Delta Function and Optical

Catastrophe Models

H. Vic Dannon vic0@comcast.net February, 2011

Abstract

We propose models for the propagation of intense electromagnetic

ultrashort pulses.

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; 42.55.Vc; 42.60.Fc; 42.62.Be; 42.65.Jx; 42.65.Re;

42.65.Tg;

1

2000 Mathematics Subject Classification 35C08; 35Q51;

37K40; 74J35; 76B25; 35K55; 78A60;

26E35; 26E30; 26E15; 26E20; 26A06; 26A12; 03E10; 03E55;

03E17; 03H15; 46S20; 97I40; 97I30;

Contents

Introduction

1. Hyper-real Line

2. Hyper-real Integral

3. Delta Function

4. The Fourier Transform

5. The Laplace Transform

6. Optical Catastrophe Model for the Free Electron Laser Pulses

7. Optical Catastrophe Model for Optical Solitons

References

2

Introduction

Intense electromagnetic pulses can serve as bullets that demolish

matter. Such pulses have been used as scalpels in surgery [W-4],

and as shells by the military to shoot down missiles [W-5].

While dissipation is not a concern for surgical pulses, it requires

that the military pulses be very powerful pulses in order for them

to be effective on a remote target.

In [Dan10], we showed that the Fundamental Optical Soliton can

be considered a Delta Function, and here we will show that the

model that serves to describe its propagation, guarantees no

dissipation.

The Delta Function is not the limit of a Delta sequence as

presented in Engineering, and in Physics, and its singularity does

not disappear when it is presented as a Generalized Functional in

Mathematics.

We have shown that the Delta Function is a Hyper-real Function

defined on the hyper-real line, an infinite dimensional line that

has room for infinitesimals, and their reciprocals, the infinite

hyper-reals.

In the next sections, we sum up the main points necessary for the

definition, and the application of the Delta function

3

1.

Hyper-real Line Each real number α can be represented by a Cauchy sequence of

rational numbers, so that . 1 2 3( , , ,...)r r r nr α→

The constant sequence ( is a constant hyper-real. , , ,...)α α α

In [Dan2] we established that,

1. Any totally ordered set of positive, monotonically decreasing

to zero sequences constitutes a family of

infinitesimal hyper-reals.

1 2 3( , , ,...)ι ι ι

2. The infinitesimals are smaller than any real number, yet

strictly greater than zero.

3. Their reciprocals (1 2 3

1 1 1, , ,...ι ι ι ) 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

4

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 non-

constant 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

5

to the infinite hyper-reals, or to the non-constant hyper-

reals.

16. No neighbourhood of a hyper-real is homeomorphic to

an ball. Therefore, the hyper-real line is not a manifold. n

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.

6

2.

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 2

[ ,dx dxx x− +2], 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 , and end at x b , a= =

[ , ]

( )x a b

f x dx∈∑ .

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 ,

to x , and denote it by

x a=

b=

( )x b

x a

f x dx=

=∫ .

7

If the hyper-real is infinite, then it is the integral over [ , , ]a b

If the hyper-real is finite,

( ) real part of the hyper-realx b

x a

f x dx=

=

=∫ .

2.1 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, , and

we have

2CardCard =

2 2( ) .... 2 2 ...CardCardCard Card= = = = = ≡ ∞ .

In particular, we demonstrated that the real numbers may be

well-ordered.

Consequently, there are countably many real numbers in the

interval [ , , and the Integration Sum has countably many terms. ]a b

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 2 2

[ ,dx dxx x− + ]

8

2.2 2 2[ , ]

inf ( )dx dxx t xx a b

f t dx− ≤ ≤ +∈

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠∑

The Upper Integral is the Integration Sum where ( )f x is replaced

by its largest value on each interval 2 2

[ ,dx dxx x− + ]

2.3 2 2[ , ]

sup ( )dx dxx t xx a b

f t dx− ≤ ≤ +∈

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∑

If the integral is a finite hyper-real, we have

2.4 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.

9

3.

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

hyper-real line into the set of two hyper-reals 1

0,dx

⎧ ⎫⎪⎪⎨⎪ ⎪⎪ ⎪⎩ ⎭

⎪⎪⎬. The

hyper-real is the sequence 0 0,0, 0,... . The infinite hyper-

real 1dx

depends on our choice of dx .

2. We will usually choose the family of infinitesimals that is

spanned by the sequences 1n

,2

1

n,

3

1

n,… It is a

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

negative sign. Therefore, 1dx

will mean the sequence n .

