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Gauge Institute Journal
H. Vic Dannon
Optical Soliton, and
Propagating Delta Function
H. Vic Dannon
[email protected]
February, 2011
Abstract
A detailed derivation of Optical Solitons, and experimental data
suggest that an Optical Soliton may be modeled by a propagating
Delta Function.
Keywords: Optical Temporal Solitons, Nonlinear Phase, Group
Velocity Dispersion, Nonlinear Optics, Nonlinear Schrödinger
Equation, Infinitesimal, Delta Function, Fourier Transform,
Laplace Transform, Ultrashort Light Pulses, Ultrafast Optics,
Optical Catastrophe, Free Electron Laser.
2010 Physics Astronomy Classification Scheme 42.81.Dp;
42.55.Wd; 42.60.Fc; 42.65.Jx; 42.65.Re; 42.65.Tg;
2000 Mathematics Subject Classification 35C08; 35Q51; 37K40;
74J35; 76B25; 35K55; 78A60;26E35; 26E30; 26E15; 26E20; 26A06; 26A12;
03E10; 03E55; 03E17; 03H15; 46S20; 97I40; 97I30;
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Gauge Institute Journal
H. Vic Dannon
Contents
Optical Solitons
1. Non-Linear Phase Function
2. Non-Linear Schrödinger Equation
3. Dispersion and phase modulation of a Gaussian pulse
4. Non-Linear Schrödinger equation for Solitons
5. The Soliton’s Envelope Equation
6. The Fundamental Soliton solution
Delta Function
7. Hyper-real Line
8. Hyper-real Integral
9. Delta Function
10. The Fourier Transform
11. The Laplace Transform
12. Pulse Compression and Delta
13. Soliton and Delta Function
References
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H. Vic Dannon
Optical Solitons
Optical Solitons are pulses enveloping an optical frequency carrier
that keep their shape as they propagate through the optical fibers.
A wave packet fed into an optical fiber, evolves into a soliton if its
group velocity dispersion is balanced by its self-phase modulation
in the glass.
That balance is attained when the frequency ω of the carrier
under the pulse, remains the same before the peak of the pulse,
and beyond the peak of the pulse.
The frequency ω , is the time derivative of the Non-linear phase
function of the pulse in the optical fiber.
the phase function is necessary to obtain an expression for the
wave packet that is used to derive the Non-Linear Schrödinger
Equation, the Soliton’s equation
The phase function depends on the frequency ω , and on the
refractive index n(ω, E 0 ) , where E 0 is the Amplitude of the wave
packet, and may be approximated by a Taylor polynomial in the
variables ω , and E 0 .
That approximation is nowhere to be found in the literature.
authors seem to know about Taylor polynomials in one variable,
but not in two variables.
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Gauge Institute Journal
H. Vic Dannon
Consequently, the literature has no substantiated derivation of
the Non-Linear Schrödinger Equation.
We aim to show that Solitons may be represented by a Delta
function, but this requires that generation of Solitons be clearly
understood.
In section 1, we obtain the Taylor approximation for the nonlinear
phase function, and the expression for the wave packet
In section 2, we derive the Non-Linear Schrödinger equation.
In section 3, we discuss dispersion and phase modulation of a
Gaussian pulse, in order to gain insight about pulses in general,
and solitons in particular.
In section 4, we obtain the Non-linear Schrödinger equation
modified for solitons, and in section 5 the equation for the Soliton’s
envelope equation.
in section 6, we present the known fundamental Soliton solution,
in a form that will enable us to establish that it may be
represented by a Delta function.
We do that in the second part of the paper, in sections 7-13.
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Gauge Institute Journal
H. Vic Dannon
1.
Non-Linear Phase Function
Consider a wave packet of light frequencies propagating in an
optical fiber,
ω =∞
∫
ψ(z , t ) =
A(ω)e i(ωt −βz )d ω .
ω =−∞
The wave packet may be written as
ω =∞
ψ(z , t ) = e
−i ω0t
A(ω)e i(ω0t −βz )e i ωtd ω
∫
ω =−∞
ω =∞
Then, it is an envelope
∫
A(ω)ei(ω0t −βz )e i ωtd ω
modulating a
ω =−∞
carrier with frequency ω0 .
The phase function is
ω0t − βz .
In the air, β is a constant β0 .
In the optical fiber, β is
a
nonlinear function of ω , and of the light intensity I . We aim to
express β in terms of ω , and I .
G
G
Let E , and B be the electromagnetic fields of the wave packet.
G
G
By Faraday’s Law, ∇ × E = −∂t B ,
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Gauge Institute Journal
H. Vic Dannon
G
G
∇ × (∇ × E ) = ∇ × (−∂t B ) ,
G
G
G
∇(∇ ⋅ E ) − ∇2E = −∂t ∇ × B ,
G
G
= −∂t ∇ × (M + μ0H )
G
ρ
Assuming a scalar dielectric coefficient ε = ε0 εr , and ∇ ⋅ E = .
ε
G
Assuming no electric charges, ρ = 0 , and ∇ ⋅ E = 0 .
G
Assuming no magnetic charges, M = 0 , and
G
G
−∇2E = −μ0∂t ∇ × H
G
G
G
By Ampere’s Law, ∇ × H = J + ∂t D .
G
G
G
Assuming no current, J = 0 , ∇ × H = ∂t D , and
G
G
∇2E = μ0∂t2D
G
= μ0∂t2 (ε0εr E )
G
2
= ε0μ0∂t (εr E )
Assuming a Plane Electromagnetic wave propagating in the z
G
direction, E = ⎡⎢ E (z , t ), 0, 0 ⎤⎥ , and
⎣
⎦
∂z2E = ε0 μ0∂t2 (εr E ) .
The refractive index is
n =
εr ,
β =
ω0
n,
c0
and
where
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Gauge Institute Journal
H. Vic Dannon
1
c0 =
ε0μ0
is the speed of the light in the vacuum.
Since
εr = n 2 ,
and since
εr = 1 + χ ,
where χ is the susceptibility of the optical fiber, we have
n2 = 1 + χ .
In a Non-Linear Optical Medium, χE may depend on powers of
the electric field E
χE = χ(1)E + χ(2)E 2 + χ(3)E 3 + ... + χ(N )E N ,
where the coefficients χ(1), χ(2)...χ(N ) are very small. In Optical
Fibers, we put χ(2) = 0 , and assume a cubic nonlinearity
χE = χ(1)E + χ(3)E 3 .
Therefore,
n 2E = (1 + χ)E
= (1 + χ(1) )E + χ(3)E 3 .
Substituting a harmonic Electric field
E = E 0 (z , t )cos(ωt − kz ) ,
n 2E 0 cos(ωt − kz ) = (1 + χ(1) )E 0 cos(ωt − kz ) + χ(3)E 03 cos3 (ωt − kz )
Since E 0 ≠ 0 ,
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Gauge Institute Journal
H. Vic Dannon
n 2 cos(ωt − kz ) = (1 + χ(1) )cos(ωt − kz ) + χ(3)E 02 cos3 (ωt − kz )
Now,
cos3 (ωt − kz ) =
=
1
2
3
1
3
2
(ei(wt −kz ) + e −i(wt −kz ) )3
(e 3i(wt −kz ) + 3ei(wt −kz ) + 3e−i(wt −kz ) + e−3i(wt −kz ) )
In Optical Fibers, the third harmonics 3(wt − kz ) is negligible, and
cos3 (ωt − kz ) ≈
=
3
3
2
(e i(wt −kz ) + e−i(wt −kz ) )
3
cos(ωt − kz )
4
Substituting into the equation for n 2 ,
n 2 cos(ωt − kz ) = (1 + χ(1) )cos(ωt − kz ) +
3 (3) 2
χ E 0 cos(ωt − kz )
4
Since cos(ωt − kz ) does not vanish identically,
n 2 = (1 + χ(1) ) +
3 (3) 2
χ E0 .
