Difference between revisions of "Hermite Gauss mode"

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[[Hermite Gauss mode]] refers to the specific solution $F=F(x,z)$ of equation
 +
[[Hermite Gauss mode]] refers to the specific solution \(F=F(x,z)\) of equation
   
−
$ F''+ 2 \mathrm i \dot F=0$
 +
\( F''+ 2 \mathrm i \dot F=0\)
   
 
where prime differentiates with respect to the first argument, and dot differentiates with respect to the last argument.
  
where prime differentiates with respect to the first argument, and dot differentiates with respect to the last argument.
−
This equation appears at the paraxial propagation of monochromatic beam. If $k$ is wavenumber,
 +
This equation appears at the paraxial propagation of monochromatic beam. If \(k\) is wavenumber,
−
then, assuming some large positive $M$, expression $M^2 z/k$ has sense of the coordinate along the propagation of wave, and $M x/k$ has sense of the transversal coordinate.
 +
then, assuming some large positive \(M\), expression \(M^2 z/k\) has sense of the coordinate along the propagation of wave, and \(M x/k\) has sense of the transversal coordinate.
   
 
The [[Hermite Gauss mode]] is expressed in terms of the [[Hermite polynomial]]
  
The [[Hermite Gauss mode]] is expressed in terms of the [[Hermite polynomial]]
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The [[Elegant Hermite Gauss]] can be written as follows:
  
The [[Elegant Hermite Gauss]] can be written as follows:
   
−
$\displaystyle
 +
\(\displaystyle
 
F_m(x,z)=\frac{ w^{m/2} (w+\mathrm i z)^{-\frac{m}{2}-\frac{1}{2}}
 
F_m(x,z)=\frac{ w^{m/2} (w+\mathrm i z)^{-\frac{m}{2}-\frac{1}{2}}
 
}
  
}
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\,
  
\,
 
\exp\! \left(-\frac{x^2}{2 (w+\mathrm i z)}\right)
  
\exp\! \left(-\frac{x^2}{2 (w+\mathrm i z)}\right)
  +
\)
−
$
  
   
−
where $w$ refers to the with of the beam in the narrowest position, and $H_m$ is the [[Hermite polynomial]].
 +
where \(w\) refers to the with of the beam in the narrowest position, and \(H_m\) is the [[Hermite polynomial]].
   
 
<poem><nomathjax><nowiki>
  
<poem><nomathjax><nowiki>
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The [[Classical Hermite Gauss]] can be written as follows:
  
The [[Classical Hermite Gauss]] can be written as follows:
   
−
$f_m(x,z)=
 +
\(f_m(x,z)=
 
\displaystyle
  
\displaystyle
 
\frac{ \left(o^2\!+\!z^2\right)^{-1/4}}{\sqrt{2^m\, m!\, \pi}}\,
  
\frac{ \left(o^2\!+\!z^2\right)^{-1/4}}{\sqrt{2^m\, m!\, \pi}}\,
Line 49: Line 49:
 
- \mathrm i \left( m\!+\!\frac{1}{2} \right) \tan^{-1}\left(\frac{z}{o}\right)
  
- \mathrm i \left( m\!+\!\frac{1}{2} \right) \tan^{-1}\left(\frac{z}{o}\right)
 
\right)
  
\right)
  +
\)
−
$
  
   
−
where $o$ has the similar meaning as $w$ above, it characterises the waste of the beam.
+
where \(o\) has the similar meaning as \(w\) above, it characterises the waste of the beam.
 
<poem><nomathjax><nowiki>
  
<poem><nomathjax><nowiki>
 
In Mathematica, this solution can be verified as follows:
  
In Mathematica, this solution can be verified as follows:
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</nowiki></nomathjax></poem>
  
</nowiki></nomathjax></poem>
   
−
The [[Classical Hermite Gauss]] with waist $o$ can be obtained with scaling $x \rightarrow ox$, $z \rightarrow o^2z$,
+
The [[Classical Hermite Gauss]] with waist \(o\) can be obtained with scaling \(x \rightarrow ox\), \(z \rightarrow o^2z\),
−
as $\frac{1}{o} \tilde f_m(ox,o^2z)$
 +
as \(\frac{1}{o} \tilde f_m(ox,o^2z)\)
   
 
where
 
where
   
−
$\tilde f_m(x,z)= f_m(x,z)_{\left| \displaystyle \begin{array}{c} \\ o \rightarrow 1 \end{array}\right. }$ $=
 +
\(\tilde f_m(x,z)= f_m(x,z)_{\left| \displaystyle \begin{array}{c} \\ o \rightarrow 1 \end{array}\right. }\) \(=
 