Alternatively, we may choose the family spanned by the

sequences 1

2n,

1

3n,

1

4n,… Then, 1

dx will mean the

sequence 2n . Once we determined the basic infinitesimal

10

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 ∞

4. We define, 2 2,

1( ) ( )dx dxx x

dxδ χ⎡ ⎤−⎢ ⎥⎣ ⎦

≡ ,

where 2 2

2 2,

1, ,( )

0, otherwisedx dx

dx dxxxχ⎡ ⎤−⎢ ⎥⎣ ⎦

⎧ ⎡ ⎤⎪ ∈ −⎢ ⎥⎪ ⎣ ⎦= ⎨⎪⎪⎩.

5. Hence,

for , 0x < ( ) 0xδ =

at 2dx

x = − , jumps from to ( )xδ 01dx

,

for 2 2

,dx dxx ⎡ ⎤∈ −⎢ ⎥⎣ ⎦ , 1

( )xdx

δ = .

at , 0x =1

(0)dx

δ =

at 2dx

x = , drops from ( )xδ1dx

to . 0

for , . 0x > ( ) 0xδ =

( ) 0x xδ =

6. If 1n

dx = , 1 1 1 1 1 12 2 4 4 6 6

[ , ] [ , ] [ , ]( ) ( ),2 ( ), 3 ( )...x x xδ χ χ χ− − −= x

7. If 2n

dx = , 2 2 2

1 2 3( ) , , ,...

2 cosh 2cosh 2 2cosh 3x

x x xδ =

8. If 1n

dx = , 2 3[0, ) [0, ) [0, )( ) ,2 , 3 ,...x x xx e e eδ χ χ χ− − −

∞ ∞ ∞=

11

9. . ( ) 1x

x

x dxδ=∞

=−∞

=∫

12

4.

The Fourier Transform

In [Dan6], we defined the Fourier Transform and established its

properties

1. { }( ) 1xδ =F

2. = the inverse Fourier Transform of the unit function 1 ( )xδ

12

i xe dω

ω

ω

ωπ

=∞

=−∞

= ∫

, 2 ixe dν

π

ν

ν=∞

=−∞

= ∫ 2ω π= ν

3. 0

12

i x

x

e ddx

ωω

ω

ωπ

=∞

=−∞ =

=∫1

= an infinite hyper-real

0

0i x

x

e dω

ω

ω

ω=∞

=−∞ ≠

=∫

4. Fourier Integral Theorem

1( ) ( )

2

kik ikx

k

f x f eξ

ξ

ξ

ξ ξπ

=∞ =∞−

=−∞ =−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ d e dk

does not hold in the Calculus of Limits, under any

conditions.

13

5. Fourier Integral Theorem in Infinitesimal Calculus

If ( )f x is hyper-real function,

Then,

the Fourier Integral Theorem holds.

( )x

i x

x

f x e dxα=∞

−

=−∞∫ converges to ( )F α

1( )

2i xF e d

αα

α

απ

=∞−

=−∞∫ α converges to ( )f x

14

5.

The Laplace Transform

In [Dan7], we have shown that

1. The Delta function , for is represented by Sequence

.

( )tδ 0t ≥

[0, )( ) ( )ntn t ne tδ χ

∞

−=

2. If 1ni n= ,

[0, ) [0, ) [0, )

2 3( ) ( ),2 ( ), 3 ( ),...t t tx e t e t e tδ χ χ χ∞ ∞ ∞

− − −=

3. { }( ) 1tδ =L

4. the inverse Laplace Transform of the unit function 1 ( )tδ =

12

s ist

s i

e dsiπ

= ∞

=− ∞

= ∫ .

5. 0

12

s ist

s i t

e dsi d

γ

γπ

= + ∞

= − ∞ =

=∫1t= an infinite hyper-real

0

0s i

st

s i t

e dsγ

γ

= + ∞

= − ∞ ≠

=∫ .

6. Laplace Integral Theorem

If ( )f t is hyper-real function,

Then,

the Laplace Integral Theorem holds.

15

0

1( ) ( )

2

s ist s

s i

f t e e fi

γ ττ

γ τ

τ τπ

= + ∞ =∞−

= − ∞ =

⎛ ⎞⎟⎜ ⎟⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫ ∫ d ds

, converges to 0

( )t

st

t

e f t dt=∞

−

=∫ ( )F s

1( )

2

s ist

s i

e F s dsi

γ

γπ

= + ∞

= − ∞∫ converges to ( )f t .

16

6.

Optical Catastrophe Model for the

Free Electron Laser Pulses

In the Free electron Laser [W-5], an electron beam accelerated to

almost light speed, is forced into a wiggling sinusoidal path by a

transverse magnetic field with alternating poles.

