse Laser Emission

Parameters α, β, and γ were initially estimated from the total group delay td(ω) of a fiber(including the self-phase modulation). td(ω) is fitted to the form of Eq. (36), and α, β, and γ
are obtained. Then the negative values of these fitted parameters were initially imposed by
the SLM to satisfy td,SLM(ω) + td(ω) = constant (zero in our case) in the entire frequency range.
Theoretically, this condition should produce the shortest pulses. Use of the Taylor expansion
in Eq. (35) is found to be the most suitable way to perform fine adjustment of the phase.
Parameters α, β, and γ are fine-tuned by a computer to yield the shortest pulse, with theconstant group delay dispersion optimized as a function of wavelength. Figure 8 shows thecalculated group delay after the pulse propagation of a fiber (solid curves) and thecompensated group delay (dashed curve) with the negative value of the optimized groupdelay applied by the SLM. The group delay after fiber propagation is added by applying its
negative value to the SLM, and then the total of these values is shown as a dashed line forwhich the shortest pulse (a nearly transform-limited pulse) is obtained for the condition
where λ0 is set to 800 nm. The temporal intensity profile of this shortest pulse is shown in
Fig. 9. The pulse width (FWHM) is estimated to be 6.9 fs. This pulse width agrees with thecompressed pulse of 7.1 fs (Nakamura et al, 2002a) previously obtained experimentally byusing a combination of prism compression and a 256 pixel SLM. This finding indicates thatthe combination of this extended FDTD and group delay compensation simulation is robust
for estimating the shortest pulse characteristics after the propagation of a fiber, and usefulfor constructing a fiber-compression system for obtaining a single-cycle pulse.

400
Before compensation

Group Delay (fs)

After compensation
200

0

–200
700

800

900

1000

Wavelength (nm)Fig. 8. Calculated group delay after pulse propagation of a fiber (solid curves) and thecompensated group delay (dashed curve) with the negative of the optimized group delay
applied by the SLM. (Nakamura et al., 2005a)

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Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
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465

Intensity (a. u. )1

6.9 fs

0.5

0
–100

0

100

Time (fs)Fig. 9. Temporal time profile compensated optimally by SLM after propagation of a fiber.
The pulse width (FWHM) is 6.9 fs. (Nakamura et al., 2005a)
3.3 Summary of single pulse propagation
We have extended the nonlinear FDTD method with a JGTH algorithm to one with exact
Sellmeier’s fitting values in order to compare the experimental and calculated results ofnonlinear femtosecond ultrabroadband-pulse propagation in a silica fiber, and haveanalyzed the temporal and spectral characteristics of the propagated pulse. This extended
method is robust against the breakdown of the SVEA. The spectrum, temporal intensity
profile, and pulse width after phase compensation results as calculated by the extended
FDTD method agree well with those obtained in our previous experiment. These findings
indicate that the combination of this extended FDTD method and the group delay
compensation simulation is robust for estimating the spectrum, the temporal pulse profile
after propagation through a fiber and the shortest pulse obtained by compensation, and isuseful for estimating the shortest pulse when the fiber-compression system is designed toobtain a single-cycle pulse (Yamane et al., 2003).
When the extended FDTD method is established as the ultrashort optical pulse propagation
analysis technique, its application to the characteristic analysis of an ultrahigh-speed opticalswitch, the propagation characteristic analysis of the optical pulse in a photonic crystal, andnonlinear propagation analysis in a photonic crystal fiber or a taper fiber are expected.

4. Dual wavelengths femtosecond pulses propagation
4.1 Induced-phase modulation
Recently, there has been significant interest in the generation of single-cycle optical pulses
by optical pulse compression of ultrabroadband light produced in fibers. The spectrabroadening by the induced-phase modulation (IPM) is one of these methods. There havebeen some reports of experiments on ultrabroadband-pulse generation using IPM in a silica
fiber (Xu et al., 1999), (Karasawa et al., 2001b), (Shibata et al., 2002), (Yamashita et al., 2004)
and an Ar-gas-filled hollow fiber, (Karasawa et al., 2000b) and also on optical pulse

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Coherence and Ultrashort Pulse Laser Emission

compression by nonlinear chirp compensation. (Yamashita et al., 2004) For theseexperiments on generating few-optical-cycle pulses, characterizing the spectral phase ofultrabroadband pulses analytically as well as experimentally is very important.
In IPM calculation, details of the phase characteristics at the connecting point of the twopulses are not clearly understood because the beam propagation method (BPM) has beenused to describe the propagation of an optical pulse in a fiber. (Agrawal, 1995) In the BPM,
each phase curve is obtained independently as a function of frequency for each center
frequency of the two pulses. (Agrawal, 1995) The two phase curves are connected at thesame phase value, as reported in refs. (Yamashita et al., 1998) and (Yamashita et al., 1996).
Recently, we extended (Nakamura et al., 2002b) the finite-difference time domain (FDTD)
method with nonlinear polarization PNL involving the Raman response function to 12 fs
ultrabroadband-pulse propagation in a silica fiber with the consideration of linear
polarization PL, including all exact Sellmeier-fitting values of silica with three resonant
frequencies, in order to compare the calculated results with our experimental results.
(Nakamura et al., 2002a), (Karasawa et al., 2000) This extended FDTD method (Nakamura etal., 2002b) is more accurate than previous FDTD methods (Joseph & Taflove, 1997), (Kalosha& Herrmann), (Goorjian et al., 1992), (Joseph et al., 1993), (Taflove & Hagness, 2000).
Recently, a similarly accurate algorithm, which assumes that the linear and nonlinear
dispersions are modeled by three and one Lorentzian resonances, respectively, wassuggested by Goojian and Cundiff (Goojian & Cundiff, 2004). On the point of phaseobservation, the extended FDTD is superior to BPM (Yamashita et al., 1998) and (Yamashita
et al., 1996) because the extended FDTD method includes no assumption for the two-pulse
case, which implies that it can be used to calculate two different pulses simultaneously. We
have recently numerically obtained the phase when two different pulses were simultaneously
injected into a fiber and IPM occurred, and we clarified how the two pulse phases maintaintheir connection with each other (Nakamura et al., 2004a). We did not observe the four-wave
mixing but only SPM and IPM with the simultaneous injection of two-color pulses into afiber (Nakamura et al., 2004a). In this study, we observed the anti-Stokes light of thedegenerate four-wave mixing (DFWM) in the IPM spectral phase obtained by the extended
FDTD method with an initial delay of -80 fs. The chirped first and second pulses broadening
with dispersion interact with each other, and the nonlinear interaction length is prolonged
with this initial delay. Then the phase matching condition for DFWM at the end of the fiber
is satisfied with an initial delay of -80 fs, which corresponds to the situation of two pulses
meeting at the end of the fiber via a different group velocity of the two pulses. This is thefirst observation of the DFWM with IPM by FDTD. Furthermore, we compared the extended
FDTD method (Nakamura et al., 2002b), (Nakamura et al., 2004b), (Nakamura et al., 2005)
with experimental results (Xu et al., 1999) for nonlinear propagation with the IPM effect of
two-color sub-picosecond laser pulses in a silica fiber that have different initial delays between
the two pulses. To the best of our knowledge, this is the first comparison between FDTD
calculation and experimental results for nonlinear propagation with the IPM effect of twocolor 120 fs laser pulses in a silica fiber. The extended FDTD method clarifies how the twopulse phases maintain their connection with each other with three initial time delays.
4.2 Experimental setup
A schematic of our experimental setup (Xu et al., 1999) is shown in Fig. 10. The first pulse
was generated from a Ti:sapphire regenerative amplifier at a 1 kHz repetition rate.

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Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation

Idler
Pump
Mode–locked
OPA Signal
Ti:sapphire
Laser and640 nmRegenerative
M1
NF2
Amplifier
M4
System
795 nm
P1
M2

467

M5
Spectrometer
NF1 Fiber
DM R1

R2

M6

M3

Fig. 10. Layout of the experimental setup.(Xu et al., 1999) M, silver-coated plane mirrors; R1
and R2, reflective objectives (36x); DM, dichroic mirror; NF1 and NF2, valuable opticalattenuators; OPA, optical parametric amplifier.
The pulse duration was about 120 fs full width at half maximum (FWHM) with a bandwidthof 12 nm FWHM centered at 795 nm, and the output energy was 750 μJ, henceforth called
the fundamental pulse. The main output energy from the regenerative amplifier was used togenerate a continuum wave and pump an optical parametric amplifier (OPA) for the second
pulse generation. The tuning range in the OPA was from 480 to 740 nm with an output
energy of over 10 μJ at 1 kHz. In this experiment, we set the wavelength of the second pulse
at 640 nm with a 120 fs FWHM in the pulse duration and a 12 nm FWHM in the spectral
width, henceforth called the signal pulse. Since the continuum wave was generated by the
SPM of the fundamental pulse under the single-filament condition and was selectively
amplified using its second harmonic pulse by the optical parametric process with the phasematching condition, the phase of the signal wave should have a relationship to the phase of
the fundamental wave. This condition is critical for generating a stable combined spectrumwith IPM in a single-mode glass fiber, which will be confirmed later. The calculated walk-off
length (Agrawal, 1995) Lw between the signal pulse and the fundamental pulse was about
2.7 mm. The reason the second pulse wavelength was selected to be 640 nm lies in the factthat the walk-off length Lw of these two pulses in a fused silica fiber ( Lw = 2.7 mm) isaround the length that we are able to cut easily, about 3 mm, and the IPM effect between thetwo pulses can be observed clearly at this length. The two pulses were combined by adichroic mirror, DM, which has a high transmission for the fundamental pulse and a highreflection for the signal pulse. The optical path of the fundamental pulse was varied by an
optical delay stage formed by M2 and M3 so that the delay of the two pulses in the fiber
could be adjusted. N1 and N2 are variable neutral-density filters and the energies of the twopulses can be controlled independently. The combined two pulses, which have the same
linear polarizations, were coupled into a 3 mm single-mode polarization-preserving fusedsilica fiber (2.7 μm core diameter, 550 ± 50 nm cutoff wavelength) by a 36 × reflective
objective, R1. The output from the fiber was collimated by the same type of reflective
objective, R2, to a spectrometer that monitored the output spectrum.
4.3 Numerical results
In our calculation for a fused silica fiber, the parameters in eq. (10) are set as b1 = 0.6961663,
b2 = 0.4079426, b3 = 0.8974794, λ1 = 0.0684043 μm, λ2 = 0.1162414 μm, and λ3 = 9.896161 μm(Agrawal, 1995), where λi = 2πc/ωi and c is the velocity of light in vacuum. We used thewww.intechopen.com

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Coherence and Ultrashort Pulse Laser Emission

value of the nonlinear refractive coefficient n2I = 2.48 × 10-20 m2/W from ref. (Taylor et al.,1996), and the third-order susceptibility χ(3) was found to be χ(3) = 1.85 × 10-22 m2/V2 at 800

nm, given by χ(3) = (4/3)ε0cn(ω0)2 n2 , where ω0 is the center angular frequency of the opticalI

Intensity [arb. units]

pulse. The parameters α, τ1, and τ2 in eq. (15) are set to be α = 0.7, τ1 = 12.2 fs and τ2 = 32 fs.
(Agrawal, 1995) A single time step of the finite difference is set as Δt = 7.116034310 × 10-17 s.
The time step of Δt is defined as the single optical cycle divided by 60. In the extended
FDTD calculation for two-pulse propagation in a silica fiber, we set all other parameters tobe the same as those in the previous IPM experiment. (Xu et al., 1999) Both pulses had thesame pulse width of 120 fs and the same energy of 20 nJ. The fiber was 3-mm long with acore diameter of 2.7 μm. The fiber length of 3 mm nearly equals the walk-off length of 2.7
mm. The effective core area Aeff was 5.47 μm2. Two laser pulses at the center wavelengths of
795 and 640 nm (the fundamental and signal pulse) propagated in a 3 mm silica fiber withthe delay of the signal pulse for the fundamental pulse, Td = 0, -40, -80 fs. The delay of –40 fs
corresponds to the situation in which the two pulses are launched simultaneously into thesame fiber with an optimum initial delay between the two pulses that meet at the center ofthe fiber, to be discussed later, because of the IPM effect of the two pulses, the two spectra
meet each other. The delay of –80 fs corresponds to the situation which two pulses meet atthe end of the fiber. Since every delay case is performed close to the walk-off length, thenonlinear interaction length is sufficient.

10

12 nm

12 nm

Intensity [arb. units]

0
500
500

900
900

600
600

700
800
700
800
Wavelength [nm]

1

12 nm

12 nm

0
500

600

700

800

900

Wavelength [nm]Fig. 11. Spectra obtained (a) experimentally (Xu et al., 1999) and (b) numerically (Nakamura
et al., 2005b) for two pulses at center wavelengths of 795 nm (solid line) and 640 nm (dashedline) before propagation in a 3 mm silica fiber.

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469

Intensity [arb. units]

Figure 11 shows (a) the experimentally measured and (b) the numerically assumed spectraof the input fundamental (solid line) and signal (dashed line) pulses. The FWHM spectral
width of Δλ is 12 nm for both fundamental and signal pulses. The Fourier transform of a
Gaussian spectrum with the FWHM spectral width of Δλ =12 nm for the fundamental andsignal pulses leads to a pulse width of 80.0 and 50.0 fs, respectively, which does not agreewith the input pulse width of 120 fs. This fact indicates that the input pulses are chirped,
which accords with the second derivation of the spectral phase of ∂ 2ϕ (ω ) / ∂ω 2 |ω =ω0 = 4651
and 2382 fs2 for the fundamental and signal pulses, respectively. Fig. 11(b) is assumed by the
Fourier transformation of the pulse width of 120 fs with the chirp values. Due to the groupvelocity difference between the two optical pulses propagating in the fiber, they travel at
different speeds and after some propagation distance they are separated. We measured the
IPM-induced spectra with different initial time delays between the two pulses.
1

(a)

0
800
100

(b)

Phase [rad]Intensity [arb. units]

600
1

0
500

600

700

800

Wavelength [nm]0900Fig. 12. (a) Experimentally (Xu et al., 1999) and (b) numerically (Nakamura et al., 2005b)
obtained spectra induced by SPM in 3 mm single-mode fused-silica fiber. The thin solid line
represents the spectrum induced by the SPM of the pulse at the center wavelength of 795 nm
and the thin dashed line represents the spectrum induced by the SPM of the pulse at thecenter wavelength of 640 nm at the pulse duration of 120 fs and the energy of 20 nJ in the 3
mm fiber for each pulse. The thick lines in (b) are spectral phases obtained numerically as afunction of wavelength for the 3 mm fiber-propagated pulses at center wavelengths of 795
nm (thick solid line) and 640 nm (thick dashed line).
Fig. 12 depicts the spectra of the two pulses with SPM alone at the energy of 20 nJ in thefiber for each pulse, which are obtained (a) experimentally and (b) numerically. The solidline represents the SPM-induced spectrum of the fundamental pulse and the dashed line
represents the SPM-induced spectrum of the signal pulse. From Figs. 12(a) and 12(b) weobserve the separated spectra for the two pulses with only the SPM effect at 20 nJ energy
levels. Here, we define the wavelengths in which the spectral intensity becomes 1/100 of thewww.intechopen.com

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Coherence and Ultrashort Pulse Laser Emission

maximum intensity as λmin and λmax for the shortest and longest wavelength, respectively.
Then, we define the spectral width as Δλ1/100 = λmax - λmin. The spectral widths Δλ1/100 of thedashed lines in Figs. 12(a) and 12(b) are 93.8 and 83.7 nm, respectively. The spectral widths
Δλ1/100 of the solid lines in Figs. 12(a) and 12(b) are 117.5 and 110.9 nm, respectively. The
spectral widths in Fig. 12 caused by only SPM are smaller than those caused by SPM andIPM of Figs. 13 and 14, this will be discussed later. The difference in the spectral width
between the experimental and numerical results is less than 10 nm (<10%), which indicatesthat the difference is small. Blue shift due to self-steepening is not observed but breakdownof the spectral symmetry is observed. The intensity of the spectrum is higher forwavelengths longer than the center wavelengths in Fig. 12(b). In particular, the solid line
(fundamental pulse) in Fig. 12(b) shows asymmetrical behavior and three- or four-peak
behavior, which are similar to those in the solid line in Fig. 12(a). The extended FDTD result
and the experimental result in Fig. 12 are similar from the point of view of spectral widths.
The thick lines in Fig. 12(b) show the numerically obtained spectral phase wherein thedashed line shows the phase of the fundamental pulse and the solid line shows that of thesignal pulse. For both the fundamental and signal pulse, the phase curve is a symmetricparabolic curve. The spectral phases of the two pulses cross around the wavelength of 713
nm, where the phase value is 48.0 rad.
6(a) Td = 0 fs

4Intensity [arb. units]

2
0
500

600

700

800

2

900

1000

(b) Td = – 40 fs

1
0
500

600

700

800

900

1000

3
(c) Td = – 80 fs

2
1
0
500

600

700

800

900

1000

Wavelength (nm)Fig. 13. Experimentally measured spectra (Xu et al., 1999) induced by IPM for two pulses at
center wavelengths of 795 and 640 nm with different initial time delays after propagation in
3 mm silica fiber. (a) Spectrum when two pulses meet at entrance of fiber, (b) spectrum
when two pulses meet at center of fiber and (c) spectrum when two pulses meet at end offiber.

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Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation

Intensity [arb. units]1471(a) Td = 0 fs

0
500
1

600

0
500
1

600

700

800

900

1000

800

900

1000

800

900

1000

(b) Td = – 40 fs

700(c) Td = – 80 fs

0
500

600

700

Wavelength [nm]Fig. 14. Numerically obtained spectra induced by IPM for two pulses at center wavelengthsof 795 and 640 nm with different initial time delays after propagation in 3 mm silica fiber. (a)Spectrum when two pulses meet at entrance of fiber, (b) spectrum when two pulses meet at
center of fiber and (c) spectrum when two pulses meet at end of fiber. (Nakamura et al.,2005b)
Figures 13 and 14 show the IPM-induced spectra obtained by the previous experiment (Xuet al., 1999) and the extended FDTD calculation with different initial time delays between
the two pulses, these figures indicate spectra after propagation of 3 mm fiber with the initialdelays Td of (a) 0, (b) –40, and (c) –80 fs, respectively. The pulse energies after fiber
propagation of the two pulses are the same value, 40 nJ. The initial time delay between thetwo pulses, (a) Td = 0 fs, represents that the two pulses coincide at the fiber entrance, and thedelays, (b) Td = -40 fs and (c) Td = -80 fs, represent that the two pulses meet at the center andthe end of the fiber, respectively. Figures 13 and 14 indicate that the spectra are all broader
than that of Fig. 12, because not only SPM but also IPM occurs. The spectral bandwidth ofFig. 13(a) is Δλ1/100=λmax-λmin= 851.1 - 568.9 = 282.2 nm. The spectral bandwidth of Fig. 14(a)is Δλ1/100=λmax-λmin= 883.8 - 571.5 = 312.3 nm. Quantitatively, there is a difference between
the experiment [Fig. 13(a)] and the calculation [Fig. 14(a)]. However, a larger gap between
the two spectra is observed in both cases. This shows that the calculation qualitatively
agrees well with the previous experimental results. In our case, when the two pulses meet atthe entrance of the fiber Td = 0 fs, the leading edge of the signal pulse interacts with thetrailing edge of the fundamental pulse because the fundamental pulse travels faster than thesignal pulse in the fused silica fiber. As a result, both the signal pulse and the fundamental
pulse have IPM-induced chirp. This leads to a larger modulation of the signal pulse
spectrum at shorter wavelengths and a larger spectral modulation of the fundamental pulse

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Coherence and Ultrashort Pulse Laser Emission

at longer wavelengths. Hence, the modulation of the signal pulse at longer wavelengths andof the fundamental pulse at shorter wavelengths is depressed. Then, the larger gap between
the fundamental and signal spectra is formed. In contrast, when the two pulses meet at theend of the fiber Td = -80 fs, the trailing edge (longer wavelengths) of the signal pulse mainlyinteracts with the leading edge (shorter wavelengths) of the fundamental pulse. As a result,the IPM-induced spectrum of the signal pulse shifts towards its longer wavelengths and the
IPM-induced fundamental pulse spectrum shifts towards its shorter wavelengths and alarger overlap of the two pulse spectra is observed in both Figs. 13(c) and 14(c). This showsthat the calculation qualitatively agrees well with the previous experimental results. The
spectral bandwidth of Fig. 13(c) is Δλ1/100=λmax-λmin= 861.8 - 582.9 = 278.9 nm. The spectral
bandwidth of Fig. 14(c) is Δλ1/100=λmax-λmin= 895.9 - 579.1 = 316.8 nm. It is found that when
the two pulses meet at the center of the fiber, the fundamental pulse passes through thesignal pulse in a symmetric manner. As a result, the spectra are broadened by the IPM effectto shorter and longer wavelengths simultaneously for both pulses and the broadest
combined spectrum is generated in both Figs. 13(b) and 14(b). This shows that thecalculation qualitatively agrees well with the previous experimental results. The spectral
bandwidth of Fig. 13(b) is Δλ1/100=λmax-λmin= 887.3 - 567.3 = 320.0 nm. The spectral
bandwidth of Fig. 14(b) is Δλ1/100=λmax-λmin= 896.6 - 577.6 = 319.0 nm. Figures 13(b) and 14(b)generally have the same value of Δλ1/100. This case is referred to as the optimum initial timedelay to produce the broadest spectrum. In Figs. 13 and 14, it is the same feature that themost broadened spectra are Fig. 13(b) and Fig. 14(b) with an initial delay of Td = -40 fs.
Consequently, the extended FDTD results and our previous experimental results have thesame behavior of spectral bandwidth, which indicates the same initial delay dependence.
That is, the extended FDTD calculation qualitatively agrees well with the previous
experimental results although it does not agree quantitatively. This quantitative
disagreement might be due to the low sensitivity of the detector of the spectrometer atwavelengths longer than 800 nm because the detector is made of Si. We suggest that InGeAs
should be used for the detector, which is highly sensitive in the detection of the infrared
region light.
Figure 15 shows the spectral phase obtained numerically as a function of wavelength formixed two pulses at center wavelengths of 795 and 640 nm after simultaneous
copropagation with an initial delay of (a) 0, (b) -40, and (c) -80 fs in a 3 mm silica fiber.
Around the center wavelength (795 nm) of the fundamental pulse, the phase curve isparabolic and its symmetry axis is (a) 774.14, (b) 749.45, and (c) 775.58 nm. Around thecenter wavelength (640 nm) of the signal pulse, the phase curve is parabolic and its
symmetry axis is (a) 673.99, (b) 676.90, and (c) 687.78 nm. The symmetry axis of every
parabolic curve in Fig. 15 shifts toward the wavelength of 720 nm, which is the center
wavelength between 795 and 640 nm. This feature is seen when the IPM effect occurs, sothat this feature is not found only in the SPM case of Fig. 12. In addition, it is observed inFig. 15 that the spectral phases of the two pulses at the wavelength λ of (a) 711.7, (b) 713.17,
and (c) 713.17 nm are continuously connected at a phase value of φ (λ ) = (a)17.76, (b) 10.46,
and (c) 17.94 rad without any discrete point. Furthermore, regarding wavelengths shorter
than 640 nm, one more parabolic curve is found between 500 and 570 nm. This curve isfound to be that of the anti-Stokes light (theoretically 535.58 nm) generated by the induced
DFWM where the pump light is 640 nm and the Stokes light is 795 nm. The chirped first andsecond pulses broadening by dispersion interact each other, and the nonlinear interaction
length is prolonged with an initial delay of -80 fs. The DFWM is robust for chirping.
(Cundiff et al., 1999), (Geraghty et al., 1998)

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Phase [rad]Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation

600(a) Td = 0 fs

400
200
0

Phase [rad]500

Phase [rad]

473

600

700

800

900

(b) Td = – 40 fs

400
200
0
400

500

600

700

800

900

(c) Td = – 80 fs
200
0

500

600

700

800

900

Wavelength [nm]Fig. 15. Spectral phase obtained numerically as function of wavelength for mixed two pulses
at center wavelengths of 795 and 640 nm after simultaneous copropagation with an initialdelay of (a) 0, (b) -40, and (c) -80 fs in 3 mm silica fiber. (Nakamura et al., 2005b)
When the two pulses meet at the end of the fiber Td = 80 fs, the trailing edge of the signal
pulse mainly interacts with the leading edge of the fundamental pulse and the chirped
pulses gradually become overlapping at the places where the pulses are temporally
broadened; therefore, the interaction length is long. As a result, the four-wave mixing effectis enhanced and a larger parabolic phase curve of the anti-Stokes light in the spectral phase
is observed. Then the phase matching condition for DFWM at the end of the fiber is satisfied
with an initial delay of -80 fs, which corresponds to the situation in which two pulses meetat the end of the fiber via a different group velocity of the two pulses. Looking again at Fig.
15, the anti-Stokes light appears faintly in the phase with an initial delay of -40 as well, but
no anti-Stokes light appears with an initial delay of 0 fs. The anti-Stokes light of DFWM
appears when long chirped pulses finally overlap at the end of the fiber, the IPM effect isstrong when short pulses overlap at the beginning of the fiber, and IPM and DFWM crossinteract and spectral broadening is maximum when the pulses overlap at the center of thefiber. Such precise information on the spectral phase of the pulse propagated in a fiber isvery important for the wavelength conversion with spectral broadening or the generation ofa monocycle pulse using a spatial light modulator (SLM) that can compensate the spectralphase of a superbroadened continuum. The shortest pulse attainable by phase correction ofthis ultrabroad spectrum is obtained by the Fourier transform of the spectrum of Fig. 14 andassuming a constant spectral phase. This yields a transform-limited pulse. Figure 16 shows
the temporal profiles of the transform-limited pulses obtained numerically for the mixed

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474

Coherence and Ultrashort Pulse Laser Emission

Intensity [arb. units]

Intensity [arb. units] Intensity [arb. units]

two-pulse spectra shown in Fig. 14 at the center wavelengths of 795 and 640 nm after
simultaneous copropagation with an initial delay of (a) 0, (b) -40, and (c) -80 fs in a 3 mm
silica fiber. The FWHM pulse widths are (a) 4.1, (b) 4.0, and (c) 4.7 fs. The pulse width under
an initial delay of –40 fs is 4.0 fs, which is the same value, 4 fs (Xu et al., 1999), obtained by
Fourier transform of the experimental result of Fig. 15(b) reported by Xu et al.

(a)Td = 0 fs

1

4.1 fs

0
–50
1

0

50

(b)Td = – 40 fs
4.0 fs

0
–50
1

0

50

(c)Td = – 80 fs
4.7 fs0–500Time [fs]50

Fig. 16. Temporal profiles of transform-limited pulses obtained numerically for mixed twopulse spectra shown in Fig. 6 at center wavelengths of 795 and 640 nm after simultaneous
copropagation with initial delay of (a) 0, (b) -40, and (c) -80 fs in a 3 mm silica fiber.
(Nakamura et al., 2005b)
4.4 Summary of dual wavelengths femtosecond pulses propagation
The ultrabroad spectrum generation based on the IPM of two optical pulses copropagating
in a single-mode fused-silica fiber has been numerically demonstrated, which is comparedwith our previous experimental results. To the best of our knowledge, this is the firstcomparison between the extended FDTD calculation and the experiment. In the extended
FDTD, it was found that the spectral phases of the two pulses around the wavelength of 710

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Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation

475

nm are continuously connected at a phase value of φ (λ ) = 10-20 rad without any discrete
point. This fact was unknown when BPM was used previously. In the extended FDTD
simulation, two 120 fs pulses with wavelengths at 640 and 795 nm were coupled into a 3 mmsingle-mode fiber. At a pulse energy of 20 nJ coupled into the fiber for both pulses, anultrabroad coherent spectrum induced by IPM which covered the range from 567 to 897 nm
was obtained and was compared with our previous experiment. This result generally agrees
well with the previous experimental results at the optimum initial delay of -40 fs, whichcorresponds to the situation in which two pulses meet at the center of the fiber, where thespectral width is Δλ1/100 = 320 nm. The Fourier transform of the spectrum can yield a 4.0 fs
transform-limited pulse that is the same pulse width of the Fourier-transform-limited pulseof the previous experimental results. It opens a path for generating a single-cycle highenergy optical pulse in the near future. It is necessary to use not only fundamental (795 nm)
and signal (640 nm) pulses, but also the idler pulse (1067 nm) of the OPA in order to obtain asingle-cycle (2.66 fs at the center wavelength of 795 nm), which means three-color pulse
propagation in a fiber.
Finally, we obtained the spectral phase after fiber propagation with the calculation. The
extended FDTD method clarified how the two pulse phases maintain their connection witheach other because it includes no assumption for the two-pulse case, which implies that themethod can be used to calculate two different pulses simultaneously with three delays. We
reconfirmed that the spectral phases of two different pulses are connected continuously in
any case of three initial delays. We found in the IPM spectral phase that the DFWM occurs
with an initial delay of -80 fs, which corresponds to the situation in which two pulses meetat the end of the fiber. To the best of our knowledge, this is the first simultaneous
observation of DFWM and IPM by FDTD simulation.

5. Slowly varying envelope approximation breakdown in fiber propagation
5.1 Slowly varying envelope approximation breakdown
There has recently been significant interest in the generation of single-cycle optical pulses by
optical pulse compression of ultrabroad-band light produced in fibers. There have beensome experiments reported on ultrabroad-band-pulse generation using a silica fiber(Nakamura et al., 2002a), (Karasawa et al., 2000) and an Ar-gas-filled hollow fiber (Karasawa
et al., 2001), and optical pulse compression by nonlinear chirp compensation (Nakamura etal., 2002a), (Karasawa et al., 2001) . For these experiments on generating few-optical-cycle
pulses, characterizing the spectral phase of ultrabroad-band pulses analytically as well as
experimentally is highly important.
Conventionally, the slowly varying envelope approximation (SVEA) in the beam
propagation method (BPM) has been used to describe the propagation of an optical pulse ina fiber. (Agrawal, 1995) However, if the pulse duration approaches that of the optical cycle
regime (<10 fs), this approximation becomes invalid. (Agrawal, 1995) It is necessary to use
the finite-difference time-domain (FDTD) method (Joseph & Taflove, 1997), (Kalosha &Herrmann, 2000) without SVEA. (Agrawal, 1995) In previous reports, Goorjian andcoworkers (Goorjian et al., 1992), (Joseph et al., 1993)), Joseph and coworkers (Joseph &Taflove, 1997) (Goorjian et al., 1992), (Joseph et al., 1993)), Taflove and coworkers (Joseph &Taflove, 1997) (Goorjian et al., 1992)-(Taflove & Hagness, 2000) and Hagness and coworkers(Goorjian et al., 1992), (Taflove & Hagness, 2000), (JGTH) proposed an excellent FDTD
algorithm considering a combination of linear dispersion with one resonant frequency andnonlinear terms with a Raman response function.

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Coherence and Ultrashort Pulse Laser Emission

We performed an experiment of 12 fs optical pulse propagation, as described in §4. In order
to compare FDTD calculation results with the experimentally measured ultrabroad-band
spectra of such an ultrashort laser pulse, we extend the JGTH algorithm to that considering
all of the exact Sellmeier fitting values for ultrabroad-band spectra. Because of the broad
spectrum of pulses propagating in the fiber, it becomes much more important to take anaccurate linear dispersion into account. It is well known that at least two resonant
frequencies are required for the linear dispersion to accurately fit the refractive index data.
In a recent report, Kalosha and Herrmann considered the linear dispersion with two
resonant frequencies and the nonlinear terms without the Raman effect.(Kalosha &Herrmann, 2000). For the single-cycle pulse-generation experiment, we must use at least the
shortest pulse of 3.4 fs (Yamane et al., 2003) or sub-5 fs (Karasawa et al., 2001), (Cheng et al.,1998) or 7.1 fs (Nakamura et al., 2002a) or the commercially available 12 fs pulses. Such atime regime is comparable to the Raman characteristic time of 5 fs (Agrawal, 1995) in a silica
fiber. Therefore, it is very important to consider not only an accurate linear dispersion ofsilica but also the Raman effect in a silica fiber in the few-optical-cycles regime. In addition,
because of the high repetition rate and pulse intensity stability in particular, ultrabroadband supercontinuum light generation and few-optical-cycles pulse generation by nonlinear
pulse propagation in photonic crystal fibers (Ranka et al., 2000) and tapered fibers (Birks etal., 2000), which are both made of silica, have attracted much attention. We have extended
the FDTD method, (Nakamura et al., 2002b) with nonlinear polarization PNL involving theRaman response function (JGTH-algorithm) to 12 fs ultrabroad-band-pulse propagation in asilica fiber with the consideration of linear polarization PL, including all exact Sellmeier-fitting
values of silica with three resonant frequencies, in order to compare the calculation results with
our experimental results(Nakamura et al., 2002a), (Karasawa et al., 2000b). In this section,
we describe the details of the calculation algorithm of the extended FDTD method(Nakamura et al., 2002b) and we also compare the extended FDTD method (Nakamura etal., 2002b) with BPM by applying the split-step Fourier (SSF) method, which is the solutionof a modified generalized nonlinear Schrödinger equation (MGNLSE) (Sone et al., 2002),
with SVEA, precisely considering the same Raman response function as that of the extended
FDTD method, and up to the fifth-order dispersion. Then, in the calculations, the pulse
width is gradually shortened from 12 fs to 7 fs to 4 fs. Moreover, the soliton number N isestablished as 1 or 2. To the best of our knowledge, this is the first observation of the
breakdown of SVEA as a function of the laser pulse width and soliton number bycomparison between the extended FDTD (Nakamura et al., 2002b) and BPM calculations forthe nonlinear propagation of a very short (< 12 fs) laser pulse in a silica fiber.
5.2 Beam propagation method
Conventionally, BPM for solving the generalized nonlinear Schrödinger equation (GNLSE)
including SVEA has been used to describe the ultrashort-laser-pulse propagation in an
optical fiber. The GNLSE is expressed asj∂A∂2 A 1∂3 A= − β2+ β3 3
2
∂z
2! ∂T
3! ∂T
⎡
j ∂
α
2
− a A + jN 2 ⎢ A A +
2
ω0 ∂T
⎢⎣