\displaystyle
  
\displaystyle
−
\frac{ \left(1 \!+\!z^2\right)^{-1/4}}{\sqrt{2^m\, m!\, \pi}}\, $ $\displaystyle
 +
\frac{ \left(1 \!+\!z^2\right)^{-1/4}}{\sqrt{2^m\, m!\, \pi}}\, \) \(\displaystyle
−
H_m\!\left(\frac{x}{\sqrt{1\!+\!z^2}}\right) \,$ $\displaystyle
 +
H_m\!\left(\frac{x}{\sqrt{1\!+\!z^2}}\right) \,\) \(\displaystyle
 
\exp\! \left(
  
\exp\! \left(
 
\frac{-x^2}{2 (1\!+\!\mathrm i z)}
  
\frac{-x^2}{2 (1\!+\!\mathrm i z)}
 
- \mathrm i \left( m\!+\!\frac{1}{2} \right) \tan^{-1}(z)
  
- \mathrm i \left( m\!+\!\frac{1}{2} \right) \tan^{-1}(z)
 
\right)
  
\right)
  +
\)
−
$
  
   
 
===Comparison===
  
===Comparison===
−
Similiarities in expressions for $F_m(x)$ and $f_m(x)$ causes ceduction to set
 +
Similiarities in expressions for \(F_m(x)\) and \(f_m(x)\) causes ceduction to set
−
$o=w$ and to scale the solutions, in such a way, that
+
\(o=w\) and to scale the solutions, in such a way, that
   
−
$F_m(x,z)=A_m f_m(x,z)$
 +
\(F_m(x,z)=A_m f_m(x,z)\)
   
−
where $A_m$ is scaling factor, that may depend on $m$.
+
where \(A_m\) is scaling factor, that may depend on \(m\).
−
However, this is not a case, because the scale of the Hermite polynomials is different, even at $z=0$.
 +
However, this is not a case, because the scale of the Hermite polynomials is different, even at \(z=0\).
 
So, these are different solutions of the same equation.
  
So, these are different solutions of the same equation.
   
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The two caces above can be combined with generalised Hermite Gauss
  
The two caces above can be combined with generalised Hermite Gauss
   
−
$\displaystyle
 +
\(\displaystyle
−
G_m(x,z)=$ $\displaystyle
 +
G_m(x,z)=\) \(\displaystyle
−
H_m\left(\frac{x}{w \sqrt{1+\mathrm i z/w^2} \sqrt{2-c \left(1+\mathrm i z/w^2\right)}}\right) $
 +
H_m\left(\frac{x}{w \sqrt{1+\mathrm i z/w^2} \sqrt{2-c \left(1+\mathrm i z/w^2\right)}}\right) \)
−
$\displaystyle
 +
\(\displaystyle
 
\exp\left(
  
\exp\left(
 
-\frac{x^2}{2 w^2
  
-\frac{x^2}{2 w^2
 
\left(1+\mathrm i z/w^2\right)}\right)
  
\left(1+\mathrm i z/w^2\right)}\right)
  +
\)
−
$
  
−
$\displaystyle
 +
\(\displaystyle
 
\exp
  
\exp
 
\left(\frac{m}{4} \ln \left(4+c^2
  
\left(\frac{m}{4} \ln \left(4+c^2
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\log \left(\frac{z}{w^2}-\mathrm i\right)
  
\log \left(\frac{z}{w^2}-\mathrm i\right)
 
\right)
  
\right)
  +
\)
−
$
  
   
−
where $w$ is size of the waist of the beam (it is placed at zero) and the real parameter $c<2$ determines the scale for the Hermite polynomials;
 +
where \(w\) is size of the waist of the beam (it is placed at zero) and the real parameter \(c<2\) determines the scale for the Hermite polynomials;
−
the larger is $c$, the denser are oscillations of the Hermite function at the waist.
 +
the larger is \(c\), the denser are oscillations of the Hermite function at the waist.
   
