misalignment sensitivity of the cat’s eye cavity he–ne laser
TRANSCRIPT
www.elsevier.com/locate/optcom
Optics Communications 265 (2006) 270–276
Misalignment sensitivity of the cat’s eye cavity He–Ne laser
Zhiguang Xu *, Shulian Zhang, Wenhua Du, Yan Li
The State Key Laboratory of Precision Measurement Technology and Instruments, Department of Precision Instruments,
Tsinghua University, Beijing 100084, China
Received 18 January 2006; received in revised form 17 February 2006; accepted 22 February 2006
Abstract
A concave mirror and a cat’s eye reflector acting as a resonator mirror form the cat’s eye cavity. Misalignment sensitivities of the cat’seye cavity and conventional resonator are both analyzed in geometric method and matrix optics with misalignment sensitivity parameter.Valuable conclusions are drawn: in full-external He–Ne laser, cat’s eye cavity can improve the laser stability up to about 60 times betterthan the conventional one; diffraction loss introduced by the misalignment of the cat’s eye cavity attributes to the straight-line displace-ment vertical to the laser bore of the cat’s eye reflector; and with the convex lens center of the cat’s eye reflector secured immobile, theultra-stable and adjustment-free cat’s eye cavity He–Ne 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 cat’s eye cavity lasers.� 2006 Elsevier B.V. All rights reserved.
PACS: 42.55.Lt
Keywords: He–Ne laser; Cat’s eye cavity; Cat’s eye reflector; Misalignment sensitivity
1. Introduction
The misalignment problem of laser resonators has beenalways concerned, which is very serious especially in He–Ne lasers, due to their correspondingly low gain. Whenthere 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 He–Ne lasers.
Cat’s eye reflector is now usually applied in interfer-ometer systems as an auxiliary component to reflect lightback [1–3]. Recent years there have been some efforts toapply the cat’s eye reflector as resonator mirror. As amethod to select laser transverse mode, not an approachto settle the laser stability problem, the cat’s eye cavitywas introduced in Li and Smith’s laser [4]. Some valu-
0030-4018/$ - see front matter � 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.optcom.2006.02.047
* Corresponding author. Tel.: +86 1062 788120; fax: +86 1062 784691.E-mail address: [email protected] (Z. Xu).
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 itsapplication very limited [6]. With cylindrical lens andgrating cat’s eye resonator was also equipped in anexternal-cavity semiconductor laser [7]. Recently weapply cat’s eye cavity into He–Ne lasers and find won-derful application prospects. Comprehensive experimentsare carried out both in a half-external cavity and afull-external cavity He–Ne laser [8]. Furthermore weset up a new convenient real-time means for thecontrol and selection of the laser transverse mode bycat’s eye cavity [9].
Although the stability advantage of the cat’s eye cavityhas been accepted and applied in some fields, its theoreticanalysis has not been reported in detail. In this paper wecalculate and analyze the misalignment sensitivity of thecat’s eye cavity compared with conventional resonator soas to apply theoretic foundation for the design and applica-tion of cat’s eye cavity lasers.
(2)
(1)
(3)
α
α α
Fig. 2. Obliquely incident paraxial beam with angle a to a cat’s eyereflector (1), a plane mirror (2) and a concave mirror (3).
Z. Xu et al. / Optics Communications 265 (2006) 270–276 271
2. Cat’s eye cavity
For a long time, the long He–Ne lasers are alwaysconstructed in full-external configuration 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 reflecting mirror it is.Therefore we built up a cat’s eye cavity in a full-externalHe–Ne laser to display its stability advantage.
The configuration 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 reflector M2, i.e. a plane or concavemirror, form the laser resonator. Draft (2) of Fig. 1 showsthe structure of the cat’s eye cavity, in which a cat’s eyereflector M3 displaces M2 to form the laser resonator.Our cat’s eye reflector is composed of a convex lens (withreflection reducing coating on both surfaces) and a concavemirror (with high-reflection 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 He–Ne laser bore is verysmall, no need to consider spherical aberration, an einzellens is selected as the convex one. We mount the wholecat’s eye reflector in a mechanic component to make itfunction as a resonator mirror.
