LCLS-TN-4- Setting The Vale And Polarization Mode Of The Delta Undlator Zachary Wolf, Heinz-Dieter Nhn SLAC September 4, 04 Abstract This note provides the details for setting the longitdinal positions of the for qadrants of the Delta ndlator in order to prodce the desired wavelength and polarization mode of the emitted light. Introdction The longitdinal positions of the for qadrants of the Delta ndlator determine the characteristics of the light it prodces for given electron beam parameters. In this note, we stdy the connection between the qadrant positions and the light characteristics. The note breaks into parts the stdy of how the qadrant positions determine the light characteristics. First, the relations between qadrant positions and the magnetic eld in the ndlator are determined. Then the relation between the magnetic eld in the ndlator and the electric eld of the light is derived. After that, the relation between the electric eld of the light and the polarization ellipse is derived. The relation between electric eld and the Stokes parameters is also derived. Finally, the relation between the polarization ellipse and the Stokes parameters is derived. Overview. Parameters The Delta ndlator has 4 movable qadrants. An overall shift of all 4 qadrants prodces a phase shift of the electron beam relative to a light wave. This featre will not be sed since a separate phase shifter will be placed before the ndlator. So three parameters, three relative qadrant positions, mst be determined in order to set the vale and polarization mode of the ndlator. The forth qadrant position is set by making the average of the qadrant positions eqal to zero. The three qadrant positions determine three parameters of the magnetic eld in the ndlator. The magnetic eld is eqivalent to the eld from two crossed planar adjstable phase ndlators, as will be shown below. The three parameters of the magnetic eld are the vale relative to the maximm vale 0, the ratio of the eld strengths from the two crossed planar ndlators, and the phase di erence of the elds in the two crossed ndlators. The vale determines the wavelength of light prodced by the ndlator. The relative strengths of the two crossed planar ndlators and the phase di erence of the magnetic elds determine the light polarization characteristics. The Work spported in part by the DOE Contract DE-AC0-76SF0055. This work was performed in spport of the LCLS project at SLAC.
reqired vale of 0 is determined experimentally, and so it is a parameter pt in by hand after measrement. A charged particle going throgh the ndlator prodces light. The electric eld from the light is de ned by three parameters. The rst is the wavelength of the light. Next is the ratio of the transverse electric eld strengths in two perpendiclar directions. The last parameter is the phase di erence between the electric eld components in the two perpendiclar directions. The combination of the ratio of the eld strengths and the phase di erence determine the polarization characteristics of the light. The phase di erence also determines the rotation direction of the electric eld in a transverse plane at a xed location as time evolves. It also determines the helicity of the electric eld wave, how it rotates in space along the direction of propagation at xed time. The electric eld polarization can be characterized by the polarization ellipse. The ellipse is characterized by the ratio of the minor and major axes, the angle of the ellipse relative to an axis of the transverse plane, and the rotation direction of the eld arond the ellipse. The rotation direction is reqired to resolve a sign ambigity of the phase di erence between the components of the electric eld, which the angle of the ellipse and the ratio of the axes does not give. In essence, the polarization ellipse pls the rotation direction ( eld helicity) gives two parameters describing the eld, namely the ratio of the eld components and the phase between them. The third parameter reqired to describe the light is the wavelength, which mst be measred separately. The electric eld can also be characterized by the Stokes parameters. There are for Stokes parameters and three of them are independent. We se ratios of the parameters to determine the polarization mode. Three ratios are calclated, bt they are related by a constraint. The third ratio is sed, even with the constraint, to resolve a sign ambigity. The ratios of the Stokes parameters pls the wavelength of the light give three parameters needed to characterize the electric eld.. Relative Phases A topic which will come p repeatedly is the rotation direction of a vector as a parameter w evolves. Sppose the components of the vector are given by x A cos(w) () y B cos(w + ') () where ' is the phase of y relative to x. In the x-y plane, as w increases, the point with coordinates (x; y) rotates clockwise when 0 < ' <, and the point with coordinates (x; y) rotates conterclockwise when < ' < 0. This is easily seen by looking at the initial step at w 0 for a positive increment w. x 0 (3) y B sin(')w (4) y is negative, giving clockwise rotation, when sin(') > 0, or when 0 < ' <. y is positive, giving conterclockwise rotation, when sin(') < 0, or when < ' < 0. We adopt the convention that phase angles and geometrical angles are in the range [ ; ]. Imposing this restriction will give niqe vales to all qantities, whether we are calclating radiation properties from row positions, or whether we are calclating row positions from radiation properties. We now go throgh in detail the steps leading from setting the qadrant positions to determining the radiation properties.
Figre : Coordinate system for the scalar potential of a single qadrant. 3 Magnetic Field On The Undlator Beam Axis 3. Scalar Potential For The Undlator In a previos technical note the scalar potential from a single magnet array was derived. coordinate system shown in gre, the scalar potential has the form In the 0 exp ( k s) cos (k (z z 0 )) (5) where 0 is a constant, k where is the ndlator period, z is the coordinate down the ndlator axis, and z 0 gives the qadrant position along z. We work close to the beam axis where the variation of the scalar potential in the r-direction is small, and we ignore the r dependence. We se this form of the potential to calclate the magnetic scalar potential in the ndlator by rotating the for qadrants and their scalar potentials. The Delta is positioned as shown on the left side of gre. In the laboratory, z is along the beam direction, y L is p, and x L makes a Figre : The left side of the gre shows the Delta ndlator in its con gration in the tnnel where y L is p, z is in the beam direction, and x L makes a right handed system. For or calclations, we se the rotated coordinate system on the right, where x is along the line pointing from qadrant 3 to qadrant, y is along the line pointing from qadrant 4 to qadrant, and z makes a right handed system. right handed system. For or calclations, it is more convenient to se the rotated system x, y, z, Z. Wolf, "A Calclation Of The Fields In The Delta Undlator", LCLS-TN-4-, Janary, 04. 3
where x is in the direction from qadrant 3 to qadrant, y is in the direction from qadrant 4 to qadrant, and z is in the beam direction. Using eqation 5, the scalar potential for each of the qadrants in the x, y, z system is (x; y; z) 0Q exp (k x) cos (k (z z 0 )) (6) (x; y; z) 0Q exp (k y) cos (k (z z 0 )) (7) 3 (x; y; z) 0Q exp ( k x) cos (k (z z 03 )) (8) 4 (x; y; z) 0Q exp ( k y) cos (k (z z 04 )) (9) where z 0i is the longitdinal shift of qadrant i, and 0Q is the amplitde of the scalar potential of all the identical qadrants on the axis of the ndlator where x 0 and y 0. Qadrants 3 and 4 are loaded with opposite polarity magnets as qadrants and in order to make a vertical eld planar ndlator in the laboratory frame when all the rows are aligned. This acconts for the mins signs, 0Q, in the potentials for qadrants 3 and 4. In order for the eld from each qadrant to go throgh the relative phase range [ ; ], the z 0i mst have the range ;. Changing z0i by a mltiple of does not change the eld from that qadrant. This can be sed to restrict the z 0i to the range ;. We do this in order to determine niqe qadrant positions from given radiation properties. The scalar potential for the ndlator is the sm of the scalar potentials for the qadrants. Qadrants and 3 both depend on x, and qadrants and 4 both depend on y. We rst add the scalar potentials for qadrants and 3, and then add the scalar potentials for qadrants and 4, and then add the sms to get the scalar potential for the whole ndlator. We will interpret this as forming the entire ndlator from two crossed planar adjstable phase ndlators. The scalar potential for the combination of qadrants and 3 is given by Let 3 0Q exp (k x) cos (k (z z 0 )) 0Q exp ( k x) cos (k (z z 03 )) (0) z 0 Z 3 + 3 z 03 Z 3 3 So Z 3 z 0 + z 03 is the average z-position of the qadrants, and () () (3) 3 z 0 z 03 (4) is the z-shift between the qadrants. With these de nitions, the scalar potential for the pair of qadrants becomes 3 3 0Q sinh (k x) cos k cos (k (z Z 3 )) 3 + 0Q cosh (k x) sin k sin (k (z Z 3 )) (5) This is the scalar potential for a planar adjstable phase ndlator 3. The fll range of amplitdes and phases are covered if the range of 3 incldes ; and the range of Z3 incldes, both of which are covered by the range of the z0i given above. ; 3 Z. Wolf, "Variable Phase PPM Undlator Stdy", LCLS-TN--, May, 0. 4
Let Similarly, the scalar potential for the combination of qadrants and 4 is given by 4 0Q exp (k y) cos (k (z z 0 )) 0Q exp ( k y) cos (k (z z 04 )) (6) z 0 Z 4 + 4 z 04 Z 4 4 So Z 4 z 0 + z 04 is the average z-position of the qadrants, and (7) (8) (9) 4 z 0 z 04 (0) is the z-shift between the qadrants. With these de nitions, the scalar potential for the pair of qadrants becomes 4 4 0Q sinh (k y) cos k cos (k (z Z 4 )) 4 + 0Q cosh (k y) sin k sin (k (z Z 4 )) () This is again the potential for a planar adjstable phase ndlator. The scalar potential for the ndlator is the sm of the scalar potentials for the qadrant pairs. 3 + 4 () Performing the sm, we nd 3 0Q sinh (k x) cos k cos (k (z Z 3 )) 3 + 0Q cosh (k x) sin k sin (k (z Z 3 )) 4 + 0Q sinh (k y) cos k cos (k (z Z 4 )) + 0Q cosh (k y) sin 4 k sin (k (z Z 4 )) (3) By ptting in the varios vales for Z 3, Z 4, 3, and 4, we get the scalar potential for the varios ndlator modes at di erent vales. 3. Magnetic Field On The Undlator Axis The magnetic eld in the ndlator is given by B r. Taking the gradient and setting x 0 and y 0 on the ndlator axis, we nd 3 B x (0; 0; z) 0Q k cos k cos (k (z Z 3 )) (4) 4 B y (0; 0; z) 0Q k cos k cos (k (z Z 4 )) (5) 5
If we let B 0 0Q k, then on the ndlator axis 3 B x B 0 cos k cos (k (z Z 3 )) (6) 4 B y B 0 cos k cos (k (z Z 4 )) (7) In order to simplify these formlas frther, let 3 B x0 B 0 cos k (8) 4 B y0 B 0 cos k (9) x0 k Z 3 (30) y0 k Z 4 (3) Note that positive and negative vales of 3 and 4 prodce the same eld amplitde becase of the cosine dependence. With these sbstittions, the elds become B x B x0 cos (k z + x0 ) (3) B y B y0 cos k z + y0 (33) We adopt the convention that eld amplitdes are positive, which reqires that we restrict the range of 3 and 4 to ;. Each z0i has the range ;, however, giving a possible range to ij of [ ; ]. Sppose ij > so cos k ij is negative. Sbtract from z 0i and add to z 0j. This changes ij by, which changes the sign of cos k ij, making it positive, and does not change the vale of Z ij. If ij < so cos k ij is negative, add to z 0i and sbtract from z 0j. This changes ij by +, which changes the sign of cos k ij, making it positive. We can change the origin of z so that B x has zero phase. In this case, we write the elds as B x B x0 cos (k z) (34) B y B y0 cos (k z + ) (35) y0 x0 k (Z 3 Z 4 ) (36) As noted previosly, we adopt the convention that be in the range [ ; ]. Mltiples of can be added to or sbtracted from. This amonts to changing both the Z ij, one by and the other by, so the di erence changes by. We do this by changing z 0i and z 0j both by so ij doesn t change. If >, or Z 3 Z 4 >, sbtract from z 0 and z 03 and add to z 0 and z 04. This changes Z 3 Z 4 by, which changes by. If <, or Z 3 Z 4 <, add to z 0 and z 03 and sbtract from z 0 and z 04. This changes Z 3 Z 4 by +, which changes by +. This places in the range [ ; ]. This convention for the range of will be sed when we discss the helicity of the magnetic eld. 3.3 Magnetic Field Helicity Using eqations 34 and 35, we can plot the point (B x ; B y ) as one moves down the ndlator. The point (B x ; B y ) traces ot an ellipse as one moves in z. This is shown in gre 3 for eqal strength 6
Figre 3: Plots of the point (B x ; B y ) as one moves down the ndlator. negative helicity elds. Negative vales of give positive helicity elds. Positive vales of give B x and B y, and 4 and 4. In general, positive vales of, 0 < <, give clockwise rotation as z increases, giving left hand or negative helicity magnetic elds. Negative vales of, < < 0, give conterclockwise rotation as z increases, giving right hand or positive helicity magnetic elds. 3.4 Relation Between The Magnetic Field Parameters And The Qadrant Positions nowing the ndlator qadrant positions lets s determine the magnetic eld parameters. The vale of the ndlator relative to the maximm vale 0 is given by q Bx0 + B y0 p (37) 0 B 0 + B0 s p cos 3 k + cos 4 k s p cos z 0 z 03 k + cos k z 0 z 04 (38) (39) 7
The ratio of the eld strengths B y0 B x0 is given by The phase of B y relative to B x is B y0 4 cos k B x0 cos k 3 (40) cos k z0 z 04 (4) cos k z 0 z 03 k (Z 3 Z 4 ) (4) k [(z 0 + z 03 ) (z 0 + z 04 )] (43) Similarly, knowing the magnetic eld parameters lets s determine the qadrant positions. need, however, one additional constraint in order to determine the for qadrant positions. choose the extra constraint to be that the average z-position of the qadrants is zero. The three other eqations are We We z 0 + z 0 + z 03 + z 04 0 (44) z 0 z 03 3 (45) z 0 z 04 4 (46) z 0 z 0 + z 03 z 04 k (47) The row position di erences 3 and 4 are fond in terms of 0 and B y0 B x0 to be 0v 3 cos B 0 k @ t C By0 A (48) + B x0 0v 4 By0 cos B 0 B x0 k @ t C By0 A (49) + B x0 There is a sign ambigity in 3 and 4 since both positive and negative angles have the same cosine. Either sign can be chosen for 3 and 4 since changing the sign of 3, for instance, only interchanges the vales of z 0 and z 03, and all the eqations for the z 0i remain valid. The same is tre for 4. In order to make cos single valed, we take the range cos to be [0; ], then 3 and 4 are positive. There was an additional sign ambigity when we took the sqare root in the argment of the cos fnction. We chose the positive sign. This restricts the range of cos to ;. The range of ij is then ;. We have already seen that this is reqired in order for eld amplitdes to be de ned as positive. 8
The soltion for the row positions in terms of the eld parameters is z 0 + 3 k z 0 + 4 k z 03 3 k z 04 4 k where 3 and 4 are expressed in terms of the eld parameters in eqations 48 and 49. ij in the range ; and in the range [ ; ], the range of the z0i is ;. (50) (5) (5) (53) With 4 Radiation Field 4. Electric Field Of The Radiation In the previos section we fond the magnetic eld on the ndlator axis in terms of the row positions of the for qadrants. We now nd the electric eld from a charge moving down the axis of the ndlator and being accelerated by the magnetic eld. The general form for the far electric eld from an accelerating charge is 4 n [(n ) ] 3 E(t) q 4 0 c 6 4 r( n ) 3 where q is the charge of the radiating particle, n is the direction from the particle to the observation point, is the velocity of the particle divided by the speed of light, and r is the distance from the particle to the observation point. The qantity in the bracket is evalated at the retarded time t r de ned sch that t t r + r(t r )c. We want the electric eld at the fndamental freqency from an electron in a long ndlator. To nd the eld, we take the Forier transform of eqation 54. The Forier transform is 5 i!q E(!) p 40 c r Z [n (n (t r ))] e 7 5 ret (54) i!(tr+ r(tr ) c ) dt r (55) In this expression for the Forier transform, we assme a long ndlator which lets s integrate from t r to +. This simpli es the calclations withot a ecting the reslting polarization mode of the eld. We also assme that in the ndlator, the particle stays very close to the beam axis, so the observation point can be taken in the forward z-direction so that n e z (56) where e z is a constant nit vector in the z-direction. Frthermore, we assme that the observation point is far away so r can be taken as constant in the denominator. This assmption is not made in the phase term in the exponent, however. With n e z, the triple cross prodct in the integral is given by n (n ) e x x e y y (57) 4 A. Hofmann, The Physics of Synchrotron Radiation, Cambridge University Press, Cambridge, 004. 5 Ibid. 9
To nd, we start with the Lorentz force law dp dt q (v B) (58) where p mv,, and m is the particle rest mass. The energy of the particle is constant in the magnetic eld of the ndlator, except for radiation losses which we neglect. With constant energy, is constant. The Lorentz force law gives Using we nd q ( B) (59) m B x B x0 cos (k z + x0 ) (60) B y B y0 cos k z + y0 (6) x qb y0 sin k z + mck y0 (6) y qb x0 mck sin (k z + x0 ) (63) To perform the integral in the Forier transform, we need to know z as a fnction of the retarded time. In order to simplify the analysis, we take the z-position of the charge at time t to be z v z t, where v z is the average velocity in the z-direction, and we ignore small forward velocity deviations. This means that the z-position of the charge at time t r is z v z t r. In the exponent of the Forier transform, we need the distance r from the observation point to the charge. Let z o be the z-position of the observation point. Then r(t r ) z o z(t r ) z o v z t r. Inserting these expressions into the Forier transform formla and performing the integral gives E(!) q p 40 mc k r!e i!zoc v f(k v z!( zc ))[ e x B y0 e i y0 + ey B x0 e i x0 ] v (k v z +!( zc ))[ e x B y0 e i y0 + ey B x0 e i (64) x0 ]g To nd the electric eld as a fnction of time, we perform the inverse Forier transform given by To do the integral, we make se of the identity E(t) p Z E(!)e i!t d! (65) f(x)(ax b) jaj f( b a ) (66) We nd E(t) q 4 0 mc r ( fe x B y0 cos t e y B x0 cos t v z v zc ) z o k v z c z o k v z c v zc v zc + y0 + x0 g (67) 0
From this formla, we see that the radiation anglar freqency! r and wave nmber k r are! r k r c k v z v ( zc ) (68) where where k r r and r is the radiation wavelength. Sbstitting the vales for the anglar freqency and wave nmber gives E(t) q 4 0 mc r ( v z v zc ) fe x B y0 cos(! r t k r z o + y0 ) e y B x0 cos(! r t k r z o + x0 )g (69) From eqation 68, we have Using this expression in the formla for E(t) gives E(t) q 4 0 mv z r r ( v zc v zc ) k r k r (70) ex B y0 cos(! r t k r z o + y0) e y B x0 cos(! r t k r z o + x0) (7) Eqation 68 is also derived by considering the light slipping ahead of the charge by one radiation wavelength per ndlator period c v z + r (7) which leads to eqation 70. v Solving for ( zc ) in terms of, x, and y for a relativistic particle gives v z c + x + y Eqations 6 and 63 for x and y can be expressed as x x sin k z + y0 (73) (74) y y sin (k z + x0 ) (75) where x qby0 mck and y qbx0 mck. x is the parameter for the -4 planar ndlator, and y is the parameter for the -3 planar ndlator. Averaging the velocities over time gives an expression for the average forward velocity of a charge in an ndlator v z c + x + y (76) Setting v z ' c in the nmerator of 70 and sing 76 gives r + x + y (77) This expression for the spontaneos radiation wavelength in an ndlator is the same as the expression for the resonance condition in an FEL. This means that at resonance, the FEL radiation wavelength is the same as the spontaneos radiation wavelength.