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A A − TR A
2

2 ,
∂A ⎤
⎥
∂T ⎥
⎦

(37)

Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation

N 2 = γ P0T0 2 β 2 , γ = n2ω0 (cAeff ) ,

477

(38)

where A is the slowly varying amplitude of the pulse envelope of an electric field, which isnormalized by N, α a is the absorption coefficient, j is an imaginary unit, T0 is the incident
pulse width which is expressed using the full width at half maximum (FWHM) pulse width
tp as T0 = tp (2 ln 2 ) for a Gaussian pulse and T0 = tp 2 ln(1 + 2 ) for a sech2 pulse, c isthe velocity of light in vacuum, ω0 is the center angular frequency of the incident pulse, n2
is the nonlinear refractive index (m2/W), P0 is the incident pulse peak power, Aeff is theeffective core area, TR is the Raman time constant and TR =5 fs. (Agrawal, 1995) β n(n = 1, 2, 3,A ) is the n-th-order derivative of the propagation constant β 0 at an incident
center angular frequency ω0 is as follows:{}⎛ dn β ⎞.
n⎟⎟⎝ dω ⎠ω =ω0

β n = ⎜⎜

(39)

T is defined as
T =t−

νg
z

= t − β1z ,(40)where ν g is the group velocity of the optical pulse, t is time and z is the distance from theincident edge of a fiber. In our previous letter (Nakamura et al., 2002b), we also used this
GNLSE for comparison with the extended FDTD method. However, a linear approximationof the actual Raman gain curve using the Raman time constant of TR = 5 fs (Agrawal, 1995),
which is related to the slope of the Raman gain, and a dispersion approximation up to only
the third-order dispersion terms are included in GNLSE. In this section, we use a more
precise version of the nonlinear Schrödinger equation, MGNLSE (Sone et al., 2002), (Gross &Manassah, 1992), that includes a Raman response function which is also found in theextended FDTD and up to fifth-order dispersion as follows:j∂A∂2 A 1∂3 A j∂4A 1∂5A= − β2+ β3 3 + β4
− β5 5
2
4
∂z
2! ∂T
3! ∂T
4! ∂T
5! ∂T
,
⎡ 2
⎤
j ∂
α
2
2
A A + ℑ−1 ⎡ χ R ℑ ⎡ A ⎤ ⎤ A ⎥− a A + jN 2 ⎢ A A +⎢⎣⎣⎦ ⎥⎦ ⎦
2
ω0 ∂T
⎣

(

)

(41)

where the symbols ℑ and ℑ−1 are the operators of Fourier transformation and inverse
Fourier transformation, and χ R is the generalized Raman-scattering susceptibility. The
generalized Raman-scattering susceptibility (Gross & Manassah, 1992) can be approximatedin the harmonic oscillator model for the molecular vibrations asχ R ( Ω) =

χ 0 (ΩR ΓR )

ΩR − Ω − jΓR ΩR
2

,

(42)

where Ω = (ω − ω0 )T0 is the angular frequency normalized to the pulse width, andΩR (=13.2 THz × 2π T0 ) (François, 1991) is the Raman shift normalized to the pulse width,

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478

Coherence and Ultrashort Pulse Laser Emissioni.e., the molecular vibrational frequency multiplied by T0, and Γ R (=15.44 THz × 2π T0 )(François, 1991) is the normalized phenomenological linewidth.
5.3 Experiment
Figure 1 shows the setup used for the our experiments. (Nakamura et al., 2002a), (Karasawa,
et al., 2000). The 12 fs and 10 nJ pulses (sech2) at the center wavelength of 800 nm weregenerated from a mode-locked Ti:sapphire laser (Femtolaser GmbH, Femtosource M-1). The
incident pulse width was measured by a fringe-resolved autocorrelator (FRAC). The 12 fs
pulses were coupled into a 2.5 mm silica fiber by a 36x reflective objective (Ealing). The
advantage of this kind of reflective objective is that no additional group-delay dispersion(GDD) or third-order dispersion (TOD) is introduced to the pulses. The peak power of the
input pulse was 175 kW. A single-mode fused silica fiber (Newport F-SPV) with a core
diameter of 2.64 μm was used. The output from the 2.5 mm fiber was collimated by anotherreflective objective and measured by a spectrometer (Ocean Optics, S-2000). The obtained
input and output spectra of the fiber are shown in Fig. 2.
5.4 Numerical results
A. Comparison between experimental and numerical results

In our calculations for a fused silica fiber, the parameters in eq. (10) are set as b1 = 0.6961663,
b2 = 0.4079426, b3 = 0.8974794, λ1 = 0.0684043 μm, λ2 = 0.1162414 μm, and λ3 = 9.896161 μm(Agrawal, 1995), where λi = 2πc/ ωi and c is the velocity of light in vacuum. We use the value ofthe nonlinear refractive coefficient n2 = 2.48×10-20 m2/W from ref. (Taylor et al., 1996), andthe third-order susceptibility χ(3) is found to be χ(3) = 1.85×10-22 m2/V2 at 800 nm, as given byχ(3) = (4/3)ε0cn(ω0)2n2, where ω0 is the center angular frequency of the optical pulse. The
parameters α, τ1, and τ2 in Eq. (15) are set to be α = 0.7, τ1 = 12.2 fs and τ2 = 32 fs (Agrawal, 1995).
A single time step of the finite difference is set as Δt = 4.4475215×10-17 s at the wavelength of
800 nm. The time step of Δt at 800 nm is defined as the optical cycle at 800 nm of 2.6666667 fs
divided by 60. The time step for the wavelength of 1550 nm is defined by the same rule as
described above.
In the extended FDTD calculation, we set all parameters to be the same as those in our
experiment (Nakamura et al., 2002a), (Karasawa et al., 2000). We compare the results of theextended FDTD calculation and the SSF calculation with the experimental result for thepulse peak power of 175 kW in order to generate an ultrabroad spectrum which can finallybe compressed to 7.1 fs (Nakamura et al., 2002a) or shorter. The total fiber length of L = 2.5mm corresponds to 136,500 spatial steps, which means that L = 136,500×Δz, where Δz is aunit spatial step in the z direction. We need 293,000 time steps to measure the electric field
up to the complete passage of the pulse tail. The peak power of the input pulse is set to be
175 kW (soliton number N = 2.09). The initial temporal pulse form is assumed to be Fouriertransformed sech2 because the mode-locked pulse generally has the sech2 shape. Hence theinput spectrum is naturally assumed to have a sech2 shape. The initial pulse width is 12 fs(FWHM). The effective core area Aeff is set to be 5.47 μm2.
Figure 17(a) shows the results calculated by the extended FDTD Maxwell equation method(A), the solution of MGNLSE obtained using the SSF method with SVEA (B) (up to the fifthorder dispersion terms with the Raman term using the Raman response function), and ourpreviously reported experimental result (Karasawa et al., 2000) (C). It is seen that withwww.intechopen.com

Intensity (arb. units )Intensity (arb. units )Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation

1 FDTD (A)

479

(a)

SVEA (B)Exp. (C)
0

1

600

800
1000
Wavelength (nm)1200

FDTD (D)No Raman(b)