Obviously, normal incident paraxial beam will bereflected back by our cat’s eye reflector along theentrance way. Even for the obliquely incident paraxialbeam our cat’s eye mirror can still provide high parallel-ism for the incident and the reflected beam, which isimpossible for any traditional laser resonator mirror(Fig. 2). That is why our cat’s eye reflector acting as aresonator mirror can improve the laser stability—if thereis tiny disturbance which make the mirror sway for asmall angle, this advantage can help reduce the banefulinfluence.
T M1
W
M2
W
T M1M3
W W
(1)
(2)
Fig. 1. A full-external He–Ne laser with a plane mirror and a cat’s eyereflector as the reflecting mirror, respectively.
3. Geometric method to analyze the misalignment sensitivityof the cat’s eye cavity and conventional resonator
3.1. Conventional resonator
Take the configuration (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 O1O2
in the paper plane. C1 and C2 are, respectively, the spherecenters of M1 and M2 which can be considered a concavemirror with infinite curvature radii, and the new resonatoraxis—line 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 ¼ ðR1 þ R2 � LÞu ð1Þh1 ¼ R1u ð2Þh2 ¼ ðR1 � LÞu ð3Þ
here h1 should be mainly considered because h1 > h2. UsingEqs. (1) and (2), we can know that:
h1 ¼R1R2h
R1 þ R2 � Lð4Þ
To make sure the operation of fundamental mode in theresonator, the following condition must be satisfied [10]:
h1 6D2�
ffiffiffipp
x1 ð5Þ
where D represents the diameter of laser bore, and x1 is thelight spot size on M1 which has the form [11]:
x1 ¼ffiffiffiffiffiffikLp
rR2
1ðR2 � LÞLðR1 � LÞðR1 þ R2 � LÞ
� �1=4
¼ffiffiffiffiffiffikLp
rR2
1
LðR1 � LÞ
� �1=4
ð6Þ
C1
R1
R L1 R2
C2
O2 O1
M1 M
ϕ
2
R2
h2h1
θ
+ −
Fig. 3. Analysis of the misalignment sensitivity of the conventional resonator.
272 Z. Xu et al. / Optics Communications 265 (2006) 270–276
The combination of Eqs. (4)–(6) leads to the largest mis-alignment angle hmax:
hmax ¼ðD=2�
ffiffiffipp
x1ÞðR1 þ R2 � LÞR1R2
¼ D=2�ffiffiffipp
x1
R1
¼ 0:46 ðminÞ ð7Þ
3.2. Cat’s eye cavity
Fig. 4 is the general structure draft of the cat’s eye cav-ity composed of the concave output mirror M1 and a cat’seye reflector M3 wholly assembled in a mechanic compo-nent G with the fixed point P. Distance between point O(the center of convex lens in the cat’s eye reflector) andthe fixed point P is defined as q being 15 mm in our exper-iments, and the misalignment distance of point O verticalto the laser bore will be d when the reflector has amisalignment 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 cat’s eye reflector is:
T ¼�1 0
0 �1
� �ð8Þ
Therefore the incident beam will be reflected back by ourcat’s eye reflector along the entry direction and the incidentand reflected 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 misalignment
along B will be reflected 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 cat’s eye reflector.
The single-trip loss factor d introduced by the cat’s eyereflector is [12]:
d ¼ I0 � Ic
2I0
¼ S0 � Sc
2S0
ð9Þ
When h is small enough and equation d = qh is satisfied,Eq. (9) can be rewritten through simple geometrical opera-tion as:
d ¼ 4ðD=2Þd2pðD=2Þ2
¼ 4ðD=2Þqh
2pðD=2Þ2¼ 4qh
pDð10Þ
Single-trip gain G in the laser cavity follows that famousempirical formula:
G ¼ 3� 10�4 LD¼ 3� 10�4 � 1100
3:2¼ 0:103 ð11Þ
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:
d < G� d0 ¼ 0:050 ð12Þ
The largest misalignment angle h0max of the cat’s eye reflec-tor is obtained as:
h0max ¼pDðG� d0Þ
q¼ 28:80 ðminÞ ð13Þ
sensitivity of the cat’s eye cavity.