4
Denoting
1 + χ(1) ≡ n0 ,
we have
1
⎛
n
3 χ(3) 2 ⎞⎟⎟2
= ⎜⎜⎜ 1 +
E ⎟ .
⎜⎝
n0
4 n02 0 ⎠⎟
Since χ(3) is very small,
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H. Vic Dannon
3 χ(3) 2
≈1+
E .
8 n02 0
Thus,
3 χ(3) 2
n ≈ n0 +
E .
8 n0 0
The dependence of n on the frequency ω is implicit, and we shall
expand
n = n(ω, E 0 )
in a Taylor Polynomial of the second order about the point
ω = ω0 ,
n(ω, E 0 ) = n(ω0 , 0) +
1 ∂2n
+
2 ∂ω 2
∂n
∂ω
ω = ω0
E0 =0
E0 = 0 .
(ω − ω0 ) +
∂2n
(ω − ω0 ) +
∂ω ∂ E 0
ω = ω0
2
E0 =0
∂n
∂E 0
ω = ω0
E0 =0
E0 +
1 ∂2n
(ω − ω0 )E 0 +
2 ∂E 02
ω = ω0
E0 =0
Here,
∂n
∂E 0
∂2n
∂ ω ∂E 0
ω = ω0
E0 =0
=
3 χ(3)
E
4 n0 0
ω = ω0
E0 =0
=
3 ⎛⎜ χ(3) ⎞⎟
⎟E
⎜∂
4 ⎜⎝⎜ ω n0 ⎠⎟⎟ 0
1 ∂2n
2 ∂E 02
3 χ(3)
=
8 n0
ω = ω0
E0 =0
ω = ω0
E0 =0
Therefore,
9
= 0,
ω = ω0
E0 =0
=0
ω = ω0
E0 =0
E 02
Gauge Institute Journal
n(ω, E 0 ) = n(ω0 , 0) +
Multiplying by k 0 =
k0n(ω, E 0 ) = k0n(ω0 , 0) +
β ( ω ,E 0 )
β0
H. Vic Dannon
∂n
∂ω
ω = ω0
E0 =0
(ω − ω0 ) +
1 ∂2n
2 ∂ω 2
ω = ω0
E0 =0
(ω − ω0 )2 +
3 χ(3) 2
E
8 n0 0
ω0
,
c0
∂(k0n )
ω 3 χ(3) 2
1 ∂2(k0n )
(ω − ω0 ) +
(ω − ω0 )2 + 0
E
∂ω ω = ω0
2 ∂ω 2 ω = ω0
c0 8 n 0 0
E0 = 0
E0 = 0
∂β
≡ β1
∂ω
∂ 2β
∂ω 2
≡ β2
Denoting
3 χ(3)
≡ n2 ,
4 c0ε0n02
we have,
1
1
β(ω, E 0 ) = β0 + β1(ω − ω0 ) + β2 (ω − ω0 )2 + ω0ε0n0n2E 02 .
2
2
Denoting
I ≡
1
c0 ε0n 0E 02 ,
2
the refractive index is
n(ω, I ) ≈ n0 + n2I ,
and
β(ω, I ) = β0 + β1(ω − ω0 ) +
ω
1
β2 (ω − ω0 )2 + 0 n2I .
2
c0
Therefore, the phase function is
⎡
⎤
ω
1
ω0t − ⎢ β0 + β1(ω − ω0 ) + β2 (ω − ω0 )2 + 0 n2I ⎥ z ,
⎢
⎥
2
c0
⎣
⎦
and the wave packet is
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Gauge Institute Journal
H. Vic Dannon
ω =∞
ψ(z , t ) = e−i ω0t
∫
A(ω)e
⎛
⎞
ω
1
i ⎜⎜ ω0t −[ β0 + β1 ( ω −ω0 )+ β2 (ω−ω0 )2 + 0 n2I ]z ⎟⎟⎟
⎜⎝
c0
2
⎠⎟ i ωt
e dω .
ω =−∞
This wave packet leads to the Non-Linear Schrödinger equation.
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H. Vic Dannon
2.
Non-Linear Schrodinger Equation
The wave packet
ω =∞
ψ(z , t ) = e −i ω0t
∫
A(ω)e
⎛
⎞
ω
1
i ⎜⎜ ω0t −[ β0 + β1 ( ω−ω0 )+ β2 (ω −ω0 )2 + 0 n2I ]z ⎟⎟⎟
⎜⎝
c0
2
⎠⎟ i ωt
e dω
ω =−∞
ω =∞
= ei(ω0t −β0z )
∫
A(ω)e
⎛
⎞
ω
1
i ⎜⎜ (t −β1z )( ω−ω0 )− β2 (ω−ω0 )2 z − 0 n2Iz ⎟⎟⎟
c0
2
⎝⎜
⎠⎟
dω
ω
=−∞
E 0 (z ,t )
is an envelope
ω =∞
E 0 (z , t ) =
∫
A(ω)e
⎛
⎞
ω
1
i ⎜⎜ (t −β1z )( ω−ω0 )− β2 ( ω−ω0 )2 z − 0 n2Iz ⎟⎟⎟
⎟
⎜⎝
c0
2
⎠
dω
ω =−∞
modulating the carrier e i (ω0t −β0z ) .
We show that the envelope satisfies the Non-Linear Schrodinger
Equation.
2
1
1
∂z E 0 + β1∂t E 0 − i β2∂tt E 0 + i ω0ε0n0n2 E 0 E 0 = 0
2
2
ω =∞
∂z E 0 (z , t ) = ∂z
∫
A(ω)e
⎛
⎞
ω
1
i ⎜⎜ (t −β1z )( ω−ω0 )− β2 ( ω−ω0 )2 z − 0 n2Iz ⎟⎟⎟
⎜⎝
c0
2
⎠⎟
dω
ω =−∞
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Gauge Institute Journal
ω =∞
= −i
∫
A(ω)e
H. Vic Dannon
⎛
⎞
ω
1
i ⎜⎜ (t − β1z )( ω − ω0 )− β2 ( ω − ω0 )2 z − 0 n2Iz ⎟⎟⎟
⎜⎝
c0
2
⎠⎟
ω =−∞
ω =∞
= −β1 i
∫
(ω − ω0 )A(ω)e
⎡
⎤
ω
⎢ β1[ω − ω0 ] + 1 β2[ω − ω0 ]2 + 0 n2I ⎥ d ω
⎢
⎥
2
c0
⎣
⎦
⎛
⎞
ω
1
i ⎜⎜ (t −β1z )( ω −ω0 )− β2 (ω −ω0 )2 z − 0 n2Iz ⎟⎟⎟
⎜⎝
c0
2
⎠⎟
dω
ω =−∞
∂t E 0
⎛
⎞
ω
1
i ⎜⎜ (t −β1z )( ω−ω0 )− β2 ( ω−ω0 )2 z − 0 n2Iz ⎟⎟⎟
⎜⎝
c0
2
⎠⎟
ω =∞
1
+i β2 i 2 ∫ (ω − ω0 )2 A(ω)e
dω
2
ω =−∞
∂tt E 0
ω =∞
⎛
⎜
1
2
ω0
⎞⎟
i ⎜ (t −β1z )( ω −ω0 )− β2 ( ω −ω0 ) z − n2Iz ⎟⎟
ω0
⎜⎝
2
c0
⎠⎟
ω
−i
n 2I
A
(
)
e
dω
∫
c0
=−∞
ω
1
ωnn E
2 0 0 2 0
2
E0
Therefore,
2
1
1
∂z E 0 = −β1∂t E 0 + i ∂tt E 0 − i ω0ε0n 0n2 E 0 E 0 . ,
2
2
We proceed to explore the possibility of a Soliton solution for the
Non-Linear Schrödinger equation.
