Characterising output beams of unstable laser ...

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CHARACTERISING OUTPUT BEAMS OF UNSTABLE LASER RESONATORS EMPLOYING MODAL ANALYSIS (Invited Paper) Saiedeh Saghafi Laser Research Institute, Shahid-Beheshti Universiw Phone: +98 21 241 3350 Fax:+98 21 2418698 Email: s-saphafiiii~bu.cc.ac.ir Abstract The modal analysis, based on the physical optics theory, is a strong tool to characterise a wide range of laser beams. From the simplest (fhdamental) structure to the more complicated forms such as uniform cross sectional intensity profile (commonly called flattop or top-hat beams), with or without a central shadow (usually associated with output from edge-coupled unstable resonators). It has been shown that in some cases “pure” modes, and in others the superposition of such modes, can be used to analyse laser beams of arbitrary cross sectional profile. In this paper, the so-called FGBHM (Flattened Gaussian Beams with a Hole in the Middle) model that is the superposition of standard modes is employed to characterise the output beams produced by unstable laser resonators. Using this model, the effects of the size of central hole on the beam quality will be discussed.

INTRODUCTION ,

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Operation of the hrst laser in 1960 enhanced the significance of phyiical optics and created extensive opportunities for developing new concepts. A theoretical approach to laser mode structures (i.e. laser resonator modes) was published in 1961 by Fox and Li. [I]. They demonstrated that an electromagnetic field after several reflections between two mirrors adopts a stable form (that is a stable cross-sectional intensity pattem), which we know as a transversal mode. The lowest order mode (TEMoo mode) for a stable resonator has a bell shaped (Gaussian) intensity profile with lowest diffraction loss. The propagation of this beam is easily depicted with the scalar wave equation [2-4]. However, it must be noted that most lasers do not operate in fundamental modes and their beam profiles are more complex, thus we need another approach to characterise these beams. This may be achieved by employing higher-order solutions of the paraxial wave equation. The eigenmodes for a stable confocal resonator with square mirrors can be expressed in terms of Hermite-Gaussian functions in the practical limit of small losses or large mirrors, assuming paraxial [2]. The analysis was extended to symmetric concave spherical reflectors, and later to mirrors with circular cross section, producing modes described by Laguerre-Gaussian functions [3]. Mathematically, the Hermite-Gaussian modes in a Cartesian coordinates, can be directly transformed to Laguerre-Gaussian modes in cylindrical coordinates. These modes, Hermite- and Laguerre- Gaussian modes, are the so-called standard solutions of the paraxial wave equation in Cartesian and cylindrical coordinates. As multi-elements laser resonators producing output beams with complicated nonGaussian modal structures have been developed, new concepts for characterising these

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CAOL 2003, 16-20 September 2003. Alushfa. Crimea. Ukraine

beams are required. The altemative solutions of the paraxial wave equation, the so-called Elegant solutions, expressed as Hermite or Laguerre polynomials involving a Gaussian function for which the argument can be complex in both parts, have been reported by Zucker [ 5 ] , Siegman [6], Saghafi et. al. [7-8]. Beam propagation through systems such as parabolic gain-guided lensguides can not be described using normal modes [9]. In laser resonators with transversely varying mirror amplitude reflectivity [VRM], the same problems are detected [ 5 ] . Furthermore, an aperture with a parabolic transmission profile or mirror with a parabolic reflectivity profile has the same properties as a thin lens with complex focal length and spherical mirrors with a complex radius of curvature, respectively. All these problems were addressed by developing a complex Gaussian approach with a complex argument. In 1965, Siegman reported the first theoretical studies of unstable resonalors, which demonstrated very different optical properties to those of stable resonators [IO]. It was shown that the optical field in unstable resonators can not be described by the usual Hermitian operator and: therefore, does not assume the normal modes or the Standard solutions. Later experimental research developed the concept of positive and negative branch unstable resonators. It has been shown that the quality of the output beam can be significantly improved by employing high magnification unstable resonators. High gain lasers (e.g. Copper Vapour and XeCI excimer lasers), using confocal positive or negative branch unstable resonators, can produce output beams with uniform cross sectional intensity profile (flat-top beams) with or without a central shadow. Based on the truncated version of standard Laguerre-gaussian modes, Gori introduced a new type of axially symmetric beams called the Flattened Gaussian Beams (FGB) in 1994 [ I I]. Based on the FGB model, a new analytical expression called Flattened Gaussian Beam with a Hole in the Middle (FGBHM) was introduced by Saghafi et. a1.[12]. This model can characterise the free propagation of the output beams generated by unstable laser resonators. A complementary approach in characterising an arbitrary laser beam is a mathematical method called the moment theorem. This theorem in physical optics is called the intensity moment formalism method (IMFM), and allows two standard parameters the so-called the beam propagation parameter (M') and kurtosis factor (k) to be defined [5-61. These standard parameters provide simple indices of beam quality (i.e. the angle oT divergence, the beam diameter, and beam shape at different transversal planes). The intensity profiles for a variety of modal structures corresponding to real laser beams are then used to develop analytical or numerical formulae for these standard parameters and their correspondence. From the mathematical point of view this is easy for simple mode structures such as the standard fundamental mode. Indeed, M2 and k are defined as 1 and 3 respectively for pure Gaussian profile [13-141. Defining the correspondence between these two parameters for pure higher order modes is not difficult. However, for cases involving more complicated mathematical forms such as superposition of modes, this is more challenging. A mathematical approximation theory called the Pad6 method can be employed to develop the correspondence between Mz and k or these two parameters and the mode number, which shows the maximum number of modes involved in modal structure [ 121 for flat-top beams. In this paper, the FGBHM model and standard parameters (A? and k ) are employed to characterise the output beams produced by unstable laser resonators [12]. A particular parameter in FGBHM model that is a unit-less value controls the radius of shadow. By employing IMFM, it will be demonstrated that the size ofshadow can directly affect the beam quality.

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THEORY An optical field can be characterized using the wave optics bhysical optics) theory. Recalling the Huygens’ principle, an arbitrary incoherent source can be described as the incoherent superposition of independent spherical waves. The solutions of the familiar wave equation provide adequate information about the beam phase and amplitude at any point. The scalar wave equation can be written as V 2 y ( r , t ) =&pa’y(r,t), a*,

(1)

where E and p are the permittivity and permeability of the medium. In physical optics, the exact solution of wave equations (e.g. Helmholtz);

where k = o / c = 2 d d , is generally impractical, thus approximations are used. the paraxial approximation, the wave equation changes to a simplified form; (02

+ k 2 )y(x,y,z)= 0

Various laser modes can be approximated by the solutions of the paraxial wave equation.

GAUSSIAN MODAL STRUCTURES (TEM m) One of the objectives of many industrial applications employing lasers is the accommodation of maximum possible energy in a given region of space in a bell-shaped distribution. This bell-shaped irradiance distribution of lasers, is called fundamental mode (TEMoo mode) and can be described mathematically by Gaussian function. It has been discussed in details that as a Gaussian beam propagates along the z-axis, the transverse profile of the beam remains Gaussian. For more complicated cases, the higher-order solutions of the paraxial wave equation are used. The Hermite- and Laguerre-Gaussian modes, are the so-called standard solutions of the paraxial wave equation in Cartesian and cylindrical coordinates. In the standard solution, where the paraxial approximation is applied, the Gaussian part has a complex argument, while the Hermite or Laguerre part has a purely real argument [ 15-161. At the waist, the SHG modes are given by;

and is plotted in Fig.]

, ' CAOL 2003,16-20 Sepfember 2003.Alushta. Crimea. Ukraine

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Figure 1

At the waist, the standard Laguene-Gaussianmodes are given as;

Other particular solution with a complex argument, the so-called Elegant higherorder modes [15] have the same complex scaling factor in the argument of both the Gaussian and Hermite or Laguerre functions. These solutions are not orthogonal in the usual sense, they are biorlhogonal. Thus an adjoined junction, using a particular operator, can be found to form a biorthogonal set that satisfies the orthogonality formula. At the waist, this altemative solutions of the paraxial wave equation, the so-called Elegant solutions in Cartesian and cylindrical coordinates, are given respectively as;