 
This solution can be verified with the Mathematica code below:
  
This solution can be verified with the Mathematica code below:
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</nowiki></nomathjax></poem>
  
</nowiki></nomathjax></poem>
   
−
Parameter $w$ is kept in equations above to make allowance with the tradition.
 +
Parameter \(w\) is kept in equations above to make allowance with the tradition.
−
In order to boost the deduction, one may set $w=1$; then in the final expression, this parameter can be recovered with
 +
In order to boost the deduction, one may set \(w=1\); then in the final expression, this parameter can be recovered with
−
replacement $x\rightarrow x/w$, $z\rightarrow z/w^2$.
 +
replacement \(x\rightarrow x/w\), \(z\rightarrow z/w^2\).
   
−
At $c=0$, the solution $G_m$ can be considered as the [[elegant Gaussian mode]] and
 +
At \(c=0\), the solution \(G_m\) can be considered as the [[elegant Gaussian mode]] and
−
at $c=1$, the solution can be considered as the [[classic Gaussian mode]]. In this sense, parameter $c$ can be called "classicality".
 +
at \(c=1\), the solution can be considered as the [[classic Gaussian mode]]. In this sense, parameter \(c\) can be called "classicality".
   
 
==Modifications==
  
==Modifications==
 
However, the solution can be transformed, using the symmetries of the equation: two translations, one tilt and one scaling.
  
However, the solution can be transformed, using the symmetries of the equation: two translations, one tilt and one scaling.
−
For this reason, it is sufficient to consider the case $w=1$, id est,
 +
For this reason, it is sufficient to consider the case \(w=1\), id est,
   
−
$\displaystyle
 +
\(\displaystyle
−
G_m(x,z)=$ $\displaystyle
 +
G_m(x,z)=\) \(\displaystyle
−
H_m\left(\frac{x}{\sqrt{1+\mathrm i z} \sqrt{2-c \left(1+\mathrm i z\right)}}\right) $
 +
H_m\left(\frac{x}{\sqrt{1+\mathrm i z} \sqrt{2-c \left(1+\mathrm i z\right)}}\right) \)
−
$\displaystyle
 +
\(\displaystyle
 
\exp\left(-\frac{x^2}{2 \left(1+\mathrm i z\right)}\right)
  
\exp\left(-\frac{x^2}{2 \left(1+\mathrm i z\right)}\right)
  +
\)
−
$
  
−
$\displaystyle
 +
\(\displaystyle
 
\exp
  
\exp
 
\left(\frac{m}{4} \ln \left(4+c^2
  
\left(\frac{m}{4} \ln \left(4+c^2
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\log \left(z-\mathrm i\right)
  
\log \left(z-\mathrm i\right)
 
\right)
  
\right)
  +
\)
−
$
  
   
 
Solution with realistic width can be recovered with transform
  
Solution with realistic width can be recovered with transform
   
−
$\displaystyle
 +
\(\displaystyle
−
G_m(x,z) \rightarrow \frac{1}{\sqrt{w}}\, G_m\!\left(\frac{x}{w}, \frac{z}{w^2}\right)$
 +
G_m(x,z) \rightarrow \frac{1}{\sqrt{w}}\, G_m\!\left(\frac{x}{w}, \frac{z}{w^2}\right)\)
   
 
The solution allow the straightforward generalisation to the multidimensional case.
  
The solution allow the straightforward generalisation to the multidimensional case.
Line 174: Line 174:
   
 
The Editor is not successful to write a Mathematica code to verify such a "solution"; the "solutions" seem to be just wrong.
  
The Editor is not successful to write a Mathematica code to verify such a "solution"; the "solutions" seem to be just wrong.
−
The origin of the error seems to be confusion of the sense of parameter $w$ that scales the argument of the Hermite polynomial.
 +
The origin of the error seems to be confusion of the sense of parameter \(w\) that scales the argument of the Hermite polynomial.
   
 
==References==
  
==References==
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[[Hermite polynomial]],
  
[[Hermite polynomial]],
 
[[HermiteH]],
  
[[HermiteH]],
−
[[Oscillator function]]
 +
[[Oscillator function]],
  +
[[Quantum mechanics]]
   
 
[[Category:Atomic physics]]
  
[[Category:Atomic physics]]

Latest revision as of 18:43, 30 July 2019

Hermite Gauss mode refers to the specific solution \(F=F(x,z)\) of equation

\( F''+ 2 \mathrm i \dot F=0\)

where prime differentiates with respect to the first argument, and dot differentiates with respect to the last argument. This equation appears at the paraxial propagation of monochromatic beam. If \(k\) is wavenumber, then, assuming some large positive \(M\), expression \(M^2 z/k\) has sense of the coordinate along the propagation of wave, and \(M x/k\) has sense of the transversal coordinate.

