## A new approach to compute Overlap efficiency in axially pumped Solid State Lasers

Optics Express, Vol. 5, Issue 6, pp. 125-133 (1999)

http://dx.doi.org/10.1364/OE.5.000125

Acrobat PDF (216 KB)

### Abstract

In this work, we are reporting a new approach to compute the overlap efficiency of end pumped solid-state laser systems. Unlike existing methods in which the overlap integral is computed with a linearize approximation near the threshold, in this method the inverse of the overlap integral is computed numerically in the above threshold regime for several values of circulating fields. Now by fitting a linear curve to this data the overlap efficiency is obtained. The effect of the beam quality factor is also taken into account. It is demonstrated that the linearized approximation near the threshold can give rise to 50% error in overlap efficiency. The method was used to estimate the overlap efficiency in different types of axially pumped lasers.

© Optical Society of America

## 1. Introduction

*M*

^{2}can affect the efficiency of an axially pumped Solid State Laser. We have also tried to find out the effect of pump waist size on the overlap efficiency for a constant

*M*

^{2}value.

## 2. Theoretical Analysis

*I*

_{p}(

*z*) is the incident pump intensity at any plane inside the gain medium and the reflectivity of the two end mirrors is equal to unity, then in steady state, if we neglect the standing wave effects, the saturated gain coefficient

*g*(

*z*) at that plane can be given as[8]

*α*

_{a}is the absorption coefficient of the gain medium at the pump intensity and

*I*

_{sat}is the saturation intensity of the gain medium,

*η*

_{q}is the product of quantum efficiency and quantum defect of the system. In most of the four level solid state lasers the quantum efficiency is approximately equal to one therefore

*η*

_{q}is given by

*ω*

_{m}is the laser frequency and

*ω*

_{p}is the pump beam frequency. Eq.1 is valid only for the plane wave approximation, in real lasers the pump intensity as well as the mode intensity inside the gain volume, is a function of position. Therefore, the gain coefficient inside the gain volume is also a function of position. If we neglect the saturation of the pump power absorption, then the spatial distribution of pump intensity inside an axially pumped gain volume can be defined as

*P*

_{p}is the incident pump power,

*f*

_{p}(

*x*,

*y*,

*z*) is the spatial profile function at a plane

*z*inside the gain medium and

*A*

_{p}(

*z*) is the pump area at the same plane. If

*P*

_{circ}is the total circulating power inside the resonator, the

*I*

_{circ}(

*x*,

*y*,

*z*) at any point inside the gain medium can similarly be defined as

*f*

_{m}(

*x*,

*y*,

*z*) is the spatial mode profile function at a plane

*z*inside the gain medium and

*A*

_{m}(

*z*) is the mode area at the same plane. If

*g*(

*x*,

*y*,

*z*) is the value of the saturated gain coefficient at a particular position (

*x*,

*y*,

*z*) inside the gain volume, then the change in circulating power in one round trip is defined as

*l*is the length of the gain medium. Now for the small gain approximation the total saturated round trip gain

*G*can be defined as

*P′*

_{circ}is defined in terms of saturation intensity

*I*

_{sat}as

*I′*

_{circ}is the normalized circulating intensity, 〈

*A*

_{m}〉 is the average mode area inside the gain medium and is defined as

*R*

_{1}and

*R*

_{2}are the reflectivities of the two mirrors, and

*T*

_{1}and

*T*

_{2}are the corresponding transmissions.

*α*

_{0}is the absorption coefficient at the lasing wavelength. All the intracavity losses are clubbed in term

*L*. If

*T*

_{2}is the output coupler transmission, the output power from a laser is given as

*C*and

*D*are constants. By comparing Eq.13 with Eq.8, the value of the integral in Eq.8 will be given as

*C*will have inverse of length dimensions and

*D*is a dimensionless quantity. Now from Eq.11 and Eq.13, we get the following relation

*T*=

*T*

_{1}+

*T*

_{2}. The circulating power, that must built up inside the resonator in order to saturate the gain factor down to where it can just be equal to total cavity losses, is given by

*η*

_{o}as follows

*η*

_{a}is the absorption efficiency of pump power inside the gain medium. The output power equation can be rewritten as

*P*

_{th}can be given as

*m*can also be obtained from Eq.19.