With the expressions given above, the electric eld of the light from a relativistic charge in an ndlator is E(t) q 3 0 mc r [ + ( x + y)] fe x B y0 cos(! r t k r z o + y0 ) e y B x0 cos(! r t k r z o + x0 )g (78) We see that the electric eld has the form E x F B y0 cos(! r t k r z o + y0 ) (79) E y F B x0 cos(! r t k r z o + x0 ) (80) where F is a common factor that depends on the vale of the ndlator. Let E xo F B y0 and E y0 F B x0. We drop the o for the observation point and write z instead of z o. We can also add to the phase of E y to accont for the mins sign in front. and E y0 are positive magnitdes, jst as B x0 and B y0 are positive magnitdes. We choose or time origin to make the phase of E x eqal to zero. With these sbstittions, the electric eld is written as E x cos(! r t k r z) (8) E y E y0 cos(! r t k r z + ) (8) where x0 y0 +. Expressing the phase di erence in terms of the phase di erence of the magnetic elds, we get +, and in terms of ndlator row shifts, k (Z 3 Z 4 ) +. Note that mltiples of can be added or sbtracted from. We se this to adopt the convention that is in the range [ ; ]. This convention will be sefl when we discss the helicity of the electric eld. At xed z, the electric eld evolves in time as E x cos(! r t) (83) E y E y0 cos(! r t + ) (84) Positive prodces an electric eld which rotates clockwise. which rotates conterclockwise. At xed time, the electric eld behaves as Negative prodces an electric eld E x cos( k r z) (85) E y E y0 cos( k r z + ) (86) Taking the negative of the argments of the cosines in these formlas gives E x cos(k r z) (87) E y E y0 cos(k r z ) (88) Positive prodces a right handed wave. Negative prodces a left handed wave. This will be shown in more detail below. 4. Electric Field Helicity The helicity of the electric eld is fond by looking at how the eld varies with z at xed time. Eqations 87 and 88 are sed in gre 4 to show how the elds evolve in z. Note that positive vales of give a right hand, or positive helicity eld. Negative vales of give a left hand, or negative helicity eld.
Figre 4: Positive vales of give positive helicity elds. Negative vales of give negative helicity elds. The helicity of the electric eld is opposite to the helicity of the ndlator magnetic eld. Since +, we have E y E y0 cos(k r z + ) (89) The extra in the phase changes the helicity of the electric eld compared to the magnetic eld. In order to gain insight why the helicity of the electric eld is opposite to the helicity of the magnetic eld, we start by considering the helicity of the motion of a charge in the magnetic eld. The left part of gre 5 shows a charge in an ndlator whose magnetic eld has right handed helicity. The charge moves in z with right handed helicity, as we will show. The magnetic eld given in eqations 34 and 35 gives the charge velocity given in eqations 6 and 63. The velocity can be rewritten as x qb y0 mck cos (k z) (90) y qb x0 mck cos (k z ) (9) The phase of y is the negative of the phase of the magnetic eld B y, bt with an added. So the helicity of the charge velocity is the same as the helicity of the magnetic eld. This is expected since the force and the de ection direction are at right angles to the eld and follow the eld. A projection 3
Figre 5: A charge in a magnetic eld with right handed helicity prodces an electric eld with left handed helicity. onto a screen of the charge motion shows the charge moving in a conterclockwise direction and tracing ot an ellipse when viewed from +z looking toward z. This is shown in the middle of the gre. The charge acceleration at several locations is also shown. The electric eld from the charge is in the direction of the charge acceleration. This comes from the triple cross prodct in eqation 54. n [(n ) ] ( v z c ) (9) The right part of gre 5 shows the electric eld from an electron as a fnction of z. When the electric eld points away from s, it is marked with a dotted line, and when it points toward s, it is marked with a solid line. We start where the charge is moving down, and every eighth of a revoltion we indicate the electric eld direction in the charge acceleration direction. The key point is that becase the velocity of light is greater than the charge velocity, the initial point we consider has its electric eld at the greatest z location. Sbseqent points, which are at larger z for the charge, are at smaller z for the electric eld. This e ectively reverses the z direction of the seqence of charge positions compared to electric eld positions. This reverses the helicity of the electric eld compared to the charge motion. The electric eld is left handed as shown in the gre. In smmary, the helicity of the electric eld is opposite to the helicity of the ndlator magnetic eld. 4.3 Relation Between The Electric Field Parameters And The Magnetic Field Parameters The parameters describing the electric eld are the radiation wavelength r, the ratio of the strength of E y to E x, namely E y0, and the phase of E y relative to E x. The radiation wavelength is given by eqation 77. The term x + y is eqal to given in eqation 37. Using eqation 37, we write the wavelength as " r + # 0 (93) We measre 0 dring the initial magnetic measrements of the ndlator. wavelength is determined by the ratio 0. Since we fond that E xo F B y0 and E y0 F B x0, we have 0 Once it is known, the E y0 B x0 B y0 (94) 4
Finally, we have already fond the relation between the phases + (95) Mltiples of can be added or sbtracted from. We se this to adopt the convention that is in the range [ ; ]. An additional parameter which is interesting is twice the time average vale of the sqare of the electric eld strength I he Ei (96) E x0 + E y0 (97) The parameter I is proportional to a fnction of and so is not an independent parameter. I / [ + ] 4 (98) I is proportional to the power of the radiation from the ndlator. The reverse transformation from electric eld parameters to magnetic eld parameters is fond by solving the above eqations for the magnetic eld parameters. 0 s 0 r (99) B y0 B x0 E y0 (00) + (0) Mltiples of can be added or sbtracted from. in the range [ ; ]. We se this to adopt the convention that is 5 Polarization Ellipse 5. Eqations Of The Polarization Ellipse The polarization ellipse is fond by considering the radiation wave at a xed z location as a fnction of time. The eqations for the electric eld are E x cos(! r t) (0) E y E y0 cos(! r t + ) (03) These eqations can be expressed as a relation between E x and E y that does not involve time. This is the polarization ellipse 6. Using the identity cos(! r t + ) cos(! r t) cos() sin(! r t) sin() (04) and setting we nd Ex cos(! r t) E x (05) Ey Ex Ey + cos() sin () (06) E y0 E y0 5
Figre 6: Ellipse with major axis a, minor axis b, rotated throgh an angle. This eqation is analogos to the eqation for a rotated ellipse as shown in gre 6. In the rotated primed coordinate system, the ellipse has eqation x 0 a + y0 b (07) and in the nprimed system, the eqation is x a cos ( ) + b sin ( ) +y a sin ( ) + b cos ( ) +xy a b cos( ) sin( ) (08) Becase of the symmetry of the ellipse, we take to be in the range ;. We can identify the major axis, minor axis, and rotation angle of the polarization ellipse throgh the following eqations. Ex0 sin () a cos ( ) + b sin ( ) (09) E y0 sin () a sin ( ) + b cos ( ) (0) cos() E y0 sin () a b cos( ) sin( ) () The electric eld amplitdes and phase reqired to prodce a given polarization ellipse are given by solving for, E y0, and in terms of a, b, and. After some algebra, we nd r [(a + b ) + (a b ) cos ( )] () r E y0 [(a + b ) (a b ) cos ( )] (3) s cos() sin( ) (a b ) j sin( )j a 4 + b 4 + a b tan + cot (4) 6 E. Hecht, Optics, 3 rd edition, Addison Wesley Longman, Reading, Massachsetts, 998. 6
Note that the ellipse parameters do not distingish between positive and negative vales of. The helicity of the eld mst be speci ed in addition to the ellipse parameters. For a right handed eld, is positive. For a left handed eld, is negative. A di erent ambigity in the sign of cos() was resolved by sing eqation. The sign of cos() is the same as the sign of sin( ) since a > b. So the sign when taking the sqare root was replaced by sin( )j sin( )j. The polarization ellipse parameters can be fond in terms of the electric eld parameters. After some algebra, we nd tan( ) E y0 E x0 E y0 a q Ex0 + Ey0 + b q Ex0 + Ey0 cos() (5) Ex0 Ey0 + 4E x0 Ey0 cos (") Ex0 Ey0 + 4E x0 Ey0 cos (") (6) (7) As noted above, the ellipse angle does not depend on the sign of. The fnction tan( ) has the same vale for mltiple vales of. We will resolve this ambigity below when we solve for. A sign ambigity in front of the sqare root in a and b was resolved by making a > b. 5. Relation Between The Polarization Ellipse Parameters And The Electric Field Parameters The polarization ellipse parameters sefl for de ning the ndlator mode are the ratio ba of minor axis to major axis, the direction of the major axis relative to the x-axis, and the rotation direction of the eld point arond the ellipse. The ratio ba is given by q b a + Ey0 Ex0 Ey0 + 4E x0 Ey0 cos (") q (8) Ex0 + E y0 + Ex0 Ey0 + 4E x0 Ey0 cos (") Expressed in terms of the eld ratio parameter E y0 and the relative phase, this relation becomes v r b + E y0 E y0 E a Ex0 E + 4 y0 cos x0 E (") x0 r t (9) + E y0 E + y0 E Ex0 E + 4 y0 cos x0 E (") x0 An expression for the angle of the major axis of the ellipse was fond in 5. Expressed in terms of the eld ratio parameter E y0 and the relative phase, the expression for becomes 0 tan @ E y0 Ey0 Ex0 cos() A (0) The tan fnction is mltivaled. We restrict its range to ; where it is single valed, and se other criteria to add or sbtract to get to the next branch of the tan fnction in order to determine over its entire range. Becase of its symmetry, an ellipse angle greater than is eqivalent to a negative angle. Becase of this, the range of is restricted to be ;. When E y0 and 0 jj, the tan fnction is in the range 0;, and is in the range 0; 4. When Ey0 and jj, the tan fnction is in the range ; 0, and is in 7