The Hermite Gauss mode is expressed in terms of the Hermite polynomial and the exponent.

Notations

In the literature, there is no unified notations for the Hermite Gauss mode. Among them, the more often are so-called Elegant Hermite Gauss and Classical Hermite Gauss

Elegant

The Elegant Hermite Gauss can be written as follows:

\(\displaystyle F_m(x,z)=\frac{ w^{m/2} (w+\mathrm i z)^{-\frac{m}{2}-\frac{1}{2}} } {\sqrt{\pi\, 2^m\, m!}} \, H_m\!\left(\frac{x}{\sqrt{2} \sqrt{w+\mathrm i z}}\right) \, \exp\! \left(-\frac{x^2}{2 (w+\mathrm i z)}\right) \)

where \(w\) refers to the with of the beam in the narrowest position, and \(H_m\) is the Hermite polynomial.


In Mathematica, the solution above can be verified with the code below:
F[x_] = ((w+I z)^(-1/2-m/2) HermiteH[m,x/(Sqrt[2] Sqrt[w+I z])]) Exp[-x^2/(2(w+I z))]
em = Simplify[ F''[x] + 2 I D[F[x],z] ]
Simplify[ReplaceAll[em, m -> 2]]
Simplify[ReplaceAll[em, m -> 3]]
Simplify[ReplaceAll[em, m -> 4]]

Classical

The Classical Hermite Gauss can be written as follows:

\(f_m(x,z)= \displaystyle \frac{ \left(o^2\!+\!z^2\right)^{-1/4}}{\sqrt{2^m\, m!\, \pi}}\, H_m\!\left(\frac{x}{\sqrt{o+z^2/o}}\right) \, \exp\! \left( -\frac{x^2}{2 (o+\mathrm i z)} - \mathrm i \left( m\!+\!\frac{1}{2} \right) \tan^{-1}\left(\frac{z}{o}\right) \right) \)

where \(o\) has the similar meaning as \(w\) above, it characterises the waste of the beam.


In Mathematica, this solution can be verified as follows:
f[x_]=E^(-x^2/(2(o+I z)) - (I/2) (1+2m) ArcTan[z/o]) HermiteH[m,x/Sqrt[o+z^2/o]])/(1+z^2/o^2)^(1/4)
em = Simplify[f''[x] + 2 I D[f[x], z]]
Simplify[ReplaceAll[em, m -> 2]]
Simplify[ReplaceAll[em, m -> 3]]
Simplify[ReplaceAll[em, m -> 4]]

The Classical Hermite Gauss with waist \(o\) can be obtained with scaling \(x \rightarrow ox\), \(z \rightarrow o^2z\), as \(\frac{1}{o} \tilde f_m(ox,o^2z)\)

where

\(\tilde f_m(x,z)= f_m(x,z)_{\left| \displaystyle \begin{array}{c} \\ o \rightarrow 1 \end{array}\right. }\) \(= \displaystyle \frac{ \left(1 \!+\!z^2\right)^{-1/4}}{\sqrt{2^m\, m!\, \pi}}\, \) \(\displaystyle H_m\!\left(\frac{x}{\sqrt{1\!+\!z^2}}\right) \,\) \(\displaystyle \exp\! \left( \frac{-x^2}{2 (1\!+\!\mathrm i z)} - \mathrm i \left( m\!+\!\frac{1}{2} \right) \tan^{-1}(z) \right) \)

Comparison

Similiarities in expressions for \(F_m(x)\) and \(f_m(x)\) causes ceduction to set \(o=w\) and to scale the solutions, in such a way, that

\(F_m(x,z)=A_m f_m(x,z)\)

where \(A_m\) is scaling factor, that may depend on \(m\). However, this is not a case, because the scale of the Hermite polynomials is different, even at \(z=0\). So, these are different solutions of the same equation.

