KR0100829 KAERI/TR-1661/2000 · 2005. 1. 17. · (TEMox) modes 69 Figure 20k. Diffractional losses...

KR0100829

KAERI/TR-1661/2000

Optical Resonator Theory

Korea Atomic Energy Research Institute


2000\! 10-i

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Jaegwon YooSung Oh Cho

Yung Uk JeongByung Chul Lee

Jongmin Lee

Laboratory for Quantum Optics

October 25, 2000

Korea Atomic Energy Research Institute

Abstract

In this report we present a theoretical study of bare optical resonators having in

mind to extend it to active resonators. To compute diffractional losses, phase

shifts, intensity distributions and phases of radiation fields on mirrors, we coded

a package of numerical procedures on bases of a pair of integral equations. Two

numerical schemes, a matrix formalism and an iterative method, are programmed

for finding numeric solutions to the pair of integral equations. The iterative

method had been tried by Fox and Li, but it was not applicable to cases for

high Fresnel numbers since the numerical errors involved propagate and

accumulate uncontrollably. In this report, we implemented the matrix method to

extend the computational limit further. A great deal of case studies are carried

out with various configurations of stable and unstable resonators. Our results

presented in this report show not only a good agreement with the results

previously obtained by Fox and Li, but also a legitimacy of our numerical

procedures in high Fresnel numbers.

Table of Contents

I. INTRODUCTION 1

II. ELECTROMAGNETIC WAVES IN OPTICAL SYSTEMS 411.1 Ray Tracing in Optical Systems 411.2 Wave Optics 711.3 Longitudinal Modes in Resonant Cavities 811.4 Transverse Modes in Resonant Cavities 1011.5 Diffraction of Light in Resonant Cavities 12

III. STABLE RESONATORS 14111.1 Formulation of Numerical Computation 14111.2 Description of Code System 17111.3 Rectangular Plane Mirror System 19111.4 Cylindrical Mirror System 20111.5 Circular Spherical Mirror System 21

IV. UNSTABLE RESONATORS 23IV.l Asymmetric Mirror Unstable Resonators 23IV.2 Nonuniform Reflectivity Output Coupling of Unstable Resonators 25

V. CONCLUDING REMARKS 29

REFERENCES 30

List of Figures

Figure 1. Optical cavity and lens-waveguide equivalent tothe mirror system. 31

Figure 2. Stability diagram of optical cavities. 31Figure 3a. Ray-tracing results with a typical stable mirror system. 32Figure 3b. Ray-tracing results with a typical unstable mirror system. 32Figure 4. Farby-Perot type optical cavity. 33Figure 5a. Transmittances of etalon for several different reflectances. 34Figure 5b. Transmittances of etalon for several different reflectances. 34Figure 5c. Transmittances of etalon for several different angles. 35Figure 5d. Transmittances of etalon for several different wavelength

variations 35Figure 6. TEM modes in rectangular system. 36Figure 7. TEM modes in cylindrical system. 37Figure 8a. The Gaussian beam radius of curvature for 5 starting beam

waists of Ar laser. 38Figure 8b. The Gaussian beam waists for 5 starting beam waists

of Ar laser. 38Figure 8c. The Gaussian beam radius of curvature for 3 wavelengths. 39Figure 8d. The Gaussian beam waists for 3 wavelengths 39Figure 9. Source radiation fields on observation surface. • 40Figure 10. Tests of kernel function. 41Figure 11. Convergence tests. 42Figure 12. Sample computation shows mode formation as pass number

increases. 43Figure 13. Schematic of OC_MODE code system. 44Figure 14. Flow chart of EIGENV package. 45Figure 15. Flow chart of EIGENF package. 46Figure 16. Mirror configurations of various resonator systems. 47Figure 17a. Diffractional losses per transit for infinite strip plane mirror. 49Figure 17b. Phase shifts per transit for infinite plane mirror. 49Figure 17c. Normalized intensity distributions on mirror for m=0 mode. 50Figure 17d. Phases of radiation on mirror for m=0 mode. 50

Figure 17e. Normalized intensity distributions on mirror for m=l mode 51Figure 17f. Phases of radiation on mirror for m=l mode. 51Figure 18a. Diffractional losses per transit for symmetric cylindrical mirror

(m=0 mode). 52Figure 18b. Phase shifts per transit for symmetric cylindrical mirror

(m=0 mode). 52Figure 18c. Diffractional losses per transit for symmetric cylindrical mirror

(m=l mode) 53Figure 18d. Phase shifts per transit for symmetric cylindrical mirror

(m=l mode) 53Figure 18e. Diffractional losses per transit for asymmetric cylindrical mirror

(m=0 mode) 54Figure 18f. Phase shifts per transit for asymmetric cylindrical mirror

(m=0 mode) 54Figure 18g. Diffractional losses per transit for asymmetric cylindrical mirror

(m=l mode) 55Figure 18h. Phase shifts per transit for asymmetric cylindrical mirror

(m=l mode) 55Figure 18i. Diffractional losses per transit for asymmetric cylindrical mirror

(m=0 mode) 56Figure 18j. Phase shifts per transit for asymmetric cylindrical mirror

(m=0 mode) 56Figure 18k. Diffractional losses per transit for asymmetric cylindrical mirror

(m=l mode) 57Figure 181. Phase shifts per transit for asymmetric cylindrical mirror

(m=l mode) 57Figure 18m. Diffractional losses per transit vs g parameters for symmetric

cylindrical mirrors 58Figure 18n. Phase shifts per transit vs g parameters for symmetric

cylindrical mirrors 58Figure 18o. Normalized intensity distributions on symmetric cylindrical

mirror for m=0 modes 59Figure 18p. Phases of radiation on symmetric cylindrical mirror

for m=0 modes 59Figure 18q. Normalized intensity distributions on symmetric cylindrical

mirror for m=l modes 60

Figure 18r. Phases of radiation on symmetric cylindrical mirrorfor m=0 modes 60

Figure 19a. Diffractional losses per transit for circular plane mirrorTEMoo mode 61

Figure 19b. Phase shifts per transit for circular plane mirror TEMoo mode 61Figure 19c. Diffractional losses per transit for circular plane mirror

TEMio mode 62Figure 19d. Phase shifts per transit for circular plane mirror TEMio mode 62Figure 19e. Normalized intensity distributions on circular plane mirror for

TEMoo mode 63Figure 19f. Phases of radiation on circular plane mirror for TEMoo mode 63Figure 19g. Normalized intensity distributions on circular plane mirror for

TEMio mode 64Figure 19h. Phases of radiation on circular plane mirror for TEMio mode 64Figure 20a. Diffractional losses per transit for circular curved mirror

TEMoo mode 65Figure 20b. Phase shifts per transit for circular curved mirror TEMoo mode 65Figure 20c. Diffractional losses per transit for circular curved mirror

TEMoi mode 66Figure 20d. Phase shifts per transit for circular curved mirror TEMoi mode 66Figure 20e. Diffractional losses per transit for circular curved mirror

TEMio mode 67Figure 20f. Phase shifts per transit for circular curved mirror TEMio mode 67Figure 20g. Diffractional losses per transit for circular curved mirror

TEMn mode 68Figure 20h. Phase shifts per transit for circular curved mirror TEMn mode 68Figure 20i. Diffractional losses per transit for symmetric circular curved

mirror (TEMox) modes 69Figure 20j. Phase shifts per transit for symmetric circular curved mirror

(TEMox) modes 69Figure 20k. Diffractional losses per transit for symmetric circular curved

mirror (TEMix) modes 70Figure 201. Phase shifts per transit for symmetric circular curved mirror

(TEMix) modes 70Figure 20m. Normalized intensity distributions on circular curved mirror for

TEMoo mode 71Figure 20n. Phases of radiation on circular curved mirror for TEMoo mode 71

Figure 20o. Normalized intensity distributions on circular curved mirror for

TEMio mode 72

Figure 20p. Phases of radiation on circular curved mirror for TEMio mode 72

Figure 21a. Diffractional losses per transit for negative branch

unstable resonator (TEMoo) modes 73

Figure 21b. Diffractional losses per transit for negative branch

unstable resonator (TEMio) modes 73

Figure 21c. Diffractional losses per transit for negative branch


Figure 21d. Diffractional losses per transit for negative branch


Figure 21e. Normalized intensity distributions on mirror #1 of negative

branch unstable resonator for TEMoo mode 75

Figure 21f. Normalized intensity distributions on mirror #2 of negative


Figure 21 g. Normalized intensity distributions on mirror #1 of negative


Figure 21h. Normalized intensity distributions on mirror #2 of negative


Figure 21i. Normalized intensity distributions on mirror #1 of negative

branch unstable resonator for TEMio mode 77

Figure 21 j . Normalized intensity distributions on mirror #2 of negative


Figure 21k. Normalized intensity distributions on mirror #1 of negative


Figure 221. Normalized intensity distributions on mirror #2 of negative


Figure 22a. Diffractional losses per transit for positive branch


Figure 22b. Diffractional losses per transit for positive branch


Figure 22c. Diffractional losses per transit for positive branch


Figure 22d. Diffractional losses per transit for positive branch


Figure 22e. Normalized intensity distributions on mirror #1 of positive


Figure 22f. Normalized intensity distributions on mirror #2 of positive


Figure 22g. Normalized intensity distributions on mirror #1 of positive


Figure 22h. Normalized intensity distributions on mirror #2 of positive


Figure 22i. Normalized intensity distributions on mirror #1 of positive


Figure 22j. Normalized intensity distributions on mirror #2 of positive


Figure 22k. Normalized intensity distributions on mirror #1 of positive


Figure 221. Normalized intensity distributions on mirror #2 of positive


Figure 23a. Mode U(r) and output I(r) computed from reflectance R(r). 85

Figure 23b. Output I(r) and reflectance R(r) computed from

Super-Gaussian U(r). 85

Figure 23c. Mode U(r) and reflectance R(r) for yielding output I(r). 86

I. INTRODUCTION

A laser is a quantum mechanical device based on stimulated emissionsfrom population inversion between quantum states of bound electrons in atomicor molecular systems. Unlike other sources of light, the laser systems can emitvery bright and directional light beam because of their ability of producinghighly monochromatic and coherent radiations with very low angular divergence.With these peculiar properties the lasers have played key roles in industrial andmedical applications, holography and fiber optic communications, as well asimportant research tools in basic sciences. Furthermore, since the intensity oflaser beam is high enough that the optical electric field is much stronger thanthat in atoms or dielectrics, the responses of optical media to the intensive laserbeam exhibit nonlinear behaviors, for examples; intensity dependence of theindex of refraction, optical second harmonic generations, optical parametricoscillations, stimulated Raman scattering, optical phase conjugation, two-photonabsorption, self-focusing, and bistability. Mode-locked lasers can generate veryshort laser pulses in precisely regular pulse trains or as a single high intensityburst. These pulses of light can be used to generate very short bursts ofx-rays, phonons, electrons and electrical pulses, and to explore subpicosecondand femtosecond events such as molecular dynamics, semiconductor physics,chemical reactions.

In general a laser consists of three components for its operation; an activemedium with energy levels that can be selectively populated; a pumping systemto produce population inversion between some of these energy levels; and aresonant optical cavity containing the active medium, which stores the emittedradiation and provides feedback to maintain the coherence of the radiation field.Quantum mechanical picture of matter tells us that the gain medium iscomposed of atoms and that electrons bounded in the atoms can have onlycertain allowed energy levels. The electron in an excited state emits a photonby making a spontaneous decay to a lower energy state. The photon energy,hv, is equal to the energy difference, AE, between the two states. In thermal

equilibrium, however, most atoms are in their lowest energy state, since theenergy distribution of the population is proportional to e~E/kT, where E, T, andk are the energy level, the temperature, and the Boltzmann constant,respectively. A pumping system is, thus, required to promote the atoms intotheir higher energy states. The gain materials can be pumped in many ways,

_ 1 _

such as gas discharges, optical pumping, chemical reactions, direct electricalpumping. The externally applied field can cause an electronic transition betweenthe energy states by emitting or absorbing a photon. The stimulating radiationfields of frequency v can coherently induce the absorption or emission ofphotons. If the population of the higher energy state exceeds that of the lowerenergy state, then population inversion takes place, leading to break thermalequilibrium to a sufficient degree. The population inversion enables the activemedium to amplify the radiation fields coherently. Hundreds of different types oflasers have been developed since the first laser was successfully operated in1960. Types of laser beam generators can be classified in terms of the gainmaterial; solid state lasers, gas lasers, liquid lasers, semiconductor lasers, andother types such as x-ray lasers, particle beam pumped lasers, and free electronlasers.

Operation of a laser requires a resonant cavity to increase stimulatedemissions induced by the light passing through the gain medium many times.Since the closed conventional closed resonators are not suitable for operating thelaser generators in the range of optical frequencies, most resonators have opencavity structure that is composed of mirrors at the ends of the gain medium.Since the length of a laser resonator is, in most cases, much larger than thelateral dimension of the mirrors, the radiation fields in a resonant cavity havethe transverse and longitudinal modes. Types of resonant cavities can becategorized into stable, marginally stable, and unstable resonators depending ontheir mirror configurations. The stable resonators have the lowest diffractionlosses, while marginally stable resonators have higher losses, and the unstableresonators have the highest losses. The optical resonant cavity determines thefrequency and spatial distribution of the laser beam. The frequencies of thelongitudinal modes are separated by the inverse of the round trip time for thelight beam in the resonator. Since the gain of a laser is peaked at a transitionfrequency determined by the energy levels of the gain medium, operation of thelaser tends to occur at the longitudinal mode frequency closest to the gain peak.In the open resonator the transverse mode structure is determined by axialmirrors. The resonator maintains characteristic configuration of the radiationfield with low loss which is compensated for by the gain from stimulatedemission as the light passes through the gain medium. Thus it is very gooddesign strategy to make the mirrors fill all the volume of the active mediumcontained between the mirrors. The optical resonators must be designed to meetrequirements of low angular divergence and high efficiency by taking intoconsideration of many aspects in laser physics such as the geometry of the gainmedium, the desired cavity length, the diffractive properties of the radiation, and

_ 9 _

the single pass gain.Operational efficiency of a laser generator depends on the quality of mirror

systems demanding the highest possible reflectance for one mirror and aloss-free output mirror. For high power lasers, the resonator can be severelydamaged if the mirror has some absorption coefficiency. For metal coatedmirrors, the reflectance depends on the index of refraction, on the absorptioncoefficient and on the angle of incidence. Very high reflectance can be obtainedwith silver and gold films for the infrared region, while the loss in onereflection in visible range can cause the absorption damage of the mirror,especially this is the case for a pulsed high power laser. The mirrors used inthe resonator are coated with multiple layers of dielectric materials that arehighly reflective and resistive to damage compared with metal coated mirrors.Thus it is necessary to find adequate film materials and film depositiontechniques for fabricating mirror structures with extremely low losses suitablefor specific applications of laser beams.

Our research target is in understanding the behavior of the laser beamunder influences of the active material in the optical resonators, which are veryimportant in designing high power laser systems, such as metal vapor lasersand free electron lasers. In this report, however, we carried out a preliminarystudy of bare resonators. We begin section II with the simplest approach basedon ray optics; optical rays are traced as they reflect in the resonator and thegeometrical conditions for stabilities are determined. The modes of theresonators are determined by making use of wave optics which enables us toexplain the interference and diffraction in resonator. Beam optics is very usefulfor understanding the optical characteristics of spherical mirror resonators.Propagation and diffraction of electromagnetic waves based on Fourier optics arenecessary for investigating the effect of the finite size of the resonator mirroron the loss and on the spatial distribution of the modes. In section III,discussing numerical procedures for calculating diffraction integrals, we brieflydescribe the structure of code system developed for numerical computation. Wepresent typical case studies with stable resonators for understanding the effectof the finite size of the resonator's mirrors on its loss and on the spatialdistribution of the modes. In section IV, we investigate unstable resonatorshaving in mind for applications in high power laser systems. Finally we makeour concluding remarks on the optical resonant cavities in section V.

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II. ELECTROMAGNETIC WAVESIN OPTICAL SYSTEMS

Light is an electromagnetic wave phenomenon described by Maxwellequations that govern all theories in electromagnetic radiation. Some opticalphenomena can be nicely explained with a ray tracing theory based ongeometrical optics, such that lenses focus light, mirrors direct and reflect it, andit is subjected to bending and refraction at the intersurface between two opticalmedia having different indices of refraction. Since the electromagnetic radiationobeys Maxwell equations, however, there are some peculiar cases whichexclusively belong to wave optics realm, such as the diffraction and interferenceof light. Even though the electromagnetic wave is of a form of two mutuallycoupled vector waves, it is possible to model optical phenomena using a scalarwave theory. The geometrical optics is an approximate model of light on thebasis of that a small beam of electromagnetic energy having very highfrequencies does not spread to any degree. This enables one to follow rays oflight as they traverse on optical path. It, however, does not describe theintensity of the light because the ray can not carry any information about theradiation fields. Assuming a beam of light having a finite transverse dimensionand oscillating in longitudinal direction, we can find an approximate expressionof the wave equation, i.e., paraxial Helmholtz equation from the Maxwellequations. The Gaussian beam is a solution of this equation that exhibits thecharacteristics of the light beam. Typical configuration of the laser beam outputis a transverse electromagnetic mode (TEM). Since the higher transverse modesof the beam are spatially broader than the lower ones, the most focusable beamspot has the lowest mode (TEMoo), which is a round mode with a Gaussianprofile in cross-section.

II.l Ray Tracing in Optical SystemsLaser optics essentially requires an analysis of laser beam propagating

through a system of optical components such as lenses and mirrors. Geometricaloptics is the simplest theory of light, where the light is described by raystraveling at small inclination around an optical axis. Traveling of the rays isassumed to be confined within a single plane, so that the formalism is applicableto systems with plane geometry and to meridional rays in circularly symmetric

- 4 -

systems. Ray transfer matrices of optical systems can describe the propagationof paraxial rays, the stability of the optical structure and the loss of unstableresonators. Ray optics also approximates the electromagnetic wave by groups ofrays. In isotropic media, optical rays are associated with the direction of theflow of optical energy, so that the density of rays are proportional to the energydensity of the light. So tracing rays through an optical system makes it possibleto determine the optical energy crossing a given surface element.

A ray tracing method uses a 2x2 matrix to trace the light through anoptical system, where the location and slope of the rays are represented as thefrist and second component of a vector, respectively. For a ray traveling fromtransverse plane 1 to transverse plane 2, a ray matrix equation can be writtenas

A BCD (1)

where the physical meaning of the matrix elements, A, B, C, and D can berelated to the lateral magnification, the effective thickness, the reciprocal length,and the angular magnification, respectively. For examples; the ray matrix for adistance d between two planes 1 and 2 in free space is given by

l d0 l

(2)

the ray matrix for a thin lens of focal length / is given by

1 0 (3)

the ray matrix for a spherical mirror with radius of curvature R is given by

1 0

•*- l(4)

The optical resonator can be modeled as a periodic deployment of thinlenses with a distance d spacing the lenses as illustrated in Fig. 1. The raymatrix for a single cell of this periodic structure can be read as

- 5

1 dd

T1 , d

A l~Tx

d

h h

l -

\ 1 , dA h Ah

= (AB\ln

T 2d d , £1 A h+ Ah J

(5)

Notice that the determinant of the ray matrix is unity. The ray matrix for pperiodic sequences of the single cell can be read by applying Eq. (5)successively

{H)P[: (6)

If the ray is contained in the periodic structure, the matrix elements, A and D,satisfy a condition of

A + D2 (7)

Introduction of a parameter, Q= cos i(A + D)/2, reduces the pth power of aunimodular matrix to a simple form of

A B\p = _l—( Asinpe-sm(p-l)8CD) sinfl\ Csinpd

Bsinpd \Dsinpd-sin(p-l)d)

(8)

If the ray has escaped the periodic structure, then the optical system does notmeet the condition, Eq. (7).

For a laser resonator with spherical mirrors of unequal curvature, thestability condition can be read from Eq. (7) after writing it in more specificform with lens parameters

0< 1 - d 2/2< 1 (9)

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Since a resonant cavity can be modelled to be a periodic lens system, thestability condition for a resonator can be read by replacing the focal lengths oflenses, f\ and k, with curvatures of spherical mirrors, R-J2 and Ro/2, in Eq. (9);

a o )

In general, the stability condition for a resonator composed of variouscombination of mirrors can be determined graphically on the basis of Eq. (10).The stable and unstable regions are illustrated in Fig. 2. Typical examples ofstable and unstable mirror configurations are plotted in Figs 3a and 3b.

II.2 Wave OpticsLight is an electromagnetic wave governed by Maxwell equations. In

section II.l, we exploited the rays of ray optics orthogonal to the wavefronts ofwave optics. Ray optics is the limit of wave optics when the wavelength isinfinitesimally small, i.e., A-+0, or much shorter than the dimension of theobjects encountered in optical systems. Wave optics enables one to describeoptical phenomena that are outside the realm of ray optics such as interferenceand diffraction of light. In this section, we describe light on the basis of thescalar wave equation which is a simplified version of the Maxwell equations.Since light wave propagates with speed of light, c, in free space, we read thewave equation as follows

(11)

where v 2 is the Laplacian operator, and the u is a real scalar wavefunction.Since a monochromatic wave is represented by a wavefunction with

harmonic time dependence,

u( r, t) = a( r) cos[2xvt+

so that the Eq. (13) must satisfy the wave equation, Eq. (11) and the sameboundary conditions. Substitution of Eq. (13) into Eq. (11) yields a timeindependent differential equation called the Helmholtz equation

= 0 (14)

where the wavenumber k denotes 2KVIC. The absolute square of the complex

amplitude is the optical intensity averaged over a time longer than an optical

period. Since the wavefronts are defined as the surfaces of equal phase, the

gradient of the phase at position r, v^( r ) , is normal to the wavefront at that

point. The simplest solutions of the Helmholtz equation, Eq. (14), in a

homogeneous medium are Aexp(— ik- r), the plane wave, and Ar-1exp(— ikr),

the spherical wave.

II.3 Longitudinal Modes in Resonant CavitiesAmong infinitely many possible wavelengths inside a resonant cavity, a

laser can lase only those wavelengths for which an integral multiple of halfwavelengths precisely match with the cavity length, i.e., a set of longitudinalmodes. Furthermore, the operating frequency of the laser is determined by theintersection between the gain profile of the laser gain material and the set oflongitudinal modes. The frequencies of the longitudinal modes are a function ofthe optical cavity length,

(15)

where m=l, 2, 3, . . ., n and L are the index of refraction and the cavity lengthof the resonator, respectively. A small deviation in the length of the cavity cancause large variations in the frequency location of the mode, since m will be avery large number in a typical laser generator. The longitudinal mode spacing isespecially of interest in designing and operating a laser generator,

An etalon is an optical component consisting of two plane parallel reflectivesurfaces as illustrated in Fig. 4. This particular configuration is a very usefuldevice in laser physics, since it can be used as a very narrow linewidth filter to

isolate a single longitudinal mode. The stable single longitudinal mode operation

of a laser is critical in many applications. Utilizing a piezoelectric crystal to

change its length, the etalon can also be used as a very precise instrument to

measure the spectral character of laser lines. An ideal etalon consists of an ideal

plane parallel plate of glass of thickness L and of index ng immersed in a

background of air at index rh.

Suppose that an optical beam of amplitude Am is incident at the first

air-glass intersurface of the etalon. Then there are total reflected wave of

amplitude Ar in incident side and the transmitted wave of amplitude At in other

side of the etalon. The optical path difference, JMtt, between two adjacent rays

is given by

A p a t k = 2ngLco$.6t • (17)

where 6t is an angle between the refracted ray and the line normal to the

intersurface. Then the phase difference, S, can be calculated by multiplying the

wavenumber, k=2xlA, by the optical path difference,

(18)

The amplitudes of the reflected and transmitted waves are determined by the

phase difference of Eq. (18) and the reflectance, R, and the transmittance, T,

A (leVT? ,r~ (l-ReiS) in (19)

TA'= (l-Re

iS) Ain ( 2 0 )

Since the intensity of the radiation is defined the absolute square of the

amplitude, the reflectance and the transmittance of the etalon are given as

_\Ain\

2 ( l - \ r ^ 2 )2 + 4\T^sin2(5/2) (2D

The transmittances of specific etalon configurations are plotted in Figs. 5a~5d to

illustrate optical characters of longitudinal modes in a resonator.

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Optical character of an etalon is determined by the quality factor Q defined

as the resonant frequency of the cavity, v, divided by the frequency linewidth,

M at FWHM

or the finesse F defined as the free spectral range divided by the frequencylinewidth at FWHM

{24)Av

Even though it is common to consider a laser resonator as an etalon, thefinesse of a laser resonator is typically much smaller than the finesse of atypical etalon, and the free spectral range of the etalon is typically much largerthan that of the resonator.

II.4 Transverse Modes in Resonant CavitiesA transverse electromagnetic mode (TEM) is a typical configuration of the

laser beam output. The transverse mode structure can be calculated from theparaxial approximation to the electromagnetic wave equation that describes awave propagating in the z direction in free space,

(25)

The resonator structure determines the cross sectional shape of the spot and thepropagation mode of the laser beam. Analytic solutions to the Eq. (25) can befound for two special cases; circular transverse modes and rectangulartransverse modes shown in Figs. 6 and 7 respectively. Since the highertransverse modes of the beam are spatially broader than the lower ones, themost focusable beam spot has the lowest mode (TEMoo), which is a round modewith a Gaussian profile in cross-section. However, a laser generator can beoperated on a wide variety of other transverse mode configurations.

In cylindrical coordinates, the azimuthal part can be easily separated outfrom the Eq. (25). The radial part of the Eq. (25) can be read as

- 10 -

dz U

The lowest order solution can be written in a form of

(27)

Substitution of Eq. (27) into Eq. (26) enables one to relate the unknown beamparameters to the beam waist, w(z), and the beam radius of curvature, R(z),

1 1 Mq(z) ~ R(z) nnw{z)

ip(z)=]n(l-i-r-\ (29)

Taking the origin of the z axis at the position where the beam has theminimum waist, w0, the beam waist can be expressed in a hyperbolic equation,

(30)mtur0

The far-field diffraction angle can be defined as an asymptotic angle of thehyperbola

(31)nnw0

In the far-field region, the beam waist propagation is linearly proportional to the

wavelength and the axial distance from the beam waist, i.e., w{z) = Az/7cnw0.

The radius of curvature of the wavefront is given by

Introducing the Rayleigh range defined as a distance of which a transition fromnear-field to far-field takes place,

_ xnw20ZR=—\— (33)

- 11 -

For various beam waists, the radii of curvatures are plotted in Figs. 8a~d. Notethat the radius of the curvature is rapidly decreasing from infinity for distancessmaller than the Rayleigh range, R(z). In the far-field region, the radius ofcurvature is approximately equal to the axial distance from the beam waist, i.e.,

II .5 Diffraction of Light in Resonant CavitiesAs electromagnetic waves propagate around obstacles, diffraction is an

exclusive phenomenon belonging to the realm of the wave optics, inexplicablefrom the viewpoint of geometrical rays of light. Thus Huygens' principle playsa key role in describing diffraction. For simplicity, we confine ourselves to ascalar-wave theory of electromagnetic waves.

The Huy gens' principle says that one can imagine every point on a wavefront to be a point source for a spherical wave. Based on this picture we cancalculate the field on the observation plane, S, a distance r away from theknown field in the source plane, S', as shown in Fig. 9. Setting up coordinatesystems (x,y,z) and {x',y',z') on each plane, we write a contribution of thesource field on a field at observation plane

ikr

j ^ (34)

where r=v(x—x')2 + (y— y')2+z2, and As' = dx'Ay is an infinitesimal surfaceelement surrounding the point (x',y',z') on the source plane. Since the completefield at the point (x,y,z) can be obtained by summing up the contribution fromall the Huygens' spherical waves

B{x,y,z) = - f \X i s T (35)

\x ,y ,z')E(x ,y ,z')ds

where K{x,y,z',x',y',z') represents a propagator of the scalar-wave equation.

When a resonator consists of two mirrors having a mirror spacing, L, and linear

dimensions of a\ and at, the Fresnel number of the resonator, NF= a-[a2lAL,

plays an important role in the integral kernel. Note that the kernel depends on

the shape of mirror and on the approximations employed in the evaluation of the

fields. Depending on the approximations adopted such as zyyk(x'2 + y'2)mBX or

- 12 -

zzy>k{(x— x')2 + (y— y)2]iax, the diffraction scheme is named as Fraunhoferdiffraction or Fresnel diffraction respectively. Since the z dependent part of thekernel K(x,y,2',x',y',z') is in a form of exponential function under one of aboveapproximations, it satisfies the paraxial wave equation of Eq. (25), slightlymodified from the Helmholtz equation of Eq. (14),

(v \+ 2ik-^ )K(x, y,z,x',y, z) = 0 (36)

where v2T denotes a transverse Laplacian operator, and the kernel of the

integral equation is identical to the Green function of the Eq. (36).In the case of the light in a resonator, we can apply Eq. (35) repetitively

to accomodate bouncing waves between two mirrors in a single integralequation such as

E(x',y',0) = \K(x,y;x,y)E(x,y,z)ds

= f (MX ,y;x",y")K(x",y";x,y)E(x,y,O)dsds (37)

= [Mx',y';x,y)E(x,y,O)ds

where the kernel of Fredholm integral equation of the first kind is a product ofindividual kernel such as

'R{x,y;x,y)= \K(X ,y';x",y")K(x",y";x,y)ds" (38)

The eigenfunctions of Eq. (38) represent the eigenmodes of the resonator, andthe eigenvalues are related to the diffractional power loss and the phase shiftper transit. Analytic solutions to the integral equation have been obtained onlyin the case of the confocal resonator. We discuss numerical methods for findingeigenvalues and eigenfunctions of the integral equation in section III.l.

- 13

STABLE RESONATORS

The simplest configuration of an optical resonator consists of two mirrorsfacing each other to confine the laser beam inside the cavity for amplifying itby means of an active medium at a narrow band of frequencies. In this section,the calculations of the eigenmode and eigenvalue for various resonator structuresare discussed without considering effects of an active saturable-gain medium.For simplicity, the resonator has been assumed perfect mirror reflectivities, eventhough the real resonator suffers some energy losses because of imperfectmirror reflectivities and transmissive output coupling through one of mirrors.Mathematical formulation for the resonator has been done on the basis of theintegral equation modelled by a travelling wave reflecting back and forthbetween the two mirrors. Although the eigenvalue problem of the integralequation is equivalent to the boundary value problem of the partial differentialequation, Eq. (24), the eigenfunction of the integral equation has a uniqueadvantage in determining transverse field distributions at the mirror edges overthe solution to the differential equation. For stable resonators, the transversefield distribution plays an important role in determining diffractional losses. Eventhough various geometrical symmetries involved in resonator system make itpossible to describe the problem with a set of one dimensional equations,analytic solutions are not available for the general cases.

TTT-1 Formulation of Numerical ComputationOur programming target is to find the eigenmodes and eigenvalues of the

integral kernel. Two different approaches are available depending on computationprocedures; (i) firstly calculate the eigenmode iteratively, then find theeigenvalue from a simple integration, (ii) find the eigenvalue first, then theeigenmode by solving the integral equation. Note that the former approach ismemory intensive, while the later one is CPU intensive, especially, for cases ofhigh Fresnel numbers. Thus assessments of each approach should be takenseriously to use the limited computer resources and CPU times efficiently.

For a specific case of a resonator consisting of two infinite plane mirrorsseparated by a distance L, the problem can be reduced to one dimensionalbecause of the symmetry in the y direction. To exploit the iterative method, wewrite the Fresnel integral equation of Eq. (35) in a recursive form expressing

- 14

the field at each mirror in terms of the reflected field at the other

-777 \V LA Js2 ( 3 9 )

uq+2{xx) = - 7 7 7 \ K{xi,x2)uq+l(x2)ds2V LA Js2

uq+3{x2) = 7777 \c.K(x2;xi)ug+2(x1)ds1V L,A JS,

= -JJJ js K(xl;x2)uQ+z(x2)ds2

where Si's represent the mirror surfaces with coordinates Xi on each surface, S\and S2, respectively. To solve a set of the recursive integral equations, we beginthe numerical integration with a properly chosen arbitrary trial function for

uo{x\), where the 'arbitrary' means that any function can be the trial function,

on the other hand, the 'properly' limits that the number of nodal points of the

trial function must be the same as that of the eigenmode that we seek for.

Since the trial function can be represented by a linear superposition of

eigenmodes, the resultant function will converge to a single eigenmode after

many iterative integrations. Then the corresponding eigenvalue can be calculated

by comparing numerical results between each iterative step, i.e., y(1)= «,+3/a9+1and /(2) = uq+Jug+2 at arbitrary chosen position on each mirror. The average

diffractional loss and phase shift per each transit are given by 1 — |ya)7(2)| and

{phase of r ( i ) + phase of r(2))/2 respectively.To employ a method based on the eigenvalue problem, we write the

Fresnel integral equation of Eq. (35) in a matrix equation,

AxKijUj^yui (40)

where 7 denotes the eigenvalue of the kernel and discrete representations of

eigenfunction and kernel function with a stepsize of Ax are used, i.e.,

Ks =EXxi,xi> (41)

A x = \ x j — x i \

Note that the problem domain deals with complex representation of the kerneloperator and the eigenvector in Eq. (40). The real representation of the problem

- 15 -

domain can be done by increasing the ranks of kernel operator and thedimensions of the eigenvector as

Re(Ki;) -

(42)

U} [Imiuj)

In general, the resonator system requires to compute the eigenvalues for each

transit, since the real representation of the kernel operator, Kih is not a

transposed operator of if,,. For a known eigenvalue, /, the corresponding

eigenfunction can be easily computed from Eq. (42). The average diffractional

loss and phase shift per each transit can be obtained from 1 — |y(1V2)l and

{phase of r{l) +phase of>(2))/2 respectively.

In general, the kernel function has an oscillatory part whose period dependson the Fresnel number as plotted in Fig. 10. The number of periods can beestimated from the argument of the kernel,

xAx^l (43)

The kernel operator must include at least O(4NF) matrix elements, the totalmemory allocation for the kernel operator is 8 times of the number of matrixelements since each element demands 8 bytes in heap memory. Accordingly,demands on the computing resources grow exponentially as the Fresnel numberincreases, since the numerical procedure is supposed to cover all of the highlyoscillating region. Thus, the matrix method has its own merit in computingeigenvalues, while it is not competent in calculating eigenfunctions. For caseswith high Fresnel numbers, the iterative method is advantageous over thematrix approach in the usage of physical memories. If the numerical errorsoccurred in each computational step are small enough not to cause computationalinstabilities, the output of the sequence of Eq. (39) is getting closer to theeigenfunction as the iteration goes on. Results obtained from samplecomputations are plotted in Fig. 11. The typical example illustrated in Fig. 12shows that the iterative method can be employed in finding the eigenmodes andeigenvalues. However, the iterative method turned out to be very unfavorable in

- 16 -

viewpoints of CPU time management, since the number of iterations requiredincreases exponentially in the cases of higher Fresnel numbers. Furthermore,since the numerical error involved in each iteration keeps propagatinguncontrollably, repeating iterations many times makes the case even worse.Besides the CPU time management problem, for higher Fresnel numbers, theiterative method does not enable one to calculate the eigenvalues based on thoseinaccurate eigenfunctions.

Ill .2 Description of Code SystemIn the viewpoints of mathematics, the calculation of the eigenvalue is

closely related with that of the eigenmode, however, in numerical computations,different numerical techniques can be used for calculating each of them. Ourcomputer codes are written in C++ programing language on the basis of twomethods; (i) the eigenvalues are calculated by making use of the matrixapproach, (ii) the eigenfunctions are approximately computed from the iterativemethod. Numerical computations are carried out with a personal desktopcomputer (Pentium III 600 MHz, 128 MB RAM). This configuration of amachine enables one to solve the eigenvalue problem up to the Fresnel numberof 360, the maximum Fresnel number the physical memory of the desktopcomputer can accomodate. Even though the machine limit may be relieved byexploiting virtual memories swapped to computer's hard disk drive, we have notconsidered this trick because of the unrealistic demand on CPU times.

The code system, OC_MODE, is composed of two packages; EIGENV forcomputing the eigenvalues, and EIGENF for calculating the eigenmodes, asillustrated in Fig. 13. Note that the eigenvalue computation for asymmetricresonator takes twice of the CPU time compared with that for symmetric case,since the EIGENV package is composed of two almost identical subroutines todeal with the asymmetric configuration. Each package requires its own inputparameters, such as the mirror shape, the mirror size, the gap distance betweenmirrors, the wavelength. Notice that the EIGENV must be run before executingthe EIGENF package since the latter requires the corresponding eigenvalue asone of input parameters. The key mathematical routines employed in writing theOC_MODE are adopted from Ref.D after making minor modifications.

Figure 14 illustrates the computational scheme of numerical proceduresimplemented in the EIGENV package for calculating the eigenvalues of thekernel operator. The QR factorization has been known as the most efficient andwidely used general method for the calculating all of the eigenvalues of amatrix. Since the QR method is CPU time intensive when repeated many times,

- 17 -

the matrix, however, should be reduced to a simpler form for which the QRfactorization is much less expensive. Notice that the real matrix representationof the kernel operator, Eq. (42), is a nonsymmetric matrix. A nonsymmetricmatrix can be reduced to a Hessenberg matrix which is upper triangular exceptfor a single nonzero subdiagonal. Since the eigenvalues of a nonsymmetricmatrix are very defective as well as sensitive to small changes in the matrixelements, no numerical procedure can overcome these intrinsic difficulties, andthe presence of rounding error can make the situation even worse. To reducethe sensitivity of eigenvalues to rounding errors, the matrix is balanced by aprocedure, that makes corresponding rows and columns of the matrix havecomparable norms, reducing the overall norm of the matrix while leaving theeigenvalues unchanged. The procedure of balancing must track down theaccumulated similarity transformation of the original matrix to compute theeigenvectors, however, it is extremely complicated to implement and requiresexcessive computing resources.

Figure 15 describes the computational scheme of numerical proceduresimplemented in the EIGENF package for calculating the eigenmode of theresonator system iteratively. The integrand is composed of two parts! the one isthe kernel function of the Fredholm integral equation which is a highlyoscillating function for cases of the high Fresnel numbers, the other is the fieldfunction which is a smooth function. Since the recursive representation of theintegral equation takes the previous result of the integration as an approximateeigenmode, it should be properly normalized before inserting in the right handside of the integral equation and the discreteness in the representation of theapproximate eigenmode should be relieved by making use of interpolationprocedure or by averaging two adjacent values. For the cases of the higherFresnel number, the iterative numerical integration must be performed as preciseas possible to make the result converge to the proper eigenmode. Verysophisticated routines developed for numerical integration that have capability ofchecking the convergence and adaptive adjustment of stepsize are veryexpensive in viewpoints of the CPU time management. On the other hand, theroutines based on the Gaussian quadrature are straightforward to implement.Note that the Gaussian quadrature formula works accurately only for cases withthe smooth integrand. Even though the integrands are highly oscillatingfunctions for cases of high Fresnel numbers, the range of integration can bedivided by subranges estimated in Eq. (44) in which the integrand becomesmooth. The convergence is checked by comparing the normalized result withthe previous normalized one. Note that the eigenvalues can be obtained from theEIGENF package by implementing four sets of storage vectors that can keep

- 18 -

unnormalized results for comparison purposes.

Ill.3 Rectangular Plane Mirror SystemThe resonator composed of rectangular plane mirrors can be reduced to a

one-dimensional problem of infinite strip mirrors. Figure 16a illustrates theresonator configuration composed of two identical plane mirrors parallel to thexy plane. Assuming that a21 LA is much less than (a/L)2, the integral equationcan be written as

J — a(45)

Because of the symmetric configuration the kernel is symmetric and continuous

(46)

The products of the one-dimensional eigenmodes of the infinite strip mirrors inx and y directions enable one to find the eigenfunctions and eigenvalues forrectangular plane mirrors,

vm»{x, y) = vx, m(x) vy_ n(y) (47)

Yam ~ Yx. m Yy, n (4o/

The loss and phase shift are given by 1 — lr»J2 and tan~1(Im[r»,n]/Re[ymn]).

Figure 17a and 17b show that the loss and phase shift are getting smalleras A'F increases. For one and the same value of JV> they increase when themode order increases, that is, the higher modes suffer much larger spill-overloss. The eigenfunctions of the integral equation turn out to be complexfunctions. For the lowest mode, the amplitude at the edges is smaller than onthe axis. Figure 17c shows that the field intensity at the edge of the mirror issmaller for the larger the NF. However, the amplitude distributions are almostthe same for sufficiently large NF, which means that the propagation of theshort wavelength waves is very directional, while the long wavelength wavessuffer from diffraction. Figure 17d shows that the amplitude distribution of theodd symmetric normal modes, and that, for the same values of NF, theamplitude at the edge is higher than for the lowest mode. The phase has a

- 19 -

waving trend, but is substantially different from the value on the axis only innarrow marginal zones of the mirrors. Though the shape of field distributiondepends on NF, one can say that it is a function of x/a, rather than of x and aseparately, so that the region size of the mirrors where the field is appreciablydifferent from zero (field spot) varies almost proportionally to a. The parallelplane mirrors have good filling, because the light rays can fill the entire volumebetween the mirrors. However, the parallel plane resonator is not used forpractical lasers, since it becomes unstable with only slight misalignment of themirrors. The resonators of most lasers have at least one spherical mirrorsurface.

For large NF, asymptotic expressions for the eigenfunctions are given by

(f f 1 + m + y 2 V 2 ^ ) even modes, (49)cos

= sin (f f 1+ff(1 + 0/2V2^V) °dd m°d e S ' (50)

where /? is a fitting constant. The power loss per transit, d, and the resonant

frequencies, u, are asymptotically given by

(51)

where q = arg / / it.

HI.4 Cylindrical Mirror SystemFor sufficiently large mirror sizes, the properties of curved resonators

depend on the g parameters which measures the mirror curvatures relative tothe mirror spacing. Figure 16b illustrates the resonator configuration composedof two identical cylindrical mirrors facing each other. The integral equation hasthe same form of Eq. (45), but the kernel for cylindrical resonators can be read

K(xi,X\) = j ^L e (53)

where the g,- is a measure of flatness of each mirror, gj=l — L/Rj, These

- 20 -

resonators can be divided into two classes, depending on the signs of gi, stableand unstable resonators.

In comparison with the cases of parallel plane mirrors, the loss and phaseshift decrease much faster as NF increases, and, for one and the same value ofNF, they decrease for decreasing \g,\, as shown in Fig. 18a~l. For the samevalues of NF and gi, the loss increases for increasing mode number. Theeigenfunctions of the integral equation turn out to be complex functions exceptfor confocal resonator (gv=O). For the lowest mode, the amplitude at the edges issmaller than on the axis. Figure 18o~r shows that the distribution of fieldintensity is getting much sharper for decreasing \g\, which reflects that thecurved mirror system is very convergent. Figure 18o~r shows the amplitudedistribution of the odd symmetric normal modes. For the same values of NF, theamplitude at the edge is higher than for the lowest mode. Though the shape offield distribution depends on NF, the same as for cases of parallel plane mirrors,it depends on x only, not on x/a, so that the region size of the mirrors wherethe field is appreciably different from zero (field spot) does not vary much whena varies. Because of the symmetry of the mirror system in the rectangularcoordinates, for reversal of signs of gi, the eigenvalues are complex conjugatesof each other, i.e., the same losses for gi=~gi as illustrated in Fig. 18m~n.

For stable configurations of curved mirrors, the diffractional losses forlarge NF axe extremely small enough to be neglected. The resonance frequenciesare given by

^ (54)

where a=(l/a)cos"V (l + L/J?i)(l-L//?2).

in.5 Circular Spherical Mirror SystemFigure 16c illustrates the resonator configuration composed of two identical

circular plane mirrors parallel to the xy plane. Since the mirror system has anaxial symmetry, the azimuthal mode can be easily separated as e'1""*, and, forthe nh mode, the radial integral equation can be written in a symmetric form

7nvn(.r2){T2= Jo Kn(r2;r1)vn(r1)^fr1drl (55)

Notice that the eigenfunction of the radial integral equation is vn{r)'fr for the

radial field function, vn(r). The radial kernel of the nth mode can be expressed as

- 21 -

(56)

where Jn is an ntb order Bessel function of the first kind. Figure 16d illustrates

the resonator configuration composed of two identical circular spherical mirrorsfacing each other. The kernel function can be read from Eq. (56) by includingthe curvatures of each mirror

(57)

The radial integral equation for the nth mode has the same form of Eq. (55).Because the mirror systems have finite circular aperture, the loss of the

circular mirror system is slightly larger than that of the rectangular system.The loss and phase shift of the circular spherical mirror systems decrease muchfaster than those of the circular plane mirror systems as NF increases, and, forone and the same value of NF, they decrease for decreasing \gt\, as shown inFig. 20a~l. For the same values of NF and gi, the loss increases for increasingmode number. Since the systems have the axial symmetry, the eigenfunctions ofthe integral equation are the Gaussian-Laguerre type, and they turn out to becomplex functions except for confocal resonator (gi=0). For the lowest mode, theamplitude at the edges is smaller than on the axis. Figure 20m~n shows thatthe distribution of field intensity is getting much sharper for decreasing \g,\,

which reflects that the curved mirror system is very convergent. Figure 20o~pshows the amplitude distribution of the odd symmetric normal modes. For thesame values of NF, the amplitude at the edge is higher than for the lowestmode. Though the shape of field distribution depends on NF, the same as forcases of parallel plane mirrors, it depends on r only, not on r/a, so that theregion size of the mirrors where the field is appreciably different from zero(field spot) does not vary much when a varies. Notice that, for the symmetricmirror system, the eigenfunctions that satisfy the integral equations are complexconjugates of each other for reversal of signs of gi, and that the eigenvalues arealso complex conjugates of each other, i.e., the losses are symmetric about g=0as illustrated in Fig. 20i~l. Since the mode volume occupied by the fundamentalmode of the confocal resonator is generally very small, in practice, it isdesirable to use a quasi-hemispherical resonators to enhance the spot size onthe mirrors. This type of resonator has the additional advantage that it does notrequire a critical adjustment.

- 22 -

IV. UNSTABLE RESONATORS

Optical power loss inside the resonant cavity occurs due to the imperfectmirror reflectivites, the transmissive output coupling through one of the mirrors,in addition to scattering and absorption. Unstable resonators suffer much largerpower loss associated with the escape of radiation past the mirrors than stableresonators, as indicated by ray tracing in Fig. 16f. Since the diffraction losses ofunstable resonators are sizable but not intolerable, for sustaining laseroscillations, unstable resonators typically require active media with higher gainfor compensating for this additional loss factor. Conventional devices such ascommercial He-Ne lasers employ stable resonators. However, stable resonatorsare not suitable for high power laser devices because of their small modevolumes and mirror damages. On the contrary, unstable resonators have severaladvantages for certain high-power lasers such as large mode volumes, smallthermal distortion and damage of mirrors, and diffractional output coupling. Theidea of diffraction coupled resonators is based on geometrically unstable cavities,in particular, the output mirror incorporates a small reflecting dot and the outputmode was a very characteristic doughnut shape. In addition, the unstablegeometry is much less sensitive to misalignment. In general the high powerlaser generators employ one of three types of resonators, i.e., plane-plane,concave-plane asymmetric and confocal positive and negative branch unstableresonators.

IV.l Asymmetric Mirror Unstable ResonatorsFigures 16e~h show the mirror configurations of unstable resonators losing

large radiation fields. Place a set of points Pi and Pz that are virtual objects forthe other mirror as shown in Fig. 16f. Suppose that Pi is located r\L frommirror 1 and (n+l)L from mirror 2, then the reflected wave appears to comefrom the object point Po and the image and object distance satisfy mirrorformulas,

Applying the same arguments to the P2, one can find the image and object

- 23 -

distance similar to Eq. (58)

(59)

Introducing the concept of dimensional magnification, one can compute thesize of the beam on the other mirror. The magnification factors of mirror 1 and2 are calculated from g parameters,

,f_M\=

V -1+- 1 -

V 1 }

The effective reflection of the power at mirror 2 is proportional to the ratio of

the wave emitted from mirror 1 and intercepted by mirror 2 as well as the

reflectivity of mirror 2, l\,

By the same token, the effective reflection of the power at mirror 1 is given as,

Then the reflection of the power for the round trip is given by

(64)

Note that the net power gain should be larger than the loss, i.e., GR>\. The

mean reflection for one-way trip can be expressed as

_ >/ 8182-^ 8182-I

- 24 -

where I-T^l is absorbed in p. Since the p should be positive and less than

unity, there are two branches for unstable resonators, i.e., a positive and a

negative branch, depending on gig2- The equiloss contours satisfy

5riSr2 = (l + p)2/P f° r positive branches, and gig2 = —(l — p)2/4p for negative

branches. The plots of the equiloss contours are hyperbolas on the stability

diagram.Since the integral equation method for optical resonators is general enough

to cover any cavity configurations, stable or unstable, it can generate numericaldata for the fields and the loss per transit for asymmetric cavities shown inFig. 21a~d and 22a~d. Numerical computations are carried out with asymmetricresonators having perfect reflectivity on each mirrors. Figure 18a~c shows thatthe symmetric resonators with \g^>l have oscillatory losses as NF increases.The cusps in the oscillatory loss curves are due to mode crossings at whichtwo different modes have exactly the same diffraction losses. However, only themagnitudes and not the phase angles of the eigenvalues are equal at thesecrossing points. Figure 18o and 18q show that diffraction effects push the peakof intensity outward. The higher order modes have substantially largerdiffraction spread, and thus much increased diffraction losses for the TEMiomode. The general features will always be very much the same for either thesymmetric or the asymmetric mirror. For asymmetric mirror resonators, theintensity profiles of mirror 1 reflect large spill-over outside the mirror edge,while those of mirror 2 are negligibly small as illustrated in Figs 21e~l and22e~l. The complex field amplitude falling outside the mirror edges in thosefigures will be coupled out of the resonator past the mirror edges.

IV.2 Unstable Resonators with Nonuniform Reflectivity Output CouplingA positive branch confocal unstable resonator is similar to Fig 16h, but the

mirrors have the same aperture sizes. The lossless and partially transmitting

feedback mirror has a reflectivity, IX r), depending on the distance from the

optical axis. Basically the integral equation can be written in the same form of

Eq. (39) except for some modification in the amplitude on mirror 1 due to the

reflection coefficient such that W^^lX^lu^. Then the output intensity is

given by

(66)

where U(r) is the mode intensity incoming on the feedback mirror. The

- 25 -

resonator equation can be written in the form

R(r) U(k\r) = M2 4 U(k){Mr) (67)

where U{k\r) denotes the intensity of a mode of the resonator and ak is its

eigenvalue given by

(68)

where po=p(0). Since the k=0 is the dominant mode, Eq. (67) can be written

for the dominant mode after dropping out all higher modes,

p(r)U(r) = PoU(Mr) (69)

where U{r) denotes the dominant mode. Substituting Eq. (69) into Eq. (66)yields

I(r)=U(r)-p0U(Mr) (70)

The form of the integral equation suggests that the main contribution to

the integral comes from the regions near rx = r2/M, because for intervals of rxmust different than that value, the exponential term in the kernel oscillates

violently through all complex values of 1. A power series expansion can be used

to solve Eq. (69),

(71)

Substituting Eq. (71) into Eq. (66) and Eq. (70) and comparing the coefficientsof the same order of r yield a set of recursive equations,

nUH-k (72)

= Un{\-PoMn) (73)

- 26 -

Suppose that the reflectivity of mirror 1 is given as

P0cos2(-f)

0 r>3(74)

For U0=l, po = 0.4, and M=3, the first few nonzero coefficients of the

reflectivity, the mode and the output intensities calculated from Eqs. (71)~(73)

are listed in Table 1,

Table 1. Expansion coefficients

n

0

2

4

6

8

10

Pn

0.400000

-0.986904

0.811743

-0.267053

0.047066

-0.005161

Un1.000000

-0.308425

0.034880

-1.895E-36.083E-5

-1.295E-6

In

0.600000

0.801905

-1.095232

0.550730

-0.159579

0.030590

The series solutions for U(r), p(r), and I(r) are plotted in Fig. 23a. Note thatthe mode and output intensities are the same for 3


n

048121620

Pn

0.400000-6.0000004.500E+1

-2.25E+28.434E+2

-2.756E+3

Un

1.000000-1.0000000.500000

-0.1666670.041667

-8.333E-3

In

0.6000005.400000

-5.070E+12.729E+2

-1.092E+33.495E+3

Assume that the output intensity is specified as

/(r) = /0(l + 5>-3)exp(-2r3)

For /o = 0.6, £0 = 0.4, and M=2>, the first few nonzero coefficients of the

reflectivity, the mode and the output intensities calculated from Eqs. (71)~(73)

are listed in Table 1,


n

036912

Pn

0.400000-1.9102044.459058

-4.3498352.726146

Un

1.000000-0.1836730.016518

-6.605E-41.694E-5

In

0.61.8

-4.85.2

-3.6

The series solutions for U(r), p(r), and I(r) are plotted in Fig. 23c. Note thatthe mode and reflectivity profiles are not spatially limited even though theoutput intensity is spatially limited.

- 28 -

V. CONCLUDING REMARKS

The integral equation method for optical resonators is general enough tocover any cavity configurations, stable or unstable, it can generate numericaldata for the fields and the loss per transit for various optical cavities. Theintegral equation method turned out to be incompetent for dealing with thecases of high Fresnel number because of limited computing resources andnumeric accuracy. Also the technique of the integral equation method should beextended to cover cases of resonators containing gain media.

The use of optical cavity sustaining high-(2 resonances offers a number offeatures in operating lasers; (i) the Q of the fundamental mode determines thepump power threshold for laser oscillations, (ii) the number of high-0resonances per unit band are small, (iii) resolving the high-0 resonances bysimple devices can improve the mode selectivity and the monochromaticity ofthe laser beam, (iv) optimal output coupling, (v) optimal mode volumedetermines the amount of gain medium for amplifying light.

The use of unstable resonators offers a number of advantages in theoperation of high power lasers; (i) a greater portion of the gain mediumcontributing to the laser output power as a result of the availability of a largermodal volume, (ii) higher output powers attained from operation on thelowest-order transverse mode, rather than on higher-order transverse modes,and (iii) high output power with minimal optical damage to the resonatormirrors, as a result of the use of purely reflective optics that permits the laserlight to spill out around the mirror edges (this configuration also permits theoptics to be water-cooled and thereby to tolerate high optical powers withoutdamage).

- 29 -

REFERENCES

1. O. Svelto, Principles of Lasers, 2nd e

Mirror #2 Mirror #1

A

u [< Unit Cell =•]

Figure 1. Optical cavity and lens-waveguide equivalent to the mirrorsystem.

Q 2

Unstable (negative branch) Unstable (positive branch)

Plane parallel

Concentric

Unstable (positive branch) Unstable (negative branch)

Figure 2. Stability diagram of optical cavities.

- 31 -

1 irr

or

.2

C

ocgtoo0.

4 -

2 -

0 -

-2 -

-4 -

-

-

A. A A n. A A A A / \ A - * T ^

"\/ \AA / \ / \ AA A A /V A / \ A Av \ ' ^ v v v w \ / v \ / ^ ' ^ \ / \ /v v * V_

-

10 20 30Single Pass Number

40

Figure 3a. Ray-tracing results with stable mirror system.

10

Single Pass Number

Figure 3b. Ray-tracing results with unstable mirror system.

- 32 -

Figure 4. Farby-Perot type optical cavity.

- 33 -

1.2

1.0 -

g 0.8 Hcro

at

I 0.4 H

0.2-

0.0

R=50%/50%R=90%/90%

0.51440 0.51445 0.51450

Wavelength (urn)0.51455

Figure 5a. Transmittances of etalon for several different reflectances.

1.0-

1.0-

O 0.8-oSS

| 0 . 4 -

0.2-

0.0-I \_ j

0.51440

6,=0.0 radian 6,=0.005 radianO-nnnQrarilan fl -d 119̂ rariian

i

if

•»

!;

;;

.A}.}£)

—

I Jm

-

-

-

-

0.51445 0.51450 0.51455

Wavelength (jim)

Figure sc. Transmittances of etalon for several different angles.

1.2

1.0 -

0.8-

=CO

ra

0.6-

0 .4 -

0.2-

0.0

AL=0.0 nm -AL=10 nm —

AL=1 nmAL=100 nm

JK.0.51440 0.51445 0.51450

Wavelength0.51455

Figure 5d. Transmittances of etalon for several different wavelength variations.

- 35 -

Figure 6. TEM modes in rectangular system.

- 36 -

0,t mode

Figure 7. TEM modes in cylindrical system.

- 37 -

to£o

3

1000

800 -

600 -

400 -

200 -

0 •-600

wo=2 mmwo=4 mm-wo=6 mmwo~8 mmw =10 mm

-400 -200 0 200

Distance (m)400 600

Figure 8a. The Gaussian beam radius of curvature for 5 starting beamwaists of Ar laser.

10

to'ns

£TO

m2 -

0 •-150 -100 -50 0 50

Distance (m)100 150

Figure 8b. The Gaussian beam waist for 5 starting beam waists of Arlaser.

- 38 -

50

40 -

30 -

oo

3

-ato

en

20 -

10 -

• ••• X=514.5 nmX=632.6 nmX=1064 nm

4 6

Distance (m)10

Figure 8c. The Gaussian beam radius of curvature for 3 wavelengths.

3.8

EJE

.1?to

ECD(X>

CD

4 6

Distance (m)10

Figure 8d. The Gaussian beam waists for 3 wavelengths.

- 39 -

Figure 9. Source radiation fields on observation surface.

- 40 -

Retetve Fasten en Mncr Relative Position on Mirror

Real PartImaginary Part

Relative Pos-tort on Mirror

5 0 .

» •

10.20

• »

50-

f-1000

MMffiiff!InSHIfpfl

Real P i rtknaQiraryPart

Ijpy||j5Mffiflli

Rdaliw Pcsaon on Mrror Relative Position on Mirror

Figure 10. Tests of kernel function.

- 41 -

Number oJ Transits

Number ot Trinrfs Number of Transits

Number o* Trans**Nurrbef of Transits

Figure 11. Convergence tests

- 42 -

1.0-

0.8-

0.0-

-1.0 -0.5— i —0.0

—I—

0.5 1.0

Normalized Position on Mirror

Figure 12. Sample computation shows mode formation aspass number increases

- 43 -

STARTOC.MODE

INPUT DATAmirror sizes

g parameterswavelength

etc.

DISPLAY- X LIST OF INPUT

PARAMETERS

EIGENV

DISPLAYCOMPUTATION

PHASE

STOREEIGENVALUE

inEIGENV.DAT

EIGENF

DISPLAYEIGENMODE andCONVERGENCEjor each transit

STOREEIGENMODE

inEIGENF.DAT

Figure 13. Schemaic of OC_MODE code system

- 44 -

INPUT DATAresonator symmetry

mirror sizesmirror spacing

g parameterwavelength

DISPLAYLIST of INPUTPARAMETERS

ALLOCATE VECTORSALLOCATE MATRIX

CALLINTEGRAL KERNEL

FUNCTION

r

BALANCE MATRIX

iTRANSFORM

KERNEL MATRIXinto

HESSENBERG MATRIX

CALLINTEGRAL KERNEL

FUNCTIONi

ISET SYMM=TRUE

YES

STOREEIGENVLAUES

inEIGENV.DAT

Figure 14. Flow chart of EIGENV package

- 45 -

STARTEIGENF

INPUT DATAmirror sizesg parameter

mirror spacingwavelength

DISPLAY- X LIST of INPUT

.PARAMETERS

ALLOCATE VECTORSCALL INTEGRAL KERNEL

INITIALIZE VECTORS

SAVE OLD RESULTSDO INTEGRATION

INCREASE FLAG #REPLACE INTEGRAND

FIND MAXIMUMNORMALIZE VECTORS

COMPARENEW and OLD VECTORS

CONVERGENT?

STROEEIGENMODE

inE1GENF.DAT

Figure 15. Flow chart of EIGENF package

- 46 -

+a

-a

(a) Infinite StripMirros

- a

(b) CylindricalMirrors

(c) CircularMirrors.

(d) Curved CricularMirrors

Figure 16. Mirror configurations of various resonator systems

- 47 -

Mirror 2

Mirror 1

(e) Unstable resonator

(f) Magnification factor

(g) Negative branch

(h) Positive branch

Figure 16. (Continued)

- 48 -

COcCD

1 -

H 0.1-

CD

m 0.01 i

1E-3-;

2b

1E-4-

1E-50.1 1 10

Fresnel Number

• m=0 ,m = 1m=2m=3

100 1000

Figure 17a. Diffractional losses per transit for infinite strip plane mirror.

1 10Fresnel Number

100 1000

Figure 17b. Phase shifts per transit for infinite strip plane mirror.

- 49 -

1.1

1.0

0.9

^ 0.8-

1 °7j= 0.6-

"§ 0.5

^ 0.4-

0.2

0.1

0.0-1.0 -0.5 0.0 0.5

Normalized Position on Mirror

1.0

Figure 17c. Normalized intensity dirstributions on mirror for m=0 mode.

-1.0

(Ra

dia

n)

ha

se

a.

1.0-

0.8-

0.6 -

0.4 -

0.2 -

0.0 -

-0.2 -

-0.4 -

• • t

f=0.1 ...1=10 —

1 f=100

K% -

• •• f=1.0— f=50

-0.5 0.0 0.5Normalized Position on Mirror

1.0

Figure 17d. Phases of radiations on mirror for m=0 mode.

- 50 -

0.0-1.0 -0.S 0.0 0.5

Normalized Position on Mirror1.0

Figure 17e. Normalized intensity dirstributions on mirror for m=l mode.

ro

2.0

1.5 -

1.0 -

0.5 -

0.0 -

-0.5 -

-1.0 -

-1.5 -

-2.0

f=0.1 f=1.0f=10 f=50f=100

-1.0 -0.5 0.0 0.5


Figure 17f. Phases of radiations on mirror for m=l mode.

- 51 -

COCID

Q.

cocoO

COco

10

0.1 -

0.01 -

1 E " 4 '

1E-5

m = 0 mode

9,=g,=0-0

, 29,=9i=0.79,=9,=0.8g,=g,=0.9

12

10

Fresnel Number

Figure 18a. Diffractional losses per transit for symmetric cylindrical mirror(m=0 mode).

c5CD

cCO

CDQ .

CO

CDCOCO

- CCL

0.1 -

m = 0 mode9,=92=0.7

o5

0.1 10

Fresnel Number

Figrue 18b. Phase shifts per transit for symmetric cylindrical mirror (m=0 mode).

- 52 -

toc

10

0.1

toO 0.01

CDcoT5CO

1E-3-

1E-5

m = 1 mode

10

Fresnel Number

Figure 18c. Diffractional losses per transit for symmetric cylindrical mirror(m=l mode).

CD

CO

V)c(0

s:COCDCOTO.cQ.

1 -

0.1 -

0.01 -

m = 1 mode9,=3,=0.7g,=gj=o.5g,=g,=o.8g,=g,=o.9g,=g2=i.2

0.1

Fresnel Number

Figrue 18d. Phase shifts per transit for symmetric cylindrical mirror (m=l mode).

- 53 -

tocCO

0.1 -

CD 001Q.t/>

8 1E-3_ 1

*co

.2 1 E - 4

CO£ 1E-S5

1E-6

m = 0 modeg,=1.0g;=0.9 ig^i.Og^O.8g,=1.0g;=0.7g,=1.0g;=0.5.

0.1 10

Fresnel Number

Figure 18e. Diffractional losses per transit for asymmetric cylindrical mirror(m=0 mode).

cCO

CO

incCO

a.

sz(f)

g,=1.0g,=0.9g,=1.0 9jg,=1.0g,=0.7g,=1.0g,=0.5

1E-610

Fresnel Number

Figure 18g. Diffractional losses per transit for asymmetric cylindrical mirror(m=l mode).

c

TO

TO

OT

V)TO

1 -

0 . 1 •:

0.01 -.

m = 1 mode

, 2,=g,=0.9

0.1 10

Fresnel Number

Figure 18h. Diffractional losses per transit for asymmetric cylindrical mirror(m=l mode).

- 55 -

i ran

s

CL

COCO

o_lCO

ooCO

0.1

0.01

1E-3

1E-4

1E-5

m = 0 modeg,=0.0g,=0.9g,=0.0g!=0.8g,=0.0g,=0.7g,=0.0g2=0.5

0.1

Fresnel Number

Figure 18i. Diffractional losses per transit for asymmetric cylindrical mirror(m=0 mode).

•o

a:

tt)

CO

a>toCD

0 . 1 -

0.01

m = 0 mode

g,=0.0g2=0.7g,=0.0gz=0.5g,=0.0gj=0.8g,=0.0g,=0.9

0.1

Fresnel Number

Figure 18j. Diffractional losses per transit for asymmetric cylindrical mirror(m=0 mode).

- 56 -

•••• g,=0.0 gj=0.- - g,=0.0g2=0.7— g,=0.o gs=0.5

1E-6

Fresnel Number

Figure 18k. Diffractional losses per transit for asymmetnc cylindrical mirror(m=l mode).

-aCO

a:COc05

Q .

SICO

m

gg9

i i i i—I i 1 1 1 1 1 1—r—I 1

= 1 mode

-

,=0.0g2=0.7,=0.0 g2=0.5,=0.0 g2=0.8,=0.0 g2=0.9 .

0.1

Fresnel Number

Figure 181. Diffractional losses per transit for asymmetric cylindrical mirror(m=l mode).

- 57 -

V)cro1 0 . 8 -

1.2

1.0 -

•o

1.2

0.8 -

•o

— 1 -u>cCD

H 0.1,

Q.

S 0.01o

_ l

1 1E-3-=|o

2 1E-4-

1E-5

0.1

0,0 -0,10,20,3

1 10

Fresnel Number100 1000

Figure 19a. Diffractional losses per transit for circular plane mirror(TEM00) mode.

CD

CC

CD

CDQ .

JZ.

CD

Q.

0.1

1 -

0.1 -

0.01 -

1 ' ' ' I '

\i ll

V

• I

V

-1—1 1 I 1 111 1 1—1 1 1 1 1IJ 1 1—1 1 I

0 0••"- - . . • • • • . 0 , 1

0,2 -v : •••-, ' - . 0 , 3 \

r •* * * . * • .

^ \ ^ ^ ' • • - . ' • • • :

1 10

Fresnel Number100

Figure 19b. Phase shifts per transit for circular plane mirror (TEM00)mode.

- 61 -

CO£=CO

CDD .

0300O

03

g

CO

1 -

0.1 -

0.01 -

1E-3-

^ \ ^ ;

m=1.3 .

*"*•./*•. m-1,0 [

' ...

.i i

. .

0.1 1 10


Figure 19c. Diffractional losses per transit for circular plane mirror(TEM10) mode.

CO01

GOcCO

Q .

COCD

00

CO

0.01 -

1 10


Figure 19d. Phase shifts per transit for circular plane mirror (TEM10)mode.

- 62 -

COc

sJZ

•a

N

£o

1.0 -

0.8 -

0.6 -

0.4 -

0.2 -

0.0 -

TEM00 mode

f=0.1f=1.0f=10f=50

-

0.00 0.25 0.50 0.75 1.00Normalized Radial Position on Mirror

1.25

Figure 19e. Normalized intensity dirstributions on circular plane mirrorfor TEMOO mode.

CO

T3

a:

inCD

0.00

0.90 -

0.75 -

0.60 -

0.45 -

0.30 -

0.15 -

0.00 -

-0.15 -

-0.30 -

-0.45 -

-0.60 -

-0.75 -

-0.90 -

- •"'

S ^ " " — ~ ~ ~ v ^ ^

- - _

/ TEMTOmodeJ

/ f=o.i :/ f=1.0 •

/ f=10 -f=50 •

"y'l •

0.25 0.50 0.75 1.00Normalized Radial Position on Mirror

1.25

Figure 19f. Phases of radiations on circular plane mirror for TEMOOmode.

- 63 -

c

z

o

ize

03

E5z

1.0 -

0.8 -

0.6 -

0.4 -

0.2 -

0.0 -

/ / • ' ' , /

f/iJ //

, - • • "

• ' ' \y - • .

i\

V

\

TEMlamodef=0.1f-1 of=10f-50f=100

-

-

0.00 0.25 0.50 0.75 1.00Normalized Radial Position on Mirror

1.25

Figure 19g. Normalized intensity dirstributions on circular plane mirrorfor TEM10 mode.

0.30

0.25

0.20

0.15

0.10

0.05

0.00

0.05

0.10

0.15

0.20

0.25

-0.30I.OO 0.25 0.50 0.75 1.00

Normalized Radial Position on Mirror1.25

Figure 19h. Phases of radiations on circular plane mirror for TEM10mode.

- 64 -

mcto

Q-

ooGOo

tocg"oCD

! * =

b

1 •:

0.1 -

0.01 -

1E-3-.

TEM00 mode

B,=g,=o.og,=g,=o.5g,=g,=o.7g,=gi=o.8g,=g;=o.9B = g = i 2

0.1 10

Fresnel Number

Figure 20a. Diffractional losses per transit for symmetric circular curvedmirror (TEMOO) mode.

73CD

tr

Q .

CO

CDCOCD

1 -

0.1 -

01 -

TEM00

g,9,g,g,g,

mode

=gjWs=g,=o.8=g;=o.9=92=1.2

- — — .

,-•"'•'

-I

\

' • •

/

*

1

Ii

\•

1 /i /

i ;

V

^\!

\

0.1

Fresnel Number

Figure 20b. Phase shifts per transit for symmetric circular curved mirror(TEMOO) mode.

- 65 -

10

1 -

0.1 -U>O

TO

. ^ 0.01 -

1D 1E-3-J

TEM01 modeg,=g,=o.o

g|=9;=0.9

0.1

Fresnel Number

Figure 20c. Diffractional losses per transit for symmetric circular curvedmirror (TEMOl) mode.

cTO

TJTO

a:

10

0.1

"to 0 0 1c2 1E-3

5 1E-4Q.

£ 1E-5

5 1E-6

J5 1E-7a.

1E-8

mode

g,=9,=o.7g,=g,=o.9g,=g2=o.8

-- g!=g]=i.2

0.1 10

Fresnel Number

Figure 20d. Phase shifts per transit for symmetric circular curved mirror(TEMOl) mode.

- 66 -

risit

CO

o_1racot5ro

1.2

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

- 3 . 0 - 2 . 5 - 2 . 0 - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5

Geometrical Parameter g2.0 2.5 3.0

Figure 20i. Diffractional losses per transit for symmetric circular curvedmirror (TEMOx) modes.

ro

ro

COcroI—

CDQ.

"tz

CO

1.2incCD

0)CL

WO

rocooro

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5Geometrical Parameter g

2.0 2.5 3.0

Figure 20k. Diffractional losses per transit for symmetric circular curvedmirror (TEMlx) modes.

Q.

-1.5 -1.0 -0.5 0.0 0.5 1.0

Geometrical Parameter g

Figure 201. Phase shifts per transit for symmetric circular curved mirror(TEMlx) modes.

- 70 -

IS It

ized

In

ter

Mor

mal

1.0 -

0.8 -

0.6 -

0.4 -

0.2 -

0.0 -

' ' ' ' '

_ _ _ ^ ^ _

" - " " ^ : b 5 > > < s r """•••-.

FN=1.0 -TEM00 mode-

9,=9,=0.0 .

g|=g!=o.9\ - g,=9,=1-2 •

....

- - . . ' • • • ' . "

0.0 0.2 0.4 0.6 0.8 1.0


Figure 20m. Normalized intensity dirstributions on symmetric circularcurved mirror for TEMOO mode.

cra73ro

IDCOnjx:O_

o

0

0

-0

4 -

2 -

0 -

2 -

FN=1

T E M 0 0

g99

- 9

0/

mode / .-•"

i = 9II n

i = 9i = 9

-

,=o.o / .-•

^=0^8,=0.9

0.0 0.2 0.4 0.6 0.8 1.0


Figure 20n. Phases of radiations on symmetric circular curved mirror forTEMOO mode.

- 71 -

1.2

1.0 -

0.8 -

0.6 -

t 0.4 -

0.2 -

0.0

FN=1.0

0.0 0.2 0.4 0.6 0.8 1.0


Figure 20o. Normalized intensity dirstributions on symmetric circularcurved mirror for TEM10 mode.

- oas

•f -0

0

8 -

6 -

4 -

2 -

0 -

2 -

4 -

,6 -

8 -

0

TEM10 mode— 9,=92=0.0

g,=g2=o.5o8

F =1.0

0.0 0.2 0.4 0.6 0.8


Figure 20p. Phases of radiations on symmetric circular curved mirror forTEM10 mode.

- 72 -

wcCD

ino

toco

1 -

TEMO0 modea2=1.2a,g,=0.95

1 10

Fresnel Number100

Figure 21a. Diffractional losses per transit for negative branch unstableresonator (TEM00) modes.

sit

CO

H 1 -

Q.

O1

CDCo

CD

Itt

Q

1

— - — ^ ^

g, = -5.50g,--5.75g, = -6.00g, = -6.25n - R 50

TEM10 modea2=1.2a,g2=0.95

Vi <

1"

ia

1

--

0.1 1 10

Fresnel Number100

Figure 21b. Diffractional losses per transit for negative branch unstableresonator (TEM10) modes.

- 73 -

05cTO

a.

wo

1.2

1.0 -

0.2 -

0.0

TEMo0 modeg2=0.95

0.0 0.2 0.4 0.6 0.8 1.0

Normalized Position on Mirror #11.2

Figure 21e. Normalized intensity dirstributions on mirror #1 of negativebranch unstable resonator for TEMOO modes.

w

iten

73

hz

£o•z.

1.0 -

0.8 -

0.6 -

0.4 -

0.2 -

0.0 -

I

I \\1 ifi1 A'<

f -\I:Ws * v

! '- ^>;

TEM00 modeg2=0.95a2=1.5a,

g, = -5.50g, = -5.75g, = -6.00 .gv = -6.25g, = -6.S0 -

• / A i '•• J\ A- '• A ' V •' '•• ' v

V f

0.0 0.2 0.4 0.6 0.8 1.0


Figure 2If. Normalized intensity dirstributions on mirror #2 of negativebranch unstable resonator for TEMOO modes.

- 75 -

1.2

TEM10 m ode g,=-5.50g,=-5.75g, = -6.00'g, = -6.25

- g,=-6.50

0.0 0.2 0.4 0.6 0.8 1.0

Normalized Position on Mirror#11.2

Figure 21i. Normalized intensity dirstributions on mirror #1 of negativebranch unstable resonator for TEM10 modes.

-5"VJd

ity

N

rma

l

o2

1.0 -

0.8 -

0.6 -

0.4 -

0.2 -

0.0 -

1

1 1 •' ' I 1 ••

,4' rf; 1• IU /

W

• • i

TEM10 modeg2=0.95g,=-6.o -

a ~1 2a '

a,=1.5a,aj^i^a, •

A

A \

r\ k'irr* ** \ i i l ' l

\ lAf 4/ ' 1

0.0 0.2 0.4 0.6 0.8 1.0Normalized Position on Mirror #1

1.2

Figure 21k. Normalized intensity dirstributions on mirror #1 of negativebranch unstable resonator for TEM10 modes.

0.0

TEM00 mode-


1.2

Figure 211. Normalized intensity dirstributions on mirror #2 of negativebranch unstable resonator for TEM10 modes.

- 78 -

IDC

a)

Weno

roco

Q

1 -

0.1

g, = 1.10g,=1.n9, = 1.139, = 1-159, = 1.20

0.1

^

^dr-

TEM00 modeg2=0.95

a2=1,2a,

- / : ' . • • • • . . / " , - • ' S , '

1 10

Fresnel Number100

Figure 22a. Diffractional losses per transit for positive branch unstableresonator (TEM00) modes.

cTO

O

(0tzooro

0.1 1 10

Fresnel Number100

Figure 22b. Diffractional losses per transit for positive branch unstableresonator (TEMIO) modes.

- 79 -

IDCCO

IDCL

v>o

(0coo

1 -

0.1

TEM00 modeg,= 1.15

0.1 1 10

FresnelNumber100

Figure 22c. Diffractional losses per transit for positive branch unstableresonator (TEM00) modes.

rot -

mo

roco

0.1

TEM10 modeg,=i. i5

0.1 1 10

Fresnel Number100

Figure 22d. Diffractional losses per transit for positive branch unstableresonator (TEM10) modes.

- 80 -

1.2

0.2 -

0.00.0

TEM00 modeg,= 0.95

0.2 0.4 0.6 0.8 1.0Normalized Position on Mirror #1

1.2

Figure 22e. Normalized intensity dirstributions on mirror #1 of positivebranch unstable resonator for TEMOO modes.

1.2

0.0

TEM00 modeg2=0.95a2=2.0a,

g, = 1.11


1 .2

Figure 22f. Normalized intensity dirstributions on mirror #2 of positivebranch unstable resonator for TEMOO modes.

- 81 -

5 ity

tens

c

"OCDNTO

Eoz

1.0 -

0.8 -

0.6 -

0.4 -

0.2 -

0.0 -

i • i • i

1 1 1 1 • 1 .

TEM00 modeg2=0.95 '_

. •

a2=1.8a, •a,=1.9a,a2=2.0a, "

a2=2.2a,—i • 1 1

0.0 0.2 0.4 0.6 0.8 1.0Normalized Position on Mirror#1

1.2

Figure 22g. Normalized intensity dirstributions on mirror #1 of positivebranch unstable resonator for TEMOO modes.

>,

v>c

c

T3CON

COLUJ

OZ

i .£. -

1.0 -

0.8 -

0.6 -

0.4 -

0.2 -

0.0 -

i • i • i i

\ R'\ /A \.\ i^-#V'•\ ••/••.' f^'-'S. •

• . - • / / /

\"\>/-' / \ ••

u 1M /

\ ; /sy\\ \ / _\••• \ \ i '-' V - V .\\.-^5>r. \ \

'.. ' .••. . , ' ••- ,• . . . \ A

' \ / " '"-••'V-- \ - -

t E M '00 modeg2=0.95g, = i .15

a =1 8a •a2=1.9a] .a2=2.0a,a2=2.1a, •a2=2.2a,

-

^ •

0.0 0.2 0.4 0.6 0.8 1.0


Figure 22h. Normalized intensity dirstributions on mirror #2 of positivebranch unstable resonator for TEMOO modes.

- 82 -

1.2

1.0 -

O 0.8 -

0.00.0

TEM10 modeg2=0.95

g,=1.i3g,=1.15g,= 1.20

0.2 0.4 0.6 0.8 1.0


Figure 22i. Normalized intensity dirstributions on mirror #1 of positivebranch unstable resonator for TEM10 modes.

>•

1.2

1.0 -

0.0

TEM10 modeg2=0.95

0.0 0.2 0.4 0.6 0.8 1.0


Figure 22k. Normalized intensity dirstributions on mirror #1 of positivebranch unstable reson

KR0100829 KAERI/TR-1661/2000 · 2005. 1. 17. · (TEMox) modes 69 Figure 20k. Diffractional losses...

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Transcript of KR0100829 KAERI/TR-1661/2000 · 2005. 1. 17. · (TEMox) modes 69 Figure 20k. Diffractional losses...