Introduction to polarization of light

Chapter 2 Introduction to polarization of light This Chapter treats the polarization of electromagnetic waves. In Section 2.1 the concept of light polarization is discussed and its Jones formalism is presented. Net, Section 2.2 shows the coupling between the Jones vector and the parameters of polarization ellipses, its orientation, and the ratio of the major and minor aes. The last Section 2.3 analzes the interaction of a plane linear polarized wave with the liquid crstal used in spatial light modulators. 2.1 Description of polarization b a Jones vector The polarization of light is due to the orientation of the electrical field vector. Since an electromagnetical wave is a three dimensional object, it can be described b superposition of its two orthogonal components orientated in a plane perpendicular to the propagation direction 56. E = E 0 e i(kz ωt+ϕ ) E = E 0 e i(kz ωt+ϕ ) (2.1) These components are oscillating in time with the same frequenc, however the amplitudes and phases ma differ. Eq. 2.1 can be rewritten in a vector form with the fast oscillating term separated from the both components 5

E = E 0 e iϕ + E 0 e iϕ e i(kz ωt+ϕ ) = Ẽ0e i(kz ωt+ϕ) (2.2) where Ẽ 0 = E 0 e iϕ + E 0 e iϕ. (2.3) The epression in the bracket is no longer dependent from the fast oscillating component, but from phase shifts and amplitudes. It can be epressed in form of a vector. E0 e Ẽ 0 = iϕ E 0 e iϕ (2.4) The vector 2.4 is called Jones vector and reflects the polarization state. Since the relative amplitudes and phases full determine the state of polarization, the Jones vector is a complete description of it. To comprehend the phsical meaning of comple amplitudes one can consider Eq. 2.1 with z = 0, ϕ = 0, ϕ = ɛ and take the real part of it. E = E 0 cos (ωt) E = E 0 cos (ωt + ɛ) (2.5) This has the form of restricted Lissajous figures with ɛ as the relative phase between the components. The restriction is due to the same factor ω placed in front of the argument t, so in consequence, three possible tpes of figures are allowed. A linear polarization is obtained if the relative phase is set to 0 ± 180 where n Z. An elliptical polarization will result for an other set of amplitudes and relative phases, including a circular case, when the amplitudes are equal and the relative phase is 90 ± 180. Figure 2.1 illustrates the influence of the phase difference on the resulting Lissajous figures for fied, nonequal amplitudes. It is interesting to notice that for given amplitudes one can get a linear polarization in two directions depending on the phase difference set to 0 or 180, as it is shown in Figure 2.1. The difference of 180 in phase means that instead of a maimum of an electric field, a minimum is encountered, 6

0.5 0.5 0.5 0.25 0.25 0.25-1 -0.5 0.5-0.25-1 -0.5 0.5-0.25-1 -0.5 0.5-0.25-0.5-0.5-0.5-0.75-0.75-0.75 0.5 0.5 0.5 0.25 0.25 0.25-1 -0.5 0.5-0.25-1 -0.5 0.5-0.25-1 -0.5 0.5-0.25-0.5-0.5-0.5-0.75-0.75-0.75 Figure 2.1: Eample Lissajous figures plotted for various relative phase shifts ɛ and an amplitude ratio of 0.75. As indicated on top of the graphs, in the upper row, from left ɛ = 0 is plotted, which corresponds to linear polarization. Net, two plots are shown for ɛ = 45 and ɛ = 45 with the same tilted ellipse in both cases. The first plot in the lower row presents an ellipse with the major ais aligned along the X ais of the coordinate sstem as a result of ɛ = 90. The following plot shows the ellipse similar to the one for ɛ = 45 but with different orientation. The last plot shows the second possible linear polarization for the fied amplitudes. 7