FDTD (A)Exp. (C)
0

600

800
Wavelength (nm)1000

Fig. 17. (a) Spectra of 12 fs laser transmission through a 2.5 mm silica fiber calculated using(A) the extended FDTD Maxwell equation considering all orders of dispersions and theRaman response [α = 0.7 in eq. (15)] and (B) the solution of MGNLSE obtained using the SSF
method with SVEA (considering up to 5th-order dispersion terms and the Raman termusing the Raman response function), and (C) our previously reported experimental result(Karasawa et al., 2000), where the incident laser intensity corresponds to the soliton numberof 2.09. (b) Spectra calculated by the FDTD Maxwell equation method (D) without the
Raman response [α = 1 in eq. (15)] and (A) with the Raman response. (A) and (C) are thesame as those in (a). (Nakamura et al., 2004)
SVEA (B), the spectral intensity at short wavelengths and at long wavelengths is muchhigher and slightly higher than those for FDTD (A) and the experimental result (C),respectively. The shortest wavelengths (intensity of 1%) of the spectra of FDTD (A), SVBA
(B) and the experimental result (C) are 600 nm, 560 nm and 600 nm, respectively. The
longest wavelengths of FDTD (A), SVBA (B) and the experimental result (C) are 1160 nm,
1240 nm and 1016 nm (maximum measurable wavelength of spectrometer), respectively.
The FWHM spectral bandwidths Δλ of FDTD (A), SVBA (B) and the experimental result (C)are 172 nm, 214 nm and 136 nm, respectively. Thus the spectral bandwidth of theexperimental result (C) is narrow and that of FDTD (A) is closer to the experimental result(C) than is that of SVEA (B). This indicates that the extended FDTD directly solving Maxwell
equation is superior to BPM in which MGNLSE (Sone et al., 2002) is solved by SSF with
SVEA. The Raman response function (Gross et al., 1992), which is also included in theextended FDTD, and up to the fifth-order dispersion terms are accurately included in BPM
for solving MGNLSE (Sone et al., 2002) by SSF with SVEA (B). However, in BPM for solving
MGNLSE by SSF with SVEA, the second derivative of the electric field with respect to z,
22∂ Ey / ∂z , is neglected, which corresponds to neglecting the backward-propagating wave.
On the other hand, our extended FDTD Maxwell equation method (A) accurately includes
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Coherence and Ultrashort Pulse Laser Emission

and does not require SVEA. Thus, the difference between (A) and (B) is considered to be dueto the higher order (more than 6th order) dispersion effect, or the backward-propagatingwave. For wavelengths longer than 800 nm in Fig. 17(a), the extended FDTD result and the
BPM result are similar but the intensity of the experimental data is lower than those of both
calculated results. We assume the sensitivity of the spectrometer detector to be low at
wavelengths longer than 800 nm because the detector is made of silicon. We believe our
extended FDTD result is accurate. We suggest that it is better to use an IR detector made of amaterial such as InGaAs for the spectrometer.
Next, in order to clarify the importance of the Raman response, we performed a calculation
using (D) the FDTD Maxwell equation method without the Raman response [α = 1 in eq.
(15)], as shown in Fig. 17(b), where (A) and (C) show the same data as those in Fig. 1(a). In
Fig. 17(b), the spectrum for case (A) is closer to the experimental result (C) than that of thecase of FDTD in which the Raman effect is not considered (D). It is evident that by including
the Raman term (A), the spectral intensity at a shorter wavelength is lower, and theagreement between the experimental and calculated results becomes better than that in thecase of (D). For example, the spectral intensity at 700 nm in (D) is 48% higher than that in(A), which is almost the same as that in the experimental result (C). On the other hand, at alonger wavelength, for example, 850 nm, the spectral intensity of (A) is 15% higher than thatof (D). This feature of (A) shows a tendency analogous to that of (C) because there is ahigher peak at 850 nm than at the center wavelength of 800 nm in (C). These tendencies ofthe spectral characteristics indicate that it is important to include the Raman term.
B. Observations of breakdown of SVEA

We calculate the time profiles and spectra in silica fibers with lengths of 0 to 2LD, where LD is
the dispersion length, with Gaussian input pulses of 12 fs, 7 fs and 4 fs duration (FWHM) at
the center wavelengths of 800 nm and 1550 nm which simulate the normal group-velocity
dispersion (GVD) and anomalous GVD, respectively. The dispersion lengths LD at 800 nm
and 1550 nm are 1.4364 mm and 1.8586 mm for the 12 fs pulse, respectively. Power regionsin this calculation are selected to be sufficiently small to enable comparison with results
based on the previous theory (Karasawa et al., 200b), (Agrawal, 1995), which means that wenormalize the peak power of the input pulse by a soliton parameter of N = 1.
Figures 18(a) and 18(b) show the time profiles of the output pulses from the silica fiber withincident pulse width of 12 fs and fiber lengths of 0 to 2LD, where LD is the dispersion length(Agrawal, 1995), calculated by the extended FDTD and BPM with SVEA, respectively. The
dispersion length LD for 12 fs pulses at 800 nm is 1.4364 mm. The pulse widths simply
broaden with propagation in the normal dispersion regime, as shown in Figs. 18(a) and18(b), because both GVD and self-phase modulation (SPM) (Agrawal, 1995) produce upchirp in normal dispersion. Figures 18(c) and 18(d) show the Fourier-transformed spectra ofFigs. 18(a) and 18(b), where ν and ν 0 are the frequency and the center frequency of thepulse, respectively. The symmetric spectral broadening due to the SPM effect is seen in thenormal dispersion regime, as shown in Figs. 18(c) and 18(d).
Figure 19 show the case of the pulse width of 7 fs under the same definitions of (a) to (d) as
in Fig. 18, where the dispersion length LD for 7 fs pulses at 800 nm is 0.48939 mm. This figure
indicates that pulse width broadening by positive GVD and spectral broadening by the SPM
effect are greater than those in Fig. 18, and there is no difference between the results
calculated using FDTD [Figs. 19(a) and 19(c)] and BPM [Figs. 19(b) and 19(d)]. This means

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Femtosecond Laser Pulse Propagation

481

that there is no breakdown of SVEA. Figure 20 shows the case of the pulse width of 4 fs
under the same definitions of (a) to (d) as in Figs. 18 and 19, where the dispersion length LD
for 4 fs pulses at 800 nm is 0.15980 mm. This figure also indicates that pulse width
broadening and spectral broadening effects are greater than those in Figs. 18 and 19, andthere is again no difference between the results calculated using FDTD [Figs. 20(a) and 20(c)]and BPM [Figs. 20(b) and 20(d)]. However, focusing on the case of z = 2LD in Fig. 20(a)calculated by the extended FDTD, we see that the main peak is slightly doubled or slightly
dipped. On the other hand, only one peak exists in the results of BPM in Fig. 20(b). In thiscase, a very small difference is found but it need not be taken into account. That is, there isno breakdown of SVEA even in the case of a pulse width of 4 fs. It is very useful to knowthat we can use BPM even for extremely short pulses of 4 fs without breakdown of SVEA in
the case of a soliton number of 1 in the normal GVD region.

(a) FDTD

Z=0
Z=LD
Z=2LD

T FWHM = 12 fs
λ0 =800 nm

0.5

1Intensity (arb. units)

Intensity (arb. units)

10(b) BPM

0.5

0
–100

0

100

–100

Time T (fs)
1 (c) FDTD
ν0 =375.0 THz
λ0 =800 nm

Z=0
Z=LD
Z=2LD

0.5

0

Frequency ν−ν0 (THz)

–100

0

0

100

Time T (fs)100Normalized Intensity (arb. units)

Normalized Intensity (arb. units)

Z=0
Z=LD
Z=2LD

T FWHM = 12 fs
λ0 =800 nm

1(d) BPM

Z=0
Z=LD
Z=2LD

ν0 =375.0 THz
λ0 =800 nm

0.5

0
–100

0

100

Frequency ν−ν0 (THz)

Fig. 18. (a), (b) Temporal profiles and (c), (d) spectra numerically obtained by (a), (c) the
extended FDTD method and (b), (d) BPM for 12 fs, 800 nm laser propagation through a silica
fiber of up to twice the dispersion length LD, where the laser intensity corresponds to thesoliton number of 1. (Nakamura et al., 2004)

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Coherence and Ultrashort Pulse Laser Emission(a) FDTD

1

Z=0
Z=LD
Z=2LD

T FWHM = 7 fs
λ0 =800 nm

Intensity (arb. units)

Intensity (arb. units)

1

0.5

0
–50

0

(b) BPM

0.5

0
–50

50

0

(c) FDTD

Z=0
Z=LD
Z=2LD

ν0 =375.0 THz
λ0 =800 nm

0.5

0
–200

–100

0

50

Time T (fs)100Frequency ν−ν0 (THz)200

Normalized Intensity (arb. units)

Normalized Intensity (arb. units)Time T (fs)
1

Z=0
Z=LD
Z=2LD

T FWHM = 7 fs
λ0 =800 nm

1(d) BPM

Z=0
Z=LD
Z=2LD

ν0 =375.0 THz
λ0 =800 nm

0.5

0
–200

–100

0

100

Frequency ν−ν0 (THz)200

Fig. 19. (a), (b) Temporal profiles and (c), (d) spectra numerically obtained by (a), (c) the
extended FDTD method and (b), (d) BPM for 7 fs, 800 nm laser propagation through a silica
fiber of up to twice the dispersion length LD, where the laser intensity corresponds to thesoliton number of 1. (Nakamura et al., 2004)
On the other hand, in the anomalous dispersion regime, it is seen in Figs. 21(a) and 21(b)that the pulse widths do not vary with pulse evaluation because the soliton effects(Agrawal, 1995) occur as a result of interplay between SPM and anomalous GVD. The
dispersion length LD for 12 fs pulses at 1550 nm is 1.8586 mm. However, the details of Figs.
21(a) and 21(b) show that the pulse peaks are delayed and appear shifted toward the trailingside with propagation. This is due to self-steepening. (Agrawal, 1995) There are three small
peaks following the main peak. These are formed by third-order or higher dispersions. In
Figs. 21(a) and 21(b), the second largest peaks slightly differ from each other, and thisdifference also exists in Figs. 22(a) and 22(b). Figures 21(c) and 21(d) show the Fouriertransformed spectra of Figs. 21(a) and 21(b), where ν and ν 0 are the frequency and thecenter frequency of the pulse, respectively. There is no obvious change of spectral
broadening, as shown in Figs. 21(c) and 21(d), because the fundamental soliton is generated,
as already described.