Z. Xu et al. / Optics Communications 265 (2006) 270–276 273
The 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 cat’seye cavity makes the stability of the full-external cavityHe–Ne 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. Cat’s 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 cat’s eye reflector. With theassumptions of paraxial approximation and small mis-alignment, optical vector of the incident ray is transferredto:
x01h01
� �¼
x1
h1
� ��
d
�h
� �ð14Þ
in the after-misalignment coordinate. Then the emergentray reflected by the reflector follows that:
x02h02
� �¼�1 0
0 �1
� �x01h01
� �ð15Þ
Transforming the upper vector back into the before-misalignment coordinate system we obtain:
x2
h2
� �¼
x02h02
� �þ
d
h
� �ð16Þ
then:
x2
h2
� �¼�1 0
0 �1
� �x1
h1
� ��
d
�h
� �� �þ
d
h
� �
¼�1 0
0 �1
� �x1
h1
� �þ
2 0
0 0
� �d
�h
� �ð17Þ
Therefore the misalignment augmented matrix of the cat’seye reflector can be expressed as:
x2
h2
1
1
26664
37775 ¼
�1 0 2d 0
0 �1 0 0
0 0 1 0
0 0 0 1
26664
37775
x1
h1
1
1
26664
37775 ð18Þ
Assume the misalignment linear and angular displace-ments on the cat’s eye reflector are, respectively, x11 andh11, and those on the concave output mirror are x21 andh21. First, with the cat’s eye reflector as the reference initialposition, the ray will represent itself after a round trip,which is written as:
x11
h11
1
1
26664
37775 ¼
1 0 2d 0
0 �1 0 0
0 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 0
0 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
x11
h11
1
1
26664
37775ð19Þ
One readily obtains x11 = d. Then with the concave outputmirror as the reference starting position, we will havesimilarly:
x21
h21
1
1
26664
37775 ¼
1 0 0 0
�2=R1 1 0 0
0 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 0
0 �1 0 0
0 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
x21
h21
1
1
26664
37775ð20Þ
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 configurationparameters. High value of U means large diffraction lossintroduced and high misalignment sensitivity [13]. The U
in our case now is:
U ¼ 1
hx11
x1
� �2
þ x21
x2
� �2" #1
2
ð21Þ
where x1x2 denote the light spot sizes on the cat’s eyereflector and output mirror, which are given as [11]:
x1 ¼ffiffiffiffiffiffikLp
rR2
2ðR1 � LÞLðR2 � LÞðR1 þ R2 � LÞ
� �1=4
¼ffiffiffiffiffiffikLp
rR1 � L
L
� �1=4
ð22Þ
x2 ¼ffiffiffiffiffiffikLp
rR2
1ðR2 � LÞLðR1 � LÞðR1 þ R2 � LÞ
� �1=4
¼ffiffiffiffiffiffikLp
rR2
1
LðR1 � LÞ
� �1=4
ð23Þ
Substituting Eqs. (22) and (23) into (21) we obtain:
U ¼ dh
pk
� �12 ð2R1 � LÞ2
LðR1 � LÞ3
" #14
¼ qpk
� �12 ð2R1 � LÞ2
LðR1 � LÞ3
" #14
ð24Þ
Fig. 5. Laser power variation with the plane mirror, respectively, rotatedin up-and-down and right-and-left directions.
Fig. 6. Laser power variation with the cat’s eye reflector, respectively,rotated in up-and-down and right-and-left directions.
274 Z. Xu et al. / Optics Communications 265 (2006) 270–276
4.2. 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 �2h
0 0 1 0
0 0 0 1
26664
37775 ð25Þ
Presuming the misalignment linear and angular displace-ment on the plane mirror are, respectively, x011 and h011,and those on the concave output mirror are x021 and h021,in the same means we obtain that: x011 ¼ ðL� R1Þh andx021 ¼ �R1h. Then the value of misalignment sensitivityparameter U 0 in the conventional resonator is further givenby:
U 0 ¼ pk
� �12 ð2R1 � LÞ2ðR1 � LÞ
L
" #14
ð26Þ
The first important conclusion can be readily drawn bycomparing Eq. (24) with (26):
UU 0¼ q
R1 � Lð27Þ
In our actual application q is much less than R1 � L,so in cat’s eye cavity the diffraction 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 cat’s eye cavity is not so sen-sitive to misalignment and much more stable than theconventional.
The second valuable conclusion concerning the cat’s eyecavity is based on the relation between d (the diffractionloss introduced by misalignment) and the parameter U,which follows [13]:
d / ðhUÞ2 ¼ d2pð2R1 � LÞk½L1=2ðR1 � LÞ3=2�
ð28Þ
Eq. (28) tells us the diffraction loss in the cat’s eye reso-nator introduced by the misalignment completely attributesto the straight-line displacement d vertical to the laser boreof the cat’s eye reflector. Making the center O of the convexlens assured immobile, i.e. q = 0 then d = 0, the misalign-ment of cat’s eye reflector will not cause any laser powervariation.
5. Experiments and results
We carry out several experiments to verify our theoreticanalysis.
5.1. Experiments
The two experiments in Figs. 5 and 6 are to record thepower output of full-external He–Ne lasers applying twokinds of resonators while rotating their reflecting 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 cat’s eye reflector M3 areabout 60 min in cat’s eye cavity, about 60 times of theother, which coincides with the theoretical calculationvery well.
5.2. Adjustment-free He–Ne laser experiments
All the analysis above is based on the common configu-ration of cat’s eye cavity. Now we design a new specialmechanic retainer for the cat’s eye reflector fixing it juston the position of the convex lens center to make sureq = 0 mm. Fig. 7 shows the laser power when rotatingthe cat’s eye reflector from �15� to 15� with the convex lenscenter assured immovable, which indicates little fluctuationin the whole process.
This conclusion implies an important method for theachievement of ultra-stable and adjustment-free cat’s eye
Fig. 7. Laser power variation with the cat’s eye reflector rotated with theconvex lens center of the cat’s eye reflector immovable.
Z. Xu et al. / Optics Communications 265 (2006) 270–276 275
cavity He–Ne laser. If the cat’s eye reflector is fixed bythe mechanic component just on the position of the con-vex lens to ensure the lens center immovable, the laserpower will not be influenced by the misalignment of thecat’s eye reflector, based on which we succeed in theinvention of the adjustment-free He–Ne laser. An auda-cious experiment is performed to show its high stabil-ity—without any mechanical system for the cat’ eyereflector to be mounted, we just grasp it by tweezers asthe resonator mirror and make the laser working contin-uously with high and stable power output, which is abso-lutely impossible for a conventional cavity. In Fig. 8, thelight radiation can be obviously observed on the receivingbessel and the Brewster window with the cat’s eyereflector held by tweezers. In another word, withoutany precise adjustment, the cat’s eye cavity laser can keepworking. This adjustment-free function is quite useful forHe–Ne lasers in bad working environment or requiringease to adjust.
Every experiment above is repeated several times andthe same conclusions are reached. These conclusions arealso referential in any other type of laser such as Nd:YAGor CO2 laser and so on. In many conditions which requir-ing high laser stability, the cat’s eye cavity will be a satisfy-ing choice. Considering its additional advantage in controlof the laser output transverse mode [9], cat’s eye cavity mayturn into one of the main resonators in many industry andmeasurement fields.
Fig. 8. Making laser working with the cat’s eye reflector held by tweezersas the resonator mirror.
6. Misalignment sensitivity of the cat’s eye cavity with three
kinds of dimension errors
6.1. Error in focus length of the convex lens
In the producing procedure the focal length of the con-vex lens is the most possible element to introduce error,since there are three parameters need to be manufacturedprecisely: two curvature radiuses of two surfaces and theirdistance. We will analyze the influence on the misalignmentsensitivity by that inevitable error.
In this case the transformation matrix of the cat’s eyereflector is:
1 0
�1=f 1
� �1 l
0 1
� �1 0
�2=r 1
� �1 l
0 1
� �1 0
�1=f 1
� �
¼r¼l 6¼f�1 0
2=Req �1
0B@
1CA ð29Þ
where
Req ¼ f � ll� f
ð30Þ
Still following the method above we get the misalign-ment linear and angular displacements on the cat’s eyereflector and the output concave mirror, respectively, thenthe value of misalignment sensitivity parameter. Whenf > l, we have Req < 0, here the cat’s eye reflector actsas a convex mirror which can introduce large loss intocavity and is rarely used in He–Ne lasers whose gain isrelatively low. Therefore we only need to research the caseof f 6 l.
In our experiments there is r = l = 19 mm, and Fig. 9illustrates the relation between misalignment sensitivityparameter U and the focal length of the convex lens f, fromwhich we can get a conclusion that U will increase gradu-ally with f enlarging away from 19 mm, and the cavity sta-bility will be reduced. In a quite large range of f with error,however, the cat’s eye cavity still has an advantage instability because its U is much less than that of the conven-tional resonator all the same.
6.2. Error in the distance between the convex lens and theconcave mirror
Error is introduced ineluctably in the assembling processof the cat’s eye reflector, which can make the distancebetween the convex lens and concave mirror l away fromthe ideal value, whose influence on the misalignment sensi-tivity is analyzed below.
Now the transformation matrix of the cat’s eye reflectoris:
1� 2lr
1� l
f
� �� l
f 2l� 2l2
r
� 1� lf
� �2f þ 1
r 1� lf
� �h i1� l
f
� �1� 2l
r
� l
f
0B@
1CA ð31Þ
Fig. 9. Misalignment sensitivity parameter U vs. focal length of theconvex lens f.
276 Z. Xu et al. / Optics Communications 265 (2006) 270–276
We have f = r = 19 mm in our system, and the relationbetween misalignment sensitivity parameter U vs. distancel is shown in Fig. 10. A conclusion can be drawn that U willincrease gradually with l enlarging away from 19 mm, andthe cavity stability will be reduced. As is the same as case of(1), in a quite large range of l with error the cat’s eye cavitystill has an advantage in stability comparing with the con-ventional resonator.
6.3. Error in the radius of curvature of the concave mirror
This case is calculated similarly as (1) and (2) withf = l = 19 mm, and Fig. 11 demonstrates the influence bythe error on the misalignment sensitivity.
Fig. 10. Misalignment sensitivity parameter U vs. distance between theconvex lens and the concave mirror l.
Fig. 11. Misalignment sensitivity parameter U vs. curvature radius of theconcave mirror r.
An important conclusion can be obtained from the threeparts of analysis above: the cat’s eye reflector is to be idealwhen the three parameters, the focal length of the convexlens, the radius of curvature of the concave mirror andthe distance between them, equal to one another. Underthese circumstances the cat’s eye cavity can reach its high-est stability.
7. Conclusions
Laser stability can be improved by cat’s eye cavity upto about 60 times better than the conventional one infull-external He–Ne laser. Diffraction loss introduced inthe cat’s eye resonator by the misalignment attributes tothe straight-line displacement vertical to the laser boreof the cat’s eye reflector. With the convex lens center ofthe cat’s eye reflector assured immobile, the ultra-stableand adjustment-free cat’s eye cavity He–Ne laser is pre-sented and the adjustment-free operation is demonstrated.Cavities with three kinds of dimension errors are also cal-culated, and the cat’s eye reflector will be perfect with thefocal length of the convex lens, the radius of curvature ofthe concave mirror and the distance between them, allidentical. This paper proposes several principles, as thetheoretic foundation for the design and application ofcat’s eye cavity lasers.
Acknowledgements
This work is supported by the National Nature ScienceFoundation of China and Tsinghua University.
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