*α*

_{a}

*C*. If we compare this factor with the match function which describes the spatial overlap of pump beam and resonator modes [7

7. Y.F. Chen, T.S. Liao, C.F. Kao, T.M. Huang, K.H. Lin, and S.C. Wang, “Optimization of fiber coupled laser diode end pumped lasers: Influence of pump beam quality,” IEEE J. Quantum Electron. ,**vol.QE - 32**, pp. 2010–2016, 1996. [CrossRef]

*F*can be given as

*TEM*

_{00},

*f*

_{m}(

*x*,

*y*,

*z*) can be given as

*ω*

_{mx}and

*ω*

_{my}are the radii of the beam along the x-axis and y-axis respectively. The pump profile function

*f*

_{p}(

*x*,

*y*,

*z*) for a diode laser can be written as

*ω*

_{px}and

*ω*

_{py}are the radii of the beam along the x-axis and y-axis respectively. The expression for the beam radius can be given as

*I′*

_{circ}. In the linearized approximation near the threshold, both sides of Eq.14 are expanded into a series. The values of the constant

*C*and

*D*are obtained only from the first two terms of the series. While in the proposed method no such approximation is made. The value of the inverse of the integral can be plotted as a function of

*I′*

_{circ}and by fitting a linear curve the values of the constant

*C*and

*D*can be obtained. There after the value of overlap efficiency and threshold power for a given setup can be computed with the help of Eq.18 and Eq.20. Although by neglecting the higher order terms of the series in the linearized approximation near the threshold, the computed value of the threshold pump-power [5

5. Paolo Laporta and Marcello Brussard, “Design criteria for mode size optimization in diode pumped solid state lasers,” IEEE J. Quantum Electron. ,**vol.QE - 27**, pp. 2319–2326, 1991. [CrossRef]

7. Y.F. Chen, T.S. Liao, C.F. Kao, T.M. Huang, K.H. Lin, and S.C. Wang, “Optimization of fiber coupled laser diode end pumped lasers: Influence of pump beam quality,” IEEE J. Quantum Electron. ,**vol.QE - 32**, pp. 2010–2016, 1996. [CrossRef]

## 3. Simulations

*TEM*

_{00}mode profile were considered. The absorption coefficient of a 0.5

*mm*. thick gain medium is

*α*

_{a}=4.2

*mm*

^{-1}. The pump beam parameters along both axes were

*ω*

_{px}=100

*µm*,

*ω*

_{py}=10

*µm*, and

*e*

^{2}) are

*ω*

_{mx}=40

*µm*, and

*ω*

_{my}=50

*µm*. With these parameters, we computed the variation of the match function and overlap efficiency with the position of the pump-beam focal plane in the gain medium. The computations were done with both the linearized approximations near the threshold and the proposed methods. Fig.1(a) and Fig.1(b) show the variation of the match function and overlap efficiency with focal plane position. The threshold pump-power of a laser is inversely proportional to the match function. Therefore, it can be seen that the values of the match function obtained with both methods are almost the same. However the values of overlap efficiency computed with the linearized approximation near the threshold are about 25% less than the values obtained with our method. This corresponds to about 50% error. We also plotted in Fig.2 the variation of the inverse of the integral in Eq.14 with respect to

*I′*

_{circ}for different pump beam position. It can be seen that in all the cases the curves fit to the straight lines with a regression coefficient not less than 0.9986. This validates our assumption of the linear approximation for the overlap integral in Eq.13. Now we applied this method to a 0.5

*mm*. thick axially pumped Nd:

*YVO*4 microchip laser with absorption coefficient

*α*

_{a}=4.2

*mm*

^{-1}and

*n*=2.165. The variation of the overlap efficiency with respect to the mode radius was computed. The laser mode was taken as a circular

*TEM*

_{00}mode. The pump-beam was considered to be elliptic with two different spots sizes

*Nd*:

*YAG*laser. A crystal of thickness 5

*mm*with absorption coefficient

*α*

_{a}=0.6

*mm*

^{-1}and

*n*=1.82 was considered. The computations were done for a circular pump beam with different values of beam quality factor

*M*

^{2}and different values of laser-mode waist radius

*ω*

_{m}

_{0}. The results are shown in Fig.4. Values of overlap efficiency with

*M*

^{2}=1 correspond nearly to the constant pump-beam spot along the gain medium.

