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ireye cavity and conventional resonator are both analyzed in geometric method and matrix optics with misalignment sensitivity parameter.Valuable conclusions are drawn: in full-external HeNe laser, cats eye cavity can improve the laser stability up to about 60 times better

always concerned, which is very serious especially in HeNe lasers, due to their correspondingly low gain. When

to settle the laser stability problem, the cats eye cavitywas introduced in Li and Smiths laser [4]. Some valu-

application very limited [6]. With cylindrical lens andgrating cats eye resonator was also equipped in an

has been accepted and applied in some elds, its theoreticanalysis has not been reported in detail. In this paper wecalculate and analyze the misalignment sensitivity of thecats eye cavity compared with conventional resonator soas to apply theoretic foundation for the design and applica-tion of cats eye cavity lasers.

* Corresponding author. Tel.: +86 1062 788120; fax: +86 1062 784691.E-mail address: xuzhiguang99@mails.tsinghua.edu.cn (Z. Xu).

Optics Communications 265there is subtle disturbance on resonator mirrors, the laserpower output will vary greatly, even disappear. So somekinds of useful means need to be found to solve this prob-lem in HeNe lasers.

Cats eye reector is now usually applied in interfer-ometer systems as an auxiliary component to reect lightback [13]. Recent years there have been some eorts toapply the cats eye reector as resonator mirror. As amethod to select laser transverse mode, not an approach

external-cavity semiconductor laser [7]. Recently weapply cats eye cavity into HeNe lasers and nd won-derful application prospects. Comprehensive experimentsare carried out both in a half-external cavity and afull-external cavity HeNe laser [8]. Furthermore weset up a new convenient real-time means for thecontrol and selection of the laser transverse mode bycats eye cavity [9].

Although the stability advantage of the cats eye cavitythan the conventional one; diraction loss introduced by the misalignment of the cats eye cavity attributes to the straight-line displace-ment vertical to the laser bore of the cats eye reector; and with the convex lens center of the cats eye reector secured immobile, theultra-stable and adjustment-free cats eye cavity HeNe laser is obtained. The analysis matches the experiment results very well. Cavitieswith three kinds of dimension errors are also calculated. This paper could be used as theoretic foundation for the design and applicationof cats eye cavity lasers. 2006 Elsevier B.V. All rights reserved.

PACS: 42.55.Lt

Keywords: HeNe laser; Cats eye cavity; Cats eye reector; Misalignment sensitivity

1. Introduction

The misalignment problem of laser resonators has been

able research was done by Dimakov in a CO2 laser [5],but the requirement for extremely high accuracy in man-ufacture procedure of conic optics components makes itsMisalignment sensitivity of th

Zhiguang Xu *, Shulian Z

The State Key Laboratory of Precision Measurement Tech

Tsinghua University,

Received 18 January 2006; received in revised fo

Abstract

A concave mirror and a cats eye reector acting as a resonator m0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.optcom.2006.02.047cats eye cavity HeNe laser

ng, Wenhua Du, Yan Li

gy and Instruments, Department of Precision Instruments,

ijing 100084, China

17 February 2006; accepted 22 February 2006

ror form the cats eye cavity. Misalignment sensitivities of the cats

www.elsevier.com/locate/optcom

(2006) 270276

2. Cats eye cavity

For a long time, the long HeNe lasers are alwaysconstructed in full-external conguration for adjustmentconvenience. Because two resonator mirrors are both sepa-rate with the gain tube, the stability problem is more criti-cal, especially in bad environment. The longer the laser is,the more sensitive to the tilt of its reecting mirror it is.Therefore we built up a cats eye cavity in a full-externalHeNe laser to display its stability advantage.

The conguration of a full-external conventional reso-nator is shown in (1) of Fig. 1. There is a window platesW adhered on every end of the gain tube T. The concaveoutput mirror M1 (radius of curvature R1) and a conven-tional resonator reector M2, i.e. a plane or concavemirror, form the laser resonator. Draft (2) of Fig. 1 shows

h2 R1 Lu 3here h1 should be mainly considered because h1 > h2. UsingEqs. (1) and (2), we can know that:

h1 R1R2hR1 R2 L 4

To make sure the operation of fundamental mode in theresonator, the following condition must be satised [10]:

h1 6 D2 p

px1 5

where D represents the diameter of laser bore, and x1 is thelight spot size on M1 which has the form [11]:

x1 kLp

rR21R2 L

LR1 LR1 R2 L 1=4

r2 1=4

Z. Xu et al. / Optics Communicthe structure of the cats eye cavity, in which a cats eyereector M3 displaces M2 to form the laser resonator.Our cats eye reector is composed of a convex lens (withreection reducing coating on both surfaces) and a concavemirror (with high-reection coating). The focal length ofthe convex lens, the radius of curvature of the concavemirror, and the distance between the two elements are allequal. Since the diameter of our HeNe laser bore is verysmall, no need to consider spherical aberration, an einzellens is selected as the convex one. We mount the wholecats eye reector in a mechanic component to make itfunction as a resonator mirror.

Obviously, normal incident paraxial beam will bereected back by our cats eye reector along theentrance way. Even for the obliquely incident paraxialbeam our cats eye mirror can still provide high parallel-ism for the incident and the reected beam, which isimpossible for any traditional laser resonator mirror(Fig. 2). That is why our cats eye reector acting as aresonator mirror can improve the laser stabilityif thereis tiny disturbance which make the mirror sway for asmall angle, this advantage can help reduce the banefulinuence.

T M1

W

M2

W

T M1 M3

W W

(1)

(2)

Fig. 1. A full-external HeNe laser with a plane mirror and a cats eyereector as the reecting mirror, respectively.3. Geometric method to analyze the misalignment sensitivityof the cats eye cavity and conventional resonator

3.1. Conventional resonator

Take the conguration (1) of Fig. 1 as an example, inwhich the curvature radius R1 of output mirror M1 is3000 mm, and M2 is a plane mirror. The diameter of laserbore is about 3.2 mm and the whole resonator length L is1100 mm. As is shown in Fig. 3, line O1O2 designates theoriginal resonator axis when there is no misalignment,and M2 is tilted by an angle h with respect to line O1O2in the paper plane. C1 and C2 are, respectively, the spherecenters of M1 and M2 which can be considered a concavemirror with innite curvature radii, and the new resonatoraxisline C1C2 has an included angle u with line O1O2.The misalignment displacement of the intensity patternon mirrors of M1 and M2 are denoted severally by h1 andh2.

It can be easily obtained that:

R2h R1 R2 Lu 1h1 R1u 2

(2)

(1)

(3)

Fig. 2. Obliquely incident paraxial beam with angle a to a cats eyereector (1), a plane mirror (2) and a concave mirror (3).

ations 265 (2006) 270276 271 kLp

R1LR1 L 6

to the laser bore will be d when the reector has a

C1

R1

R L1 R2

C2

O2 O1

M1 M

2

+

sens

272 Z. Xu et al. / Optics Communications 265 (2006) 270276misalignment angle h in the paper plane. Lines A and Brepresent, respectively, top and bottom brims of the laserbore, whose symmetrical images about point O areexpressed as A 0 and B 0.

The transformation matrix of the cats eye reector is:

1 0 The combination of Eqs. (4)(6) leads to the largest mis-alignment angle hmax:

hmax D=2p

px1R1 R2 LR1R2

D=2p

px1

R1 0:46 min 7

3.2. Cats eye cavity

Fig. 4 is the general structure draft of the cats eye cav-ity composed of the concave output mirror M1 and a catseye reector M3 wholly assembled in a mechanic compo-nent G with the xed point P. Distance between point O(the center of convex lens in the cats eye reector) andthe xed point P is dened as q being 15 mm in our exper-iments, and the misalignment distance of point O vertical

R2

h1

Fig. 3. Analysis of the misalignmentT 0 1 8

Therefore the incident beam will be reected back by ourcats eye reector along the entry direction and the incidentand reected rays are symmetrical about point O. That isto say the incident ray along the up brim A of laser borewill return back along A 0, and identically the incident ray

Fig. 4. Analysis of the misalignmentalong B will be reected back along B 0. Considering theright draft in Fig. 4, there are only rays in section Sc canreturn back into the laser bore when rays in section S0 enterthe cats eye reector.

The single-trip loss factor d introduced by the cats eyereector is [12]:

d I0 Ic2I0

S0 Sc2S0

9

When h is small enough and equation d = qh is satised,Eq. (9) can be rewritten through simple geometrical opera-tion as:

d 4D=2d2pD=22

4D=2qh2pD=22

4qhpD

10

Single-trip gain G in the laser cavity follows that famousempirical formula:

G 3 104 LD 3 104 1100

3:2 0:103 11

The whole transmission, absorption and scattering lossesof the output mirror and window plate in a single-tripare calculated to be 0.013, therefore to make the laser radi-ate d must meet the equation:

h2

itivity of the conventional resonator.d < G d0 0:050 12The largest misalignment angle h0max of the cats eye reec-tor is obtained as:

h0max pDG d0

q 28:80 min 13

sensitivity of the cats eye cavity.

h k k

unications 265 (2006) 270276 273The maximal adjustable angles of the resonator mirrorare used to indicate laser stability directly and convinc-ingly. Comparing Eq. (13) with (7) we obtain that the catseye cavity makes the stability of the full-external cavityHeNe laser improved about 63 times better than theconventional one.

4. Matrix optics method with misalignment sensitivity

parameter to evaluate the misalignment sensitivity of two

resonators

Now we make another precise analysis with matrixoptics method.

4.1. Cats eye cavity

Set up the before-misalignment and after-misalignmentcoordinate systems. Now suppose an incident ray with anoptical vector of [x1 h1]

T in the before-misalignment coor-dinate system reaches the cats eye reector. With theassumptions of paraxial approximation and small mis-alignment, optical vector of the incident ray is transferredto:

x01h01

x1

h1

dh

14

in the after-misalignment coordinate. Then the emergentray reected by the reector follows that:

x02h02

1 0

0 1

x01h01

15

Transforming the upper vector back into the before-misalignment coordinate system we obtain:

x2h2

x

02

h02

d

h

16

then:

x2h2

1 0

0 1

x1h1

dh

d

h

1 00 1

x1h1

2 0

0 0

d

h

17

Therefore the misalignment augmented matrix of the catseye reector can be expressed as:

x2h21

1

26664

37775

1 0 2d 00 1 0 00 0 1 0

0 0 0 1

26664

37775

x1h11

1

26664

37775 18

Assume the misalignment linear and angular displace-ments on the cats eye reector are, respectively, x11 andh11, and those on the concave output mirror are x21 andh21. First, with the cats eye reector as the reference initialposition, the ray will represent itself after a round trip,

Z. Xu et al. / Optics Commwhich is written as:x11h111

1

26664

37775

1 0 2d 0

0 1 0 00 0 1 0

0 0 0 1

26664

37775

1 L 0 0

0 1 0 0

0 0 1 0

0 0 0 1

26664

37775

1 0 0 0

2=R1 1 0 00 0 1 0

0 0 0 1

26664

37775

1 L 0 0

0 1 0 0

0 0 1 0

0 0 0 1

26664

37775

x11h111

1

26664

3777519

One readily obtains x11 = d. Then with the concave outputmirror as the reference starting position, we will havesimilarly:

x21h211

1

26664

37775

1 0 0 0

2=R1 1 0 00 0 1 0

0 0 0 1

26664

37775

1 L 0 0

0 1 0 0

0 0 1 0

0 0 0 1

26664

37775

1 0 2d 00 1 0 00 0 1 0

0 0 0 1

26664

37775

1 L 0 0

0 1 0 0

0 0 1 0

0 0 0 1

26664

37775

x21h211

1

26664

3777520

The solution is obtained as x21 = dR1/(R1 L).Introduce the concept of misalignment sensitivity

parameter U, which is a number characterizing any reso-nator with respect to its sensitivity against mirror tilting,and is totally determined by the resonator congurationparameters. High value of U means large diraction lossintroduced and high misalignment sensitivity [13]. The Uin our case now is:

U 1h

x11x1

2 x21

x2

2" #1221

where x1x2 denote the light spot sizes on the cats eyereector and output mirror, which are given as [11]:

x1 kLp

rR22R1 L

LR2 LR1 R2 L 1=4

kLp

rR1 L

L

1=422

x2 kLp

rR21R2 L

LR1 LR1 R2 L 1=4

kLp

rR21

LR1 L 1=4

23

Substituting Eqs. (22) and (23) into (21) we obtain:

U d p 1

2 2R1 L23

" #14

q p 1

2 2R1 L23

" #14

24

LR1 L LR1 L

5.1. Experiments

The two experiments in Figs. 5 and 6 are to record thepower output of full-external HeNe lasers applying twokinds of resonators while rotating their reecting mirrorscontinuously in two vertical directions: up-and-down andright-and-left directions. In conventional resonator fromthe maximal power to the minimal the adjusting anglesin two directions of the plane mirror M2 are both about1.0 minute, while those of the cats eye reector M3 areabout 60 min in cats eye cavity, about 60 times of theother, which coincides with the theoretical calculationvery well.

5.2. Adjustment-free HeNe laser experiments

All the analysis above is based on the common congu-ration of cats eye cavity. Now we design a new specialmechanic retainer for the cats eye reector xing it juston the position of the convex lens center to make sureq = 0 mm. Fig. 7 shows the laser power when rotatingthe cats eye reector from 15 to 15 with the convex lenscenter assured immovable, which indicates little uctuationin the whole process.

This conclusion implies an important method for theachievement of ultra-stable and adjustment-free cats eye

unications 265 (2006) 2702764.2. Conventional resonator

Now, consider our second case of misalignment by planemirror M2 with tilting angle h in the conventional resona-tor in (1) of Fig. 1, where the misalignment augmentedmatrix of the plane mirror is given by:

1 0 0 0

0 1 0 2h0 0 1 0

0 0 0 1

26664

37775 25

Presuming the misalignment linear and angular displace-ment on the plane mirror are, respectively, x011 and h

011,

and those on the concave output mirror are x021 and h021,

in the same means we obtain that: x011 L R1h andx021 R1h. Then the value of misalignment sensitivityparameter U 0 in the conventional resonator is further givenby:

U 0 pk

12 2R1 L2R1 L

L

" #14

26

The rst important conclusion can be readily drawn bycomparing Eq. (24) with (26):

UU 0

qR1 L 27

In our actual application q is much less than R1 L,so in cats eye cavity the diraction loss introduced bymisalignment is less than in traditional resonator quite afew. Although this ratio does not represent the powerratio of two resonators, for there are many other factorsrelative to the loss in laser needing to be considered, atleast we can assure that the cats eye cavity is not so sen-sitive to misalignment and much more stable than theconventional.

The second valuable conclusion concerning the cats eyecavity is based on...