*F* ∼= *F*<sup>∞</sup> ∼ 15.5*µ*m

This *F* value is of the same order of the observed interval and/or large front size of the cavities [11], [23].

## **Author details**

Yoshito Shuto

Ofra Project, Iruma, Japan

### **References**

32 Optical Fiber

core layer is given by [63], [101]

Substitution of Eq. (36) into Eq. (35) gives

the focal length *F* is given by [63], [101]

K−<sup>1</sup> [5], *F* is given by

cavities [11], [23].

**Author details**

Ofra Project, Iruma, Japan

Yoshito Shuto

<sup>∆</sup>*T*(*r*, *<sup>t</sup>*) = *<sup>α</sup><sup>P</sup>*

4*πλ* ln 1 + 8*Dt ω*<sup>0</sup> 2 

*<sup>n</sup>* <sup>∼</sup><sup>=</sup> *<sup>n</sup>*<sup>0</sup>

*<sup>F</sup>*(*t*) = *<sup>n</sup>*0*πλω*<sup>0</sup>

*tc* <sup>=</sup> *<sup>ω</sup>*<sup>0</sup>

*<sup>F</sup>*<sup>∞</sup> <sup>=</sup> *<sup>n</sup>*0*πλω*<sup>0</sup>

2

 1 + *δ r ω*<sup>0</sup>

*<sup>δ</sup>* <sup>=</sup> <sup>−</sup> <sup>2</sup>*α<sup>P</sup>* 4*n*0*πλ*

If we assume that optical power in the optical fiber takes on Gaussian distribution, ∆*T* in the

where *α* is the absorption coefficient, *P* is the incident optical power, *ω*<sup>0</sup> is the spot size radius of the laser beam, and *λ* is the thermal conductivity of the silica glass, respectively. Parameter *D* = *λ*/*Cpρ*, where *Cp* and *ρ* are the specific heat and dnsity of the silica glass.

> *∂n ∂T*

When refractive index of the core layer takes a radial distribution as shown in Eq. (37), propagating laser beam will be focussed as a result of the thermal lens effect. In this case,

> <sup>2</sup> + 8*Dt*) *<sup>α</sup>Pl*(*∂n*/*∂T*)(8*Dt*) <sup>=</sup> *<sup>F</sup>*<sup>∞</sup>

where *l* is the length of the heating core, where *α* exhibits large value. In typical SM fiber, *λ* = 9.2 W/mK, *Cp* = 788 J/kgK, *ρ* = 2,200 kg/m<sup>3</sup> [5], and *ω*<sup>0</sup> ∼ 4.5 *µ*m. If we insert these values into Eqs. (39)–(41), we obtain *tc* <sup>∼</sup> 0.95 *<sup>µ</sup>*s. This means that *<sup>F</sup>* <sup>∼</sup><sup>=</sup> *<sup>F</sup>*<sup>∞</sup> when *<sup>t</sup>* is 10 *<sup>µ</sup>*<sup>s</sup> or above. If we assume *P* = 2W, *l* ∼ 40 *µ*m, *α* = 5.6 × 10<sup>4</sup> m−<sup>1</sup> [5], and *∂n*/*∂T* = 1.23 × 10−<sup>5</sup>

*F* ∼= *F*<sup>∞</sup> ∼ 15.5*µ*m

This *F* value is of the same order of the observed interval and/or large front size of the

<sup>2</sup>(*ω*<sup>0</sup>

2

*αPl*(*∂n*/*∂T*)

2 

 8*Dt ω*<sup>0</sup>

> 1 + *tc* 2*t*

<sup>4</sup>*<sup>D</sup>* (41)

, (40)

<sup>−</sup> <sup>16</sup>*Dt ω*<sup>0</sup>

<sup>2</sup> + 8*Dt*

*r*2 *ω*<sup>0</sup> 2 

, (37)

<sup>2</sup> <sup>+</sup> <sup>8</sup>*Dt*. (38)

, (39)

(36)


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