Müeller Matrix for Imperfect, Rotated Mirrors

From: Steve Scott, Howard Yuh To: MSE Enthusiasts Re: MSE Memo #0c: Proper Treatment of Effect of Mirrors Date: February 1, 004 (in progress) Memo 0c is an extension of memo 0b: it does a calculation for three good mirrors. Memo 0b is an extension of memo 0a: it improves the calculation of β by evaluating its tangent rather than its cosine. This memo also evaluates β for mirrors M and M3 for light rays that are nearly vertical and nearly horizontal, respectively. Müeller Matrix for Imperfect, Rotated Mirrors A mirror may not have the same reflectivity for S- and P- polarizations, and it may impose a phase shift between the S- and P- polarizations. This effect was evaluated in previous memos. Unfortunately, we neglected to consider that the S- and P- polarizations are defined with respect to a reference frame specific to the mirror, and this reference frame will not, in general, be the same as the coordinate system in the plasma. So a proper treatment must: 1. Determine the angle between the plasma coordinate system and the mirror coordinate system, χ.. Transform the Stokes vector that is defined in the plasma coordinate system (based on E x, E y ) into the mirror coordinate system (E s, E p ) by applying a rotation through angle χ. 3. Compute the effect of the mirrors on the incident Stokes vector in the mirror frame of reference to obtain the reflected Stokes vector. 4. Transform this Stokes vector back into the reference frame of the plasma by rotating through an angle χ. The Müeller matrix for the mirror is M m r m 1 0 0 r m+1 0 0 0 0 rm cos(δ) rm sin(δ) 0 0 r m sin(δ) rm cos(δ) r m+1 r m 1 (1) The Müeller matrix that rotates a Stokes vector through a positive counterclockwise angle χ is: 0 cos(χ) sin(χ) 0 R(χ) () 0 sin(χ) cos(χ) 0 0 0 0 1 1

and the corresponding matrix that rotates a Stokes vector in the negative direction is just R( χ): 0 cos(χ) sin(χ) 0 R( χ) (3) 0 sin(χ) cos(χ) 0 0 0 0 1 So we define an effective Müeller matrix R eff that includes both the effects of rotation and imperfect mirror parameters r m, δ as R eff R( χ)m m R(χ) (4) The result is: a b cos χ b sin χ 0 b cos χ [ a cos R eff χ + r m cos δ sin χ ] [ a r m cos δ ] sin 4χ r m sin δ sin χ [ b sin χ a rm cos δ ] sin 4χ [ a sin χ + r m cos δ cos χ ] rm sin δ cos χ 0 rm sin δ sin χ r m sin δ cos χ rm cos δ (5) where a r m + 1 b r m 1 (6) The retardance of standard mirrors is about δ 180 o. To clarify the notation s handling of small deviations from ideal behavior, we will define δ π + δ, where δ 1. Then sin δ sin δ and cos δ cos δ and so we can rewrite Eq. 5 as a b cos χ b sin χ 0 b cos χ [ a cos R eff χ r m cos δ sin χ ] [ a + r m cos δ ] sin 4χ + r m sin δ sin χ [ b sin χ a + rm cos δ ] sin 4χ [ a sin χ r m cos δ cos χ ] r m sin δ cos χ 0 r m sin δ sin χ rm sin δ cos χ r m cos δ (7) Specific Case: a Good Mirror First let s evaluate R eff in the limit of a good mirror that has ideal reflectance (r m 1, a 1, b 0) but non-ideal phase shift (δ 0). R good eff 0 [ cos χ cos δ sin χ ] [1 + cos δ sin 4χ ] + sin δ sin χ 0 [1 + cos δ sin 4χ [ ] sin χ cos δ cos χ ] sin δ cos χ 0 sin δ sin χ sin δ cos χ cos δ (8)