the range 4 ; 0. When E y0 and 0 jj, the tan fnction is in the range ; 0. We add to get to the next branch of the tan fnction making the range ; This makes in the range 4 ; : When Ey0 and jj, the tan fnction is in the range 0;. We sbtract to get to the next branch of the tan fnction making the range ;. This makes in the range ; 4. The rotation direction arond the E x, E y plane is determined by, as we have already seen. Positive prodces an electric eld which rotates clockwise. Negative prodces an electric eld which rotates conterclockwise. Positive prodces a right handed wave. Negative prodces a left handed wave. The electric eld parameters can be expressed in terms of the polarization ellipse parameters sing eqations to 4. From eqations and 3, we have q E y0 + b ) (a b ) cos ( )] q [(a + b ) + (a b ) cos ( )] () Expressed in terms of the ratio of the minor to major ellipse axis lengths and the rotation angle of the ellipse, this becomes s E y0 + b b a a cos ( ) () + b a + b a cos ( ) From 4, we nd in terms of the polarization ellipse parameters. 0 v cos B @ sin( ) b t a j sin( )j + b4 a + b 4 a tan + cot C A (3) In order to make the cos fnction single valed, its range is restricted to [0; ] and it can not distingish between positive and negative. We choose the sign of based on the rotation direction arond the ellipse. Clockwise rotation (right hand helicity) gives positive and conterclockwise rotation (left hand helicity) gives negative. The radiation wavelength, or vale from the ndlator, is not speci ed by the polarization ellipse. It mst be measred separately. Note, however, that a + b Ex0 + Ey0 I, which we have seen is a fnction of. In order to keep constant while changing the parameters of the polarization ellipse, one mst keep a + b constant. 6 Stokes Parameters 6. Eqations Of The Stokes Parameters The Stokes parameters are de ned in terms of the electric eld parameters 7. The Stokes parameters are related by E x0 + E y0 (4) S E x0 E y0 (5) S E y0 cos() (6) S 3 E y0 sin() (7) S 0 S + S + S 3 (8) 7 E. Hecht, Optics, 3 rd edition, Addison Wesley Longman, Reading, Massachsetts, 998. 8
The reverse transformation is easily fond, the electric eld parameters in terms of the Stokes parameters are given by r S0 + S (9) E y0 r S0 S (30) tan() S 3 S (3) 6. Relation Between The Stokes Parameters And The Electric Field Parameters It is sefl to normalize the Stokes parameters to the intensity. The parameters we se for de ning the polarization mode of the ndlator are the ratios S, S, and S 3. These ratios are not independent and are related by S S S3 + + (3) All three ratios are reqired, however, in order to determine the sign of. The Stokes parameter ratios do not give the wavelength of the radiation, which mst be measred separately. The ratios of the Stokes parameters can be expressed in terms of the electric eld parameter ratio E y0 and the relative phase. The ratio S is given by The ratio S is given by The ratio S 3 is given by S S S 3 + + + Ey0 Ey0 Ey0 (33) Ey0 cos() (34) Ey0 Ey0 sin() (35) The electric eld parameters can be expressed in terms of the Stokes parameter ratios. The eld parameter ratio E y0 is given by v E y0 t S (36) + S The relative phase di erence is given by (tan ) S3 ; S (37) where (tan ) is meant to represent the for qadrant inverse tangent. The for qadrant inverse tangent gives the sign of, so a separate determination of the helicity is not reqired, nlike the case for the polarization ellipse. As noted previosly, the radiation wavelength, or eqivalently the vale of the ndlator, is not determined by the Stokes parameters in an easily accessible way. is a fnction of, bt it is mch easier to measre the wavelength instead in order to determine. 9
