Eigenmode in a misaligned triangular optical cavity
aa r X i v : . [ phy s i c s . op ti c s ] O c t Eigenmode in a misaligned triangular optical cavity
F.Kawazoe , R. Schilling and H. L¨uck Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut) und LeibnizUniversit¨at Hannover, Callinstr. 38, D–30167 Hannover, Germany Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut),Fr¨ottmaninger Weg 18, D-85748 Garching, GermanyE-mail: [email protected]
Abstract.
We derive relationships between various types of small misalignments on atriangular Fabry-Perot cavity and associated geometrical eigenmode changes. We focuson the changes of beam spot positions on cavity mirrors, the beam waist position, andits angle. A comparison of analytical and numerical results shows excellent agreement.The results are applicable to any triangular cavity close to an isosceles triangle, withthe lengths of two sides much bigger than the other, consisting of a curved mirror andtwo flat mirrors yielding a waist equally separated from the two flat mirrors. Thiscavity shape is most commonly used in laser interferometry. The analysis presentedhere can easily be extended to more generic cavity shapes. The geometrical analysisnot only serves as a method of checking a simulation result, but also gives an intuitiveand handy tool to visualize the eigenmode of a misaligned triangular cavity.PACS numbers: 42.79.Gn, 42.60.Da
1. Introduction
Fabry-Perot cavities are widely used in the field of laser interferometry, and longitudinallength shifts of a cavity mirror and the resulting change in the phase of the resonatingfield is well known. However in the case where suspended mirrors are used, suchas in gravitational wave detectors, angular shifts play a crucial role in the detectorperformance; they ensure clean length control signals. Angular shifts on the cavitymirrors and resulting eigenmode changes in the circulating Gaussian beam of a planecavity were geometrically analyzed in [1], and the results are used, together with resultsfrom simulation work, to obtain error signals to control the alignment of various cavitymirrors. Recently we designed a triangular optical cavity for the purpose of frequencystabilization for the AEI 10 m Prototype [2], and in the process of designing an alignmentcontrol system, a geometrical analysis for this cavity was performed. The cavity isclose to an isosceles triangle, with the lengths of two sides much bigger than the other,consisting of a curved mirror and two flat mirrors yielding a waist equally separatedfrom the two flat mirrors. However this cavity shape is most commonly used in laserinterferometry, the results presented here can easily be extended to more generic cavity igenmode in a misaligned triangular optical cavity
2. Types of misalignments
Figure 1 shows the schematic of a triangular cavity when aligned. Two flat mirrors arerelatively close together and are labeled as M a and M c , while the curved mirror is faraway, has a radius of curvature R and is labeled as M b . The position where the beamhits the mirror M i is given by P i , as well as the waist position by P w , followed by theassociated coordinates within the x − y plane. Here, we also introduce a coordinatesystem attached to each of the flat mirrors ( y a and y c ) for convenience. The two equalangles of the beam at M a and M c and half the small angle at M b are given by γ and φ , respectively. Due to the shape of the triangle the following approximations hold andare used throughout this paper unless otherwise noted: γ ≈ π/ φ ≪ Waist P (cid:1) (L,0) φ 2dL γR M (cid:2) M (cid:3) M (cid:4) P (cid:0) (L, -d)P (cid:5) (L, d)P (cid:6) (0, 0) xyz y (cid:7) y (cid:8) Figure 1.
Schematic of an aligned triangular cavity within the x − y plane. Alsodefined are the two coordinate axes y a and y c that are fixed on the flat mirrors M a and M c , respectively. Mirror M b has a radius of curvature of R . Misalignment angles of mirror M i are given by α i and β i for horizontal (anglesaround the z -axis, sometimes also called yaw or rotation), and vertical (inclinationangle with respect to the x − y plane, sometimes also called pitch or tilt) directions, igenmode in a misaligned triangular optical cavity Table 1.