Now consider the effect of a good mirror operating on incident linearly polarized light at an angle γ to the horizontal. The Stokes vector for such light is S linear 1 cos γ sin γ 0 (9) The resultant light intensity is I good 1 cos γ(cos χ sin χ cos δ ) + 1 sin γ sin 4χ(1 + cos δ ) 1 cos γ sin 4χ(1 + cos δ ) + sin γ(sin χ cos χ cos δ sin δ (cos γ sin χ sin γ cos χ) 1 cos (γ χ) (1 cos δ ) sin χ sin (γ χ) sin (γ χ) + (1 cos δ ) cos χ sin (γ χ) sin δ sin (γ χ) (10) (11) As expected, the phase shift creates a component of circularly-polarized light with a magnitude proportional to sin δ. Specific Case: an Ideal Mirror Let s now evaluate R eff in the limit of ideal mirrors, i.e. r m 1 (so a 1, b 0) and δ 0: Reff ideal 0 cos 4χ sin 4χ 0 (1) 0 sin 4χ cos 4χ 0 0 0 0 1 Now consider the effect of an ideal mirror operating on incident linearly polarized light at an angle γ to the horizontal. The output light is given by I out 0 cos 4χ sin 4χ 0 0 sin 4χ cos 4χ 0 0 0 0 1 1 cos 4χ cos γ + sin 4χ sin γ sin 4χ cos γ cos 4χ sin γ 0 1 cos (γ χ) sin (γ χ) 0 1 cos γ sin γ 0 (13) 3

By comparison with Eq. 9, we see that I out is just light that is linearly polarized at an angle γ out χ γ γ + (χ γ) (14) so the effect of an ideal mirror is to rotate the direction of polarization through an angle (χ γ). Figure 1 illustrates the change in polarization direction that we expect for an ideal mirror, in a simplified geometry in which the initial reference frame (that defines the pitch angle γ) is the same as the s/p reference frame of the mirror. The change in polarization direction is γ ( π γ) (15) and so I conclude that the correct interpretation of β in Eq. 9 is χ π ± β (16) where β, the angle needed to rotate inital coordinate system into mirror s s/p coordinate system), is calculated in the next section. I m not sure whether it should be plus or minus in this equation. 4

Effect of Ideal Mirror on Polarization Direction incident E p E p reflected E s E s incident E p reflected E p γ in E s E s γ out γ out γ in + (π/ - γ in ) Change in polarization direction (π/ - γ in ) Figure 1: Change in polarization direction expected for an ideal mirror, in a simple geometry for which the inital reference frame is the same as the mirror s s/p reference frame. 5

Effect of three good mirrors To simplify the notation, we re-write Eq. 8, the Müeller matrix for a good mirror that has r m 1, δ 0 R good eff 0 C S cos δ CS(1 + cos δ ) S sin δ 0 CS(1 + cos δ ) S C cos δ C sin δ 0 S sin δ C sin δ cos δ with C cos χ, S sin χ. We have three mirrors, each of which is characterized by different angles (χ 1, χ, χ 3 ) and phase shifts (δ δ 1, δ, δ 3 ). We can represent the net effect by multiplying the Müeller matrices for the individual mirrors: (17) R tot eff R 3 eff R eff R 1 eff (18) where R 1 eff is the Müeller matrix for the first mirror, etc. To simplify matters still further, we will take the limit δ 1 1, δ 1, and δ 3 1, i.e. each mirror introduces only a small phase shift. We will keep only the linear terms in the δ 1, δ, δ 3, so we approximate cos δ 1, sin δ δ, and we will neglect any terms that involve more than one factor of δ. In this limit, R good eff 0 C S CS Sδ 0 CS S C Cδ 0 Sδ Cδ 1 Let s first consider the effect of three ideal mirrors. Since each mirror individually simply rotates the direction of polarization of the incident light, we expect the combined effect to be a composite rotation of some sort. As usual, the Müeller matrix for the system of three mirrors is just the product of the Müeller matrices for the individual mirrors: (19) Reff tot ideal Reff ideal 3 Reff ideal Reff ideal 1 (0) where each mirror is described by R ideal eff 0 C S CS 0 0 CS S C 0 0 0 0 1 (1) Recalling Eq. 14, light that is linearly polarized at angle γ incident on a mirror that is rotated about an angle χ will be reflected with a linear polarization at angle γ out χ γ. This becomes the incident polarization angle onto the second mirror, etc. The polarization state of the light as it moves through the three-mirror system is summarized in the Table 1. The final output angle is just γ out (χ 3 χ + χ 1 ) γ. What this means is that the effect of three ideal mirrors (which are oriented at angles χ 1, χ, and χ 3 ) on linearly polarized light is the same as a single ideal mirror that is oriented at angle χ 3 χ + χ 1. 6

Mirror Polarization Angle Polarization Angle (incident) (reflected) 1 γ χ 1 γ χ 1 γ χ (χ 1 γ) 3 χ (χ 1 γ) χ 3 (χ (χ 1 γ)) Table 1: Table 1. Polarization angles of linearly polarized light as it propagates through three ideal mirrors The final Müeller matrix for three ideal mirrors is then obtained directly from Eq. 1 by replacing χ by χ 3 χ + χ 1 : where R ideal 3 eff 0 C S C S 0 0 C S S C 0 0 0 0 1 () C cos (θ 3 θ + θ 1 ) S sin (θ 3 θ + θ 1 ) (3) Using Maple, I verified that Eq. gives the save result as one gets by multiplying the three individual Müeller matrices for the ideal mirrors in Eq. 0 I evaluated R tot eff using Eq. 18 and kept only terms linear in the deltas. The result can be expressed as: R tot eff 0 C S C S r 4 0 C S S C r 34 0 r 4 r 43 1 where each of the nonideal terms (r 4, r 34, r 4, r 43 ) are first-order in the expansion parameters δ 1, δ, and δ 3. Note that the nonideal terms appear in locations which were previously zero in the Müeller matrix for the ideal mirror in Eq.. The nonideal coefficients r 4, r 34, r 4 and r 43 involve both the mirror phase shifts and the mirror orientation angles. For example, r 4 δ 1 sin(4(χ 3 χ ) + χ 1 ) +δ sin(4χ 3 χ ) (4) +δ 3 sin(χ 3 ) (5) Unfortunately, I don t see any simple relationship among the nonideal terms r 4, r 34, r 4, and r 43 : each term is a linear combination of the three phase shifts δ 1, δ, and δ 3, with coefficients that depend on the orientation angles χ 1, χ, and χ 3. So I don t see any way to represent 7

the cumulative effect of three good mirrors (r m 1, δ 0) as a single mirror with some effective phase shift. Note that in the Müeller matrix for the good mirror, Eq. 19, there are fixed relationships amongst the terms involving the mirror s phase shift, e.g. r 4 r 4 and r 43 r 34, whereas there are no such simple relationships among the corresponding nonideal terms in the Müeller matrix for the nonideal 3-mirror system given by Eq. 4. For specificity, I evaluated Eq. 18 for an arbitrary set of orientation angles, χ 1 0.5, χ 0.35, and χ 3 0.65 radians. The result is: r 4 0.99δ 1 + 0.946δ + 0.963δ 3 r 34 0.19δ 1 + 0.33δ 0.67δ 3 r 4 0.479δ 1 0.96δ 0.783δ 3 r 43 0.877δ 1 + 0.955δ + 0.61δ 3 (6) This confirms that is isn t possible, for generalized orientations of the MSE mirrors, to define a single effective phase shift to approximate the effect of the three mirror system. Our only hope is that the actual orientations of the mirrors in the c-mod mse diagnostic lead to serendipitously simple relationships among the nonideal terms r 4, r 34, r 4, and r 43. 8

Calculation of Rotation Angle This section calculates the angle through which the Stokes vector in the plasma frame must be rotated so that it is properly oriented in the mirror coordinate system. As shown in Fig 3, the plasma coordinate system is established by the tokamak z-axis and the k-vector of the radiation. A horizontal coordinate, x, is defined as the cross product of the z-axis and the k-vector. The vertical coordinate, y, is then defined as the cross product of the x direction and the k direction. ˆx, ŷ, and ˆk form a rectangular coordinate system in the tokamak. The electric field vector will have components only in the ˆx and ŷ directions. In the tokamak coordinate system, the pitch angle of the electric field vector is defined as γ tan 1 (E y /E x ). Note that by construction, ˆx is horizontal in the tokamak frame of reference, but ŷ may not be exactly vertical (ŷ will be vertical only if ˆk lies in the tokamak horizontal midplane). ˆx ˆk ẑ ˆk ẑ ŷ ˆx ˆk (7) Similarly, there is a coordinate system for the mirror defined by the k-vector and the mirror normal, ˆn. We will call the horizontal direction Ês because this is the direction of the S-polarization, while the direction of the P-polarization is Êp: Ê s ˆk ˆn ˆk ˆn Ê p Ês ˆk (8) Linearly polarized light in the plasma coordinate system will have components E x and E y. But the reflective properties of the mirror are defined with respect to the S- and P- polarized components E s and E p. So we need to rotate from the x, y coordinates to the s, p coordinates. By reference to Fig. 3, this rotation angle, β, is given by Similarly, we can evaluate sin β: cos β Êp ŷ (9) ( ) ( ) (ˆk ẑ) ˆk (ˆk ˆn) ˆk [ [ (ˆk ˆn) ˆk] (ˆk ẑ) ˆk] ] [ ] [ˆk(ˆk ˆn) ˆn (ˆk(ˆk ẑ) ẑ ˆn ẑ (ˆk ẑ)(ˆk ˆn) (30) sin β Êp ˆx (31) 9

So ( ) ( ) (ˆk ẑ) ˆk ˆk ẑ ] [ ] [ˆk(ˆk ˆn) ˆn (ˆk(ˆk ẑ ˆn (ˆk ẑ) ˆn (ˆk ẑ) tan β ˆn ẑ (ˆk ẑ)(ˆk ˆn) (3) (33) The idl procdure mirror 3d.pro defines the mirror normals for each of the three MSE mirrors according to the actual geometry as well as the k-vectors of originating from the various spatial channel locations and then computes the local tokamak coordinate system ˆx, ŷ, ˆk from Eqs. 7 and the local mirror coordinate system Ês, Êp, ˆk from Eq. 8. Finally it determines the rotation angle by numerically computing the dot product in Eq. 9 and 33. It gets the same answer from Eq. 9. Figure shows the computed angles for mirror M1. Mirror M: The light incident on mirror M is directed almost vertically, i.e. ˆk ẑ, so we need to consider the behavior of Eq. 9 for nearly vertical k-vectors. For generality, we will consider ˆk k xˆx + k y ŷ + k z ẑ ˆn n xˆx + n y ŷ + n z ẑ (34) where we will later take the limit k z 1, k x 0, k y 0. The cross product terms are given by ˆx ŷ ẑ ˆk ẑ k x k y k z 0 0 1 and and the dot products are k y ˆx ŷk x (35) ˆx ŷ ẑ ˆk ˆn k x k y k z n x n y n z (k y n z n y k z )ˆx + ŷ(k z n x k x n z ) + ẑ(k x n y n x k y ) (36) ˆn ẑ n z ˆk ẑ k z ˆk ˆn k x n x + k y n y + k z n z (37) 10

Figure : Computed rotation angle for light incident on mirror M1. The various curves represent various assumptions about the location of the system s optical axis. Mirror M1 is assumed to lie in the tokamak horizontal midplane in these calculations (in reality it is a few centimeters high). 11

so we can now evaluate tan β: tan β n x k y n y k x n z k z (n x k x + n y k y + n z k z ) n x k y n y k x n z (1 kz) k z (n x k x + n y k y ) n x k y n y k x n z (kx + ky) k z (n x k x + n y k y ) (38) Now we impose the limitation that the normal to mirror M has very little component in the radial direction (n y n x n z ), and we take the limit of a nearly vertical k-vector, k z 1: MirrorM : tan β n x k y n z (k x + k y) n x k x k y /k x n z k xn x (1 nz kx+k y n x k x ) k y /k x (39) Note that although k x and k y are small, Eq. 39 indicates that β depends only on their ratio, which could vary significantly across the surface of the mirror. We would need to fold in ray-tracing of the rays for each mse spatial channel to assess the quantitative effect. Mirror M3: The light incident on mirror M3 is nominally horizontal, i.e. k y k z k x 1 and n z n x n y. MirrorM3 : tan β n x k y n y k x n z (kx + ky) k z (k x n x + k y n y ) n y k x n z kx k z k x n x n y (40) n x k z 1

Geometry for Rotating Incident Light into Mirror P/S Coordinate System y x k z E y β Plasma-frame coordinates x, y E p E p E p r m E p δ E p β E y k k (k n)n k x k z k z E x E s k E s θ k n k n n θ k E p E s E s E s cos β y. E p cos β y. E p E s k n k n E p E s k Figure 3: Generalized geometry for linearly polarized light incident on a flat mirror. Side View of C-Mod Mirrors and K-vectors M M3 k 4 horizontal, purely radial k 3 horizontal, purely toroidal k purely vertical (M3 is vertical) k 1 horizontal, with nonzero toroidal and radial components M1 Figure 4: Elevation view of the C-Mod MSE mirrors. 13

Plan View of C-Mod Mirrors and K-vectors k 1 M1 n 1 M k 3 purely toroidal M3 k purely vertical k n n 3 k 4 purely radial Figure 5: Plan view of C-Mod MSE mirrors. r_channel r_axis Geometry to Determine Toroidal Angle of Mirror M1 and First K-vector phi_dnb r_dnb r_mse * cos(phi_dnb) r_mse r_base r_mse * sin(phi_dnb) (toroidal direction) theta cos -1 [ (r_dnb-r_ch)/base ] Mirror M1, oriented perpendicular to the optical axis (which is pointed at r_axis) Figure 6: Algebra for calculating the k-vector for light incident on the M1 mirror. 14

Aside: Rotation Angle of Reflected Light This section calculates the angle, β through which the reflected light must be rotated from the mirror s s/p coordinate system into the frame of the reflected beam. In the analysis above we assumed that β β. But I am confused, because this analysis indicates that in general β β. After the reflection, the relative amplitude of the P-polarization may be reduced relative to the S-polarization, and there may be a phase shift. The component amplitudes are E s and E p and these are defined with respect to the exit mirror coordinate system, Ê s ˆk ˆn ˆk ˆn Ê p Ê s ˆk. (41) and we will want to project these components back onto the exit tokamak coordindate system: which will involve a rotation angle ˆx ˆk ẑ ˆk ẑ ŷ ˆx ˆk (4) cos β ŷ Ê p (43) One question of obvious interest is whether or not β β. We first evaluate the angle through which the incident rays must be rotated to put them into the mirror s reference frame: cos β Êp ŷ (44) ( ) ( ) (ˆk ẑ) ˆk (ˆk ˆn) ˆk [ [ (ˆk ˆn) ˆk] (ˆk ẑ) ˆk] ] [ ] [ˆk(ˆk ˆn) ˆn (ˆk(ˆk ẑ) ẑ ˆn ẑ (ˆk ẑ)(ˆk ˆn) The algebra for the angle through which the emergent rays must be rotated to put them back into the tokamak frame is the same, except that we replace the incident k-vector by the reflected k-vector, i.e. ˆk ˆk where ˆk ˆk (ˆk ˆn)ˆn: cos β Ê p ŷ (45) 15