The radiation is coherent, in wavelengths ranging from

microwaves to X-rays.

Since X-rays penetrate mirrors, the Free Electron Laser operates

without mirrors, hence without an oscillator, and the pulse has to

be created in a single pass of the beam.

The emitted power increases as coherence increases, but the pulse

does not shape as a Soliton, and as it propagates in the z

direction, it is subject to broadening, and loss.

We shall approximate it by a Gaussian Pulse.

Thus, [e.g. Ghatak], its power is proportional to

( )

22

14 2 22

2( )2

14 2 2 2

2

t

t

t zzt

t

e

z

σβ

σ βσ

σ β

−−

+

+

17

where

tσ is the width of the pulse,

00

10E

ω ωωβ β ==

= ∂ is the dispersion coefficient in the air

00

22

0Eω ωωβ β =

== ∂ is the second dispersion coefficient in the air.

Thus, the pulse power decays with the distance z .

Furthermore, the radiation from the electron beam disperses on

the surface of a radially expanding cylinder.

18

7.

Optical Catastrophe Model for

Optical Solitons The Solitons intensity and width qualifies them to be considered

as Delta function.

We shall model the propagation of the Soliton in the air by the

Nonlinear Schrödinger equation with the nonlinearity, and

dispersion removed, forced by the Delta function.

( , ) ( , ) ( ) ( )ct xG x t G x t x tδ δ∂ + ∂ =

7.1 Let be hyper-real in x , and in t , and have Principal

Value Derivatives with respect to x , and t , that satisfy the wave

equation

( , )G x t

1( , ) ( , ) ( ) ( )t xG x t G x t x t

cδ δ∂ + ∂ = ,

with ( , 0) 0G x = ,

Then,

( , ) ( )G x t c ct xδ= −

Proof:

In the following, all functions are Hyper-Real.

19

The Laplace Transform of the hyper-real exists with

respect to t :

( , )G x t

{ }0

( , ) ( , ) ( , )t

st

t

G x t e G x t dt g x s=∞

−

=

= ≡∫L

{ }0

( , ) ( , )t

stt t

t

G x t e G x t dt=∞

−

=

∂ = ∂∫L

{ }0

0

( , ) ( , )t

tst sttt

t

e G x t e G x t dt=∞

=∞− −=

=

= − ∂∫

00

( , 0) ( , )t

st

t

G x s e G x t dt=∞

−

==

= − + ∫

, ( , )sg x s=

{ }( ) 1tδ =L ,

Thus, the hyper-real equation 1( , ) ( , ) ( ) ( )t xG x t G x t x t

cδ δ∂ + ∂ = ,

Laplace Transforms into

1( , ) ( , ) ( )xsg x s g x s x

cδ+ ∂ = .

The Fourier Transform of with respect to is ( , )g x s x

{ }( , ) ( , ) ( , )x

i x

x

g x s e g x s dx g sω ω=∞

−

=−∞

= ≡∫F

The Inverse Fourier Transform is

20

1( , ) ( , )

2i xg x s e g s d

νω

ν

ω ωπ

=∞

=−∞

= ∫

Hence,

{ }1( , ) ( , )

2i x

x xg x s e g s dν

ω

ν

ω ωπ

=∞

=−∞

∂ = ∂∫

1( , )

2i xe i g s d

νω

ν

ω ω ωπ

=∞

=−∞

= ∫ ,

Therefore, the Fourier Transform of with respect to is ( , )xg x s∂ x

{ }( , ) ( , )xg x s i g sω ω∂ =F .

Since , the equation { }( ) 1xδ =F 1( , ) ( , ) ( )xsg x s g x s x

cδ+ ∂ = ,

Fourier Transforms into

1( , ) ( , ) 1sg s i g s

cω ω ω+ = .

Therefore,

( , )c

g ss i

ωω

=+ c

.

By the Laplace Integral Theorem, The hyper-real may be

Inverse Laplace Transformed into a hyper-real function :

( , )g sω

ˆ( , )G tω

1ˆ( , )c

G ts i c

ωω

− ⎧ ⎫⎪ ⎪⎪ ⎪= ⎨ ⎬⎪ ⎪+⎪ ⎪⎩ ⎭L ,

From Laplace Transform Tables,

i ctce ω−=Hence,

21

ˆ( , ) i ctG t ce ωω −= .

By the Fourier Integral Theorem, The hyper-real may be

Inverse Fourier Transformed into the hyper-real function :

ˆ( , )G tω

( , )G x t

( , ) ( )G x t c x ctδ= −

( )c ct xδ= −

( )x

tc

δ= − .

22

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23

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24

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[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

25