## 4. Conclusion

## References

1. | D.G. Hall, R.J. Smith, and R.R. Rice, “Pump size effects in Nd:YAG lasers,” Appl. Opt. , |

2. | D.G. Hall, “Optimum mode size criterion for low gain lasers,” Appl. Opt. , |

3. | W.P. Risk, “Modelling of longitudinally pumped solid state lasers exhibiting reabsorption losses,”J. Opt. Soc. Amer. B , |

4. | T.Y. Fan and Antonio Sanchez, “Pump source requirements for end pumped lasers,” IEEE J. Quantum Electron. , |

5. | Paolo Laporta and Marcello Brussard, “Design criteria for mode size optimization in diode pumped solid state lasers,” IEEE J. Quantum Electron. , |

6. | C. Pfistner, P. Albers, and H. P. Weber, “Influence of spatial mode matching in end-pumped solid state lasers,” Appl. Phys. |

7. | Y.F. Chen, T.S. Liao, C.F. Kao, T.M. Huang, K.H. Lin, and S.C. Wang, “Optimization of fiber coupled laser diode end pumped lasers: Influence of pump beam quality,” IEEE J. Quantum Electron. , |

8. | A. E. Siegman, “Lasers”, (University Science Book, Mill Valley, CA, 1986). |

9. | A. E. Siegman and Steven W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(140.3480) Lasers and laser optics : Lasers, diode-pumped

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 26, 1999

Published: September 13, 1999

**Citation**

Rakesh Kapoor, P. Mukhopadhyay, and Jogy George, "A new approach to compute overlap efficiency in axially pumped solid state lasers," Opt. Express **5**, 125-133 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-5-6-125

Sort: Journal | Reset

### References

- D. G. Hall, R. J. Smith, and R. R. Rice, " Pump size effects in Nd:YAG lasers," Appl. Opt. 19, 3041-3043 (1980). [CrossRef]
- D. G. Hall, "Optimum mode size criterion for low gain lasers," Appl. Opt. 20, 1579- 1583 (1981). [CrossRef]
- W. P.Risk, "Modelling of longitudinally pumped solid state lasers exhibiting reabsorption losses, "J. Opt. Soc. Amer. B 5, 1412-1423 (1988). [CrossRef]
- T. Y. Fan and Antonio Sanchez, "Pump source requirements for end pumped lasers," IEEE J. Quantum Electron. QE - 26, 311-316 (1990). [CrossRef]
- Paolo Laporta and Marcello Brussard, "Design criteria for mode size optimization in diode pumped solid state lasers," IEEE J. Quantum Electron. QE - 27, 2319-2326 (1991). [CrossRef]
- C. Pfistner, P. Albers, H. P. Weber, "Influence of spatial mode matching in end-pumped solid state lasers," Appl. Phys. B54, 83-88 (1992).
- Y. F. Chen, T. S. Liao, C. F. Kao, T. M. Huang, K. H. Lin, and S. C. Wang, " Optimization of fiber coupled laser diode end pumped lasers: Influence of pump beam quality," IEEE J. Quantum Electron. QE - 32, 2010-2016 (1996). [CrossRef]
- A. E. Siegman ,"Lasers", (University Science Book, Mill Valley, CA, 1986).
- A. E. Siegman and Steven W. Townsend, "Output beam propagation and beam quality from a multimode stable-cavity laser," IEEE J. Quantum Electron. 29, 1212-1217 (1996). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

OSA is a member of CrossRef.