Shoka function

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Explicit plot of Shoka function and two its asymptotics
$u\!+\!\mathrm i v = \mathrm{Shoka}(x\!+\mathrm i y)$

Shoka function is holomorphic function,

$ \mathrm{Shoka}(z)=z+\ln\!\Big( \exp(-z) +\mathrm e -1\Big)$

Explicit plot of the Shoka function is shown in figure at right. Complex map of the Shoka function is shown below.

Contents

Range of holomorphizm

The Shoka function is holomorphic at the complex plane with cuts along the parallel lines

$ x+ (1\!+\!2n) \mathrm i \pi ~$ at real $x$ and integer $n$.

In such a way, the countable set of cut lines are directed to the left.

Inverse function

$u\!+\!\mathrm i v= \mathrm{ArcShoka}(x\!+\!\mathrm i y)$

The inverse function $\mathrm{ArcShoka}=\mathrm{Shoka}^{-1}$ can be expressed as elementary function; for $|\Im(z)| < \pi$,

$\displaystyle \mathrm{ArcShoka}(z)= z + \ln \!\left( \frac{1\!-\!\mathrm e^{-z}}{\mathrm e \!-\!1} \right)$

Some properties of the inverse function are collected in the special article ArcShoka.

Complex maps of Shoka and ArcShoka look similar. The following relation takes place:

$\mathrm{ArcShoka}(z)=\mathrm{Shoka}\big(z-\ln(\mathrm e\! -\!1) - \mathrm i \pi\big) -\ln(\mathrm e\!-\!1) +\mathrm i \pi$

that can be verified with the direct substitution.

Shoko function

Similar function, that coincides with Shoka function along the real axis and its vicinity, but has different range of holomorphism, is denoted as Shoko function. The cut lines of Shoko function are also parallel to the real axis, but are directed toward the negative direction of the $x$ axis.

$ \mathrm{Shoko}(z) = \ln\!\Big( 1+ \exp(z)(\mathrm e -1)\Big)$

In vicinity of the real axis, the Shoka function coincides with the Shoko function,

However, the cut lines of the Shoko function are directed to the right, and the Shoko function is periodic with period $P=2 \pi \mathrm i$.

Shoka function and Shoko function coincide along the real axis and its vicinity, but has different range of holomorphism. The cut lines of Shoko function are also parallel to the real axis, but are directed toward the negative direction of the $x$ axis.

Historically, Shoko function had been constructed first; then, it happens, that for the Keller function, the iterates constructed with Shoka function look more beautiful than those constructed with Shoka function; so, in the Book Superfunctions, the Shoka function is considered as "principal", and Shoko function is treated as the modification. The last letter of name is used to distinguish identifiers of these two functions.

Tania function

The Shoka function behaves similar to the Tania function, but have countable set of cutlines in the left hand side of the complex plane, while the Tania function has only two cut lines.

Tania function is solution $f$ of equation $f'(z)= \frac{f(z)}{1+f(z)}$ with initial condition $f(0)\!=\!1$ The contour of integration of equation (1) goes from zero to the imaginary part of $z$ along the imaginary axis and then along the straight line (parallel to the real axis) to the point $z$.

Applications in Laser science

As the Shoko function, the Shoka function is superfunction of the Keller function and describes the evolution of the short pulse of light in a two-level laser medium. Properties and expansions of the Shoka function are similar to those of the Shoko function and are repeated here.

As it is mentioned above, for the real values of argument, Shoka function and Shoko function coincide; so, pictures for the Shoko function are used

Copmarison of Shoka function (thick curve) to Tania function (thin curve)

The Shoka function appears at the consideration of propagation of short pulse [1][2] in laser amplifier with simple kinetics of the laser medium.

The Shoka function is analogy of the Tania function, that describes the propagation of the continuous wave in the amplifier with permanent continuous pump, with the same simple kinetics. For real values of the argument, both Shoka function and Tania function are plotted in the figure at right. Both curves, for Tania and for Shoko, show similar behavior, exponential growth in the left hand side, that gradually changes to the almost linear growth in the right hand side; but the Shoko function grows a little bit faster than the Tania function.

in TORI, the Keller function is used as an example of the transfer function, for which the superfunction can be constructed as elementary finction, and the Shoko function is this function.

The argument of the Shoko function may have sense of the distance of propagation of a pulse in the gain medium, normalized for the inverse of the gain of a weak signal. Then the Shoko function returns the fluence at this distance, normalized for the saturation fluence.

While the fluence is small, it grows exponentially, according the the asymptotic (2).

The origin of coordinate along the direction of propagation is choosen in such a way, that $\mathrm{Shoko}(0)=1$, which corresponds to the fluence equal to the saturation fluence.

At large values of fluence, it withdraws almost all the energy, stored in the gain medium, and, therefore, the growth is almost linear, according to the asymptotic (4).

Origin of the name

Initially, the Shoko function had been implemented; that function is named after Shoko san. Then in happened, that the cut lines of the Shoko function go to the right hand side direction, that makes difficult its comparison with the Tania function in the complex plane. Name Shoka function is created as small modification of name Shoko function.

References

  1. http://www.ulp.ethz.ch/publications/paper/2001/152__APB_73__653__2001_.pdf R.Paschotta, U.Keller. Passive mode locking with slow saturable absorbers. Appl. Phys. B 73, 653–662 (2001)
  2. http://www.ulp.ethz.ch/publications/paper/2004/200__APB_79__331__2004_.pdf M.Haiml, R.Grange, U.Keller. Optical characterization of semicondutcor saturable absorbers. Appl.Phys. B 79, 331-339 (2004)

Keywords

Shoko function, Keller function, Superfunction, Laser science