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Self-focusing of an optical beam in

cold plasma

Gio Chanturia

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Intro

What do we have?

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A laser beam (ultra short). Cold plasma (collisionless).

Self-focusing of a beam in certain mediums

Due to non-linearity of medium, the beam focuses itself.

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Cold plasma and short pulse as our model

There are reasons, we use these models:

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Cool plasma model:

Ponderomotive effect;

Due to electromagnetic field.

Relativistic effect;

Due to free electrons in plasma.

NO thermal effect.

Short laser pulse model:

Short time scale;

No self-focusing process for quasineutral plasma.

Massive ions do not have time to respond

and therefore stay immobile.

Describing our system mathematically

What do we need to describe our system?

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Maxwell’s equations and equation of motion for a relativistic electrons:

Plasma current:

Electron velocity:

𝑱 = −𝑒𝑛𝑒𝒗 = −𝜔𝑝2

4𝜋𝑐

𝑁𝑒

1 + 𝐼𝑛𝑨

𝒗 =𝑷

𝑚𝛾=

𝜖

𝑚𝑐

𝑨

1 + 𝐼𝑛

Amplitude: 𝑨 = 𝑎𝑛 𝒓, 𝑡 𝑒𝑖 𝑘0𝑧−𝜔0𝑡−𝜓 𝒓,𝑡 𝒙 + 𝑖 𝒚

𝑁𝑒 = 1 +𝛿𝑛𝑒𝑛𝑜

Assumptions to deal with our calculations

Assumptions for amplitude and phase equations:

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Firstly, as we deal with axial symmetry.

That is, when none of the functions depend on 𝜃 and we’re left with only two variables:

𝑎 𝑟, 𝑧 = 𝑎 𝑟

𝜓 𝑟, 𝑧 = 𝑓 𝑧 + 𝑔(𝑟)

𝑥, 𝑦, 𝑧 → (𝑟, 𝜃, 𝑧)

Secondly, we assume, that amplitude doesn’t vary towards 𝑧 direction.

We allow phase to modulate and seek for solution as a sum of individual functions.

The first simplification

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Applying previous assumptions and separating variables, we get:

1

𝑎

𝑑2𝑎

𝑑𝑟2−

𝑑𝑔

𝑑𝑟

2

−1

𝜆𝑐2

𝑁𝑒

1 + 𝑎2= 𝐶1

𝑑2𝑔

𝑑𝑟2+

1

𝑎2𝑑(𝑎2)

𝑑𝑟

𝑑𝑔

𝑑𝑟= 0

These equations still look tricky. So, let us apply the slab limit:

{separation constant}

𝑦

𝑥

𝑦 → 0𝑟 → 𝑥

Slab limit results

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Within the slab limit we have:

1

𝑎

𝑑2𝑎

𝑑𝑥2−𝐶42

𝑎4−

1

𝜆𝑐2

𝑁𝑒

1 + 𝑎2= 𝐶1

𝑁𝑒 = 1 + 𝜆𝑐2 𝑑2

𝑑𝑥21 + 𝑎2

Which combines into:

1

𝑎

𝑑2𝑎

𝑑𝑥2−𝐶42

𝑎4−

1

𝜆𝑐2 1 + 𝑎2

−1

1 + 𝑎2

𝑑2

𝑑𝑥21 + 𝑎2 = 𝐶1

Lagrangian analogy

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The last equation can be written in a form:

We know from the least action principle, that Lagrangian of a particle is written like this:

𝐿 = 𝑔 𝑎𝑎′ 2

2− 𝑉(𝑎)

Which by Nother’s theorem gives:

𝑔 𝑎 𝑎′′ +1

2

𝑑𝑔

𝑑𝑎𝑎′ 2 −

𝜕𝑉

𝜕𝑎= 0 ≡ 𝐺(𝑎, 𝑎′, 𝑎′′)

1

𝑎

𝑑2𝑎

𝑑𝑥2−𝐶42

𝑎4−

1

𝜆𝑐2 1 + 𝑎2

−1

1 + 𝑎2

𝑑2

𝑑𝑥21 + 𝑎2 − 𝐶1 = 0 ≡ 𝐹(𝑎, 𝑎′, 𝑎′′)

Lagrangian analogy

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If for some integrating factor 𝜇 𝑎 :

Then we will be able to find the “potential” of amplitude and therefore describe the behavior of it.

𝐹 𝑎, 𝑎′, 𝑎′′ ∙ 𝜇 𝑎 = 𝐺(𝑎, 𝑎′, 𝑎′′)

Making calculations in this manner and flattening the metric by transformation

𝑎 = sinh(𝑦)

We obtain:

𝑉 𝑦 =𝐶42

2 [sinh 𝑦 ]2− cosh 𝑦 −

𝐶12[sinh 𝑦 ]2

(written in dimensionless transverse coordinate 𝜉 = 𝑥/𝜆𝑐)

Analyzing “potential”

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𝑉 𝑦 =𝐶42

2 [sinh 𝑦 ]2− cosh 𝑦 −

𝐶12[sinh 𝑦 ]2

𝐶42=0

-1<𝐶1<0

Analyzing “potential”

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𝑉 𝑦 =𝐶42

2 [sinh 𝑦 ]2− cosh 𝑦 −

𝐶12[sinh 𝑦 ]2

𝐶42=0

𝐶1>0

Analyzing “potential”

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𝑉 𝑦 =𝐶42

2 [sinh 𝑦 ]2− cosh 𝑦 −

𝐶12[sinh 𝑦 ]2

𝐶42=0

-1>𝐶1

Analyzing “potential”

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𝑉 𝑦 =𝐶42

2 [sinh 𝑦 ]2− cosh 𝑦 −

𝐶12[sinh 𝑦 ]2

1≫ 𝐶42 >0

-1<𝐶1<0

Analyzing “potential”

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𝑉 𝑦 =𝐶42

2 [sinh 𝑦 ]2− cosh 𝑦 −

𝐶12[sinh 𝑦 ]2

𝐶1>0

1≫ 𝐶42 >0

Analyzing “potential”

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𝑉 𝑦 =𝐶42

2 [sinh 𝑦 ]2− cosh 𝑦 −

𝐶12[sinh 𝑦 ]2

-1>𝐶1

1≫ 𝐶42 >0

Analyzing “potential”

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-1>𝐶1

1≫ 𝐶42 >0𝐶4

2=0

Summary:

𝐶1>0

-1<𝐶1<0

-1>𝐶1

𝐶1>0

-1<𝐶1<0

Physical values

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𝑉 𝑦 = −cosh 𝑦 −𝐶12[sinh 𝑦 ]2

휀 𝑦 > −3

2cosh 𝑦 − 𝐶1[sinh 𝑦 ]2

𝑁𝑒 > 0

Not every point of our potential corresponds to a

physical value.

To find out, a meaningful (meaning, useful for us in

this particular problem) values, we have to

remember condition:

Electron density can not be negative.

Exact solutions

𝑎 =2𝜅sech(𝜅𝜉)

1 − 𝜅2sech2(𝜅𝜉)

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𝜅2 = 1 + 𝜆𝑐2𝐶1 𝜉 = 𝑥/𝜆𝑐

Thank you!

• T.Kurki-Suonio, P.J. Morrison, T.Tajima –

“Self-focusing of an optical beam in plasma”;

• Stockholm’s Royal Institute of Technology

– “Nonlinear Optics 5A5513 (2003)”;

• Wolfram’s Mathematica (plots);

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Sources: