Self-reproducing ray-cluster and misalignment of gas laser resonator having a coaxial discharge chamber

ARTICLE IN PRESS Optics & Laser Technology 41 (2009) 217–223

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Self-reproducing ray-cluster and misalignment of gas laser resonator having a coaxial discharge chamber Yude Li a,, Mei Chen a, Jun-ping Guo a, Jing-lun Liu a, Yuan-jie Yang a, Zheng Li b, Shenggen Ju b a b

Department of Optoelectronic Science and Technology, Sichuan University, Chengdu, Sichuan 610064, China Academe of Computer Science of Sichuan University, Chengdu, Sichuan 610064, China

a r t i c l e in fo

abstract

Article history: Received 7 May 2007 Received in revised form 18 February 2008 Accepted 8 July 2008 Available online 12 September 2008

A laser resonator with two annular spherical mirrors is studied. The self-reproducing mode of this resonator is established by means of a self-reproducing ray-cluster. Its existence condition and area are analyzed and provided for the first time. The resonator’s adjusting accuracy is analyzed in detail and the relevant calculation for typical cases is illuminated. The tolerance of mirror center displacement from the axis of the coaxial discharge chamber is proved to be about 0–0.15 mm, and the tilt tolerance 0–600 , when resonator length LE1 m, the radius of curvature of the two mirrors rE5–10 L, and the loss of applicable radial width of the coaxial discharge chamber for light propagation is smaller than 1 mm. The object–image relation of the resonator and its application in adjustment of the resonator are also discussed. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Geometrical optics Laser resonators Ring

1. Introduction A coaxial discharge chamber is used to obtain high-power laser [1–5]. The active region of the discharge chamber is located in an annular gap between two tubes of radius R1 and R2, respectively. The annular gain CO2 laser is easy to realize diffusion cooling because of the large discharge surface [3,6,7]. Output powers of 2 kW have been obtained [3] from the gain length of 1.8 m. Also, the coaxial discharge chambers may be suitable for other gas media such as He–Ne media to obtain laser output, but their dimensions and resonators must be applicable to these gas media. In flash-lamp-pumped solid-state lasers [8], high excitation efficiency can also be achieved by inserting a flash lamp into the center of the annular gain medium. However, annular geometries are not compatible with simple resonators [3], and new resonators have to be found in order to extract a high-quality beam efficiently. Early attempts employed resonators consisting of two annular mirrors enclosing an annular Nd:glass, dye, or CO2 gain medium [9–11]. One of the mirrors was a coupling mirror, and an annularly shaped beam was extracted. Their drawbacks were either poor beam quality or high alignment sensitivity. Other researches used unstable resonators that utilized conical mirrors to create a compacted beam region inside the resonator cavity. These resonators include the half-symmetric unstable resonator with internal axicon [12] and the more advanced common pass decentered annular ring resonator [13]. They all have in common

 Corresponding author.

E-mail address: [email protected] (Y. Li). 0030-3992/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2008.07.001

an increased number of optical elements and strict alignment requirements. Still further, annular converging wave cavity [14], toric unstable resonator [15,16], as well as unstable [17] and stable multipass resonators [18,19] were studied. The annular converging wave cavity suffered from parasitic oscillation, and the toric unstable resonator lacked azimuthal mode discrimination. A disadvantage of the multipass resonators is the low extraction efficiency [3] because of the low mode filling factor. A stable–unstable resonator has proved suitable for annular gain areas to achieve high extraction efficiencies and good beam quality [3]. However, this resonator has three disadvantages: (i) Toric mirrors of the resonator are difficult to manufacture. (ii) The beam width and the angle between the two output beams at the coupling aperture increase with mirror tilt (see Fig. 5 in Ref. [3]), that is, the width and angle could be changed easily under operation of the laser. (iii) The beam quality is defined owing to (ii) and extracting laser output through the coupling aperture at one of the toric mirrors. In Ref. [3], experiments using a spherical unstable resonator have not yet been carried out. A Talbot cavity has been considered for extraction of highquality beams from narrow annular laser sources [5]. Such solutions and conditions are based on the adoption of a Talbot cavity to select low cavity losses, a single azimuthal distribution and a phase-filtering process. However, this method is very complex in technology. In this paper, we demonstrate that an eigenmode exists in a simple optical cavity composed of two coaxial opposite annular spherical mirrors separated by a distance L, the annular gain medium is compatible with the simple laser resonator, and the

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three disadvantages of the stable–unstable resonator [3] can be avoided by the simple laser resonator. We also give the existence condition of the eigenmode in a simple cavity. In this article, a self-reproducing mode of this resonator is described by means of a self-reproducing ray-cluster. According to the presentation of the self-reproducing mode, the resonator serves as a ring resonator if a partial surface of one of the two annular spherical mirrors is an output coupling film. In comparison with the good beam quality [3] of the stable–unstable resonator, this ring resonator can offer a good beam quality and a fixed small angle between the two small branches of output beam at the coupling section, and obviously the spherical mirrors are convenient to manufacture than the toric mirrors and the other aspherical mirrors. According to the existence condition of the self-reproducing ray-cluster and its area, based on the analysis on structure of the spherical resonator (especially including the ring resonator), we find that the efficient cross-section size of optical propagation zone in the resonator is dependent on the cavity parameters and tilt tolerances of the resonator mirrors. At last, we conclude that the analyses on the self-reproducing ray-cluster and structure of the resonator are significant for the design of the annular spherical-mirror resonator to benefit from the advantages of annular gas gain media, including media with lower gain.

Fig. 2. The conventional two-mirror resonator.

If  Al Det C

B Dl

¼0

then

l ¼ expðiyÞ

(4)

where

y ¼ arccos

AþD ¼ arccosð2g 1 g 2  1Þ 2 0

(5)

0

Therefore, in M 1M 2 resonator the eigen self-reproducing raycluster [20] with eigenvalue exp(7iy) can be given by !   ðrÞ0 r (6) ¼ expðiyÞ 0 ; 0prpR2 ðat M0 1 Þ 0 0 ðr Þ r

2. Resonator stability and eigen ray-cluster 2.1. Eigen ray-cluster The resonator formed by two annular spherical mirrors is shown in Fig. 1 and the conventional two-mirror resonator is shown in Fig. 2. r1, r2, r01, and r0 2 are the radii of curvature of the mirrors. R1 and R2 are the inner and outer radii of the annular mirrors, respectively; then d ¼ R2  R1



(1)

where d is the size of the annular gap of the discharge chamber. L and L0 are the lengths of the resonators. It is assumed that a light ray starting from M1 (or M0 1) is bounced back and forth between M1 and M2 (or M0 1 and M0 2). Let r be the ray displacement from the optic axis in the same plane of mirror M1 or M0 1, r0 be the ray   r be the initial ray-cluster [20]. Let r0 1 ¼ r1, slope, and r0 r0 2 ¼ r2, L0 ¼ L, g0 1 ¼ 1(L0 /r0 1) ¼ g1, g0 2 ¼ 1(L0 /r0 2) ¼ g2, and 0pg1g2p1. The ray transfer matrix of the conventional twomirror resonator is represented by ! !   1 0 1 L  1 0 1 L A B   (2) ¼ 2 r2 1 2 r1 1 0 1 0 1 C D Then, the eigenvalue l shall be determined by the formula !      ðr 0 Þ r A B r ¼l 0 (3) ¼ 0 0 0 ðr Þ r C D r

Fig. 1. The resonator composed of two annular spherical-reflective mirrors.

This result is coincident with that the stable electromagnetic mode exists in the resonator. Obviously, there exists a special selfreproducing ray-cluster in the M1M2 resonator, which is a part of the self-reproducing ray-cluster of M0 1M0 2 resonator and can be expressed as !   ðrÞ0 r ¼ expðiyÞ 0 (7) R1 prpR2 ðat M1 Þ 0 0 ðr Þ r

2.2. Existence condition of self-reproducing ray-cluster A trivial limit in the use of spherical mirrors with an annular bounded gain-region is that the spherical mirrors do not prevent the light rays from crossing (projecting on) the internal cylindrical surface of radius R1. If the spherical mirrors enforce projecting on the internal surface upon all light rays in the gap, no selfreproducing ray-cluster in the M1M2 resonator could exist. However, this problem can be solved (see Fig. 3), because at least an eigen self-reproducing ray-cluster does not cross the internal cylindrical surface of radius R1. A ray of this eigen ray-cluster walks [21] in a circle of radius r ¼ r0 (r04R1) around mirror 1 with the angle y between successive bounces, where r0 is a constant. Using Fig. 3, we can explain that the rays of the eigen ray-cluster do not cross the internal cylindrical surface of radius R1. In Fig. 3, the planes ABCD and AEFD are two tangent planes of the internal cylindrical surface, and they are tangential at the lines GH and IJ, respectively. The ABCD and the AEFD cross at line AD and GHJIJJAD. Obviously, this ray of the eigen ray-cluster goes from K to M after being reflected at D by the right spherical mirror; the ray does not cross the internal cylindrical surface of radius R1 because KD and DM lie on tangent planes ABCD and AEFD, respectively. In general, if rXOK or R0 1, rays travel along lines which lie on the plane OKD and are parallel to the line KD; then these rays go along lines which lie on the plane OMD and are parallel to the line DM after being reflected by the right spherical mirror, and do not cross the internal cylindrical surface. Then, in

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219

where R0 and R0 0 are initial ray displacement and slope, respectively and y is the ray rotation angle [21] between successive reflected points at M1 (see Fig. 1). Eq. (15) can be changed as ( ðR0 Þ1 ¼ ðR0 Þ0 expðiWÞ (16) ðR00 Þ1 ¼ ðR00 Þ0 expðiyÞ

Fig. 3. Diagram explaining that light rays do not cross the internal cylindrical surface when rXR1/cos(4).

general, the self-reproducing ray-cluster can be expressed as !   ðrÞ0 r y Þ (8) ; R0 1 prpR2 ðat M1 Þ ¼ expði 0 ðr 0 Þ r0 where R0 1 ¼ R1



cos

y 4

¼ R1



cos

1 arccosð2g 1 g 2  1Þ 4

(9)

Since at a position halfway between the mirrors the rays define a circle of radius [21] rc ¼ r

  g 1 þ 2g 1 g 2 þ g 2 1=2 4g 2

(10)

letting

we can get 

4g 2 g 1 þ 2g 1 g 2 þ g 2

1=2

(11)

If g1 ¼ g2, we get r ðminÞ ¼ R1



2 1þg

1=2

¼ R0 1

(12)

and Eqs. (9) and (12) are coincident in numerical calculation. Obviously, when the resonator is aligned strictly we have 0

d ¼ R2  R0 1

If closure condition ny ¼ 2pm (n and m are integers) is satisfied, the formula in Eq. (17) becomes ( ðR0 Þn ¼ ðR0 Þ0 ¼ R0 (18) ðR00 Þn ¼ ðR0 Þ00 ¼ R00 Therefore, after achieving n round-trips, the ray reappears by itself under the initial condition at a starting point. The annular-mirror resonator is coincident with a ring ! ! ðR0 Þn ðR0 Þ0 to given by resonator if the folded ray from ðR0 0 Þ0 ðR0 0 Þn Eq. (18) is the axis of a folded light beam in the resonator, and the partial surface including the start point of the initial ray on M1 is a coated output coupling film, because no standing waves are associated with this resonator [6]. In this case (see Fig. 3 and note

r cðminÞ ¼ R1

r ðminÞ ¼ R1

where (R0)0 and (R0 0)0 are the initial ray displacement and slope, respectively. The ray is bounced back and forth between M1 and M2. (R0)1 and (R0 0)1 are displacement and slope at M1 on the first round trip, respectively. In general, (R0)n and (R0 0)n are the ray displacement and slope at M1 on the nth round trip, respectively, and they can be expressed by ( ðR0 Þn ¼ R0 expðinyÞ (17) ðR00 Þn ¼ R00 expðinyÞ

ok ¼ R0 ) the ray transfer matrix of the ring resonator is given as !n   a11 a12 A B (19) ¼ a21 a22 C D where a11 a21

a12

!

a22

(13)

¼ 0 ¼@

0

where d is the applicable radial width of the coaxial discharge chamber for light propagation. If cos(y/4)p(R1/R2) in Eq. (9), i.e., R0 1XR2, then there is no selfreproducing ray-cluster in the M1M2 resonator. Therefore, the self-reproducing ray-cluster existence condition of the spherical annular-mirror resonator can be given by cos

1 R1 arccosð2g 1 g 2  1Þ4 4 R2

(14)

It is easily seen from Eq. (14) that the g (g1 ¼ g2 ¼ g) parameter whose value approaches 1 is suitable to the spherical annularmirror resonator. 2.3. Ring resonator According to geometrical optics, the middle rays of the raycluster in the range R0 1prpR2 are located in the circle of radius R0 ¼ ðR01 þ R2 Þ=2 at M1 (or M2). When r ¼ R0, Eq. (8) can be changed as ( ðR0 Þ0 ¼ R0 expðiyÞ (15) ðR00 Þ0 ¼ R00 expðiyÞ

1

L0

0

1

!

1

0

2=r 0

!

1 0 2

1 0

2

L0

!

1

1

2=r

0

0 2

1  6L =r þ 4ðL Þ =r

2L  2ðL Þ =r

4=r þ 4L0 =r2

1  2L0 =r

0 1 A

!

1 (20)

r1 ¼ r2 ¼ r L0 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

L2 þ AK ;

AK ¼ 2R0 sin

y 4

,

and ny satisfies the closure condition. For a TEMoo Gaussian beam of this ring resonator, the beam size w and radius rout of the curvature of the equiphase surface at the output coupling surface can be written as w¼



rout ¼

2lB

1=2

p 2B DA

½4  ðA þ DÞ2 1=4

(21)

(22)

l is a laser wave length in Eq. (21). The size w and the radius of curvature of the equiphase surface of the folded beam at every reflection place on the annular mirrors are the same as those at the output coupling surface, respectively, because ray transfer matrices with reference to the every reflection place are the same. For the same reason, the waist of the folded beam between reflection places on M1 and M2 is situated at the middle place

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between the two reflection places. The waist size is given by  1=2 l woo ¼ ½4  ðA þ DÞ2 1=4 (23) 2pjCj Then the far-field divergence angle (yN ¼ (l/pwoo)) may be obtained, and the resonator can offer a good quality for one branch of the output beam. The angle between the two branches of the output beam is 2arctg(R0 tg(y/2)/(L cos(y/4)) (see Fig. 3); it is fixed and can be very small.

3. Analysis on the resonator construction The structural accuracy of the annular spherical resonator should be determined by means of the annular mirror vertex center displacement from the axis of the cylindrical discharge chamber and the angles of inclination of axes of the mirrors to the axis of the discharge chamber. Generally, the greater the displacement and/or the angles, the smaller the applicable transverse dimension of the annular mirrors and the discharge chamber. Using Fig. 4, we discuss how to define the available crosssectional dimension of optical propagation zone in a misalignment annular-mirror resonator. In Fig. 4, O1 and O2, and P1 and P2 are the vertex centers, and curvature centers, respectively, O1 P 1 and O2 P 2 are axes of mirror 1 and mirror 2, respectively, while P 1 P 2 is the optic axis of the misalignment annular-mirror resonator and AA is the axis of discharge chamber. In Fig. 4(c), AA is consistent with z-axis. In Fig. 4(a), all of O1 P 1 and O2 P 2 incline to the downside of the axis. In Fig. 4(b), O1 P 1 inclines to the

downside while O2 P2 reversely. Obviously, the influence of the inclination in the same direction on cavity performance is much more serious than that of the converse inclination, for which the proof shall be given later on. Fig. 4(c) is used for analysis of the resonator. (x01,y01), (x02,y02), (x1,y1), (x2,y2) are coordinates of the mirror vertex centers and curvature centers; (x1(0),y1(0)) and (x2(0),y2(0)) are coordinates of intersection points of the mirror-vertices with P1 P2 (new optic axis). We suppose that M1 and M2 are rotated by angles a1 and a2 about its y-axis, and by b1 and b2 about x-axis, respectively. If r1, r2, Lbx1(0), y1(0), x2(0), y2(0), then (

x1ð0Þ  x01 þ r1 sin a1 þ r1 ðx02  x01 þ r2 sin a2  r1 sin a1 Þ=ðr1 þ r2  LÞ y1ð0Þ  y01 þ r1 sin b1 þ r1 ðy02  y01 þ r2 sin b2  r1 sin b1 Þ=ðr1 þ r2  LÞ

(24) and (

x2ð0Þ  x01 þ r1 sin a1 þ ðr1  LÞðx02  x01 þ r2 sin a2  r1 sin a1 Þ=ðr1 þ r2  LÞ y2ð0Þ  y01 þ r1 sin b1 þ ðr1  LÞðy02  y01 þ r2 sin b2  r1 sin b1 Þ=ðr1 þ r2  LÞ

(25) where third terms are not negligible in Eqs. (24) and (25). The annular mirror-surface operation zone on M1 for the resonator with the new axis P1 P2 is R0 1 þ ðx21ð0Þ þ y21ð0Þ Þ1=2 prpR2  ðx21ð0Þ þ y21ð0Þ Þ1=2

(26)

and the annular mirror-surface operation zone on M2 is R0 1 þ ðx22ð0Þ þ y22ð0Þ Þ1=2 prpR2  ðx22ð0Þ þ y22ð0Þ Þ1=2

(27)

If r1 ¼ r2 ¼ r, then the operation zone on M1 or M2 can be described as R0 1 þ ðx21ð0Þ þ y21ð0Þ Þ1=2 prpR2  ðx21ð0Þ þ y21ð0Þ Þ1=2 ðwhen x21ð0Þ þ y21ð0Þ 4x22ð0Þ þ y22ð0Þ Þ

(28)

or R0 1 þ ðx22ð0Þ þ y22ð0Þ Þ1=2 prpR2  ðx22ð0Þ þ y22ð0Þ Þ1=2 ðwhen x22ð0Þ þ y22ð0Þ 4x21ð0Þ þ y21ð0Þ Þ

(29)

From Eqs. (28) and (29) the available cross-section size of the optical propagation zone in the resonator is d ¼ R2  ðx21ð0Þ þ y21ð0Þ Þ1=2  ðR0 1 þ ðx21ð0Þ þ y21ð0Þ Þ1=2 Þ 00

¼ d  2ðx21ð0Þ þ y21ð0Þ Þ1=2 ðif x21ð0Þ þ y21ð0Þ 4x22ð0Þ þ y22ð0Þ Þ 0

(30)

or qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 00 0 d ¼ d  2 x22ð0Þ þ y22ð0Þ ðif x22ð0Þ þ y22ð0Þ 4x21ð0Þ þ y21ð0Þ Þ

(31)

Diffraction loss of the propagation beam in the resonator is low for the laser oscillation, because the annular spherical reflective mirrors M1 and M2 have convergent action. 4. Object–image relation

Fig. 4. Diagram for analysis of misalignment of a spherical annular-mirror resonator. (a) All of o1 p1 and o2 p2 incline to the downside of the axis AA. (b) o1 p1 inclines to the downside of the AA while o2 p2 reversely. (c) Analysis diagram of misalignment.

Installation and adjustment of the resonator shown in Fig. 1 is a difficult problem. We try finding a method for the installation and adjustment by the object–image relation between focal points of M1 and M2. In Fig. 5, F1 and F2 are the focal points of M1 and M2, respectively. The light rays from the point source at F1 converge toward F2, and vice versa. In the figure the obstacle is set for obstructing the rays along the axis of M1 and M2 system. If all points (O1, O2, F1 and F2) are located at the axis AA of the discharge chamber, then the resonator is able to operate for laser output. If the mirror M2 is tilted, and it is rotated by angle a0 2 about its y-axis or by angle b0 2 about x-axis, then the image of point source (at F1) shifts from F2 to F0 2, and the F 2 F 0 2 can

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221

Fig. 5. Point source and its image for the cavity.

be expressed by F 2 F 0 2 ¼ f 2 sin a0 2 1 ¼ r2 sin a0 2 2

(32)

or F2F02 ¼

r2 2

0

sin b 2

If a promise value of a0 2 or b0 2 is given by

a0 2 pa2 or b0 2 pb2 F2F02p

r2 2

sin a2

(34)

sin b2

(35)

or F2F02p

r2 2

Similarly, F1F

1

I:

|2g1g21|o1;

II, III:

cos(1/4)arcos(2g1g21)4(R1/R2); ( R2 XR0 1 þ 2w þ D , 2w þ D ¼ 0:1R2

III:

then

0

Fig. 6. Self-reproducing ray-cluster existence condition (g1 ¼ g2 ¼ g).

(33)

¼ r1 sin a1 or F 1 F

0

1

¼ r1 sin b1 .

(36)

5. Calculated results 5.1. Stability case of resonator This existence condition in Eq. (14) indicates that in the annular-mirror resonator there can be a stable electromagnetic mode. The corresponding electromagnetic mode of the ray-cluster with empty intermediate may not be a fundamental mode or loworder mode. However, it can offer a good beam quality when the annular-mirror resonator is coincident with a ring resonator. Calculated stable regions for a conventional two-mirror resonator and the resonator composed of two annular spherical mirrors are plotted in Fig. 6; the stable region I satisfies the condition |2g1g21|o1. The regions II and III satisfy the existence condition of self-reproducing ray-cluster of the annular resonator. Curve (2) satisfies the existence condition, and the radial width of light propagation in the coaxial discharge chamber is 2w+D for the curve. The w is half of the radial width of the ray-cluster. The D is a remainder for a low propagation loss. (2w+D)/R2 ¼ b, where b is a proper ratio. b ¼ 0.1 for curve (2).The made curve shifts and varies with variation of b in the II and III. 5.2. Misalignment case Obviously, the available cross-section size of the coaxial discharge chamber is neither the designed value d nor d0 . The available cross-section size d00 is determined by the parameters r, L, R1, R2, x01, x02, y01, y02, a1, a2, b1, and b2. The d, R0 1, d0 , x1(0) and y1(0), x2(0) and y2(0), and d00 can be calculated from these parameters by using Eqs. (1), (9), (13), (24), (25) and (30) or (31), respectively. The w, half of the annular beam radial size in the annular resonator, or radius w of output beam of the ring resonator, and the far-field divergence angle yoo for one branch of the output beam can be calculated by using Eqs. (21) and (23). The

(1): cos(1/4)arccos(2g1g21) ¼ (R1/R2), ( cos ð1=4Þ arccosð2g 1 g 2  1Þ ¼ R1 =R2 ð1  ð2w þ D=R2 ÞÞ (2): . 2w þ D ¼ 0:1R2

calculation results, for typical cases, are shown in Fig. 7. In general, when LE1 m, r ¼ 10L, and d0 d00 o1 mm, the tolerance of mirror center displacement from the axis of the coaxial discharge chamber is 00.1 mm (or 00.15 mm), and the tilt tolerance 0500 (or 0400 ), and when LE1 m, r ¼ 5L, and d0 d00 o1 mm, the displacement tolerance is 00.15 mm, and the tilt tolerance 0600 . However, the d0 at r ¼ 10L is greater than d0 at r ¼ 5L. For CO2 laser, d ¼ 6 mm [6] and d ¼ 7 mm [3]. In Fig. 7, for curves 1, 2, 3, and 4, d ¼ 7 mm, and 2w ¼ 5.5 mmpd00 when ap600 . For curve (5), d ¼ 7 mm, and 2w ¼ 4.76 mmpd00 when ap400 . Therefore, the curves can serve as a good reference for CO2 laser. A high-powered He–Ne laser having rectangular discharge cross section [22] has been made. The short side of the rectangular section of the rectangular discharging tube is b and the long side is a. b is 3–4.5 mm long. Perhaps when abb, the rectangular discharge cross section may be wound to an annular cross section. For this case the curves 6, 7, 8, and 9 in Fig. 7 may be useful to design for a He–Ne laser. 5.3. A reference for adjustment If r2 ¼ r1 ¼ r ¼ 9358 mm, a2 or b2 ¼ 400 , or a2 or b2 ¼ 600 , then F 2 F 0 2 p0:09 mm; 0:136 mm; respectively. These values are approximate reference standards for adjustment of the resonator when the object–image relation is used. Note that deriving Eqs. (24) and (25) from (34)–(36) is difficult. However, when r1 ¼ r2 ¼ rbL, x01 ¼ x02 ¼ y01 ¼ y02 ¼ 0, and a1 ¼ b1 ¼ 0, Eqs. (34) and (35) are coincident with Eqs. (24) and (25). If, therefore, F 2 F 02 4 ðr2 =2Þ sin a2 , then x016¼0, or y016¼0, or x026¼0, or y026¼0. Thus, we can recognize the misalignment state of the resonator combining Eqs. (24), (25), (34)–(36) and experimental values of F 2 F 02 . 6. Conclusions (1) In the resonator formed by two annular spherical mirrors satisfying stability requirement, there exists an eigen raycluster. If the partial zone of one of the two mirrors is partially reflective, then the annular spherical-mirror resonator can be

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Fig. 7. Calculation results (r1 ¼ r2 ¼ r, length unit: mm, angle unit: 100 ).

1: 2:

0

R1

R2

d

L

r

d

x01

Y 01

x02

Y 02

lðmmÞ

wðmmÞ

yðmradÞ

20:25 20:25

27:25 27:25

7 7

1020 1020

9358 9358

6:4 6:4

0:15 0:15

0:15 0:15

0:15 0:15

0:15 0:15

10:6 10:6

2:75 2:75

1:26 1:26

3:

20:25

27:25

7

1020

9358

6:4

0:15

0:15

0:15

0:15

10:6

2:75

1:26

4: 5:

20:25 20:25

27:25 27:25

7 7

1020 1020

9358 9358

6:4 6:08

0:1 0:1

0:1 0:1

0:1 0:1

0:1 0:1

10:6 10:6

2:75 2:38

1:26 1:41

6:

21:25

24:25

3:5

1020

9358

2:9

0:15

0:15

0:15

0:15

0:6328

0:67

0:31

7:

21:25

24:25

3:5

1020

9358

2:9

0:15

0:15

0:15

0:15

0:6328

0:67

0:31

8: 9:

21:25 21:25

24:25 24:25

3:5 3:5

1020 1020

9358 6052

2:9 2:54

0:1 0:1

0:1 0:1

0:1 0:1

0:1 0:1

0:6328 0:6328

0:67 0:608

0:31 0:33

used as a ring resonator, in which exists one surrounding eigen ray-cluster. However, selection of parameters g1 and g2 is very important, because among many spherical resonators satisfying stability requirements only the resonators with proper g1 and g2 can be used as annular spherical-mirror resonators. Especially, the g parameter approaching 1 is suitable for the annular spherical-mirror resonators when g1 ¼ g2 ¼ g. When cos(1/4) arcos(2g1g21)pR1/R2, there is no eigen ray-cluster in the annular-mirror resonator. (2) The actual radial width d00 of the light propagation region in the annular discharge gap depends on the cavity parameters L, r, R1, and R2, the displacements of two mirror-vertex centers and the mirror inclination. The calculations of d00 show that high accurately processing the optic elements and high accurately assembling the elements and the discharge chamber into assembly are necessary for constructing the annular spherical-mirror resonator, and if it is required that resonator length LE1 m, the radius of curvature of the two mirrors rE5–10L, and the loss of applicable radial width of the coaxial discharge chamber for light propagation is smaller than 1 mm, then the displacements of the vertex centers of the two annular mirrors from the axis of discharge chamber are required to be within 0–0.15 mm and the tilt tolerance of the mirror axes relative to the axis of the discharge chamber is about 0–600 .

(3) The relation between point source and its point image produced by the annular spherical-mirror resonator can give a fix on the position of the resonator at the beginning.

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