Time Time =-45 =45 Figure 2.2: 3D Lissajous figures plotted for negative (upper graph, ɛ = 45 ) and positive (bottom graph, ɛ = 45 ). As depicted, the negative ɛ results in a clockwise rotation of the electrical field vector (right hand helicit) and the positive ɛ in counterclockwise rotation (left hand helicit) in time. which results in a change of the direction of the electrical field vector. For the general case of an ellipse, a 180 change of relative phase will mirror the figure with the smmetr ais along one of the polarization component ais, as it is for the cases ɛ = 45 and ɛ = 135. Another interesting propert is that there is no alternation of the Lissajous figure if the sign of the phase shift is changed. This is not eactl true in terms of polarization of an electromagnetical wave. Adding a time ais to the Lissajous figures helps to clarif the relevance of the phase sign. = A cos (ωt) = A cos (ωt + ɛ) t = t (2.6) The parametrical equations 2.6 are plotted in Figure 2.2. For simplicit, the amplitudes A and A are chosen to be equal. The influence of the relative phase sign is apparent in Figure 2.2. The upper plot represents the Lissajous figure where ɛ is equal to 45. The negative shift results in clockwise rotation of the vector of the electrical field and the positive shift results in a counterclockwise rotation. 8

There is a more general wa to prove the direction of the rotation. One can transform Eq. 2.5 to the clindrical coordinates, where r = E 2 2 + E tan(ψ) = E E z = t and then one can directl see the behavior of the electrical field vector. ( ) E0 cos(ωt) ψ(t, ɛ) = arctan E 0 cos(ωt + ɛ) (2.7) Plotting the orientation of the electrical vector ψ as a function of phase shift ɛ and time shows the regions where ψ is rising with time, (change from black to white) and where it is falling. It is illustrated as well, b the right insets in Figure 2.3 on which one can see the ψ function plot for a fied ɛ equal 45 and 45. This eample proves that indeed a positive phase shift between the polarization components corresponds to counterclockwise rotation and negative relative phase to clockwise rotation of the electrical field vector. The plot is consistent with the obvious fact described b Eq. 2.7, that setting the phase shift to 0 ±180 will give no rotation and therefore ield the linear polarization. The transition from the Lissajous figure picture to the Jones vector formalism is quite apparent. The short summar of Jones vectors describing a few selected polarizations is presented in Table 2.1. 2.2 Parametrization of the polarization ellipse In this Section it is eplained how to calculate the resulting Jones vector from the polarization ellipse with desired major and minor aes and orientation. The calculation is based on comparing two ellipse equations, one obtained from the Lissajous figures described b Eq. ellipse equation. 2.5 and a rotated Cartesian Obtaining the first equation is done b eliminating the fast oscillating factor ωt from the Eq. 2.5. 9

ε 150 100 50 0-50 -100-150 0 50 100 150 200 250 300 350 ωt 75 50 25-25 -50-75 75 50 25-25 -50-75 ψ 50 100 150 200 250 300 350 ε=45 ψ 50 100 150 200 250 300 350 ε= 45 ωt ωt Figure 2.3: The left side of the Figure shows the contour plot of the ψ function. The relative phase ɛ is on the vertical ais and the horizontal ais is proportional to time multiplied b the factor ω. The shades of gra indicate ψ(t, ɛ) values in a wa that the change from dark to light colors depict positive change and opposite, negative change. Two graphs on the right side present the plot of ψ(t) for ɛ equal 45 and 45 and the can be regarded as horizontal cuts through the contour plot. 10

Table 2.1: Jones vectors Polarization linear, general linear, vertical (S) linear, horizontal (P) circular, (+i counterclockwise, i clockwise) elliptical, principal aes parallel to, aes elliptical, general Jones vector sin(α) cos(α) 0 1 1 0 1 ±i A ±ib A B ± ic E E 0 = cos (ωt) (2.8) E E 0 = cos (ωt + ɛ) = cos (ωt) cos (ɛ) sin (ωt) sin (ɛ) Thus E E cos (ɛ) = sin (ωt) sin (ɛ). (2.9) E 0 E 0 Since from Eq. 2.8 one can obtain sin (ωt) = the Eq. 2.9 is reformulated into 2 1 E sin 2 + (ɛ) E 0 E E 0 1 E 2, (2.10) E 0 2 2 E E 0 E E 0 cos(ɛ) 11 = 1 (2.11)

which represents the ellipse in the same wa as the Lissajous curves presented in the previous Section 56. The free parameters of the equation are the amplitudes E 0 and E 0 and the relative phase shift ɛ, which are eactl the same parameters, that are used in the Jones vector formalism. The Cartesian ellipse equation is given b 2 a + 2 2 b = 1. (2.12) 2 It describes an ellipse with principal aes a and b, with the orientation of the principal aes fied. In order to var the orientation one can transform this ellipse to rotated Cartesian coordinates. This ields a rotated ellipse equation with the rotation angle γ. ( cos(γ) + sin(γ)) 2 ( cos(γ) sin(γ))2 + = 1 (2.13) a 2 b 2 Both equations, 2.11 and 2.13, describe the same ellipse, one in basis of the electric field components E and E, and the second in the Cartesian coordinates and. Therefore the corresponding coefficients (E, E ) are equal. 1 E 2 0 sin 2 (ɛ) = cos2 (γ) + sin2 (γ) a 2 b 2 1 E 2 0 sin 2 (ɛ) = sin2 (γ) + cos2 (γ) a 2 b 2 cos(ɛ) 1 sin 2 = (ɛ) E 0 E 0 sin(γ) cos(γ) b 2 sin(γ) cos(γ) a 2 (2.14) This provides the coupling of the principal aes a and b of an ellipse, the orientation γ with the amplitudes E 0 and E 0, and the relative phase ɛ. Depending on the requirements, Eq. 2.14 can be used to solve two inverse problems. The first asks which polarization ellipse (a, b, γ) is generated with the given electric field (E 0, E 0, ɛ). The inverse problem is what electric field is needed to obtain the desired polarization state, which seems to be much more interesting, since one would like to control the polarization state b choosing proper electric field amplitudes E 0 and E 0 and the relative phase ɛ. The solution is 12

1 E 0 (a, b, γ) = 2 (a2 + b 2 + (a 2 b 2 ) cos(2γ)) 1 E 0 (a, b, γ) = 2 (a2 + b 2 + (b 2 a 2 ) cos(2γ)) ɛ(a, b, γ) = sin(2γ) sin(2γ) arccos (a 2 + b 2 ) 2 (a 4 + b 4 + a 2 b 2 (cot 2 (γ) + tan 2 (γ))) (2.15) which describe E 0, E 0, and ɛ as a function of the principal aes and the orientation of the polarization ellipse. This description of an ellipse is not ver useful in terms of light polarization since a change in the major ais results in a change of the polarization state of an electromagnetic wave together with the intensit. Other was to parameterize an ellipse are better and more intuitive in this case. Instead of this parametrization one can manipulate the ratio of the principal aes r and the intensit independentl. Substituting r = b a I = a 2 + b 2 (2.16) the relations 2.15 take the form: E 0 (I, r, γ) = E 0 (I, r, γ) = ɛ(i, r, γ) = sin(2γ) sin(2γ) arccos 1 2 I 1 2 I 1 (r2 1) (r 2 + 1) cos(2γ) 1 + (r2 1) (r 2 + 1) cos(2γ) (r 2 1) 2 (1 + r 4 + r 2 (cot 2 (γ) + tan 2 (γ))) One can see that the parameter I is the intensit since (2.17) E 2 0 + E 2 0 = I is fulfilled. Appling relations 2.17 allows to change the polarization state parameters like the ratio of principal aes and orientation in an independent wa b 13

choosing the suitable electrical fields. This relation could as well be used to calculate the Jones vectors for given polarization states. 2.3 Jones matrices for the phase retarder and the nematic crstal modulator A number of several optical elements influence the orientation of the polarization. The nature of their interaction with an optical wave allows to categorize those devices into three classes: polarizers, phase retarders, and rotators. The transmission (or reflectivit) of a polarizer depends ver strongl on the relative orientation of the polarizer and the incoming polarization of the light. In other words, a horizontall orientated polarizer will transmit all of the horizontall polarized light and none of the verticall polarized. Phase retarders do not cut out an of the polarization components, instead of that the introduce a phase retardation between them. As it was eplained in the Section before, b changing the relative phase between the components an restrained control over the polarization can be achieved. The last tpe, a rotator, turns the incident light polarization b some predefined angle. The rotators usuall take the advantage of a propert of opticall active materials like quartz, or the rotate the whole beam spatiall, so unlike retarders, the angle of rotation is not dependent from the incident polarization. Apart from these categories one has to remember that the reflectivit of ever high reflective mirror design for a nonzero incidence angle is polarization sensitive. An effect of such elements on a light wave can be described in a similar wa as the polarization state. The matrices that are multiplied b the Jones vector result in a vector describing the polarization state after the element are called Jones matrices. There are found b solving Eq. 2.18. a b i = i c d Therefore, if the influence on the incident polarizations i i i i (2.18) is known, the 14

Table 2.2: Jones matri Component Horizontal (P) polarizer PP Vertical (S) polarizer PS Phase retarder Rotator Jones matri 1 0 0 0 1 0 0 0 e iɛ 0 0 e iɛ cos(β) sin(β) sin(β) cos(β) corresponding Jones matri can be found. A set of Jones matrices for a few elements is presented in Table 2.2. Let us consider a phase retarder, which is a plate cut from a birefringent crstal. The light traveling along the phase retarder will undergo the double refraction. More precisel, there is a slow and a fast optical ais. The electrical field oscillating along the fast ais will eperience less retardation than in the case when it oscillates along the perpendicular, slow ais. The retardation of the fast and the slow ais are ɛ and ɛ, respectivel, and when the difference between them is 180, the retarder is called a half waveplate and in case of 90 a quarter waveplate. A half wave plate in a beam of linearl polarized light rotates the polarization b the angle given b twice the difference between the fast ais and the orientation of the polarization. To clarif the mechanism, one can imagine linear polarization of light as a superposition of two waves perpendicularl polarized along the fast and the slow ais of a waveplate. Introducing phase shift of a half wave (180 ) will result in a change of polarization. The angle of rotation depends on the amplitudes of these components along the waveplate aes. Appling the Jones formalism, Eq. 2.19 shows the effect of a half waveplate with the fast 15

ais orientated parallel to the table (0 ) on a parallel (P) polarized light. e i π/2 0 1 i 0 e i π/2 = (2.19) 0 0 The outcome is a vector representing eactl the same P polarization multiplied b a factor i which is due to the phase shift of 180 eperienced b passing the plate. This eample shows that if the fast ais is parallel to the polarization no rotation occurs. It is not the case for the rotated half waveplate. Corresponding Jones matrices for rotated half waveplate HW β and quarter waveplate QW β can be found b appling the rotation operator on a waveplate. HW β = QW β = cosβ sinβ sinβ cosβ cosβ sinβ sinβ cosβ e i π/2 0 cos β sin β 0 e i π/2 sin β cos β e i π/4 0 cos β sin β 0 e i π/4 sin β cos β (2.20) where β is the angle of rotation. HW 45 P = 0 i i 0 1 0 = 0 i (2.21) Multipling this matri b P polarization has the effect of rotating it 90 of the plane of the electrical field oscillation, as epected. Waveplates and nematic liquid crstals used in a spatial light modulator are ver similar in the wa of interacting with the electromagnetical waves since the crstals are birefringent as well. The crucial difference between liquid crstals and waveplates is, that an etraordinar inde of refraction of the nematic crstals can be changed in a controlled wa b appling a voltage 37, 57, 58. This is wh one can regard these crstals as waveplates with a variable phase retardance. In order to describe and analze this phenomenon let us first define a few matrices for simplifing the calculations. The rotation matri as a function of the rotation angle is given b Eq. 2.22 as cosβ sinβ Rβ =. (2.22) sinβ cosβ 16

For a nematic crstal with the etraordinar ais orientated parallel to the table one finds LCφ = e 1 2 iφ 0 0 e 1 2 iφ e 1 2 iφ (2.23) where φ denotes the retardation or the phase shift difference between the fast and the slow ais in the crstal (difference retardance), which is controlled b an eternal voltage. Having these Jones matrices we are able to show the interaction of light with crstals for a single piel in the modulator and in this wa analze shaping in frequenc domain. It is important to point out that it is ver common to use P polarized light in pulse shapers, while the efficienc of the incorporated gratings is much higher, so from this point on we will assume the incoming light polarization as P. As the eample of a waveplate orientated parallel to the polarization shows, for crstals with parallel aes onl the phase alteration will occur. 1 1 LCφ E 0 e iωt = E 0 0 e iωt e iφ 0 (2.24) Therefore this structure of the crstals is used in single arra spatial light modulators to influence the phase of the pulse. As ehibited, for the case of an incoming P polarized light, the amplitude and the polarization state remains unchanged but the phase is altered. It is relativel eas to demonstrate that for polarizations perpendicular to the crstal optical ais the phase as well as the polarization will remain unaltered. The optical aes of the crstals have to be rotated in order to influence the phase and the polarization state. The appropriate matri of this confinement is calculated in 2.25. R45 LCφ R 45 = e 1 2 iφ cos φ 2 i sin φ 2 i sin φ 2 cos φ 2 (2.25) The electric field after such a laer is found b multipling matri 2.25 b a P polarized electric field. e 1 2 iφ cos φ 2 i sin φ 2 i sin φ 2 cos φ 2 1 E 0 e iωt 0 = E 0 e iωt e 12 iφ cos φ 2 i sin φ 2 (2.26) 17