FLATTENED GAUSSIAN BEAMS (FGB) MODEL Many high-power visible lasers used in micromachining have both flat top intensity profiles (uniform energy density over a given cross section) and high beam quality (low divergence output). There is a trade-off between quality and power of output beams of these lasers. Therefore, developing sources with both these output characteristics can be technically challenging. It is not possible to improve the performance of these lasers if the beam properties are not known. For example, being able to predict the beam's profile at any arbitrary position can help researchers to improve the laser operation by comparing the theoretical results with the experimental results and solve any possible problems preventing lasers from optimum operation. Therefore, developing analytical models for lasers with complicated modal structures is important. A model was introduced in 1994 that effectively predicts the beam intensity profile, phase, spot size, and shape of uniform irradiance distributions at any point along the propagation axis [Ill. This model that was called flattened Gaussian beams (FGBs) describes a uniform beam as a finite sum of the standard Laguene-Gaussian modes

CAOL 2003.16-20 Sepiember 2003.Alurhta. Crlmea. Ukraine

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where N is the number of beam orders and

c,"= (-1)" .f m=n

(") + n

(7)

The irradiance distributions produced by the FGB model are plotted in Fig.2. v,

Figure 2

FLATTENED GAUSSIAN BEAMS WITH A HOLE IN THE MIDDLE, (FGBW) MODEL High average power, high beam quality laser sources use unstable resonator configurations that result in relatively complicated output intensity profiles. For example, copper vapor lasers (CVLs) employing positive-branch on-axis unstable resonators can produce a flat-top distribution with an axial shadow as s h o w in Fig.3.

Figure 3

The schematic diagrams of master oscillator (MO) are shown in Fig. 4 [12]. h

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Figure 4

Using the FGB expression, a new model has been developed characterising flat-top beams without or with a central shadow (usually associated with output from edge-coupled unstable resonators in high power lasers). The superposition of two FGB beams, one can describe by Eq.6 and the constructed beam using the same model given by:

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CAOL 2003,16-20 September 2003,Alushta, Crimea, Ukraine

2 where wMo = c'w,', can produce a flat-top beam with a central shadow:

where M, the beam order of the constructed FGB beam, controls the steepness of the shoulders of the hole, and 7 is a variable constant which enables various spatial intensity distributions to be achieved. The detail of this model is published in reference [4].The analytical expressions for field distribution of the flat-top beams with central shadow in near- and far-field represent the extension of the implications of the model to higher order solutions of the paraxial wave equation (Fig.5).

Figure 5

M' PARAMETER AND k FACTOR Using the second moment method, a standard parameter representing the 2 characterisation of beam propagation, called the beam propagation factor ( M ), has been proposed by a committee of the lntemational Organisation for Standardisation (ISO) [13] as:

CAOL 2003,16-20 September 2003,AIUShia. Crimea, Ukraine

The beam propagation factor can be easily measured (using special beam analyzer) and can be used to determine the angle of divergence. Additionally, a comparison between M’ (experimental value of the beam propagation factor) and Mk,, (the theoretical value of the beam propagation factor) can be performed to show the reliability of the system and improve the system. On the other hand, knowledge of the sharpness or flatness of the output laser beam is essential in order to explain the tapered wall formation and stabilized hole formation or to predict the hole shape. So, the fourth moment method provides another tool, kurtosis (k), which is employed to reveal useful information about the beam shape [14]. CXQ

Using these two standard parameters and the Pad& method [12], N in FGBHM model can be d e h e d a s ; N = 10(M2- 1) + 3(M2- 1)’ + 15(M2- 1)’ (12)

It was shown that at z=O, the kurtosis of the FGB model is related to the FGBHM

model through [ 121: ’ k , = ,k, + 2.5 (13) Therefore, by measuring the kFcnHM and knowing E , Me are able to obtain kFcn and codsequently N (Eq.12). A similar method can be used for M. Now we are able to predict the intensity distribution along the z-axis produced by lasers employing unstable resonators

EXPERIMENTAL EVALUATION OF THE F G B W A conventional 20W copper vapour laser (CVL) with active volume of 26 mm x l m was used. The Pulse repetition frequency was 1OkHz. Pulse averaged near-field intensity profiles of the green component of laser output near-field intensity profiles were recorded at different positions along the propagation axis, at z=0,0.85,2.45, 5.05 and 15.3 m, imaged onto a CCD using a 250mmfl achromat. Then, the pulse averaged far-field

intensity profiles of the green laser output were examined by imaging a low-power sample of the output and then bringin this sample to focus using a 450 mmfl achromat. Based on the data obtained, k and M are determined to be 1.71 and 1.51 respectively. From equations 11 and 12, the beam order is calculated to be N=9.