To that end, we follow the propagation of a Gaussian pulse
through the optical fiber.
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H. Vic Dannon
3.
Dispersion, and Phase Modulation
of a Gaussian Pulse
Since the wave packet is
ω =∞
ψ(z , t ) = e −i ω0t
∫
A(ω)e
⎛
⎞
ω
1
i ⎜⎜⎜ ω0t −[ β0 + β1 (ω −ω0 )+ β2 (ω −ω0 )2 + 0 n2I ]z ⎟⎟⎟
c0
2
⎝
⎠⎟ i ωt
e dω ,
ω =−∞
we have,
ω =∞
ψ(0, t ) =
∫
A(ω)ei ωtdω .
ω =−∞
Therefore,
t =∞
1
A(ω) =
ψ(0, t )e −i ωtdt .
∫
2π t =−∞
Let ψ(0, t ) be a Gaussian pulse enveloping the carrier e i ω0t ,
−
ψ(0, t ) = Ce
t2
2σt2 i ω0t
e
.
Then,
t =∞
A(ω) =
1
∫ Ce
2π t =−∞
14
−
t2
2σt2 i ω0t −i ωt
e
e
dt
Gauge Institute Journal
H. Vic Dannon
1
2π
=C
⎤
1 ⎡ t2
− ⎢⎢ −2i (ω0 −ω )t ⎥⎥
2
2
⎦⎥dt
e ⎣⎢ σt
t =∞
∫
t =−∞
t =∞
=C
1
∫
2π t =−∞
⎤2 1
1⎡ t
− ⎢ −i σt ( ω0 −ω ) ⎥ − σt2 ( ω0 −ω )2
⎢
⎥ 2
⎦
e 2 ⎣ σt
dt
⎛ t =∞ 1 ⎡ t
⎤2 ⎞
⎟⎟ −1 σ 2 (ω −ω )2
⎢ −i σt ( ω0 −ω ) ⎥
⎜
−
1 ⎜⎜
⎥
2 ⎢⎣ σt
⎦ dt ⎟
⎟⎟e 2 t 0
=C
⎜⎜ ∫ e
⎟⎟
2π ⎜ t =−∞
⎜⎝
⎠⎟
Put
τ =
t
1
− i σt (ω0 − ω)
σt
2
1
dt
σt
dτ =
Then,
1
A(ω) = C
σ
2π t
τ =∞
∫
1
1
− τ2
− σt2 ( ω0 −ω )2
2
e
dτ e 2
τ
=−∞
2π
=C
1
2π
1
− σt2 ( ω−ω0 )2
σte 2
.
Denoting
ω − ω0 ≡ ω ,
A(ω) = C
1
2π
The wave packet is
15
σt
1
− σt2ω2
e 2
Gauge Institute Journal
H. Vic Dannon
ω =∞
ψ(z , t ) = e −i ω0t
∫
A(ω)e
⎛
⎞
ω
1
i ⎜⎜ ω0t −[ β0 + β1 ( ω−ω0 )+ β2 (ω −ω0 )2 + 0 n2I ]z ⎟⎟⎟
⎜⎝
2
c0
⎠⎟ i ωt
e dω
ω =−∞
=e
⎛
⎞ ω =∞
ω
i ⎜⎜⎜ ω0t −β0z − 0 n2Iz ⎟⎟⎟
c0
⎝
⎠⎟
∫
A(ω)e
⎛
⎞
1
i ⎜⎜ (t −β1z )(ω−ω0 )− β2 (ω−ω0 )2 z ⎟⎟⎟
⎝⎜
2
⎠⎟
dω
ω =−∞
=e
⎛
⎞ ω
ω
=∞
i ⎜⎜⎜ ω0t −β0z − 0 n2Iz ⎟⎟⎟
⎟
c0
⎝
⎠
∫
A(ω)e
⎛
⎞
1
i ⎜⎜ (t −β1z )ω− β2ω 2z ⎟⎟⎟
⎜⎝
2
⎠⎟
d ω
=−∞
ω
Plugging in the A(ω) of a Gaussian Pulse,
ψ(z, t ) = C
=C
1
2π
σte
1
2π
⎛
⎞ ω =∞
ω
i ⎜⎜ ω0t −β0z − 0 n2Iz ⎟⎟⎟
⎜⎝
c0
⎠⎟
∫
⎛
⎞
1
2 ⎟
1
⎜
− σt2 ω2 i ⎜⎜ (t −β1z )ω − β2ω z ⎟⎟⎟
2
⎠
e 2 e⎝
dω
ω =−∞
σte
⎛
⎞ ω =∞
ω
i ⎜⎜⎜ ω0t −β0z − 0 n2Iz ⎟⎟⎟
c0
⎝
⎠⎟
∫
1
− ω2 σt2 +i β2z +i ( (t −β1z )ω )
.