Summary of types of misalignments and associated section numbers.Type Description Section α − Differential of the flat mirrors in horizontal 3 . .α b Curved mirror in horizontal 3 . .α + Common of the flat mirrors in horizontal 3 . .β b Curved mirror in vertical 4 . .β + Common of the flat mirrors in vertical 4 . .β + Differential of the flat mirrors in vertical 4 . . respectively. A positive angle is formed by counter-clockwise rotation around the z -axisfor horizontal misalignments, and around the y -axis, y a -axis, and y c -axis for verticalmisalignment of M b , M a and M c , respectively. We take linear combinations of these twoflat mirror misalignments to form common and differential modes: α ± = ( α a ± α c ) / β ± = ( β a ± β c ) /
2. The changes in the waist position and the beam spot positionon mirror M i are denoted by ∆ k w and ∆ k i , with k being the corresponding x or y coordinates. Ane angular change of the beam between the two flat mirrors is denoted by θ (see Fig. 3. Since we concern small misalignments, these changes are also small. Hencewe use the following approximation throughout this paper: θ ≪ O (∆ k n ) = 0 for n ≥
2. All types of misalignments are summarized and the associated section numbersare listed in Table 1.
3. Horizontal misalignments α − A misalignment in α − , i.e. contrary tilts around the z -axis, keep the cavity symmetric to the x -axis and, hence, causes a symmetric change in the eigenmode. In Fig. 2, the original and thenew eigenmodes are shown by the lighter (yellow) and darker colors (this color rule is appliedthroughout this paper), and the x and y coordinates of the spot positions on the mirrors areshown. Because of the symmetry it is obvious that ∆ x a equals ∆ x c and ∆ x w , and due tothe approximation given by Equ. 1, ∆ y a also equals ∆ x a . The angle of incidence on the flatmirrors changes by α − , as indicated by the dashed normal on one mirror surface. The largeangle γ ′ changes by − α − , yielding a change by α − in half the small angle ∆ φ . From lookingat the shaded area in Fig. 2 we get:∆ y c ≈ p L + d sin ∆ φ ≈ p L + d · ∆ φ = − p L + d · α − (3)Therefore we know the following relations between the spot position changes and themisalignment angle:∆ x a = ∆ x c = − ∆ y a = ∆ y c = ∆ x w = − p L + d · α − (4)and hence the angle deviation θ at the waist is zero.To summarize, a misalignment in +( − ) α − causes a shrink (stretch) of the eigenmodealong the x -axis, yielding the eigenmode to keep its isosceles shape, but change its shape in away that it becomes more “fat” (“thin”). As a result, the waist position shifts in x -directionby an amount that is approximately proportional to the distance between the curved mirrorand the two flat mirrors. igenmode in a misaligned triangular optical cavity γ’Δφ ½ α - ½ α - ½ α - P b (0, 0) WaistP c (L+Δx c , -d+Δy c )P a (L+Δx a , d+Δy a )P w (L+Δx w , 0) Figure 2.
Cavity eigenmodes of the aligned (lighter colored triangle) and themisaligned by α − (darker colored triangle) cases. The contrary tilts around the z -axis cause a symmetric change in the eigenmode. α b Figure 3 shows a hypothetic misaligned cavity caused by α b , i.e. a rotation of M b around thevertical axis. In this case, there is no obvious symmetry axis. One can expect changes in thepositions of the beam spots on the mirrors and of the waist, as well as an angle deviation at thewaist. We introduce a pivot, where the non-congruent side of the aligned and the misalignedeigenmodes cross, indicated by the thick circle. We start with an arbitrary location of thepivot, and will shortly show that it coincides with the bisecting point of the non-congruentside. θα b P c (L+Δx c , -d+Δy c )P a (L+Δx a , d+Δy a )P b (Δx b , Δy b ) P w (L+Δx w , Δy w )Pivot γ’’ γΦ’ γ’φ Figure 3.
Cavity eigenmodes of the aligned and the misaligned cases by α b . We startwith a general, and hence hypothetical case where the pivot and the bisecting pointof the non-congruent do not match, and later show they coincide. The changes of thetwo larger angles ( γ ) are of the same size but with opposite sign, hence the small angle φ stays unchanged. The angle of incidence on the flat mirrors changes by the same amount θ , but with oppositesign, resulting in the following changes of the large angles: γ ′ = γ + 2 θ and γ ′′ = γ − θ . Hencethe small angle stays unchanged: φ ′ = 2 φ .Looking at the flat mirrors, as shown in Fig. 4, and applying the approximation given by igenmode in a misaligned triangular optical cavity Equ. 1, one sees that ∆ x a = − ∆ y a and ∆ x c = ∆ y c . The following set of equations describethe shift of the spot positions:∆ x a = l a sin θ (5)∆ x c = l c sin θ (6)2 d = l a cos θ − l c cos θ − | ∆ y a | − | ∆ y c | (7)= ( l a − l c ) cos θ − ( | ∆ x a | + | ∆ x c | ) (8) ≈ l a − l c − ( | l a | + | l c | ) θ where l a and l c are the distances from the pivot to P a and P c along the beam, respectively. l a l c Zoom in θα b Pivot θ Δy c Δx c Δy a Δx a l c l a Figure 4.
Closer view of the two flat mirrors and the pivot. It still shows thehypothetical eigenmode where the pivot and the bisecting point do not match.
The left hand side of Equation 7 is constant, hence the right hand side must be independentof θ , yielding the following relations: l a = − l c (9) | l a | = | l c | = d/ cos θ ≡ l (10)∆ x a = − ∆ x c = − ∆ y a = − ∆ y c = l sin θ = d tan θ ≈ d θ (11)This automatically means that the pivot ( P p ) bisects the non-congruent side, as shown inFig. 5, where the changes in the location of the pivot is denoted by ∆ x p and ∆ y p . It alsoshows the details around the flat mirrors, from which the pivot location with respect to theoriginal waist is given by∆ x p = O (cid:16) θ (cid:17) = 0 (12)∆ y p = d − l cos θ + ∆ y a = − d θ (13)Connecting the beam spot on the curved mirror ( P b ) and the bisector point (the pivot, or P p ),one can see that it bisects the beam angle at M b into φ , as shown in Fig. 6. This means thatthe line passes through the center of curvature, ( P COC ), whose coordinate along the y -axis isgiven by ∆ y COC = R · α b (14)Focusing on the shaded triangles shown in Fig. 6, one can see that θ ′ = θ , and, comparingthe two triangles, one can also see that θ ′′ = θ ′ = θ . The radius vectors of the aligned and igenmode in a misaligned triangular optical cavity Bisector point
AND the pivotφ l θ
Zoom in Δy p Δy c Δx c Δy a Δx a α b dP P (L+Δx P , Δy P ) Figure 5.
Closer view of the two flat mirrors. Here, the pivot and the bisecting pointoverlap, and the y coordinate of the pivot is denoted by ∆ y p . misaligned mirror, indicated by the dotted lines in Fig. 6, cross at point P r . By focusing onthe triangle consisting of the original waist ( P w ), the pivot, ( P p ), and P r , as shown in thelower triangle in Fig. 6, one can see that the x coordinate of the point P r is given by∆ x r = d θ/ tan θ ≈ d (15) φα b P P P COC (R, Δy
COC )θ’θθ’’ Center of curvatureP r (Δx r , 0)dθ Δx r θ P r P P P w φP b (Δx b , Δy b ) Figure 6.
Radius vectors of the aligned and the misaligned cases. They cross at thepoint P r , from which the angle deviation and the pivot location are calculated. Figure 7 lists all the length information that is needed to calculate the angle θ and the spotposition on M b . These are given by the following set of equations: θ ≈ tan θ = R α b / ( R − L − d ) (16)∆ x b = O (cid:16) ∆ y b (cid:17) = 0 (17)∆ y b = − ( L + d ) tan θ ≈ − ( L + d ) θ = − R α b · ( L + d ) / ( R − L − d ) (18) igenmode in a misaligned triangular optical cavity α b θ RdL R-L-d R ・ α b P b (Δx b , Δy b ) Figure 7.
Length information needed to calculate θ and the spot position changeon M b . Having calculated the new spot positions on the mirrors, we now calculate where the newwaist is. In order for the wavefront curvature of the beam to match the radius of curvature ofthe curved mirror M b , the path lengths from the waist to the mirror M b via M a and via M c should be the same, i.e. in Fig. 8 it should be S a + D a = S c + D c = S + d . By calculating thedistances S a and S b in the following equations, we also will obtain the distances D a and D b : S a = n ( L + d θ ) + ( d + L θ ) o / (19) ≈ p L + d (cid:18) Ld θL + d (cid:19) / θ = 0 ≈ p L + d (cid:18) Ld θL + d (cid:19) Ld θL + d ≪ S + 2 d θ d /L = 0 D a = S + d − S a = d − d θ (20) P (cid:9) (0, -(L+d)θ) P (cid:10) (L-dθ , -d-dθ)P (cid:11) (L+dθ, d-dθ) D (cid:12) P (cid:13) (L+Δx (cid:14) , Δy (cid:15) )α (cid:16) D (cid:17) S (cid:18) S (cid:19) S Figure 8.
Locations of the new spot positions on the mirrors. By using them the newwaist location is calculated.
In a similar way S c = n ( L − d θ ) + ( − d + L θ ) o / = S + 2 d θ (21) D c = S + d − S b = d + 2 d θ (22)Hence, the new waist location is given by the following:∆ x w = O (cid:16) θ (cid:17) = 0 (23)∆ y w = ( d − d θ − D a ) cos θ ≈ d θ = dR α b / ( R − L − d ) (24)To summarize, a misalignment in +( − ) α b causes a clockwise (counter-clockwise) rotationof the non-congruent side around the bisecting point, yielding the long sides to rotate igenmode in a misaligned triangular optical cavity synchronously. As a result all the beam spot positions change by the amounts given by theradial distances with the bisecting point being the origin of the system of radial coordinates. α + In the case of α + = 0, there is no obvious symmetry line, thus we will start from a generalcase. Changes on the spot positions, the beam angle at the waist, and the two larger anglesare defined as shown in Fig. 9. Figure 10 focuses on the beam angle change on mirror M a . ½ α + ½ α + P c (L+Δx c , -d+Δy c )P a (L+Δx a , d+Δy a )P b (0, Δy b ) γ a γ c θ Figure 9.
Cavity eigenmodes of the aligned and the misaligned ( α + ) cases. Drawing helping lines such as the one that is parallel to the aligned beam (indicated by thelight colored thick dotted line), as well as lines that are normal to both the aligned and themisaligned mirror surfaces (indicated by the light thin, and dark thin dotted lines, respectively)one can see that half of γ a is given by γ a / γ/ θ − α + /
2. Hence, ½ α + ½ γ θ Parallel to the aligned beamNormal to the mirror surfaces ½ α + M a ½ γ a Figure 10.
Closer view on the change in one of the larger angles, γ a . γ a = γ + (2 θ − α + ) (25)In a similar manner, γ c is given by γ c = γ − (2 θ − α + ) (26)This means that the sum of the two angles stays unchanged, yielding no change in the smallangle φ . Then the line that connects P b with the center of curvature of M b (from here on thisis called the radius ), should bisect the short side, due to the fact that d ≪ L . The bisecting igenmode in a misaligned triangular optical cavity R Center of curvature½ α + ½ α + θω γ c γ a φ β Waist (aligned)Bisecting pointR-Lη Figure 11.
Ancillary angles: β , η and ω , which are use to calculate θ . point is indicated by the square point in Fig. 11. Here, we introduce some ancillary angles β and η , together with ω , which is the angle of the radius with respect to the aligned case.Focusing on the shaded area, one can see that the ancillary angles are given by β = φ + ω and (27) η = φ + β = 2 φ + ω (28) η can be expressed using γ if one focuses on the shaded triangle shown in Fig. 12, introducinga new ancillary angle γ ′ a = γ + θ , and it is given by η = π − (cid:16) γ ′ a + γ c (cid:17) = π − { ( γ + θ ) + γ − (2 θ − α + ) } = π − γ + θ − α + (29) ½ α (cid:20) η ½ α (cid:21) γ (cid:22) γ (cid:23) ’ Figure 12.
Yet another ancillary angle γ ′ a to calculate η . By comparing Equations 28 and 29 the angle ω is given by the following equations: π − γ + θ − α + = 2 φ + ω (30) ω = π − (2 γ + 2 φ ) + θ − α + = θ − α + (31)In order to gain additional information to finally calculate θ , we focus on some lengthsas shown in Fig. 13. The pivot ( P p ) is indicated by the thick circle and changes in its locationare denoted by ∆ x p and ∆ y p , and the two lengths from the pivot to the two beam spots by l a and l b . Changes in the coordinates of the beam spot position on M a are given by the followingequations: ∆ x a = − l a sin θ ≈ − l a θ (32) igenmode in a misaligned triangular optical cavity d Δx c Δy c Δx a Δy a Pivot : P P (L+Δx P , Δy P )Waist (aligned)½ π - ½ γ + ½ α + ½ α + ½ α + γl a l c ½ π - ½ γ - ½ α + θ Figure 13.
Length relations around the flat mirrors. From this the lengths l a and l c from the pivot to the beam spots on the two mirrors are calculated.) ∆ y a = − ∆ x a tan ( π/ − γ/ α + / ≈ − α + /
21 + α + / l a θ ≈ (1 − α + ) l a θ (33)and ∆ x c and ∆ y c by:∆ x c = l c sin θ ≈ l c θ (34)∆ y c = ∆ x c tan ( π/ − γ/ − α + / ≈ α + / − α + / l c θ ≈ (1 + α + ) l c θ (35)The length of the non-congruent side is then expressed by the following:2 d = l a cos θ + ∆ y a + l c cos θ + ∆ y c ≈ l a + l c − { l a − l c − ( l a + l c ) α + } θ (36)Since the left-hand side of Equation 36 does not depend on the misalignment angle θ , theangle dependent term of the right-hand side should be zero, hence, l a − l c − ( l a + l c ) α + = 0 (37)From Equations 36 and 37 the following relations can be obtained: l a = d (1 + α + ) (38) l c = d (1 − α + ) (39)With this knowledge we can calculate the location of the pivot in the following way:∆ x p = O (cid:16) θ (cid:17) = 0 (40)∆ y p = l c cos θ + ∆ y c − d ≈ d ( θ − α + ) (41)The location of the bisecting point, as shown in Fig. 14, can be calculated in a similar way,and the coordinates are given by∆ x B = O (cid:16) θ (cid:17) = 0 (42)∆ y B = d cos θ + ∆ y c − d ≈ (1 + α + ) (1 − α + ) d θ ≈ d θ (43) igenmode in a misaligned triangular optical cavity Bisecting pointP B (L+ Δx B , Δy B )d Δx c Δy c Δx a Δy a Pivot½ α + ½ α + l a l c dθ Figure 14.
Length relations around the flat mirrors, including the pivot location.From this the spot position changes on the flat mirrors are calculated.
