Polarization rotation

In ellipsometry monochromatic light such as from a He-Ne laser, is passed through a polarizer, rotated by passing through a compensator before it impinges on the interface to be studied [142]. The reflected beam will be elliptically polarized and is measured by a polarization analyzer. In null ellipsometry, the polarizer, compensator, and analyzer are rotated to produce maximum extinction. The phase shift between the parallel and perpendicular components A and the ratio of the amplitudes of these components, tan are related to the polarizer and analyzer angles p and a, respectively. The changes in A and when a film is present can be related in an implicit form to the complex index of refraction and thickness of the film. 

A future optical device exploiting these two discoveries could write, read and operate on electron spins, while using patterned magnetic regions as memory elements. The region of large nuclear polarization rotates electron spins as they pass by- one necessary operation for quantum computing."... 

Figure 9.2 Quantitative description of optical rotation. A vertically polarized electric field Em is incident on chiral system and induces vertically directed dipole moment i and magnetic moment m. Both act as sources of radiation, p, giving rise to vertically polarized field, m giving rise to horizontally polarized field. Sum of both fields is a new field E0ut with polarization rotated over angle 0.

When the excitation light is polarized and/or if the emitted fluorescence is detected through a polarizer, rotational motion of a fluorophore causes fluctuations in fluorescence intensity. We will consider only the case where the fluorescence decay, the rotational motion and the translational diffusion are well separated in time. In other words, the relevant parameters are such that tc rp, where is the lifetime of the singlet excited state, zc is the rotational correlation time (defined as l/6Dr where Dr is the rotational diffusion coefficient see Chapter 5, Section 5.6.1), and td is the diffusion time defined above. Then, the normalized autocorrelation function can be written as (Rigler et al., 1993)... 

PIECEWISE ADIABATIC PASSAGE IN POLARIZATION OPTICS AN ACHROMATIC POLARIZATION ROTATOR... 

In Figure 5.3, we show that a sequence of A = 30 birefringent crystals is sufficient to reproduce the effect of polarization rotation from H to V. It can also be seen from this figure that in the limit of a large number of birefringent crystals, the dynamic can be replaced by a continuum. This is the analog of STIRAP. 

As has been shown elsewhere [21], this sequence produces a broadband polarization rotator even for a small N, see Figure 5.5. In the limit of very large N, the two approaches are equivalent. With only a small number of pulses, the CPS works well but within a limited bandwidth, as do pi pulses. 

In this chapter, we proposed a novel design, inspired by the PAP concept introduced by Shapiro et al. in quanmm optics [5-7], for an optical polarization rotator... 

Figure B3.6.12 Depolarization of fluorescence indicates rotation of the chromophore. Monochromatic radiation from the source (S) has all but the vertically polarized electric vector removed by the polarizer (P). This is absorbed only by those molecules (see Fig. B3.6.5) in which the transition dipole of the chromophore is aligned vertically. In the case where these molecules do not rotate appreciably before they fluoresce ( no rotation"), the same molecules will fluoresce (indicated by shading) and their emitted radiation will be polarized parallel to the incident radiation. The intensity of radiation falling on the detector (D) will be zero when the analyzer (A) is oriented perpendicular to the polarizer. In the case where the molecules rotate significantly before fluorescence takes place, some of the excited chromophores will emit radiation with a horizontal polarization ( rotation ) and some with a vertical polarization. Finite intensities will be measured with both parallel and perpendicular orientations of the analyzer. The fluorescence from the remainder of the excited molecules will not be detected. The heavy arrows on the left of the diagram illustrate the case where there is rotation.

The representation of the sampling by a unipolar, single-rotation-axis, U(l) sampler of a SU(2) continuous wave that is polarization/rotation-modulated is shown in Fig. 2, which shows the correspondence between the output space sphere and an Argand plane [28]. The Argand plane, S, is drawn in two dimensions, x and v, with z = 0, and for a set snapshot in time. A point on the Poincare sphere is represented as P(t,x,y,z), and as in this representation t = 1 (or one step in the future), specifically as P(l,x,y,z). The Poincare sphere is also identified as a 3-sphere, S 1, which is defined in Euclidean space as follows ... 