e 2
dω
(
)
=−∞
ω
The exponent of the integrand is
1
− ω 2 ( σt2 + i β2z ) + i(t − β1z )ω =
2
2
1
1
⎛
⎞
−
1⎜
1 (t − β1z )2
⎟⎟
2
2
2
2
⎜
( σt + i β2z ) − i ( σt + i β2z ) (t − β1z ) ⎟ −
= − ⎜ω
⎟⎟
2 ⎜⎝
2 σt2 + i β2z
⎠
ωˆ
Denote
ˆ = ω (
ω
σt2
1
2
+ i β2z ) − i (
σt2
−
+ i β2z )
Then,
d ωˆ = (
σt2
1
2 dω
,
+ i β2z )
16
1
2
(t − β1z )
Gauge Institute Journal
H. Vic Dannon
and the wave packet is
ψ(z , t ) =
C
2π
σte
⎛
⎞
ω
i ⎜⎜⎜ ω0t −β0z − 0 n2Iz ⎟⎟⎟
c0
⎝
⎠⎟
ˆ =∞
1 ω
−
2
( σt2 + i β2z )
∫
1 (t −β1z )2
1 2
−
ˆ
− ω
2 2
e 2 d ωˆ e σt +i β2z
ˆ =−∞
ω
2π
= C σte
⎛
⎞
ω
i ⎜⎜ ω0t −β0z − 0 n2Iz ⎟⎟⎟
⎜⎝
c0
⎠⎟
−
( σt2 + i β2z )
(t −β1z )2
1 −1
2
2 e 2 σt +i β2z
Simplifying,
1
(
σt2
1
2
=
+ i β2z )
(
(
σt2
σt4
1
2
− i β2z )
+
1
2 2 2
β2 z
)
1
⎛β z ⎞ ⎞
⎛
1 −i arctan⎜⎜ 2 ⎟⎟⎟ ⎟2
⎜⎜
⎜⎜ 2 ⎟⎟ ⎟
⎝ σt ⎠ ⎟
⎜⎜ ( σt4 + β22z 2 )2 e
⎟⎟
1⎜
⎟
⎜⎝
2⎜
⎠⎟⎟
1
=
( σt4 + β22z 2 )
=
−
e
1 (t −β1z )2
2 σt2 +i β2z
−
=e
=e
e
⎛ β z ⎞⎟
1
− i arctan⎜⎜⎜ 2 ⎟⎟⎟
2
⎝⎜ σt2 ⎠⎟
(
1
2 2 4
β2 z
σt4
+
)
1 σt2 −i β2z
(t −β1z )2
2 σt4 + β22z 2
σt2
β2z
−1
1
(t −β1z )2 i
(t −β1z )2
4
2
2
4
2 σt + β2 z
2 σt + β22z 2
e
Hence,
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Gauge Institute Journal
ψ(z , t ) = C σte
H. Vic Dannon
⎛
⎞
ω
i ⎜⎜ ω0t −β0z − 0 n2Iz ⎟⎟⎟
⎜⎝
c
⎠⎟
0
( σt4 +
−σt2
=
C σte
e
⎛ β z ⎞⎟
1
− i arctan⎜⎜⎜ 2 ⎟⎟⎟
2
⎝⎜ σt2 ⎠⎟
σt4 + β22z 2
( σt4 +
(t −β1z )2
e
1
2 2 4
β2 z
1
2 2 4
β2 z
e
β2z
1 −σt2
1
(t −β1z )2 i
(t −β1z )2
2 σt4 + β22z 2
2 σt4 + β22z 2
e
)
⎛
⎛ β z ⎞⎟ 1 β z
⎞⎟
ω
1
2
i ⎜⎜⎜ ω0t −β0z − 0 n2Iz − arctan⎜⎜⎜ 2 ⎟⎟⎟+
(t −β1z )2 ⎟⎟⎟
c0
2
⎝⎜
⎝⎜ σt2 ⎠⎟ 2 σt4 + β22z 2
⎠⎟
)
The Power of the Pulse is proportional to
−2σt2
ψ(z , t )
2
=
4
2 2
C 2σt2e σt + β2 z
(t −β1z )2
1
2 2 2
β2 z
( σt4 +
.
)
Amplification along optical fibers, renders them loss-less medium,
and the pulse power does not dissipate with z . Alternatively, we
may take
β22z 2
1
σt4
.
Then,
ψ(z, t ) ≈ Ce
⎛
⎛ β z ⎞⎟ β z
⎞⎟
ω
1
−1
(t −β1z )2 i ⎜⎜⎜ ω0t −β0z − 0 n2Iz − arctan⎜⎜⎜ 2 ⎟⎟⎟+ 2 (t −β1z )2 ⎟⎟⎟
⎜⎝
⎜⎝ σt2 ⎠⎟ σt4
2
c0
σt2
⎠⎟
e
Hence,
−2
2
2 σt2
ψ(z, t ) ≈ C e
and the Gaussian Pulse power is
18
(t −β1z )2
,
.
Gauge Institute Journal
H. Vic Dannon
−2
σt2
P(t ) = P0e
(t −β1z )2
.
The intensity of the field in an optical fiber is
P(t )
Aeff
I =
−2
=
P0e
σt2
(t −β1z )2
Aeff
where Aeff is the effective area of the fiber’s cross section.
Therefore, the phase of ψ(z, t ) is
ϕ(t ) = ω0t − β0z −
ω0
βz
βz
1
n2Iz − arctan 2 + 2 (t − β1z )2 =
2
c0
σt2
σt4
−2
= ω0t − β0z −
σt2
(t −β1z )2
ω0 P0e
n
c0 2
Aeff
βz βz
1
z − arctan 2 + 2 (t − β1z )2 .
2
σt2
σt4
The frequency of the carrier is
ω(t ) = ∂t ϕ
ω
P
= ω0 − 0 n2 0
c0
Aeff
−2
⎛ 4
⎞⎟ 2 (t −β1z )2
βz
+ 2 (t − β1z )
z ⎜⎜⎜ − (t − β1z ) ⎟⎟e σt
⎜⎝ σt2
σt4
⎠⎟
−2
⎛
⎞
(t −β1z )2
⎜⎜ ω0
P0
β2z ⎟⎟⎟ (t − β1z )
σt2
ze
= ω0 + ⎜⎜ 4 n2
+
⎟
Aeff
⎜⎜ c0
σt2 ⎟⎟⎟ σt2
⎝
⎠
The frequency of the carrier is ω0 , only at the peak of the pulse
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Gauge Institute Journal
H. Vic Dannon
That is, at
t = β1z .
Otherwise, the frequency depends on the sign of
−2
2
(t −β1z )
ω0
P0
βz
σt2
ze
4 n2
+ 2 .
c0
Aeff
σt2
The two terms in the sum have opposite signs, because in Optical
Fibers
n2 > 0 ,
and
β2 < 0 .
There are three cases
(I) Group Velocity Dispersion
If
−2
2
(t −β1z )
ω0
P0
βz
σt2
ze
4 n2
+ 2 > 0.
c0
Aeff
σt2
Then,
The frequency decreases before the peak of the pulse, and the
carrier vibrations get dispersed
The frequency increases beyond the peak of the pulse, and the
carrier vibrations get compressed.
(II) Self-Phase Modulation
If
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Gauge Institute Journal
H. Vic Dannon
−2
2
(t −β1z )
2
ω
P
βz
4 0 n2 0 ze σt
+ 2 < 0.
c0
Aeff
σt2
Then,
The frequency increases before the peak of the pulse, and the
carrier vibrations get compressed
The frequency decreases beyond the peak of the pulse, and the
carrier vibrations get dispersed.
(III) Optical Soliton
If
−2
2
(t −β1z )
ω0
P0
βz
σt2
ze
4 n2
+ 2 = 0.
c0
Aeff
σt2
Then,
The frequency remains the same ω0 before the peak of the pulse,
and beyond the peak of the pulse.
the carrier vibrations dispersion is balanced by their compression.
We will modify the Non-Linear Schrödinger equation to generate a
Soliton solution.
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Gauge Institute Journal
H. Vic Dannon
4.
Non-Linear Schrodinger Equation
for Solitons
The condition for the evolution of a Gaussian pulse into an Optical
Soliton, involves a transformed time
t − β1z
σt
where σt is the standard deviation of the Gaussian distribution.
Thus, we should expect an optical Soliton time to be of the form
t − β1z
,
T
where T is the time-width of the Soliton.
But Soliton width which is not known at this point, can be
introduced in a natural way after further developments.
Rather than introduce an unknown T ,we transform the time t to
τ = t − β1z .
Then, the Soliton’s envelope moves at the group velocity
vg =
1
,
β1
The Non-Linear Schrödinger equation
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Gauge Institute Journal
H. Vic Dannon
2
1
1
∂z E 0 + β1∂t E 0 − i β2∂tt E 0 + i ω0ε0n0n2 E 0 E 0 = 0 ,
2
2
will transform to an equation that depends on T .
The modulating envelope is
E 0 (z , t ) = f (z , τ )
Then,
∂E 0
∂f ∂ z ∂ f ∂ τ
=
+
∂z
∂z ∂
∂z
z ∂τ N
N
−β1
1
= ∂z f − β1∂ τ f
∂E 0
∂f ∂ z ∂ f ∂ τ
=
+
∂t
∂t
∂z N
∂ t ∂τ N
0
1
= ∂τ f
∂2E 0
∂t 2
=
∂(∂ τ f ) ∂z ∂(∂ τ f ) ∂τ
+
∂z N
∂t
∂τ N
∂t
0
1
= ∂ ττ f
Hence, the equation transforms to
2
1
1
∂z E 0 + β1 ∂t E 0 − i β2 ∂tt E 0 + i ω0ε0n0n2 E 0 E 0 = 0
N
N
2 N
2
∂z f −β1∂ τ f
∂τ f
∂ ττ f
f
That is,
1
1
∂z f − i β2∂ ττ f + i ω0ε0n 0n2 f
2
2
23
2
f = 0.
2
f
Gauge Institute Journal
H. Vic Dannon
5.
The Soliton’s Envelope Equation
The Non-Linear Schrödinger equation
1
1
∂z f − i β2∂ ττ f + i ω0ε0n0n2 f
2
2
2
f = 0.
describes the propagation of a wave packet envelope in the optical
fiber, in a coordinate system (z, τ ) that moves with the pulse peak.
Since a Soliton’s envelope does not depend on z , a Soliton solution
has the form
f (z, τ, a ) = au(τ )eiφ(z ) ,
where
u(τ ) is the Soliton’s envelope, τ = t − β1z .
φ(z ) the Soliton’s phase, depends only on z .
φ '(τ ) is unchanged with time,
and
a is the maximal amplitude. That is,
0 < u(τ ) ≤ 1 .
Substituting in the Nonlinear Schrodinger equation,
1
1
au(τ )e iφ(z )iφ '(z ) − i β2au ''(τ )e iφ(z ) + i a 3 ω0ε0n 0n2u 2ue iφ(z ) = 0 .
2
2
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Gauge Institute Journal
H. Vic Dannon
1
1
u(τ )φ '(z ) − β2u ''(τ ) + a 2ω0ε0n0n2u 3 = 0
2
2
φ '(z ) =
1 u ''(τ ) 1 2
β2
− a ω0ε0n0n2u 2 .
2
u(τ )
2
Thus, both sides equal to a constant α . Hence,
φ(z ) = αz ,
and
α=
1 u ''(τ ) 1 2
β2
− a ω0ε0n0n2u 2
2
u(τ )
2
Therefore,
u ''(τ ) =
1
2αu + a 2ω0ε0n0n2u 3 )
(
β2
is the Soliton’s Envelope Equation.
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Gauge Institute Journal
H. Vic Dannon
6.
The Fundamental Soliton solution
To solve the Soliton envelope equation
u ''(τ ) =
1
2αu + a 2ω0ε0n0n2u 3 ) ,
(
β2
multiply it by the integration factor 2u '(τ ) .
2u ' u '' =
2
Dτ ( u ' ) =
1
4αuu '+ 2a 2ω0ε0n0n2u 3u ' )
(
β2
⎞
1 ⎛⎜
1 2
2
4 ⎟
2
+
α
D
u
a
ω
ε
n
n
D
u
) 2 0 0 0 2 τ ( ) ⎠⎟⎟
⎜
τ(
β2 ⎜⎝
Integrating both sides with respect to τ ,
2
(u ' )
=
⎞⎟
1 ⎛⎜
1 2
2
4
⎜⎜ 2αu + a ω0ε0n 0n2u + C 1 ⎟⎟ .
2
β2 ⎝
⎠
Assuming that if τ → ∞ , then u ' → 0 , and u → 0 , we have
C1 = 0 ,
and
2
(u ' )
=
⎞
1 ⎛⎜
1 2
2
4⎟
2
+
α
u
a
ω
ε
n
n
u
⎟
⎜
0 0 0 2
2
β2 ⎜⎝
⎠⎟
=
⎞
1 2 ⎛⎜
1
u ⎜ 2α + a 2ω0ε0n0n2u 2 ⎟⎟⎟
2
β2 ⎜⎝
⎠
At τ = 0 , u(τ ) peaks to u = 1 . Then, u ' = 0 , and the equation
26
Gauge Institute Journal
H. Vic Dannon
becomes
0=
⎞⎟
1 ⎛⎜
1 2
⎜⎜ 2α + a ω0ε0n0n2 ⎟⎟
2
β2 ⎝
⎠
Hence,
1
2α = − a 2 ω0ε0n 0n2 .
2
Substituting into the Soliton envelope equation
2
(u ' )
=
⎞
1 2 ⎛⎜ 1 2
1
u ⎜ − a ω0ε0n0n2 + a 2 ω0ε0n0n2u 2 ⎟⎟⎟
2
β2 ⎜⎝ 2
⎠
=
1
a 2 ω0ε0n0n2u 2 ( 1 − u 2 )
−2β2
In optical Fibers β2 < 0 , and since the maximum of u is 1 , the
square root of the right hand side is real positive, and we have
1
u' = a
ω0ε0n0n2 u 1 − u 2 .
−2β2
1
T
Denoting
a
1
1
ω0ε0n0n2 ≡ ,
T
−2β2
u' =
1
u 1 − u2 ,
T
where T > 0 .
du
u 1 − u2
Integrating both sides,
27
=
1
dτ .
T
Gauge Institute Journal
H. Vic Dannon
∫u
du
1−u
1
T
=
2
∫ dτ + C2 .
From Integration Tables, [Spiegel, p.69],
1 + 1 − u2
τ
− log
= + C2 .
u
T
At τ = 0 , u(τ ) peaks to u = 1 . Then,
1 + 1 − 12
0
− log
= + C2
1
T
That is,
0 = C2 ,
and
τ
−
1 + 1 − u2
=e T,
u
τ
−
T
(ue
2
1−u =
=u
− 1)2 ,
τ
−2
e T
2
−
τ
−
2ue T
τ
⎡ −2 τ
−
⎢
T
T
0 = u ⎢ u(e
+ 1) − 2e
⎢⎣
τ
−2
u(e T
τ
−
T
u(e
+ 1) =
τ
T
+e
u =
τ
−
2e T
)=2
2
τ
−
T
e
28
τ
T
+e
⎤
⎥
⎥
⎥⎦
+1
Gauge Institute Journal
H. Vic Dannon
1
⎛τ⎞
cosh ⎜⎜ ⎟⎟⎟
⎜⎝T ⎠
=
=
Since a
1
⎛ t − z / vg
cosh ⎜⎜⎜
⎜⎝
T
⎞⎟
⎟⎟
⎠⎟
1
1
ω0ε0n 0n2 = ,
−2β2
T
a =
−2β2
,
ω0ε0n0n2
1
T
and
1
αz = − a 2ω0ε0n 0n2z
4
=
β2
2T
2
z.
Thus, the Fundamental Soliton solution to the Non-linear
Schrödinger equation is
1
f (z , τ,T ) =
T
−2β2
ω0ε0n 0n2
1
⎛ t − z / vg
cosh ⎜⎜⎜
T
⎝⎜
⎞⎟
⎟⎟
⎠⎟
e
−i
β2
2T 2
z
.
The Soliton’s amplitude depends on its width T .
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Gauge Institute Journal
H. Vic Dannon
Delta Function
We aim to show that the Fundamental optical Soliton can be
modeled by a propagating Delta Function.
The description of the Delta Function as the limit of a Delta
sequence leads to divergent integrals, and to meaningless equality
to the inconceivable infinity.
In the Calculus of Limits, the Delta function is not defined, and its
power to describe physical singularities is restricted.
Using infinitesimals, and infinite hyper-real numbers, the Delta
Function can be defined on the hyper-real line, an infinite
dimensional line that has room for infinitesimals, and their
reciprocals, the infinite hyper-reals.