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Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation

Z=0
Z=LD
Z=2LD

TFWHM = 4 fs
λ0 =800 nm
0.5

0
–50

0

1 (b) BPM

Intensity (arb. units)

Intensity (arb. units)

1 (a) FDTD

0.5

0

Z=0
Z=LD
Z=2LD

ν0 =375.0 THz
λ0 =800 nm

0.5

0Frequency ν−ν0 (THz)

–250

0

50

Time (fs)250

Normalized Intensity (arb. units)

Normalized Intensity (arb. units)Time (fs)
1 (c) FDTD

Z=0
Z=LD
Z=2LD

TFWHM = 4 fs
λ0 =800 nm

0
–50

50

483

1 (d) BPM
ν0 =375.0 THz
λ0 =800 nm

Z=0
Z=LD
Z=2LD

0.5

0

Frequency ν−ν0

–250

0

250

(THz)

Fig. 20. (a), (b) Temporal profiles and (c), (d) spectra numerically obtained by (a), (c) the
extended FDTD method and (b), (d) BPM for 4 fs, 800 nm laser propagation through a silica
fiber of up to twice the dispersion length LD, where the laser intensity corresponds to thesoliton number of 1. (Nakamura et al., 2004)

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Coherence and Ultrashort Pulse Laser Emission(a) FDTD

1 (b) BPM

Z=0
Z=LD
Z=2LD

T FWHM = 12 fs
λ0 =1550 nm

Intensity (arb. units)

Intensity (arb. units)

1

0.5

0
–50

0.5

0
0

50

–50

Time T (fs)
ν0 =193.5 THz

1 (c) FDTD

Z=0
Z=LD
Z=2LD

λ0 =1550 nm

0.5

0Frequency ν−ν0 (THz)

–100

0

0

50

Time T (fs)100Normalized Intensity (arb. units)

Normalized Intensity (arb. units)

Z=0
Z=LD
Z=2LD

TFWHM = 12 fs
λ0 =1550 nm

ν0 =193.5 THz

1 (d) BPM

Z=0
Z=LD
Z=2LD

λ0 =1550 nm

0.5

0Frequency ν−ν0 (THz)

–100

0

100

Fig. 21. (a), (b) Temporal profiles and (c), (d) spectra numerically obtained by (a), (c) the
extended FDTD method and (b), (d) BPM for 12 fs, 1550 nm laser propagation through asilica fiber of up to twice the dispersion length LD, where the laser intensity corresponds tothe soliton number of 1. (Nakamura et al., 2004)

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Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation

1 (b) BPM

Z=0
Z=LD
Z=2LD

TFWHM = 7 fs
λ0 =1550 nm

Intensity (arb. units)

Intensity (arb. units)

1 (a) FDTD

0.5

0
–50

0

0.5

0

ν0 =193.5 THz
λ0 =1550 nm

Z=0
Z=LD
Z=2LD

0.5

0
–250

Frequency ν−ν0 (THz)050Time T (fs)250

Normalized Intensity (arb. units)

Normalized Intensity (arb. units)Time T (fs)
1 (c) FDTD

Z=0
Z=LD
Z=2LD

TFWHM = 7 fs
λ0 =1550 nm

0
–50

50

485

ν0 =193.5 THz

1 (d) BPM

Z=0
Z=LD
Z=2LD

λ0 =1550 nm

0.5

0
–250

Frequency ν−ν0 (THz)0250

Fig. 22. (a), (b) Temporal profiles and (c), (d) spectra numerically obtained by (a), (c) the
extended FDTD method and (b), (d) BPM for 7 fs, 1550 nm laser propagation through a silica
fiber of up to twice the dispersion length LD, where the laser intensity corresponds to thesoliton number of 1. (Nakamura et al., 2004)

However, it is seen that there is a small spectral continuum component from ν − ν 0 = 20 to
120 THz in Figs. 22(c) and 22(d). This feature is due to blue shift (Agrawal, 1995) by selfsteepening. This effect becomes more obvious in the case of the pulse widths of 7 fs [Figs.
22(c) and 22(d)] and 4 fs [Figs.23(c) and 23(d)], where the dispersion lengths LD for 7 fs and 4
fs pulses at 1550 nm are 0.63324 mm and 0.20677 mm, respectively. We also note that the
spectra at z = 2LD in Figs. 21(c) and 21(d) shift slightly to a lower frequency due to theRaman effect. (Agrawal, 1995) The Raman shift in Fig. 21(c) is larger than that in Fig. 21(d).
This Raman effect also become more obvious in the case of the pulse widths of 7 fs [Figs.
22(c) and 22(d)] and 4 fs [Figs.23(c) and 23(d)], and the Raman shift difference between (c)and (d) become smaller with decreasing pulse width. In terms of the Raman effect, FDTD
and BPM result in analogous behavior. On the other hand, the behavior of blue shift differs
between Fig. 23(c) and Fig. 23(d). The second largest slow perceptible peak exists from ν−ν0= 100 to 200 THz in Fig. 23(d), but the frequency region is a small continuum in Fig. 23(c).

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Coherence and Ultrashort Pulse Laser Emission

From the total viewpoint of the anomalous region, the breakdown of SVEA is tolerant in 12
fs, however, in the time region shorter than 7 fs, the breakdown cannot be ignored.
Through these calculations, we successfully demonstrated the dispersion and SPM effect(Agrawal, 1995), soliton propagation at 1550 nm, self-steepening, and the Raman effect.
Furthermore it was found that BPM and the extended FDTD results are almost the same in
the normal GVD region, and SVEA does not break down in the cases of incident pulse
widths of 12 fs, 7 fs, and 4 fs and soliton number of N = 1 in the normal dispersion frequency
region, as well as in the case of the incident pulse width of 12 fs and soliton number of N = 1
in the anomalous dispersion frequency region.

Z=0
Z=LD
Z=2LD

TFWHM = 4 fs
λ0 =1550 nm

0.5

1 (b) BPM

Intensity (arb. units)

Intensity (arb. units)

1 (a) FDTD

0
–50

0

0.5

0
–50

50

0

Z=0
Z=LD
Z=2LD

0

Frequency ν−ν0 (THz)

–250

0

50

Time T (fs)250

Normalized Intensity (arb. units)

Normalized Intensity (arb. units)Time T (fs)(c) FDTD
ν0 =193.5 THz
λ0 =1550 nm

Z=0
Z=LD
Z=2LD

TFWHM = 4 fs
λ0 =1550 nm

ν0 =193.5 THz
λ0 =1550 nm

1 (d) BPM

Z=0
Z=LD
Z=2LD

0.5

0

Frequency ν−ν0 (THz)

–250

0

250

Fig. 23. (a), (b) Temporal profiles and (c), (d) spectra numerically obtained by (a), (c) the
extended FDTD method and (b), (d) BPM for 4 fs, 1550 nm laser propagation through a silica
fiber of up to twice the dispersion length LD, where the laser intensity corresponds to thesoliton number of 1. (Nakamura et al., 2004)

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(a) FDTDTFWHM = 12 fs
1 λ0 =1550 nm
N=2

Z=0
Z=LD
Z=2LD

0.5

Intensity (arb. units)

Intensity (arb. units)Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation(b) BPMTFWHM = 12 fs
1 λ0 =1550 nm
N=2