Then, focusing on the triangle that consists of the center of curvature, the waist (in the alignedcase), and the bisecting point (indicated by the right part of the shaded area in Fig. 11), onecan obtain another relation for ω and θ which is given by ω ≈ tan ω = d θ/ ( R − L ) (44)From Equations 31 and 44 one can finally obtain the relation between θ and α + : θ = R − LR − L − d · α + (45)Using θ , the spot positions on the three mirrors (see Equations 32, 33, 34, and 35) can furtherbe calculated. This yields the following equations:∆ x a = − d (1 + α + ) θ ≈ − d ( R − L ) R − L − d · α + (46)∆ y a = (1 − α + ) ∆ x a ≈ − d ( R − L ) R − L − d · α + (47)And in similar ways,∆ x c = d ( R − L ) R − L − d · α + (48)∆ y c = d ( R − L ) R − L − d · α + (49)and ∆ x b = O (cid:16) ∆ y b (cid:17) = 0 (50)∆ y b = R · ω = R · ( θ − α + ) = d RR − L − d · α + (51)Then the waist location can be calculated in the same way as shown in equations 19 to 24,and the following can be shown: S a = n ( L + ∆ x a ) + ( d + ∆ y a − ∆ y b ) o / ≈ S − ( d θ + ∆ y p ) (52) igenmode in a misaligned triangular optical cavity D a = d + d θ + ∆ y p (53)In a similar way we obtain S b = S + ( d θ + ∆ y p ) (54) D b = d − d θ + ∆ y p (55)Therefore the new waist location is given by∆ x w = O (cid:16) θ (cid:17) = 0 (56)∆ y w = ( d + ∆ y a − D a ) cos θ ≈ − ∆ y p = − d R − L − d · α + (57)To summarize, a misalignment in +( − ) α + causes a counter-clockwise (clockwise) rotationof the non-congruent side around a point that does not coincide with the bisecting point.This yields a clockwise (counter-clockwise) rotation ω (which is very small compared to themisalignment angle α + ) of the geometrical axis of a corner reflector consisting of the two flatmirrors. As a result, the eigenmode changes in a “non uniform” way, with each spot positionchange being smaller than the misalignment case of α b .
4. Vertical misalignments
When considering vertical misalignments, it is necessary to view the cavity as a 3D body, asshown in Fig. 15. Notations of all the properties are the same as that shown in Fig. 1. zy x M b P c (L, -d, 0)P a (L, d, 0)M a M c P b (0, 0, 0) Waist P w (L, 0, 0)2dLR Figure 15.
3D view of a triangular cavity. M a and M c are the flat mirrors, and M b has a radius of curvature of R . The positions where the beam hits the mirror M i aredenoted by P i . β b A misalignment around the y -axis by β b , as shown in Fig. 16, does not affect the mirroralignment in y -direction, hence there is no change in eigenmode in that direction. Then it ispossible to project the cavity onto the x − z plane for simplicity, as shown in Fig. 17, and treatit as a plane cavity. The eigenmode of the cavity is defined by the line that is orthogonal tothe flat mirrors and passes through the center of curvature, as described in [1]. It is obvious igenmode in a misaligned triangular optical cavity β b P b (Δx b , 0, Δz b ) P a (L, d, Δz a )P c (L, -d, Δz c ) Waist P w (L, 0, Δz w ) Figure 16.
Cavity eigenmodes of the aligned (lighter colored triangle) and themisaligned by β b (darker colored triangle) cases. This type of misalignment does notaffect the mirror alignment in y -direction, hence the eigenmode only changes along the z -axis. LR M a M b M c P a (L, Δz a )P b (Δx b , Δz b ) xzy Center of curvature β b Δz coc P c (L, Δz c )P w (L, Δz w ) Figure 17.
Projection of the triangular cavity onto the x − z plane. It allows oneto view the cavity as a plane cavity. The eigenmode is defined by the line that isorthogonal to the flat mirror and passes through the center of curvature. that the eigenmode is also orthogonal to the curved mirror, yielding the shifts in z -directionof all of the spot positions to have the same size. The normal vector on the mirror M b istilted by β b , hence the center of curvature, whose z coordinate is denoted by z coc , shifts by∆ z coc = β b · R . Therefore we have the following relations:∆ x a = O (cid:16) β b (cid:17) = 0 (58)∆ z a = ∆ z b = ∆ z c = ∆ z w = ∆ z coc = β b · R (59)To summarize, a misalignment in +( − ) β b causes an upward (downward) shift of the centerof curvature along the z -axis, yielding a synchronous shift of the plane of the eigenmode byan amount proportional to the radius of curvature of the curved mirror. igenmode in a misaligned triangular optical cavity β + Similar to β b , β + has no effects in y -direction. However, since the y a -axis and the y c -axis arerotated by ± (cid:0) π − γ (cid:1) ≈ ± π around the y -axis, respectively, the projection of a misalignmentby β + / β + / √
2. Section 4.10, (page 100-102) of reference [3],gives a detailed explanation of this effect by using a vector algebra and we will not describeit in this paper. For convenience we introduce an effective misalignment angle β eff = β + / √ ½ β + ½ β + M b P c (L+Δx c , -d, Δz c )P a (L+Δx a , d, Δz a )P b (Δx b , 0, Δz b )Waist P w (L, 0, Δz w ) M a M c zy x y c -axisy a -axis Figure 18.
Cavity eigenmodes of the aligned and the misaligned ( β + ) cases. Thistype of misalignment does not affect the mirror alignment in y -direction, hence theeigenmode only changes along the z -axis. The projection of the flat mirrors are rotated by β eff / y -axis and the effect isdoubled because of the two reflections, hence, seen as a plane cavity, the misalignment angleis given by β eff , as shown in Fig. 19. The eigenmode of this cavity is defined by the line LR M a M b M c P a (L+Δx a , Δz a ) P b (Δx b , Δz b ) xzyCenter of curvature P c (L+Δx c , Δz c )P w (L+Δx w , Δz w ) β eff θ θ β eff Zoom in
Figure 19.
Projection of the triangular cavity onto the x − z plane. It allows oneto view the cavity as a plane cavity. The eigenmode is defined by the line that isorthogonal to the flat mirrors and passes through the center of curvature. In the rightpart, an enlarged cut-out around one flat mirror is shown. that passes through the center of curvature and intersects the flat mirrors orthogonally, asdescribed in [1]. The angle formed by the eigenmodes of the aligned and misaligned cases isdenoted by θ in Fig. 19, and it becomes obvious that θ = β eff when one focuses around thearea of the flat mirrors, as shown in the enlarged cut-out in the right part of Fig 19. Thereforethe following equations yield the spot position changes:∆ x a, b, c, and w = O (cid:16) β (cid:17) = 0 (60) igenmode in a misaligned triangular optical cavity ∆ z a = ∆ z c = ∆ z w = β eff · ( R − L ) = β + · ( R − L ) / √ z b = β eff · R = β + · R/ √ − ) β + causes a counter-clockwise (clockwise) tilt ofthe geometrical axis of the two flat mirrors around the center of curvature. As a result theplane of the eigenmode tilts synchronously. β − Here, mirrors M a and M c rotate around the y a and y c -axis by ± / β − , respectively, as shown inFig. 20. When the two opposite misalignment angles on mirrors M a and M c are projected onto ½ β - P c (L, -d, Δz c )P a (L, d, Δz a )Waist P w (L, 0, 0) M a M c M b P b (0, 0, 0) zy x ½ β - y c -axisy a -axis Figure 20.
Cavity eigenmodes of the aligned and the misaligned ( β − ) cases. Thebeam spot position and the waist position stays unchanged. the x − z plane, they appear as rotations around the y -axis by ± β eff , respectively, yielding nochange along the z -axis on the curved mirror M b . On the other hand, when they are projectedonto the y − z plane, as shown in Fig. 21, they both appear as rotations around the z -axisby β eff /
2, yielding shifts along the z -axis in the beam spot positions on the two flat mirrors bythe same amount, but with opposite sign. Note that here β eff ≡ β − / √
2. These spot positionchanges are symmetrical along the y -axis, thus they do not yield a change in the beam spotposition on the curved mirror along the y -axis, nor a change in the waist position (which isequidistant from the two spot positions) along the y -axis and x -axis. Hence the spot on thecurved mirror and the waist remain unchanged, indicating that the new eigenmode is formedby rotating the aligned eigenmode around the x -axis by θ , yielding no change in the lengthson any sides of the triangle.The inclination angle of the beam between the two flat mirrors with respect to the x − y plane is denoted by θ in Fig. 21. Focusing on the isosceles triangle as indicated by the shadedtriangle in the figure, whose equal angles are denoted as µ , the inclination angle is given bythe following equations: µ = π/ − β eff / θ = π − µ = β eff (64)Therefore the beam spot position shifts on the two mirrors are calculated to be∆ z a = − ∆ z c = d · α eff = d · β − / √ igenmode in a misaligned triangular optical cavity yzx dM c M a ½ β eff ½ β eff θ P c (-d, Δz c ) P a (d, Δz a )d μWaist Figure 21.