Then, the Delta function is the whole infinite Delta sequence, and
serves to model singularities.
In the next sections, we sum up the main points necessary for the
definition, and the application of the Delta function
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H. Vic Dannon
7.
Hyper-real Line
Each real number α can be represented by a Cauchy sequence of
rational numbers, (r1, r2 , r3 ,...) so that rn → α .
The constant sequence (α, α, α,...) is a constant hyper-real.
In [Dan2] we established that,
1. Any totally ordered set of positive, monotonically decreasing
to zero sequences (ι1, ι2 , ι3 ,...) constitutes a family of
infinitesimal hyper-reals.
2. The infinitesimals are smaller than any real number, yet
strictly greater than zero.
3. Their reciprocals
(
1 1 1
, ,
ι1 ι2 ι3
)
,... are the infinite hyper-reals.
4. The infinite hyper-reals are greater than any real number,
yet strictly smaller than infinity.
5. The infinite hyper-reals with negative signs are smaller
than any real number, yet strictly greater than −∞ .
6. The sum of a real number with an infinitesimal is a
non-constant hyper-real.
7. The Hyper-reals are the totality of constant hyper-reals, a
family of infinitesimals, a family of infinitesimals with
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Gauge Institute Journal
H. Vic Dannon
negative sign, a family of infinite hyper-reals, a family of
infinite hyper-reals with negative sign, and non-constant
hyper-reals.
8. The hyper-reals are totally ordered, and aligned along a
line: the Hyper-real Line.
9. That line includes the real numbers separated by the nonconstant hyper-reals. Each real number is the center of an
interval of hyper-reals, that includes no other real number.
10.
In particular, zero is separated from any positive real
by the infinitesimals, and from any negative real by the
infinitesimals with negative signs, −dx .
11.
Zero is not an infinitesimal, because zero is not strictly
greater than zero.
12.
We do not add infinity to the hyper-real line.
13.
The infinitesimals, the infinitesimals with negative
signs, the infinite hyper-reals, and the infinite hyper-reals
with negative signs are semi-groups with
respect to addition. Neither set includes zero.
14.
The hyper-real line is embedded in \∞ , and is not
homeomorphic to the real line. There is no bi-continuous
one-one mapping from the hyper-real onto the real line.
15.
In particular, there are no points on the real line that
can be assigned uniquely to the infinitesimal hyper-reals, or
32
Gauge Institute Journal
H. Vic Dannon
to the infinite hyper-reals, or to the non-constant hyperreals.
16.
No neighbourhood of a hyper-real is homeomorphic to
an \n ball. Therefore, the hyper-real line is not a manifold.
17.
The hyper-real line is totally ordered like a line, but it
is not spanned by one element, and it is not one-dimensional.
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H. Vic Dannon
8.
Hyper-Real Integral
In [Dan3], we defined the integral of a Hyper-real Function.
Let f (x ) be a hyper-real function on the interval [a, b ] .
The interval may not be bounded.
f (x ) may take infinite hyper-real values, and need not be
bounded.
At each
a ≤ x ≤b,
there is a rectangle with base [x − dx2 , x + dx2 ] , height f (x ) , and area
f (x )dx .
We form the Integration Sum of all the areas for the x ’s that
start at x = a , and end at x = b ,
∑
f (x )dx .
x ∈[a ,b ]
If for any infinitesimal dx , the Integration Sum has the same
hyper-real value, then f (x ) is integrable over the interval [a,b ] .
Then, we call the Integration Sum the integral of f (x ) from x = a ,
to x = b , and denote it by
x =b
∫
f (x )dx .
x =a
34
Gauge Institute Journal
H. Vic Dannon
If the hyper-real is infinite, then it is the integral over [a, b ] ,
If the hyper-real is finite,
x =b
∫
f (x )dx = real part of the hyper-real . ,
x =a
8.1 The countability of the Integration Sum
In [Dan1], we established the equality of all positive infinities:
We proved that the number of the Natural Numbers,
Card ` , equals the number of Real Numbers, Card \ = 2Card ` , and
we have
Card `
Card ` = (Card `)2 = .... = 2Card ` = 22
= ... ≡ ∞ .
In particular, we demonstrated that the real numbers may be
well-ordered.
Consequently, there are countably many real numbers in the
interval [a, b ] , and the Integration Sum has countably many terms.
While we do not sequence the real numbers in the interval, the
summation takes place over countably many f (x )dx .
The Lower Integral is the Integration Sum where f (x ) is replaced
by its lowest value on each interval [x − dx2 , x + dx2 ]
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Gauge Institute Journal
8.2
H. Vic Dannon
∑
x ∈[a ,b ]
⎛
⎞
⎜⎜
inf
f (t ) ⎟⎟⎟dx
⎜⎝ x −dx ≤t ≤x + dx
⎠⎟
2
2
The Upper Integral is the Integration Sum where f (x ) is replaced
by its largest value on each interval [x − dx2 , x + dx2 ]
8.3
⎛
⎞⎟
⎜⎜
f (t ) ⎟⎟dx
∑ ⎜⎜ x −dxsup
⎟
dx
≤t ≤x +
⎠⎟
x ∈[a ,b ] ⎝
2
2
If the integral is a finite hyper-real, we have
8.4 A hyper-real function has a finite integral if and only if its
upper integral and its lower integral are finite, and differ by an
infinitesimal.
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Gauge Institute Journal
H. Vic Dannon
9.
Delta Function
In [Dan5], we defined the Delta Function, and established its
properties
1. The Delta Function is a hyper-real function defined from the
⎧
⎪ 1 ⎫⎪
hyper-real line into the set of two hyper-reals ⎪
⎬ . The
⎨ 0, ⎪
⎪
⎪
dx
⎪
⎪
⎩
⎭
0, 0, 0,... . The infinite hyper-
hyper-real 0 is the sequence
real
1
depends on our choice of dx .
dx
2. We will usually choose the family of infinitesimals that is
spanned by the sequences
1
1
1
,
,
,… It is a
n
n2
n3
semigroup with respect to vector addition, and includes all
the scalar multiples of the generating sequences that are
non-zero. That is, the family includes infinitesimals with
1
will mean the sequence n .
dx
negative sign. Therefore,
Alternatively, we may choose the family spanned by the
sequences
1
n
2
,
1
n
3
,
1
n
4
,… Then,
1
dx
will mean the
sequence 2n . Once we determined the basic infinitesimal
37
Gauge Institute Journal
H. Vic Dannon
dx , we will use it in the Infinite Riemann Sum that defines
an Integral in Infinitesimal Calculus.
3. The Delta Function is strictly smaller than ∞
1
dx
δ(x ) ≡
4. We define,
where
χ
⎡ −dx , dx ⎤ (x ) ,
⎢⎣ 2 2 ⎦⎥
χ
⎧1, x ∈ ⎡ − dx , dx ⎤
⎪
⎢⎣ 2 2 ⎥⎦ .
⎪
⎡ −dx , dx ⎤ (x ) = ⎨
⎪
⎣⎢ 2 2 ⎦⎥
0, otherwise
⎪
⎩
5. Hence,
™ for x < 0 , δ(x ) = 0
™ at x = −
dx
1
, δ(x ) jumps from 0 to
,
2
dx
1
.
x ∈ ⎡⎢⎣ − dx2 , dx2 ⎤⎦⎥ , δ(x ) =
dx
™ for
™ at x = 0 ,
™ at x =
δ(0) =
1
dx
dx
1
, δ(x ) drops from
to 0 .
2
dx
™ for x > 0 , δ(x ) = 0 .
™ x δ(x ) = 0
6. If dx =
7. If dx =
8. If dx =
1
n
2
n
1
n
, δ(x ) =
, δ(x ) =
χ
χ
(x ), 2
[− 1 , 1 ]
2 2
1
,
χ
(x ), 3
[− 1 , 1 ]
4 4
2
,
(x )...
[− 1 , 1 ]
6 6
3
2 cosh2 x 2 cosh2 2x 2 cosh2 3x
,...
, δ(x ) = e−x χ[0,∞), 2e−2x χ[0,∞), 3e−3x χ[0,∞),...
38
Gauge Institute Journal
H. Vic Dannon
x =∞
9.
∫
δ(x )dx = 1 .
x =−∞
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Gauge Institute Journal
H. Vic Dannon
10.
The Fourier Transform
In [Dan6], we defined the Fourier Transform and established its
properties
1. F { δ(x )} = 1
2. δ(x ) = the inverse Fourier Transform of the unit function 1
1
=
2π
ω =∞
∫
e i ωxd ω
ω =−∞
ν =∞
∫
=
e 2πixd ν , ω = 2πν
ν =−∞
ω =∞
1
3.
e i ωxd ω
∫
2π ω =−∞
=
x =0
1
= an infinite hyper-real
dx
ω =∞
∫
ω =−∞
ei ωxd ω
=0
x ≠0
4. Fourier Integral Theorem
k =∞ ⎛ ξ =∞
⎞⎟
⎜⎜
1
⎟ ikx
−ik ξ
f (x ) =
⎜⎜ ∫ f (ξ )e d ξ ⎟⎟e dk
∫
⎟
2π k =−∞ ⎜⎝ ξ =−∞
⎠⎟
does not hold in the Calculus of Limits, under any
conditions.
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H. Vic Dannon
5. Fourier Integral Theorem in Infinitesimal Calculus
If f (x ) is hyper-real function,
Then,
ƒ the Fourier Integral Theorem holds.
x =∞
ƒ
∫
f (x )e−i αxdx converges to F (α)
x =−∞
α =∞
1
ƒ
F (α)e −i αxd α converges to f (x )
∫
2π α =−∞
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H. Vic Dannon
11.
The Laplace Transform
In [Dan7], we have shown that
1. The Delta function δ(t ) , for t ≥ 0 is represented by Sequence
δn (t ) = ne−nt
2. If in =
χ
[0,∞ )
(t ) .
1
, δ(x ) = e−t
n
χ
[0,∞ )
(t ), 2e−2t
χ
[0,∞ )
(t ), 3e−3t
χ
[0,∞ )
(t ),...
3. L { δ(t )} = 1
4. δ(t ) = the inverse Laplace Transform of the unit function 1
1
=
2πi
s =i ∞
∫
estds .
s =−i ∞
s = γ +i ∞
1
5.
2πi s
∫
e stds
= γ −i ∞
=
t =0
1
= an infinite hyper-real
dt
s = γ +i ∞
∫
s = γ −i ∞
e stds
= 0.
t ≠0
6. Laplace Integral Theorem
If f (t ) is hyper-real function,
Then,
the Laplace Integral Theorem holds.
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Gauge Institute Journal
H. Vic Dannon
s = γ +i ∞
⎛ τ =∞
⎞⎟
1
−
τ
st ⎜
s
f (t ) =
e ⎜⎜ ∫ e f (τ )d τ ⎟⎟⎟ds
∫
⎟
2πi s =γ −i∞ ⎜⎜⎝ τ =0
⎠⎟
t =∞
∫
e −st f (t )dt , converges to F (s )
t =0
s = γ +i ∞
1
e st F (s )ds converges to f (t ) .
∫
2πi s = γ −i ∞
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H. Vic Dannon
12.
Pulse Compression and Delta
Pulse Compression is used to generate Optical Solitons, with
narrower width T , and larger amplitudes proportional to
1
that
T
are used to generate even more compressed Solitons.
An infinite sequence of compressed pulses with indefinitely
narrowing widths and growing amplitudes constitutes a Delta
function.
The narrowing widths include less carrier cycles, and require
higher carrier frequencies. At X-ray frequencies the laser mirrors
have to be replaced by dielectric mirrors, and as the frequency
increases, at Attosecond (Atto=10-18 ) pulses, no mirrors will do,
and a different compression method has to be applied.
Physically, such Delta Pulse Sequence is unattainable.
However, similarly to considering the sizes of a charge or a point
light source as infinitesimal and representing them by a Delta
function, we shall ignore the physical limitations on realizing an
ideal infinitesimal-width Optical Soliton, and model the sequence
of narrowing Solitons by a Delta function.
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H. Vic Dannon
12.1 Pulse Width, and Delta
A system that combines dispersion with self-phase modulation
will generate a narrower output pulse.
In [Siegman, p.393]
an input pulse of 5,900 f sec , (femto=10-15 )
enters a 3 meter optical fiber, and is broadened to
a pulse of 10,000 f sec .
That pulse enters a diffraction grating that compresses it into
an output pulse of 200 f sec .
That pulse enters a 0.55 meter optical fiber, followed by a
diffraction grating, that compresses it into
an output pulse of 90 f sec .
Pulses of 90 f sec generated in a Dye laser have been compressed
to 30 f sec -the shortest width attained in 1986.
A Titanium-Sapphire laser [W-1] generates pulses in the range of
100 f sec , down to 10 f sec . The shortest width attained in 1999
was 5.5 f sec .
In 2008, 2.5 f sec pulses, containing only one or two cycles of the
carrier, were used in the [W-3] experiment.
Narrower pulses were generated, but not in optical fibers, and not
as Solitons with amplitude that grows as the width narrows.
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H. Vic Dannon
[W-2] describes the evolution up to 2008, of Ultrashort Pulses into
the Attosecond range (Atto=10-18 ).
Evolution of ultrashort light pulses up to 2008. (from [W-2])
The Attosecond flushes were attained, by firing ultrashort laser
pulses into a cloud of neon gas, generating carrier in the extreme
ultraviolet range.
In [W-2] such Attosecond flushes from Argon, were used to
stroboscope-light and film the motion of an electron, after it was
separated from an atom, in time period of a single carrier cycle.
By [W-3], the shortest such flush in 2011 is 80 Attoseconds.
In Comparison, an electron circles the nucleus in 150 Attosecond.
12.2 Pulse Power, and Delta
By [W-6], A carrier wave from an Argon or Nd:YVO4 laser with
average output power of 0.5 to 1.5 Watts, pumps a Titaniumsapphire laser of mode-locked oscillator type.
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Gauge Institute Journal
H. Vic Dannon
The Ti:Sapphire oscillator generates pulses with duration as short
as 5 f sec .
According to [Weiner,p.405], at 100 f sec ,
the peak power of the pulse is 104 Watts.
This Corresponds to maximum intensity 108 W/m2 =1012 W/cm2.
By [W-6], this pulse is supplied to a Titanium-sapphire laser of
Chirped-pulse amplifier type, designed to withstand the damaging
power of the pulse.
The Ti:Sapphire laser amplifier may use Chirped mirrors [W-7] to
guide the beam several times through the crystal, while keeping
the pulse width below the damage threshold, or it may use optical
switches, to insert the pulse in the cavity, and retrieve it later.
The Ti:Sapphire amplifier generates pulses with duration of 20 to
100 f sec with energy of 5 mJouls .
At 100 f sec , the peak power of the pulse is 50 GegaWatts . An
electric power plant peaks to 1 GW.
This corresponds to maximum intensity
(25)1020 W/m2 =(25)1024 W/cm2
Thus, the Delta function is suitable to describe short energetic
Optical Fiber Solitons.
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13.
Solitons and Delta Sequence
13.1 Soliton’s Delta
The Soliton
1
f (z , t,T ) =
T
−2β2
ω0ε0n0n2
1
⎛ t − z / vg
cosh ⎜⎜⎜
⎜⎝
T
⎞⎟
⎟⎟
⎠⎟
e
−i
β2
2T 2
z
,
has the envelope
1
T
−2β2
ω0ε0n0n2
1
⎛ t − z / vg
cosh ⎜⎜⎜
T
⎝⎜
⎞⎟
⎟⎟
⎠⎟
,
where
3 χ(3)
n2 =
,
4 c0ε0n02
n(ω, E 0 ) ≈ n0 + n2 ( 12 c0ε0n0E 02 ) is the refractive index,
β ≡
β2 ≡
ω0
n(ω, E 0 ) ,
c0
∂2 β
∂ω 2
,
ω = ω0
E0 =0
vg is the envelope velocity,
T is the Soliton’s width.
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Gauge Institute Journal
H. Vic Dannon
Denote
γ ≡
−2β2
,
ω0ε0n0n2
τ ≡ t − z / vg ,
T = dz .
The Delta function representing this envelope is the Hyper-real
function
1
γ
dz
For dz =
1
⎛ t − z / vg
cosh ⎜⎜⎜
⎜⎝
dz
⎞⎟
⎟⎟
⎠⎟
.
1
, this Delta function is represented by the sequence
n
γ
2γ
3γ
,...
,
,
cosh ( t − z / vg ) cosh 2 ( t − z / vg ) cosh 3 ( t − z / vg )
13.2
Each
δn (τ ) =
n
1
π cosh n τ
τ =∞
ƒ has the sifting property
∫
τ =−∞
ƒ is continuous
ƒ peaks at τ = 0 to δn (0) =
Proof:
By [Spiegel, p.88],
49
n
π
δn (τ )d τ = 1
Gauge Institute Journal
H. Vic Dannon
τ =∞
τ =∞
n
1
n2
nτ
d
arctan
(
e
)
τ
=
∫ π cosh n τ
τ =−∞
πn
τ =−∞
=
2
[arctan(∞) − arctan(0)]
π π /2
0
= 1 .,
Thus, the sequence represents the hyper-real Delta Function
13.3
δ(τ ) =
1
2
3
,
,
,...
cosh τ cosh 2τ cosh 3τ
plots in Maple, the 50th component,
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Gauge Institute Journal
H. Vic Dannon
plots in Maple the 200th component,
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Gauge Institute Journal
H. Vic Dannon
References
[Bowman] Bowman, Frank, Introduction to Bessel Functions, Dover, 1958
[Dan1] Dannon, H. Vic, “Well-Ordering of the Reals, Equality of all Infinities,
and the Continuum Hypothesis” in Gauge Institute Journal Vol.6 No 2, May
2010;
[Dan2] Dannon, H. Vic, “Infinitesimals” in Gauge Institute Journal Vol.6 No
4, November 2010;
[Dan3] Dannon, H. Vic, “Infinitesimal Calculus” in Gauge Institute Journal
Vol.7 No 3, August 2011;
[Dan4] Dannon, H. Vic, “Riemann’s Zeta Function: the Riemann Hypothesis
Origin, the Factorization Error, and the Count of the Primes”, in Gauge
Institute Journal of Math and Physics, November 2009.
[Dan5] Dannon, H. Vic, “The Delta Function” in Gauge Institute Journal
Vol.7 No 4, November 2011;
[Dan6] Dannon, H. Vic, “Delta Function, the Fourier Transform, and Fourier
Integral Theorem” September 2010, posted to www.gauge-institute.org
[Dan7] Dannon, H. Vic, “Delta Function, the Laplace Transform, and Laplace
Integral Theorem” September 2010, posted to www.gauge-institute.org
[Dan8] Dannon, H. Vic, “Polar Delta Function, and the 2-D Fourier-Bessel
Transform” October 2010, posted to www.gauge-institute.org
[Dan9] Dannon, H. Vic, “Radial Delta Function, and the 3-D Fourier-Bessel
Transform” October 2010, posted to www.gauge-institute.org
[Dirac] Dirac, P. A. M. The Principles of Quantum Mechanics, Second Edition,
Oxford Univ press, 1935.
[Gray] Gray, Andrew, and Mathews, G.B., A Treatise on Bessel Functions and
their applications to Physics, Dover 1966
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Gauge Institute Journal
H. Vic Dannon
[Hen] Henle, James M., and Kleinberg Eugene M., Infinitesimal Calculus,
MIT Press 1979.
[Hosk] Hoskins, R. F., Standard and Nonstandard Analysis, Ellis Horwood,
1990.
[Keis] Keisler, H. Jerome, Elementary calculus, An Infinitesimal Approach,
Second Edition, Prindle, Weber, and Schmidt, 1986, pp. 905-912
[Laug] Laugwitz, Detlef, “Curt Schmieden’s approach to infinitesimals-an eyeopener to the historiography of analysis” Technische Universitat Darmstadt,
Preprint Nr. 2053, August 1999
[Mikus] Mikusinski, J. and Sikorski, R., “The elementary theory of
distributions”, Rosprawy Matematyczne XII, Warszawa 1957.
[Rand] Randolph, John, “Basic Real and Abstract Analysis”, Academic Press,
1968.
[Riemann] Riemann, Bernhard, “On the Representation of a Function by a
Trigonometric Series”.
(1)
In “Collected Papers, Bernhard Riemann”, translated from
the 1892 edition by Roger Baker, Charles Christenson, and
Henry Orde, Paper XII, Part 5, Conditions for the existence of a
definite integral, pages 231-232, Part 6, Special Cases, pages
232-234. Kendrick press, 2004
(2)
In “God Created the Integers” Edited by Stephen Hawking,
Part 5, and Part 6, pages 836-840, Running Press, 2005.
[Schwartz] Schwartz, Laurent, Mathematics for the Physical Sciences,
Addison-Wesley, 1966.
[Siegman] Siegman, Anthony, Lasers University Science, 1986.
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Gauge Institute Journal
H. Vic Dannon
[Spiegel] Spiegel Murray, Mathematical Handbook of Formulas and Tables.
McGraw-Hill, 1968.
[Temp] Temple, George, 100 Years of Mathematics, Springer-Verlag, 1981.
pp. 19-24.
[Watson] Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd
Edition, Cambridge, 1958.
[Weiner] Weiner, Andrew, Ultrafast Optics, Wiley 2009.
[W-1] http://www.rp-photonics.com/titenium_sapphire_lasers.html
[W-2]
http://web.archive.org/web/20080303014710/www.atto.fysik.lth.se/research/research.html
[W-3] http://www.newscientist.com/article/dn14172-fastestever-flashgun-captures-image-of-light...
[W-4] http://en.wikipedia.org//wiki/free-electron_laser
[W-5] http://www.digitaltrends.com/international/u-s-navys-giant-laser-beam-program-achieves-...
[W-6] http://en.wikipedia.org/wiki/Ti-sapphire_laser
[W-7] http://en.wikipedia.org/wiki/Chirped_mirror
http://www.rp-photonics.com/titanium_sapphire_lasers.html
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