–100

0.5

0

–100

100

Z=0
Z=LD
Z=2LD

0.5

0
–250

Frequency ν−ν0 (THz)0250

Normalized Intensity (arb. units)(c) FDTD
ν0=193.5 THz
λ0 =1550 nm

0100Time T (fs)Time T (fs)Normalized Intensity (arb. units)

Z=0
Z=LD
Z=2LD

0

0

1

487

1

(d) BPMν0 =193.5 THz
λ0 =1550 nm

Z=0
Z=LD
Z=2LD

0.5

0
–250

Frequency ν−ν0 (THz)0250

Fig. 24. (a), (b) Temporal profiles and (c), (d) spectra numerically obtained by (a), (c) the
extended FDTD method and (b), (d) BPM for 12 fs, 1550 nm laser propagation through asilica fiber of up to twice the dispersion length LD, where the laser intensity corresponds tothe soliton number of 2. (Nakamura et al., 2004)
Although we obtained extended FDTD results which could be explained by the previous
theory (Agrawal, 1995) and could verify the validity of our calculation method, the physicalbehavior of very short pulses will be different when the peak power is four times larger thanthat in the case of N = 1, which makes the breakdown of SVEA possible even for long pulses.
We have already shown the breakdown of SVEA in a comparison of experimental andnumerical results for the positive dispersion area with N = 2.09 for 12 fs pulse propagationin the fiber. Thus, we perform calculations for N = 2 and the anomalous dispersion area.
Figure 24 shows (a), (b) temporal profiles and (c), (d) spectra numerically obtained by (a), (c)the extended FDTD method and (b), (d) BPM for 12 fs, 1550 nm laser propagation through asilica fiber of up to z = 2LD, where the laser intensity corresponds to the soliton number N of
2. This figure shows that the breakdown of SVEA occurs in BPM for even a long pulse widthof 12 fs. For the spectra shown in Figs.24(c) and 24(d), the height of the second largest peak

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488

Coherence and Ultrashort Pulse Laser Emissionin Fig. 24(d) is twice that in Fig. 10(c). In the temporal profiles of Figs. 24(a) and 24(b), themain peaks have the same form, however numerous small-peak characteristics from 20 fs to80 fs are different. The small peaks in Fig. 24(a) are larger and more obvious than those in
Fig. 24(b). These results imply that SVEA of MGNLSE in BPM, as assumed in Figs. 24(b) and24(d), breaks down in the case of N = 2 in the anomalous GVD region for 12 fs pulse
propagation.
5.5 Summary of SVEA breakdown
We have extended the nonlinear FDTD method with a JGTH algorithm to one with exact
Sellmeier fitting values in order to compare the experimental and calculated results ofnonlinear femtosecond ultrabroad-band-pulse propagation in a silica fiber, and haveanalyzed the spectral characteristics of the propagated pulse. This extended method isrobust against the breakdown of the SVEA. The spectrum obtained in our previous
experiment agrees better with the spectrum calculated by the extended FDTD method thanwith that calculated by the solution of MGNLSE obtained using the SSF method with SVEA.
Consequently, we confirmed that the various nonlinear physical phenomena in an opticalfiber can be described by FDTD methods even in a nearly single-cycle time regime, and that
SVEA in BPM is invalid in the case of pulse widths less than 4 fs with the soliton number of
1 at 800 nm, which is the normal dispersion regime, and in the case of pulse widths less than
7 fs with the soliton number of 1 at 1550 nm, which is the anomalous dispersion regime. In
the case of the soliton number of 2, the breakdown of SVEA occurs for 12 fs pulses in boththe normal and anomalous dispersion frequency regimes. To the best of our knowledge, thisis the first observation of the breakdown of SVEA as a function of laser pulse width andsoliton number by comparison of the extended FDTD and BPM calculations for the
nonlinear propagation of an ultrashort (< 12 fs) laser pulse in a silica fiber. If the extended
FDTD method can be established as an ultrashort-optical-pulse propagation analysis
technique, its application to analyses of the characteristics of an ultrahigh-speed opticalswitch, the propagation characteristic of optical pulses in a photonic crystal, and thenonlinear propagation in a photonic crystal fiber or a tapered fiber can be expected.

6. Conclusion
Conventionally, the BPM for solving the GNLSE including the SVEA has been used todescribe the ultrashort laser pulse propagation in an optical fiber. However, if the pulse
duration approaches the optical cycle regime (<10 fs), this approximation becomes invalid.
Then, it becomes necessary to use the FDTD method for solving the Maxwell equation withthe least approximation. In order to both experimentally and numerically investigate
nonlinear femtosecond ultrabroad-band-pulse propagation in a silica fiber, we haveextended the FDTD calculation of Maxwell's equations with nonlinear terms to that
including all exact Sellmeier-fitting values. We have compared the results of this extended
FDTD method with experimental results for the nonlinear propagation of a very short (12-fs)
chirped laser pulse in a silica fiber. We obtain the fiber output pulse compressed to 7 fs by
the simulation of group delay compensation under the assumption of using an SLM. To ourknowledge, this is the first comparison between FDTD calculation and experimental results
for nonlinear propagation of a very short (12-fs) chirped pulse in a silica fiber.

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Comparison Between Finite-Difference Time-Domain Method and Experimental Results for
Femtosecond Laser Pulse Propagation

489

Next, experimental results of nonlinear propagation with the IPM effect of two-color 120 fs
laser pulses with three different initial delays in a silica fiber were analyzed with FDTD
calculations, including SPM, IPM, and DFWM in the fused-silica fiber. Initial delay
dependence by the FDTD qualitatively agrees with the previous experimental results. An
ultrabroad spectrum induced by IPM mostly agrees with the previous experimental resultsthat had the optimum initial delay. A Fourier transform of this spectrum yields a 4 fs opticalpulse, which agrees with the previous experimental results. The spectral phases of two
different pulses with three different initial delays are connected continuously. The IPM
spectral phase obtained by the extended FDTD shows that the DFWM occurs. To the best of
our knowledge, this is the first observation of DFWM with IPM by the FDTD simulation.
Finally, We compared the results of this extended FDTD method with the solution of the
BPM that includes the Raman response function, which is the same as in the extended FDTD
method, up to fifth-order dispersion with the SVEA, as well as with the experimental results
for nonlinear propagation of a 12 fs laser pulse in a silica fiber. Furthermore, in only thecalculation, pulse width was gradually shortened from 12 fs to 7 fs to 4 fs to observe thebreakdown of the SVEA in detail. Moreover, the soliton number N was established as 1 or 2.
To the best of our knowledge, this is the first observation of the breakdown of the SVEA by
comparison between the results of the extended FDTD and the BPM calculations for the
nonlinear propagation of an ultrashort (< 12 fs) laser pulse in a silica fiber.

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Birks, T. A.; Wadsworth, W. J. & Russell, P. St. J. (2000). Supercontinuum generation in
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Geraghty, D. F.; Lee, R. B.; Verdiell, M.; Ziari, M.; Mathur, A & Vahala, K. J. (1998).
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Goorjian, P. M.; Taflove, A.; Joseph, R. M. & Hagness, S. C. (1992). Computational modelingof femtosecond optical solitons from Maxwell’s equations. IEEE J. Quantum.
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Goorjian, P. M.& Cundiff, S. T.(2004). Nonlinear effects on the carrier-envelope phase. Opt.
Lett., Vol. 29, No. 12, (June, 2004) 1363-1365, ISSN 0146-9592.
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Coherence and Ultrashort Pulse Laser Emission
Edited by Dr. F. J. Duarte

ISBN 978-953-307-242-5
Hard cover, 688 pages
Publisher InTech

Published online 30, November, 2010

Published in print edition November, 2010
In this volume, recent contributions on coherence provide a useful perspective on the diversity of various
coherent sources of emission and coherent related phenomena of current interest. These papers provide apreamble for a larger collection of contributions on ultrashort pulse laser generation and ultrashort pulse laser
phenomena. Papers on ultrashort pulse phenomena include works on few cycle pulses, high-power
generation, propagation in various media, to various applications of current interest. Undoubtedly, Coherence
and Ultrashort Pulse Emission offers a rich and practical perspective on this rapidly evolving field.

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Shinki Nakamura (2010). Comparison between Finite-Difference Time-Domain Method and Experimental
Results for Femtosecond Laser Pulse Propagation, Coherence and Ultrashort Pulse Laser Emission, Dr. F. J.
Duarte (Ed.), ISBN: 978-953-307-242-5, InTech, Available from: http://www.intechopen.com/books/coherenceand-ultrashort-pulse-laser-emission/comparison-beteween-fdtd-simulation-and-experimental-results-for-12-fslaser-pulse-propagation-in-a-

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