The consequence of these relations is that every proper 2n rotation on S + — in the present instance the Poincare sphere—corresponds to precisely two unitary spin rotations. As every rotation on the Poincare sphere corresponds to a polarization/rotation modulation, then every proper 2n polarization/rotation modulation corresponds to precisely two unitary spin rotations. The vector K in Fig. lb corresponds to two vectorial components one is the negative of the other. As every unitary spin transformation corresponds to a unique proper rotation of S +, then any static (unipolarized, e.g., linearly, circularly or ellipti-cally polarized, as opposed to polarization-modulated) representation on S + (Poincare sphere) corresponds to a trisphere representation (Fig. 3a). Therefore... 

Turning now to the case of polarization/rotation modulation, or continuous rotation of iT i 2rl2 corresponding to a continuous rotation of Cj through 20, there is a rotation of the resultant through 0. This correspondence is a consequence of the A 1A = 7 relation, namely, that if the unitary transformation of A or A 1 is applied separately the identity matrix will not be obtained. However, if the unitary transformation is applied twice, then the identity matrix is obtained and from this follows the remarkable properties of spinors that corresponding to two unitary transformations of, for example, 27i, namely, 471, one null vector rotation of 271 is obtained. This is a bisphere correspondence and is shown in Fig 3b. This figure also represents the case of polarization/rotation modulation—as opposed to static polarization/rotation. 

K thus defines a static polarization/rotation—whether linear, circular or elliptical—on the Poincare sphere. The 2, r representation of the vector K gives no indication of the future position of K that is, the representation does not address the indicated hatched trajectory of the vector K around the Poincare sphere. But it is precisely this trajectory which defines the particular polarization modulation for a specific wave. Stated differently a particular position of the vector K on the Poincare sphere gives no indication of its next position at a later time, because the vector can depart (be joined) in any direction from that position when only the static 2, r coordinates are given. 

In order to address polarization/rotation modulation—not just static polarization/rotation—an algebra is required which can reduce the ambiguity of a static representation. Such an algebra which is associated with 2, r, and that reduces the ambiguity up to a sign ambiguity, is available in the twistor formalism [28]. In this formalism, polarization/rotation modulation can be accomodated, and a spinor, K, can be represented not only by a null direction indicated by 2, q, or C, but also a real tangent vector L indicated in Fig. 4. 

A generalized representation of spin vectors (and thus of polarization/ rotation modulation) is in terms of components is obtained using a normalized pair, a,b, as a spin frame ... 

The controlling variables for polarization and rotation modulation are given in Table III (see page 724). We can note that the Stokes parameters (.v0,.vi AAAf) defined over the SU(2) dimensional variables, v /, %, of 2,(v(/, x) are sufficient to describe polarization/rotation modulation, and relate those variables to the SO(3) dimensional variables, oo(r,z),8, of Q (0), 5), which are sufficient to describe the static polarization/rotation conditions of linear, circular, left/right-handed polarization/ rotation. 

We can also note the fundamental role that concepts of topology played in distinguishing static polarization/rotation from polarization-rotation modulation. 

Fig. 2. Typical third-harmonic generation measurement. The beams from the laser source (here either a frequency shifted Nd YAG or a HoTmCnYAG laser) are split into a measurement and a reference beam. The polarization rotator and the polarizer serve as variable attenuator and yield the desired polarization of the input beam. The beam is focused on the sample in the vacuum chamber. The water filter removes the fundamental frequency and the attenuation filters limit the third-harmonic signal to the measurement range of the photomultiplier. The signal from the sample is divided by the reference signal and averaged with a boxcar gated integrator...

Consider now a field whose linear polarization rotates in the x-y plane ... 

An ultrafast time-resolved near- and mid-IR absorption spectrometer was designed to achieve high sensitivity, ultrafast time resolution, and broad tunability in the near- and mid-IR regions (see Fig. 2). The details of this spectrometer are described elsewhere (9). Briefly, MbCO was photolyzed with a linearly polarized laser pulse, whose polarization direction was controlled electronically by a liquid crystal polarization rotator. The photolyzed sample was probed with an optically delayed, linearly polarized IR pulse whose transmitted intensity was spectrally resolved with a monochromator and detected with either a Si photodiode (near-IR RilO cm-1 bandpass) or a liquid nitrogen-cooled InSb photodetector (mid-IR 3 cm-1 bandpass). To measure the sample transmission, this signal was divided by a corresponding signal from a reference IR pulse... 

---