Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi...
Transcript of Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi...
![Page 1: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/1.jpg)
Self-focusing of an optical beam in
cold plasma
Gio Chanturia
Free University 1
![Page 2: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/2.jpg)
Intro
What do we have?
Free University 2
A laser beam (ultra short). Cold plasma (collisionless).
![Page 3: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/3.jpg)
Self-focusing of a beam in certain mediums
Due to non-linearity of medium, the beam focuses itself.
Free University 3
![Page 4: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/4.jpg)
Cold plasma and short pulse as our model
There are reasons, we use these models:
Free University 4
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.
![Page 5: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/5.jpg)
Describing our system mathematically
What do we need to describe our system?
Free University 5
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
![Page 6: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/6.jpg)
Assumptions to deal with our calculations
Assumptions for amplitude and phase equations:
Free University 6
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.
![Page 7: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/7.jpg)
The first simplification
Free University 7
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
![Page 8: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/8.jpg)
Slab limit results
Free University 8
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
![Page 9: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/9.jpg)
Lagrangian analogy
Free University 9
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′′)
![Page 10: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/10.jpg)
Lagrangian analogy
Free University 10
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)
![Page 11: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/11.jpg)
Analyzing “potential”
Free University 11
𝑉 𝑦 =𝐶42
2 [sinh 𝑦 ]2− cosh 𝑦 −
𝐶12[sinh 𝑦 ]2
𝐶42=0
-1<𝐶1<0
![Page 12: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/12.jpg)
Analyzing “potential”
Free University 12
𝑉 𝑦 =𝐶42
2 [sinh 𝑦 ]2− cosh 𝑦 −
𝐶12[sinh 𝑦 ]2
𝐶42=0
𝐶1>0
![Page 13: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/13.jpg)
Analyzing “potential”
Free University 13
𝑉 𝑦 =𝐶42
2 [sinh 𝑦 ]2− cosh 𝑦 −
𝐶12[sinh 𝑦 ]2
𝐶42=0
-1>𝐶1
![Page 14: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/14.jpg)
Analyzing “potential”
Free University 14
𝑉 𝑦 =𝐶42
2 [sinh 𝑦 ]2− cosh 𝑦 −
𝐶12[sinh 𝑦 ]2
1≫ 𝐶42 >0
-1<𝐶1<0
![Page 15: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/15.jpg)
Analyzing “potential”
Free University 15
𝑉 𝑦 =𝐶42
2 [sinh 𝑦 ]2− cosh 𝑦 −
𝐶12[sinh 𝑦 ]2
𝐶1>0
1≫ 𝐶42 >0
![Page 16: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/16.jpg)
Analyzing “potential”
Free University 16
𝑉 𝑦 =𝐶42
2 [sinh 𝑦 ]2− cosh 𝑦 −
𝐶12[sinh 𝑦 ]2
-1>𝐶1
1≫ 𝐶42 >0
![Page 17: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/17.jpg)
Analyzing “potential”
Free University 17
-1>𝐶1
1≫ 𝐶42 >0𝐶4
2=0
Summary:
𝐶1>0
-1<𝐶1<0
-1>𝐶1
𝐶1>0
-1<𝐶1<0
![Page 18: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/18.jpg)
Physical values
Free University 18
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.
![Page 19: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/19.jpg)
Exact solutions
𝑎 =2𝜅sech(𝜅𝜉)
1 − 𝜅2sech2(𝜅𝜉)
Free University 19
𝜅2 = 1 + 𝜆𝑐2𝐶1 𝜉 = 𝑥/𝜆𝑐
![Page 20: Self-focusing of an optical beam in cold plasmafreeuni.edu.ge/sites/default/files/Giorgi Chanturia.pdfSelf-focusing of an optical beam in cold plasma Gio Chanturia Free University](https://reader030.fdocuments.us/reader030/viewer/2022040910/5e8499f7533aa946b8152d85/html5/thumbnails/20.jpg)
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);
Free University 20
Sources: