Multipass optical amplifier using a double confocal resonator geometry

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Multipass optical amplifier using a double confocal resonator geometry

R.L. Fork, F.A. Beisser, D.K. Fork

To cite this version:

R.L. Fork, F.A. Beisser, D.K. Fork. Multipass optical amplifier using a double confocal resonator geometry. Revue de Physique Appliquée, Société française de physique / EDP, 1987, 22 (12), pp.1665- 1671. �10.1051/rphysap:0198700220120166500�. �jpa-00245726�

Multipass optical amplifier using

^a

double confocal resonator geometry

R. L. Fork

(1),

^{F. A. Beisser}

(1)

^{and D. K. Fork}

(2)

(1)

^{ATT Bell} Laboratories, Holmdel, ^NJ 07733, ^U.S.A.

(2)

University ^of Rochester, Rochester, ^NY 14627, ^U.S.A.

(Reçu ^{le 29} juillet 1987, accepté le 24 août 1987)

Résumé. ²⁰¹⁴ Nous examinons des configurations d’amplificateur optique ^à multipassage ^conçu pour

l’amplification d’impulsions optiques ^{courtes. Les} configurations ^à multipassage ^{sont basées sur un résonateur}

confocal double qui ^{est obtenu en} introduisant de petites ^rotations angulaires ^{sur les} quatre miroirs formant deux résonateurs confocaux adjacents. ^D’autres petites ^{rotations et} déplacements ^{de ces} miroirs transforment le résonateur confocal double en une configuration multipassage qui fait passer ^un faisceau incident dans deux cols (« ^walst ») de faisceau à chaque boucle. Ces cols se produisent ^{en un seul endroit, ou en deux endroits,}

selon l’orientation et la position des miroirs et de l’image du faisceau incident. Des configurations ^{basées sur}

des paires ^de miroirs confocaux symétriques gardent ^{un diamètre constant aux cols} des faisceaux. Les

configurations ^{basées sur} des résonateurs confocaux asymétriques entraînent un accroissement, ^{ou une} réduction, progressive du diamètre des cols des faisceaux aux passages successifs.

Abstract. ²⁰¹⁴ We examine multipass optical amplifier configurations designed ^for amplification ^{of short} optical pulses. ^The multipass configurations ^{are based on a double confocal resonator} which is formed by introducing

small angular rotations of the four mirrors forming ^two adjacent ^confocal resonators. Further small rotations and displacements of these mirrors convert the double confocal resonator into a multipass configuration ^which

causes an incident beam to pass through ^two beam waists on each round trip. These beam waists occur at one

spatial location, ^{or at two different} spatial locations, depending ^on the orientation and location of the mirrors and the imaging of the incident beam. Configurations ^{based on} symmetric confocal mirror pairs ^{maintain a}

constant diameter at the beam waists. Configurations ^{based on} asymmetric ^confocal resonators cause a

progressive increase, ^or decrease, in the beam waist diameter on successive passes.

Classification

Physics ^Abstracts

42.60K ^- 42.60D

1. Introduction.

Recent advances in modelocked laser oscillator

design ^have produced

pulses

^{as short as 27 fs}

[1]

^and pulse compression

techniques

^have

produced

pulses

as short as 6 fs

[2].

^In this paper ^we describe a double confocal

multipass

^structure

designed specifically

^for

amplification

^of ultrashort

optical

pulses. ^The config-

uration provides

key

^features

required

^of

multiple

pass amplifiers ^{in a manner which is both} economical, ^{in the sense of}

requiring

^{a minimal}

number of optical elements, ^and flexible, ^{in that a} variety ^of amplifier types ^{can be achieved with minor} changes ^{in the} optical components.

Attention is directed towards

amplifiers

^{based on}

flowing dye

jets. ^The

configurations,

^{however, are} sufficiently general ^{to be useful in other} applications.

Some of the key features which are needed in a

multiple pass

optical pulse amplifier

^{are :}

(1)

^{at least}

one focal region for the beam traversing ^the

amplifier

REVUE DE PHYSIQUE APPLIQUÉE. - T. 22, N° 12, DÉCEMBRE 1987

which is common to all passes,

(2)

^{means for}

introducing

^and extracting ^{the beam to be}

amplified, (3)

^{means for} suppressing ^or avoiding ^excessive amplified spontaneous emission,

(4)

^{a round}

trip

time which is less than, ^or of the order of, ^{the excited}

state lifetime for the gain ^medium

(e.g., ~

^{2 ns for a} gain ^{medium based on}

optically

^excited

dye molecules), (5)

^{freedom from} excessive distortion of the

pulse

wavefront,

(6)

^{efficient transfer of the}

pump energy ^to the pulse ^{to be}

amplified, (7)

freedom from irreversible temporal broadening ^of

the amplified ^short pulse.

One of the earlier

approaches

^to

multiple

pass

amplification

^{made use of} quarter ^wave

plates

^and

polarizing

reflectors to allow introduction and extrac- tion of the beam ^to be

amplified

^{with no}

spatial

offset on successive _passes

[3].

^This

amplifier

^allowed

collinear

pumping

^{and was} efficient because of the excellent overlap of the pump and the beam to be

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:0198700220120166500

1666

amplified. ^The

amplified pulses

^also exhibited little

wave front distortion because of the on axis optics.

The number of passes which could be

conveniently

used, however, ^was limited, ^{and the use} of transmis- sion optics introduced an undesirable amount of group velocity dispersion.

A simpler way of

introducing

^and

extracting

^the

beam to be

amplified

^is by ^{means of a}

spatial

^offset

of the beam on each traversal. A correctly

imaged

incident beam enters the

amplifier,

^{makes a} pre- determined number of passes, and then exits without need for an externally ^activated switching

mechanism. Several of these

configurations

^{based on}

reflective

optics

^were

explored

ⁱⁿ

prior unpublished

work. More

recently

^{Knox et al.}

[4],

^{have described}

a

multiple

pass

amplifier

^{which uses the}

spatial

^offset

technique

^{with lenses and flat mirrors to achieve a}

six _pass

configuration

^{with a} saturable absorber introduced between the fourth and fifth _passes

through

^the

gain

medium. In this

configuration

^the

spatial

^{offsets are} sufficiently large ^{that individual}

lenses can be used on axis for

focusing.

^This

minimizes astigmatic distortion. While this

amplifier

has been very successful and widely ^{used, it is not}

clear that the particular arrangment of lenses and flats used in that work represents ^an optimal ^mul- tipass arrangement. ^{This current} paper

explores configurations

^which

require

^fewer optical ^elements

and offer different alternatives.

The work most closely ^{related to the}

configura-

tions discussed here is that of Hirlimann et al.

[5].

Hirlimann’s

design

^{uses one}

pair

^of

approximately

confocal

spherical

mirrors and one flat mirror to achieve multiple passes through ^{one focal} region

which is common to all traversals of the three mirror

arrangement. ^This configuration provides ^a simple

and elegant way of obtaining ^a large ^{number of}

passes with ^a short round

trip

delay ^{and also}

eliminates most of the

dispersive

material in the beam path. The confocal mirrors, however, ^collect spontaneous ^{emission from the}

gain

medium suffi-

ciently

^{well that}

amplified

spontaneous ^{emission can} interfere with the

amplification

process. Also the off axis reflections all occur within the same plane ^and

tend to be so

large

^{as to introduce an amount of}

astigmatic

distortion which could be

unacceptable

ⁱⁿ

some

applications.

This current work differs from

previous

^{work in} that, ^not only ^{are the}

optics primarily

reflective, ^but

the

configurations

^{are based on two confocal re-}

sonators, rather than on lenses and flats, ^{or on one}

confocal resonator and a flat. The second resonator in this double confocal

configuration

^serves

multiple

purposes. It

reimages

^{the beam to be}

amplified

^back

into the first confocal resonator with the same

direction and convergence ^as the incident beam, ^but with a

spatial

offset which is

easily adjusted.

^The

second resonator also provides ^{a second} focus which

can be located at the _same, or at a different spatial

location than the first focus. This allows, e.g., introduction of a pass

through

^a saturable absorber after each pass

through

^the

gain

^{as a means of} suppressing

amplified

spontaneous ^emission

[6].

Finally, ^{as we discuss more}

fully

below, ^{we also}

introduce asymmetric ^{confocal mirror}

pairs

^which

can be used to cause a progressive change ^{in the}

beam waists at the two foci.

2. Basic double confocal mirror configuration.

The basic double confocal mirror

configuration

discussed here consists of four

spherical

^mirrors

oriented and positioned ^{as shown in}

figure

^{1. One}

confocal

pair

^{is formed}

by

^two mirrors, ^{Ml and} M2,

which have the same radius, ^{R. Each} mirror in this

pair

^is positioned ^{at its focal} distance F ⁼

R/2

^from

the common focus A shared

by

^{the two mirrors. The}

second confocal

pair

^{is formed} by ^{two other} mirrors,

M3 and M4, which have the same radius R’ as each

other, ^{but not} necessarily ^{the same radius as the first}

confocal pair. ^Each of these mirrors is

positioned

^at

its focal distance F’ ⁼

R’/2

^from

the common

^focus

B shared by this second pair ^of

mirrors.

The mirrors

are

initially

positioned ^so that the mirror centres all lie in one common plane ^{which we define as the}

reference

plane.

The mirrors in the two confocal pairs, ^{Ml and} M2,

and M3 and M4, ^differ from the conventional confocal geometry in that each mirror is rotated by ^a

small angle ^{y, about an} axis which is normal to the reference plane ^and tangent ^to the surface of a given

mirror at its centre. The angle ^{of rotation is chosen}

so that, ^{in the} absence of the further modifications discussed below, ^{the four mirrors} support ^{a stable}

Fig. 1. ^- Double confocal mirror configuration ^{based on} symmetric ^mirror pairs ^{for the case of two foci at two} separate spatial locations. Mirrors Ml and M2 are identical

concave spherical mirrors of radius R which are located at their focal length ^{F =} R/2 from their common focus A.

Mirrors M3 and M4 are two identical concave spherical

mirrors of radius R’ which are located at F’ ⁼ R’/2 ^from

their common focus B. The radius R ^can be the same as, ^or different from R’. A beam incident over the top ^{of mirror} M4 executes multiple passes through the four mirror array

passing throught ^{beam waists at the common focus A and}

at the common focus B on each traversal and finally ^exits

under mirror Ml.

Gaussian mode. This stable mode has two beam waists which occur at the locations marked common

focus A and common focus B. The optical ^{axis for}

this mode is coincident with the solid line

lying

within the region ^bounded

by

the mirror surfaces as

shown in

figure

^{1. We} define the _{ray which} prop- agates

along

^this path ^{to be the reference} ray. As discussed more

fully

^below

multiple

passes in ^struc- tures derived from this

configuration

^{are confined to}

vertical planes ^{which are normal to the reference} plane and which intersect the reference

plane along

the path ^of the reference _ray.

Some astigmatism ^is necessarily introduced by ^the

use of off axis reflections

[7]

^{which occur both in the} plane ^of

figure

^{1 and in} planes ^{normal to the} plane ^of

the figure. For the mirror diameters and radii which

we use these angles are, however,

typically

^{small and}

we find

empirically

^{that the}

astigmatic

distortion is also small. A detailed examination of this astigmatic distortion, especially ^{for the case} of short mirror

radii, ^{would be} useful ; however, ^{such an examina-}

tion is deferred to later work.

This basic mirror configuration ^{can be modified to}

achieve a variety ^of

multipass configurations.

^For

the particular example ^{shown in}

figure

^{1 where the}

mirrors of each confocal

pair

^{have identical} radii, ^the

mirror pairs ^form symmetric ^confocal resonators.

The consequence ^for

multiple

pass

amplifiers

^based

on this

configuration

is that the beam

undergoing

amplification ^{maintains a constant beam diameter}

on successive traversals of the resonator structure.

Siegman,

in his excellent book on lasers, ^has

pointed

^{out that} symmetric ^{confocal resonators}

occupy ^a

singular,

^{or saddle}

point,

ⁱⁿ

stability diagrams

^for optical resonators. Small deviations in different directions move the resonator into stable or

unstable

regions

^{of the}

stability

plane

[8]. By analogy

the double confocal resonator based on symmetric

confocal mirror

pairs

^{which we} discuss here also

occupies ^a

unique

point ⁱⁿ

stability diagrams.

^Small departures of either pair from their symmetric

confocal relation move the resonator into stable or

unstable

regions

^{of the more} complex

stability

space of the double confocal resonator system. ^The changes ^{which we} introduce for the purpose of

creating multipass configurations necessarily

^move

this resonator into unstable

regions

^of operation. ^It is, however, precisely ^this nearly ^{stable, unstable} operation, ^{which allows us to obtain a}

relatively large

^{number of} passes without unacceptable ^wave

front distortion. The all reflective character of the structure also helps ^to minimize the amount of uncorrected group

velocity dispersion.

There are two types ^of

changes

^{which we intro-}

duce. First we rotate the mirrors M2 and M3 by

small angles ^{about axes}

lying

^{in the reference} plane.

The consequence ^{of these} rotations is that a Gaussian beam

imaged

into the array ^over the top ^{of mirror}

M4 executes a

predetermined

^number of passes

through ^two foci which are common to all passes and then exits underneath mirror Ml. The second type of modification, which also moves in the direction of

making ^{the resonator} unstable, but unstable in a

different way, is that of making ^{one, or both, of the}

confocal mirror

pairs asymmetric.

^{This has the}

consequence that the beam waist diameters at the two foci

progressively

^{increase, or decrease, on}

successive passes.

We examine four different double confocal mirror

configurations.

^Each

configuration

supports

multiple

passes through ^{two foci which are common to each}

traversal of the four mirror array. We have con-

structed each of these four

configurations

using

mirrors with diameters of 2.54 cm and radii of

curvature in the range ^{from 10 cm to} 30 cm. The spectrum ^{of the incident} pulses is centred near

0.63 microns and the beam diameters at the mirrors

are of the order of 1 to 2 mm. The

overlap

^of

successive _passes at the foci is verified by

introducing

a flowing

dye jet

^{at a}

given

beam waist location. The luminescence from the dye ^{which is excited} by ^the

beam to be

amplified

^is

easily

^{observed in the}

absence of the pump beam by ^a telescope. ^This

provides

^a

simple

guide ^for

focusing

^{the beam to be}

amplified,

^for

aligning

successive passes, and for

observing the beam waist diameter. The

dye

jet ^is

formed by ^{a nozzle} having

sapphire

plates ^which produce ^a jet ^stream with uniform flatness over the

area of the focused beam

[9]. Experiments involving

gain measurements have been initiated using pulses

of ~ 50 fs duration. We find results consistent with those

expected

^on the basis of calculations ; ^how-

ever, these

amplified pulse

measurements are not

yet complete ^{and we defer a} discussion of them to a

later paper.

3. Double confocal mirror configuration ^{based on} symmetric confocal mirror pairs : ^{two foci at two} separate spatial positions.

To convert the double confocal resonator identified above to a

multipass

arrangement ^{we introduce} rotations of mirrors M2 and M3 by ^{the small} angles ^a

and (3, respectively. ^{These rotations are made about}

axes which lie in the reference

plane

^{and are} tangent

to the mirror surface at the point ^{where the reference}

ray intersects the mirror surface. The results of these rotations is illustrated in figure ^{2 and}

figure

^3.

An equivalent

periodic

lens sequence shown in

figure

2 illustrates the consequence ^{of the} redirection of the incident beam caused by

rotating

^{mirror M2} by ^{the small} angle a ^and rotating the mirror M3

by

the small angle (3. The lenses are identical and

separated ^{from each other} by ^{twice their focal}

length.

^{An incident} beam which is

parallel

^{to the} optical axis of the lens sequence, ^but offset by ^a

1668

Fig. ^{2. -} Ray paths ^{in a} periodic lens sequence wich illustrate the change in ray path ^caused by rotating ^mirrors

M2 and M3 by ^{the small} angles ^{a and} 13, respectively,

about axes lying ^{in the} plane ^of figure ^1. The solid line indicates the ray path ^{for an} incident beam which is

imaged ^{into the} unperturbed lens sequence ^{at a} distance r off axis. The dashed line indicates the path ^{of a} ray which is redirected by ^{a small} angle ^{2 a at} focusing element 2 and

a small angle 2 Q ^at focusing element 3. These angles ^are

chosen so that the redirected beam passes through ^the optical axis of the array ^at the focus of element 3 and arrives at element 4 oriented in the same direction as the incident beam, but offset by ^an increment à in a direction normal to the optical ^axis.

Fig. 3. ^{- Beam} spots ^on the surfaces of mirrors Ml, M2, M3 and M4 for a double confocal multipass configuration

based on symmetric confocal mirror pairs ^as illustrated in

figure ^{1. The beam to be} amplified is incident at a distance

r above the midplane of the array (which ^is identical with the reference plane discussed in the text). ^The offset, 4,

on each pass has been chosen ^{to cause} four traversals before the beam walks out undemeath mirror Ml. Mirror M4 has been lowered by ^{a distance 4 so as to allow the}

incident beam to pass ^over M4 and mirror M3 has been raised by ^{a distance 4 so that it} intercepts all four passes.