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CAOL 2003,16-20 September 2003. Alushta. Crimea, Ukraine

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Figure 6

These values enable complete characterisation of laser beam propagation at any point along the z-axis.

THEEFFECTS OF THE SIZE OF THE SPOT REFLECTOR ON BEAM QUALITY In this part, the effects of the size of shadow caused by the spot reflector using h ? will be investigated. It will be demonstrated that the size of shadow can directly affect the beam quality and enables us to improve the quality of output beams generated by unstable laser resonators. As .discussed earlier, N and M are beam orders and considered to be integers that control the steepness of the shoulders of the irradiance profile and the hole in the beam. Note that decreasing^ (i.e. decreasing the size of the reflecting mirror in the cavity) causes more-diffraction and the far-field intensity distributions broadens resulting a larger beam width and M'. The effect of the size of the shadow on 111" for fixed values of N, M and 7 are shown in Fig.7.

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Figure 7

These results show that high beam quality can be achieved if the diameter of the , equates to an optimal spot reflector radius that is 15 shadow is fixed at 0.15 s e _ ~ 0 . 3 5This to 35% of the total beam radius. Similar method can be employed to investigate the effects of magnification in the FGBHM model.

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CONCLUSION Due to its projection properties [Eq.8] and the fact that it is described based on the standard Laguerre-Gaussian modes, FGBHM model can give a qualitative and quantitative description of a real laser beams of lasers employing unstable resonator. To use this model, the beam order N is required. Using two standard parameters ,the beam propagation factor and kurtosis parameter, and Eqs.11 and 12 enable researchers to simpily determine N . Finally. An excellent agreement between theoretical predictions and practical results are given.

REFERENCES 1. A. G Fox. and T. Li, “Resonant modes in a maser interferometer,” Bell-Journal of System and Technology, vol. 40, p. 453, 1961. 2. G. D. Boyd and J. P Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell-Journal of System and Technology, vol. 40, p. 489, 1961. 3. G. D. Boyd and H. Kogelnik, “Generalised confocal resonator theory,” Bell Journal of System and Technology. vol. 41, p. 1347, 1962. 4. A. E. Siegman, Lasers. University Science, Mill Valley, Calif., 1986. 5. If. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell-Journal of System and Technology, vol. 49, p. 2349, 1970. 6. A. E. Siegman, “Hermite-gaussian functions of complex argument as optical-beam eigenfunctions,” Journal of Optical Society of America A, vol. 63, p. 1093, 1973. 7. S. Saghafi, and C. J. R. Sheppard , ” Near field and far field of elegant Hermite Gaussin and Laguerre-Gaussian modes,” Journal of Modern Optics, vo1.45, p. 1999, 1998. 8. S. Saghafi, and C. J. R. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Optics Communications, vol. 153, p. 207, 1998. 9. L. W. Casperson and A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Applied Physics Letters, vol. 12, pp. 355, 1968. 10. A. E. Siegman, “Unstable optical resonator for laser applications,” Tran. of IEEE, vol. 53, p. 267, 1965. 11. F. Gori, “Flattened Gaussian beams,” Optics Communications, vol. 107, p 335, 1994. 12. Saghafi S., Withford M. J., and Piper J. A., “Characterising output beams for lasers using high magnification unstable resonators,” Journal of optical society of America-A: Special edition on Free and Guided Beams, vol. 18, p. 1.2001. 13. ISO/TC172/SC 9/WG 1 N73 “Optics and optical instruments: test methods for laser beam parameters: beam widths, divergence angle and beam propagation factor”. 14. Martinez-Herrero R., Piquero G., and Mejias P. M., Opt. Com., 115, pp. 225-232, 1995. 15. A. E Siegman, “Hermite-gaussian functions of complex argument as optical-beam eigenfunctions,” Journal of Optical Society ofAmerica A, 63, pp. 1093-4, 1973. 16. A. E Siegman., Lasers (University Science, Mill Valley, Calif., 1986)

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