Multidimensional

In the similar form, the Hermite Gauss mode can be written for the equation with two transversal coordinates [1][2]

Generailzation

The two caces above can be combined with generalised Hermite Gauss

\(\displaystyle G_m(x,z)=\) \(\displaystyle H_m\left(\frac{x}{w \sqrt{1+\mathrm i z/w^2} \sqrt{2-c \left(1+\mathrm i z/w^2\right)}}\right) \) \(\displaystyle \exp\left( -\frac{x^2}{2 w^2 \left(1+\mathrm i z/w^2\right)}\right) \) \(\displaystyle \exp \left(\frac{m}{4} \ln \left(4+c^2 \left(\frac{z}{w^2}-\mathrm i\right)^2\right)-\frac{\mathrm i\, m}{2} \tan ^{-1}\left(\frac{c}{2} \left(\frac{z}{w^2}-\mathrm i\right)\right)-\frac{m+1}{2} \log \left(\frac{z}{w^2}-\mathrm i\right) \right) \)

where \(w\) is size of the waist of the beam (it is placed at zero) and the real parameter \(c<2\) determines the scale for the Hermite polynomials; the larger is \(c\), the denser are oscillations of the Hermite function at the waist.

This solution can be verified with the Mathematica code below:


G[x_] = HermiteH[m, x/(w Sqrt[1+I z/w^2] Sqrt[2-c(1+I z/w^2)])] Exp[-x^2/(2 w^2) 1/(1+I z/w^2)] *
Exp[-(1/2) I m ArcTan[1/2 c(-I+z/w^2)] - 1/2 (1+m) Log[-I + z/w^2] +1/4 m Log[4 + c^2 (-I+z/w^2)^2] ]

em = Simplify[(G''[x] + 2 I D[G[x], z])]

Table[Simplify[em], {m, 0, 9}]

Parameter \(w\) is kept in equations above to make allowance with the tradition. In order to boost the deduction, one may set \(w=1\); then in the final expression, this parameter can be recovered with replacement \(x\rightarrow x/w\), \(z\rightarrow z/w^2\).

At \(c=0\), the solution \(G_m\) can be considered as the elegant Gaussian mode and at \(c=1\), the solution can be considered as the classic Gaussian mode. In this sense, parameter \(c\) can be called "classicality".

Modifications

However, the solution can be transformed, using the symmetries of the equation: two translations, one tilt and one scaling. For this reason, it is sufficient to consider the case \(w=1\), id est,

\(\displaystyle G_m(x,z)=\) \(\displaystyle H_m\left(\frac{x}{\sqrt{1+\mathrm i z} \sqrt{2-c \left(1+\mathrm i z\right)}}\right) \) \(\displaystyle \exp\left(-\frac{x^2}{2 \left(1+\mathrm i z\right)}\right) \) \(\displaystyle \exp \left(\frac{m}{4} \ln \left(4+c^2 \left(z-\mathrm i\right)^2\right)-\frac{\mathrm i\, m}{2} \tan ^{-1}\left(\frac{c}{2} \left(z-\mathrm i\right)\right)-\frac{m+1}{2} \log \left(z-\mathrm i\right) \right) \)

Solution with realistic width can be recovered with transform

\(\displaystyle G_m(x,z) \rightarrow \frac{1}{\sqrt{w}}\, G_m\!\left(\frac{x}{w}, \frac{z}{w^2}\right)\)

The solution allow the straightforward generalisation to the multidimensional case.

Other forms

Several authors (and citing them wikipedia), suggest another form of the Hermite Gaussian mode, where argument of the Hermite polynomial appears as real function of coordinates. [3]

The Editor is not successful to write a Mathematica code to verify such a "solution"; the "solutions" seem to be just wrong. The origin of the error seems to be confusion of the sense of parameter \(w\) that scales the argument of the Hermite polynomial.

References

  1. ↑ https://www.researchgate.net/profile/Francesco_Pampaloni/publication/2171403_Gaussian_Hermite-Gaussian_and_Laguerre-Gaussian_beams_A_primer/links/0912f50f03f0ea21c7000000.pdf Francesco Pampaloni, Jörg Enderlein. Gaussian, Hermite-Gaussian, and Laguerre-Gaussian beams: A primer. (2016)
  2. ↑ https://www.researchgate.net/profile/Francesco_Pampaloni/publication/8382690_Unified_operator_approach_for_deriving_Hermite-Gaussian_and_Laguerre-Gaussian_laser_modes/links/0912f50f03f0de26ca000000.pdf Jorg Enderlein, Francesco Pampaloni. Unified operator approach for deriving Hermite – Gaussian and Laguerre–Gaussian laser modes J. Opt. Soc. Am. A, Vol. 21, No. 8 / August 2004, p.1553–1558.
  3. ↑ https://en.wikipedia.org/wiki/Gaussian_beam

Keywords

Hermite polynomial, HermiteH, Oscillator function, Quantum mechanics

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