z Figure 2.4: Arrangement of two arras of crstal. Optical aes are orientated at 45 in the first arra and 45 in the second. For a single arra of nematic crstals orientated 45 with regard to the incoming polarization, the change of the phase is inseparabl coupled with the polarization state. When two laers of crstals rotated 45 in a first row (b) and 45 in the second (a), are combined as shown in Figure 2.4, then it is possible to influence the phase and polarization separatel. Analzing the Jones matrices in the same matter, as for a single arra, gives the Jones matri of such a modulator. R 45 LCφ b R45 R45 LCφ a R 45 = e 1 2 i(φ a+φ b ) cos (φa φ b) i sin (φa φ b) 2 2 i sin (φ a φ b ) cos (φ a φ b ) 2 2 Net, the electrical field is calculated as in the case above. (2.27) e 1 2 i(φ a+φ b ) cos (φ a φ b ) i sin (φ a φ b ) 2 2 1 i sin (φ a φ b ) cos (φ a φ b ) E 0 e iωt 0 2 2 = E 0 e iωt e 1 2 i(φ a+φ b ) cos φ a φ b 2 i sin φ a φ b (2.28) 2 This solution includes the Jones vector identifing the polarization and phase factor. Since one can independentl handle the difference of the phase shifts φ a φ b, and the sum of them φ a + φ b, autonomous manipulation of 18

polarization and phase is possible. It is important at this point to stress that A the Jones vector in the above described cases takes a form of and as ±ib shown in the Table 2.1, it corresponds to a restricted elliptical polarization with the major ais parallel or perpendicular to the incoming polarization. In terms of polarization manipulation, this solution cannot provide the full range of possibilities since it is not capable of simpl turning the linear polarization. B tuning φ a φ b one can onl change the ratio of the principal aes. Let us consider cutting off the S polarization component of the laser field described b the equation 2.28. It is eperimentall accomplished b adding a P adjusted polarizer after the modulator crstals. 1 0 0 0 E 0 e iωt e 1 2 i(φa+φ b) cos (φ a φ b ) i sin (φa φ b) 2 2 = E 0 e iωt e 1 2 i(φa+φ b) cos (φ a φ b ) 2 0 (2.29) Since there is onl one polarization component present, modulating the difference of phase shifts results in a change of the amplitude of the transmitted light. The corresponding electrical field will take the form E ph+amp = E 0 e iωt e 1 2 i(φa+φb) cos (φ a φ b ). (2.30) 2 The transmission and phase shift introduced b such a setup is given b Eq. 2.31 T = cos 2 (φ a φ b ) 2 ϕ ph+amp = 1 2 (φ a + φ b ) (2.31) Therefore combining a double laer modulator with a polarizer is a most common wa to design an independent phase and amplitude modulator. 2.4 Summar In the first Section of this Chapter the light polarization was discussed. The analog between the Lissajous curves and the polarization resulting from the two perpendicular polarization components of the light wave was shown. Net, this approach was used to present the influence of the components 19

amplitudes of the orthogonal polarization components together with their relative phase shift on the the resulting light polarization. In the net Section the polarization ellipse was described b the ratio of the major aes, the orientation, and the rotation direction. Then, the amplitudes of the orthogonal polarization components together with relative phase shift are found as a function of the ellipse parameters. It is done b comparing ellipses described in basis of electric field components with the one in Cartesian coordinates. This relation allows for the generation of the desired polarization states b choosing the proper amplitudes and the relative phase shift. The last Section uses at the beginning the Jones formalism to show the interaction of the polarization optics with light. Then, b appling the analog between waveplates and liquid crstals used in shapers, an analsis of the light interaction with the single and the double arra shaper was provided. The mechanism of phase onl, phase, and amplitude, and limited phase and polarization shaping was presented. 20