Projection of the triangular cavity onto the y − z plane. The plane ofthe cavity is rotated around the x -axis by θ , however the lengths of all the sides of thetriangle remain unchanged. To summarize, a misalignment in +( − ) β − causes no change in the spot position on thecurved mirror and a counter-clockwise (clockwise) rotation of the non-congruent side aroundthe x -axis. As a result, the plane of the eigenmode rotates synchronously.
5. Result and comparison
Tables 2 and 3 show the results from the geometrical analysis, and compare them to thesimulation results obtained by using two simulation tools. One is
OptoCad [4] and the otheris
Ifocad [5]. We used them to trace the Gaussian beam through our triangular cavity modelthat has the design parameters for the AEI 10 m Prototype reference cavity. These parametersare given as follows: R = 37 . m , L = 10 . m , and d = 0 . m . By inserting these values intoour geometrical model, we obtained the corresponding numerical values. Due to the fact thatO PTO C AD is 2-dimensional we only used it for simulating the horizontal misalignment types.
6. Conclusion
The results of the geometrical analysis are in excellent agreement with the simulation results,showing sufficient accuracy for the design of an alignment control system for a triangularcavity. We have checked that all the discrepancies between the geometrical analysis and thesimulations decrease by assigning real values for the two larger angles γ to the geometricalanalysis, instead of using γ = π/
2. This analysis can easily be extended to a cavity with moregeneral shape if one follows the equations derived in this paper and modifies the method ofapproximation properly. The geometrical analysis not only serves as a method of checking asimulation result, but also gives an intuitive and handy tool to visualize the eigenmode of amisaligned triangular cavity.
7. Acknowledgments
This work was supported by the QUEST cluster of excellence of the Leibniz Universit¨atHannover. We like to thank Gerhard Heinzel for very helpful discussions on simulationchallenges. igenmode in a misaligned triangular optical cavity Table 2.
Horizontal misalignment comparison.
Type Method M a M c M b Waist∆ x ∆ y ∆ x ∆ y ∆ x ∆ y ∆ x ∆ y AngleGeom.Analy. 0.205 -0.205 -0.205 -0.205 0 -13.970 0 0.205 -1.370 α b OptoCad
Ifocad α − OptoCad -10.051 9.902 -10.051 -9.902 0 0 -10.051 0 0
Ifocad -10.051 9.903 -10.051 -9.903 0 0 -9.9022 0 0Geom.Analy. -0.151 0.151 0.151 0.151 0 0.205 0 -0.001 1.005 α + OptoCad -0.151 0.149 0.151 0.149 0 0.206 0 -0.003 1.005
Ifocad -0.151 0.149 0.151 0.149 0 0.205 0 -0.003 1.005
Table 3.
Vertical misalignment comparison.
Type Method M a M c M b Waist∆ x ∆ z ∆ x ∆ z ∆ x ∆ z ∆ x ∆ z Angle β b Geom.Analy. 0 37.800 0 37.800 0 37.800 0 37.800 0
Ifocad β − Geom.Analy. 0 0.106 0 -0.106 0 0 0 0 0.707
Ifocad β + Geom.Analy. 0 -19.622 0 -19.622 0 -26.729 0 -19.622 0
Ifocad
References [1] Heinzel G 1999 Advanced optical techniques for laser-interferometric gravitational-wave detectors,
PhD thesis ftp.rzg.mpg.de/pub/grav/ghh/ghhthesis.pdf [2] Kawazoe F et al.
J. Phys. Conf. Ser. textbf228 012028[3] Freise A 2010 Frequency domain INterfErometer Simulation SotfwarE,
FINESSE manual [4]
OptoCad (0.90c) 2010 A Fortran 95 module for tracing Gaussian
TEM beams through an opticalset-up, written by Roland Schilling Simulation tool [5]
Ifocad2010 A framework of C subroutines to plan and optimize the geometry of laser interfer-ometers, written by Gerhard Heinzel