Radius α 0 o α 4 o β β β β 63 36.5 53.5 35.3 51.1 69 4. 65.8.9 59.1 75 1.0 78.0 11.3 60.9 81 0.0 43.7 0.0 0.0 87 11.1 78.9 10.4 61.8 Table : Computed rotation angle for incident and emergent rays for light incident on mirror M1. The first set of numbers is for incident light that lies in the horizontal midplane (α o o ). The second set of numbers is for incident light that has a slight vertical tilt, e.g. light that originates at the tokamak midplane and that hits the top of mirror M1. ˆn ẑ (ˆk ẑ)(ˆk ˆn) ˆk ẑ ˆk ˆn ˆn ẑ [ (ˆk ˆn(ˆk ˆn)) ẑ ] [ (ˆk ˆn(ˆk ˆn)) ˆn ] (ˆk ˆn(ˆk ˆn)) ẑ ˆk ˆn(ˆk ˆn) ˆn ˆn ẑ [ˆk ẑ (ˆk ˆn)(ˆn ẑ) ] [ˆk ˆn ˆk ˆn ] ˆk ẑ (ˆk ˆn)ˆn ẑ ˆk ˆn ˆn ẑ + (ˆk ˆn)(ˆk ẑ (ˆk ˆn)(ˆn ẑ)) ˆk ẑ (ˆk ˆn)ˆn ẑ ˆk ˆn The problem I have with this result is that in general, β β. This issue needs to be resolved. Three Mirrors Unfortunately, in the c-mod mse diagnostic there are three mirrors, so in principle we need to treat each one with the rotation and counter-rotation operations: M m R( χ 3 ) M 3 R(χ 3 ) R( χ ) M R(χ ) R( χ 1 ) M 1 R(χ 1 ) (46) where χ i represents the rotation angle between the coordindate sytem of the i th mirror and the plasma coordinate system, and M j represents the Müeller matrix for a flat mirror, given by Eq 1. The three mirrors were prepared in the same coating lot and so it is reasonable to assume that their reflectance and phase shifts have the same dependence on angle-ofincidence, but the distribution of angles-of-incidence for the ensemble of rays that pass through the diagnostic are not identical at the three mirrors, so in general we have M 1 M M 3. In addition, the rotation angles χ 1, χ, and χ 3 are set by the relative angle of the incident k-vector and the mirror normal, and these may be different for all of the mirrors. We have shown that the particular orientation of mirrors M and M3 is such that, in the limit of small mirror size, χ χ 3 π/ and in this limit the various rotations by R(χ) 16

will have no effect. The actual mirrors have a finite extent corresponding roughly to a range of angles of incidence of about ±10 o, and this will cause the values of χ to deviate a few degrees from π/. But we re hopeful (hoping?) that the dominant effect will be mirror M1, for which χ 1 varies from 0 degrees at the optical axis to 0-30 degrees at the extremity of the optics (channels at R 67 and R 87). 17

Orphaned stuff:... and we can also evaluate cos β: cos β n z k z (k x n x + k y n y + k z n z ) (k x + k y) 0.5 [(k y n z k z n y ) + (k x n z k z n x )) + (k x n y n x k y ) ] 0.5 n z (1 k z) k z (k x n x + k y n y ) (k x + k y) 0.5 [(k y n z k z n y ) + (k x n z k z n x ) + (k x n y n x k y ) ] 0.5 n z (k x + k y) k z (k x n x + k y n y ) (k x + k y) 0.5 [(k y n z k z n y ) + (k x n z k z n x ) + (k x n y n x k y ) ] 0.5 n z (kx + ky) 0.5 k k z (n x k x + n y k x +ky y ) k x +ky (47) [(k y n z k z n y ) + (k x n z k z n x )) + (k x n y n x k y ) ] 0.5 We can now take the limit of k z 1, k x 0, k y 0, although we will not put any constraint on the ratio of k x to k y. The first term in the numerator vanishes, leaving k n x k x + n y k x +ky k z 1 : cos β y k x +ky (48) [(k y n z n y ) + (k x n z n x ) + (k x n y n x k y ) ] 0.5 If we consider a rectangular coordinate system in which x is toroidal, y is radial, and z is vertical, then from Fig. 4 we get n x 1, n y 0, n z 1, and the expression for cos β reduces to k z 1, n y 0 : cos β (49) [kx + ky] 0.5 [ky + (1 + k x ) ] 0.5 k x 18