7 Relations Between The Stokes Parameters And The Polarization Ellipse Using eqations 5 to 7 and 4 to 7, we can easily read o the polarization ellipse parameters in terms of the Stokes parameters. These can be solved to give tan( ) S (38) S a q + S + S (39) b q S + S (40) a b tan S s s + S q S + S 0 S q S + S (4) (4) (43) As noted previosly, we restrict the range of the tan fnction to ; where it is single valed and se other criteria to add or sbtract to get to the next branch in order to determine over its entire range. When S 0 and S 0, the tan fnction is in the range 0; and is in the range 0; 4. When S 0 and S 0, the tan fnction is in the range ; 0 and is in the range 4 ; 0. When S 0 and S 0, the tan fnction is in the range ; 0. We add to get to the next branch of the tan fnction making the range ; This makes in the range 4 ; : When S 0 and S 0, the tan fnction is in the range 0;. We sbtract to get to the next branch of the tan fnction making the range ;. This makes in the range ; 4. The inverse transformation is given by expressing the Stokes parameters in terms of the electric eld parameters, and then expressing the electric eld parameters in terms of the polarization ellipse parameters. Using eqations to 4 and 4 to 7, we get 8 Laboratory Frame a + b (44) S a b cos( ) (45) S a b sin( ) (46) S 3 ab (47) The polarization parameters are measred in the laboratory frame, rather than in the rotated frame of the ndlator. A transformation is reqired to go between the laboratory frame and the ndlator frame. The polarization ellipse in the laboratory frame has the same ratio of minor to major axis. The rotation angle of the ellipse in the laboratory frame is jst 4 pls the rotation angle in the ndlator frame, since the ndlator frame is rotated by 4 in the laboratory frame. 0
The Stokes parameters change since the eld components change in the laboratory frame. transformation of the eld components is The E xl p (E x E y ) (48) E yl p (E x + E y ) (49) where the sbscript L indicates the laboratory frame. in the laboratory at a xed z-location are In complex notation, the eld components E xl p E y0 e i e i!rt (50) E yl p + E y0 e i e i!rt (5) When the electric eld is expressed in complex form, the Stokes parameters are given by E x0 + E y0 E y0 (5) S E x0 E y0 E y0 (53) S E y0 + E x0e y0 (54) S 3 i E y0 E x0e y0 (55) Applying these formlas to the elds in the laboratory frame, we nd the Stokes parameters to be L Ex0 E y0 e i E y0 e i + + E y0 e i + E y0 e i (56) Ex0 + Ey0 (57) S L Ex0 E y0 e i E y0 e i + E y0 e i + E y0 e i (58) E y0 cos() S (59) S L Ex0 E y0 e i + E y0 e i + E y0 e i + E y0 e i (60) Ex0 Ey0 S (6) S 3L i Ex0 E y0 e i + E y0 e i E y0 e i + E y0 e i (6) E y0 sin() S 3 (63) In smmary, the transformation of the Stokes parameters between the laboratory frame and the ndlator frame is L (64) S L S (65) S L S (66) S 3L S 3 (67)
9 Conclsion This note provided the path between the row positions of the ndlator and the desired polarization mode. It also provided the reverse path between the desired polarization mode and the row positions. The individal transformations from row positions to magnetic eld, from magnetic eld to electric eld, and from electric eld to polarization parameters were presented. The reverse transformations were also given. These transformations can be applied seqentially to go from row positions to polarization parameters, or from polarization parameters to row positions. 0 Smmary Of Reslts A smmary of the reslts derived above is presented here. First we show the reslts to go from given row positions to the reslting polarization mode. Then we shown the reslts to go from a given polarization mode to the reqired row positions. Restrictions on the ranges of qantities have been detailed in the text. 0. Row Positions To Polarization Mode 0.. Row Positions To Magnetic Field s p cos z 0 z 03 k + cos 0 z 0 z 04 k B y0 0.. Magnetic Field To Electric Field " z 0 z 04 cos k z B x0 cos k 0 z 03 k [(z 0 + z 03 ) (z 0 + z 04 )] r + 0 E y0 B x0 B y0 + 0..3 Electric Field To Polarization Ellipse v r b + E y0 a Ex0 r t + E y0 + Ex0 0 tan @ Ey0 Ex0 Ey0 Ex0 E y0 Ey0 Ex0 0 # E + 4 y0 cos (") E x0 E + 4 y0 cos (") E x0 cos() A Positive prodces an electric eld which rotates clockwise arond the ellipse. Negative prodces an electric eld which rotates conterclockwise.
0..4 Electric Field To Stokes Parameters S S S 3 + + + Ey0 Ey0 Ey0 Ey0 cos() Ey0 Ey0 sin() 0. Polarization Mode To Row Positions 0.. Stokes Parameters To Electric Field v E y0 t S + S (tan ) S3 ; S where (tan ) is meant to represent the for qadrant inverse tangent. wavelength r, mst be measred independently. The third parameter, the 0.. Polarization Ellipse To Electric Field s E y0 + b b a a cos ( ) + b a + b a cos ( ) 0 v cos B @ sin( ) b t a j sin( )j + b4 a + b 4 a tan + cot C A The sign of is based on the rotation direction arond the ellipse. Clockwise rotation (right hand helicity) gives positive and conterclockwise rotation (left hand helicity) gives negative. The third parameter, the wavelength r, mst be measred independently. 0..3 Electric Field To Magnetic Field 0 s 0 r B y0 B x0 E y0 + 3
0..4 Magnetic Field To Row Positions 0v 3 z 0 6 4 + cos B 0 k k @ t C7 By0 A5 + B x0 0v z 0 6 4 + By0 cos B 0 k k @ t By0 + B x0 0v 3 z 03 6 4 cos B 0 k k @ t C7 By0 A5 + B x0 0v z 04 By0 6 4 cos B 0 k k @ t By0 + B x0 B x0 B x0 3 C7 A5 3 C7 A5 4