The numerals on the mirror faces indicate the spot ^formed

on pass 1, ^{etc. Note} that the beam passes through ^the midplane, and hence the common focus, between mirrors Ml and M2, ^and again between mirrors M3 and M4. The

numbering ^{for the next} sequence is initiated ^at mirror M4.

distance r in a direction normal to the

optical

axis,

passes through ^{the lens} system ^{as indicated} by ^the

solid line in figure ^{2. This beam crosses the} optical

axis at

points

^A and B which lie,

respectively,

^{at the}

focus of lens 1 and at the focus of lens 3 in the sequence.

Without

specifying

^{the means of}

redirecting

^the

beam in this lens sequence ^we observe that a change

in direction at lens 2 by ^{2 a can be made to cause the}

beam to arrive at lens 3 with an altered slope ^{and a}

shift toward the optical ^axis by ^an increment à

[10].

The redirection of the beam at lens 3 by ^the angle

2 03B2 ^can then be chosen so that the ray ^crosses the

optical ^{axis at} point ^{B and}

finally

^{arrive at the}

beginning

^{of the next series} of four lens sequence

(lens

^{l’ in the}

figure)

oriented in the same direction

as the incident beam, ^{but offset} by ^{an increment}

delta in the direction of the optical axis. The dashed line indicates the

trajectory

^{of the} redirected ray.

Successive passes of the beam

through

^{the lens}

system ^{will be offset} by ^an additional increment 4 until the beam eventually ^{walks out} of the lens sequence.

To achieve the

corresponding multipass configura-

tion in the double confocal mirror arrangement

under discussion we lower mirror M4 by ^{an incre-}

ment of order d and raise mirror M3 by ^{a similar}

increment. We then introduce the incident beam

over the top ^{of mirror M4 in a direction} parallel ^to

the segment ^{of the reference} ray which lies between mirror M4 and mirror Ml, but offset from the reference plane by ^{a distance r.} As indicated in

figure 3, ^{the beam} spot ^on mirror Ml is also offset from the reference, ^or

midplane,

by ^{a distance r.}

The positions of the other beam spots ^{on the four} mirrors as the beam passes through ^{the four mirror}

array ^are shown in

figure

^{3. The} principal consequ-

ence of the small angular rotation of mirror M2 by ^a

is that, ^{on a} given pass, the beam arrives at mirror M3 shifted in the

upward

^direction by ^{an increment}

4. The consequence of the rotation of mirror M3 by f3 is that the beam passes through ^the reference, ^or midplane, ^{as the} point ^{B and arrives at mirror M4}

with the same orientation and direction as it would have had in the absence of the rotations a and 03B2, however, ^{with a} shift toward the reference plane ^of

magnitude

^{d. Since the} second pass

begins

^{at this} point ^{we label that} particular ^beam spot by ^a

numeral 2. The beam then continues in a direction

parallel ^to the reference plane ^until it reaches mirror Ml at the beam spot labelled 2. The beam continues

repeating ^this pattern ^{and in} doing ^so produces ^the

set of the beam spots ^on the mirrors as shown in

figure 3. The beam

finally

exists underneath mirror Ml.

One important ^use of the second focus in a structure of this type ^{is as a} location for a saturable absorber which will suppress

amplified

spontaneous emission. In such a case a shorter radius would

presumably be used for the mirrors

surrounding

^the

absorber so as to obtain a smaller beam waist at the absorber as

compared

^to the beam waist in the

gain

medium. As these mirror radii become short

astig-

matic distortion can begin ^{to be a}

problem.

^{For cases}

where a very small beam waist is

required

it may be

preferable ^{to introduce} pairs of small lenses arranged

in an on axis confocal geometry ^for each of the

spatially

^offset passes. We defer ^{a more} complete

discussion of

strategies

^{such as this to} later work.

The properties of the beam to be

amplified

^{can be}

calculated at any point ^{in this structure} by

applying

ray matrix

techniques

and the ABCD law for Gaussian beams

[6].

^We have, e.g., written a

computer program which uses ray matrices to trace the path ^{of an} incident beam

through

^{these double}

confocal structures. For the cases discussed here the calculations are

sufficiently simple

^{that the results}

can

simply

be stated. The

principal point

^{of interest}

is that if the incident beam has a large ^confocal parameter ^{and a beam waist at mirror} Ml, ^{the beam}

waists for each pass will, ^{to a} good

approximation,

occur at the common foci A and B on each pass.

4. Double confocal multipass configuration ^{based on} symmetric confocal mirror pairs : ^{two foci at one}

common point.

For some applications ^{it can be}

preferable

^{to have}

the beam to be

amplified

^{focused twice at one} point

on each round trip, rather than once at each of two

spatially separate

points.

^Minor changes ^{in the}

mirror

positions

^{and the manner in which the beam}

is

imaged

^into the four mirror array

produce

^the

configuration

^{shown in}

figure

4. Here the beam passes through ^{two different} foci which are located at one common

point.

^This allows, e.g., ^two passes

through

^a gain ^{medium on} each round trip. ^{Such an} arrangement might be desirable for the case of a

preamplifier

having ^{a low} gain ^medium.

Fig. ^{4. -} Double confocal multipass configuration ^based

on symmetric ^confocal pairs ^{for the case of two foci at one} point in space. In this example all four mirrors have the

same radius R and hence the same focal length, ^{F =} R/2.

Each of the four mirrors is positioned ^{at its} focal length, ^F,

from one common focus and all are rotated by ^an angle y.

The beam to be amplified ^{is incident} parallel ^{to the axial}

segment lying ^{between two mirrors which are not} diagonal- ly opposed. An incident beam with a large ^confocal parameter ^{and a} beam waist at mirror Ml has two smaller diameter beam waists which both occur at the common

focus.

For the configuration ^{shown in} figure ^{4 the beam}

to be

amplified

is introduced so that it is parallel ^to

the segment of the reference ray

lying

^{between two}

mirrors which are not

diagonally

opposed. ^{In this} figure ^that segment lies between mirrors M4 and Ml. The mirror radii and location are also chosen so

that each

diagonally opposed pair

^{satisfies a} symmet- ric confocal relation. It is not necessary that the radius for one confocal pair ^{be the same as the radius}

for the other confocal

pair ;

however, ^{there is no} obvious advantage ^to

having

^{the radii} different, ^so

we illustrate the case where all four mirrors have the

same radius, ^{R. The mirrors are each rotated} by ^a

small angle y about ^{an axis normal to the reference} plane ^and tangent ^to the mirror surface at the

point

intersected by ^{the reference} ray. The four mirrors

are all

positioned

^{at their}

focal length

^{F =}

R/2

^from

the common focus.

Again

mirrors M2 and M3 are

rotated by ^small

angles

^{a and} /3 ^{such that the beam}

executes a series of passes which produce ^{a beam} pattern ^{on the mirror faces as shown in}

figure

^{3. The}

beam pattern remains the same on each mirror of a

given ^number. Note, however, that the mirrors of a

given number have different positions ^{relative to}

each other in

figure

^{4 as}

compared

^to

figure

^1.

5. Double confocal multipass configuration ^{based on} symmetric confocal mirror pairs : ^{two foci at one} point in space.

It is also

possible

^{to construct}

multipass configura-

tions in which the beam waists in the

configuration

increase, ^or decrease,

progressively

^{on successive}

passes. Consider the case of one asymmetric ^confocal

pair,

^{and one}

symmetric

pair, of mirrors

sharing

^one

common focus as shown in figure 5. We have taken the

configuration

^{shown in} figure ^{4 and modified it} by replacing mirror Ml with a mirror which has a

different focal length FI ^{from the}

focal length

^{of the}

other mirrors, M2, ^M3 and M4. Let the other mirrors all have the same focal length ^{and define it}

to be F2. ^For FI ^{different from} F2 the four mirror array is unstable and the

amplified

^{beam will}

change

diameter on each pass.

A case of special interest for a multipass

amplifier

is this configuration ^{for the situation where} FI ^is

Fig. ^{5. -} Double confocal multipass configuration ^based

on asymmetric confocal mirror pairs ^{for the case of two}

foci at one point in space. The diagonally opposed ^mirrors,

Ml and M2, and M3 and M4, ^{need not} have identical radii. For the case illustrated M3 and M4 have the same

radius, ^{and the} focal length, Fl, ^{of Ml is} greater ^{than the} focal length, FZ, of M2. The beam diameter at the mirror surfaces decreases by F21F, ^on successive passes and the beam waists at the common focus increase by the inverse ratio.