Misalignment sensitivity of the cat’s eye cavity He–Ne laser

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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


    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


    (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




    T M1 M3

    W W



    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




    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



    R L1 R2


    O2 O1

    M1 M




    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



    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



    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


    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




    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:


    itivity of the conventional resonator.d <