2.1.1

Ellipse of polarization

Let us write the two electrical field orthogonal components of a plane wave propagating in the positive direction of the z axis,

where ω is the angular frequency of light, is the wavenumber (where λ is the light wavelength) and φ = φ_y–φ_x is the initial phase difference between the E_x and E_y orthogonal plane waves.

Starting from Eqs. (1) we can study the trajectory described by the electrical field as a function of time. By setting z=0 in Eqs. (1) and expanding the argument cosine by using trigonometric relations we obtain,

Replacing Eq. (2a) in (2b) and squaring the resulting relation leads to,

Afterwards, multiplying Eq. (1b) by the factor sin (φ), for z=0, and squaring.

Finally, by adding Eqs. (3) and (4) we obtain,

Note that Eq. (5) is the equation of an ellipse, meaning that the electric field temporal evolution describes an ellipse called polarization ellipse, whose characteristic parameters completely describes the state of polarization of a fully polarized beam. A representation of the polarization ellipse is given in Fig. 1(a), where it is plotted inscribed into the rectangle defined by the orthogonal basis set by its constituent electric fields E_x and E_y. In this basis, the polarization is fully described by knowing the amplitudes A_x, A_y and the retardance φ between components.

Figure 1.

(a) Ellipse of polarization; and (b) Poincaré sphere.

To evaluate what is the sense of rotation of the electric field we can do the following analysis. By taking into account Eqs. (1), for t=0 and at the plane z=0, the electric field components take the values,

Deriving the y-component of the field in Eq. (1b) as a function of time, for z=0, and evaluating it at t=0, permits us to study the evolution of the electric field. We obtain,

From Eq. (7) we see how E_y is decreasing into the interval 0<φ<π, this leading to right-handed polarization into this range, whereas E_y is growing in the interval π<φ<2π, this leading to left-handed polarization into this range. Note that the criterion we used is the following: right-handed for light coming to the observed rotating clockwise and lefthanded for light coming to the observer rotating counterclockwise.

By performing a coordinate system rotation, we can describe the polarization ellipse from the basis of its main axes (the major semi-axis and the minor semi-axis). It is illustrated in Fig. 1(a) with the x’ and y’ axes in red, which are rotated an angle α (called azimuth angle) with respect to the x-y basis. Another angle of interest is the so called ellipticity angle ε (green angle in Fig. 1(a)), which is defined as the ratio between the semi-minor and the semi-major axes of the polarization ellipse. This angle can be geometrically understood as the angle between the x’ axis and the diagonal of the rectangle in which the ellipse of polarization is inserted according its own axes (see Fig. 1(a)). As its name suggests, it provides the ellipticity of the polarization ellipse, which ranges from 0, for linear polarization, to ±π/4, for circular polarization. Importantly, by knowing the azimuth α and ellipticity ε angles, the polarization of a fully polarized beam is fully determined.

The relations between the amplitudes (A_x, A_y) and phase φ polarization description space, and the azimuth and ellipticity (α,ε; polarization angles) space are given below¹⁷:

where and q = ±1 (with minus sign, –, for right-handed and plus sign, +, for left-handed).

2.1.2

Stokes vectors and Poincaré sphere

Another way to describe the polarization of a light beam is by using the so called Stokes vector¹⁷, which comprises four intensity magnitudes, the Stokes parameters, arranged in column format. The Stokes vector parameters can be calculated from the magnitudes of the polarization ellipse described in section 2.1.1, but unlike the polarization ellipse, Stokes vectors are able to describe partially polarized or fully depolarized beams. The normalized Stokes vector (i.e., normalized by the S₀ channel) as a function of the ellipse of polarization magnitudes in its different bases is given below,

Each parameter in the Stokes vector provides particular information of the analyzed state of polarization (SoP). The S₀ parameter is the intensity of the analyzed light; the S₁ parameter estimates the contribution of linear polarization in the horizontal or vertical directions; the S₂ parameter estimates the contribution of linear polarization at 45 or at 135 degrees from the vertical; and the S₃ parameter provides the contribution of circular polarization light of the studied SoP.

As stated before, Stokes vectors are appropriate to describe partial polarized and fully depolarized beam. The Degree of Polarization (DoP) of the studied beam can be calculated as the ratio between the fully polarized content over the total intensity, as follows¹⁷,

where 0 ≤ DoP ≤ 1.

Sometimes, it is interesting to study the fully polarized and the unpolarized contributions separately. It can be done by decomposing the Stokes vector in Eq. (10) as the addition of two different contributions,

A very useful tool to visualize SoPs into a three-dimensional space is the so called Poincaré sphere¹⁷ (see Fig. 1(b)). The fully polarized SoPs are represented over the surface of the Poincaré sphere of radius and each two orthogonal polarizations are located at two diametrically opposed spots over the sphere. The linear polarizations are located at the equator of the sphere, completing a rotation of 180 degrees along the equator. The circular polarizations are located at the poles of the sphere (North Pole for right-handed polarization and South Pole for left-handed polarization). Any other point over the Poincaré sphere is related with an elliptical polarization, this being specified by its ellipticity angle ε (see Fig. 1(b)). By contrast, points located inside the sphere describes partial polarized SoPs, the sphere center corresponding to a state of polarization fully unpolarized.

2.3.1

Lyot filter

The main structure of a Lyot filter is sketched in Fig. 2(a). It consists of a series of N optical blocks where each block is composed of a uniaxial anisotropic plate inserted between two polarizers. On this basis, different versions of the Lyot filter have been proposed in literature.^18,19 In the so called Lyot filter type II^18,19, the thickness of each successive plate in the Lyot filter is half that of the preceding. Transmission axes of polarizers are parallel between them and neutral lines of plates are also parallel between them. However, transmission axes are oriented at 45° with respect to neutral lines.

Figure 2.

Optical filter sketch: (a) Lyot filter; and (b) Solc filter.

The intensity at the exit of a Lyot filter can be modeled by using the Stokes-Mueller formalism. Let us suppose that polarizers are oriented at 45° of the laboratory vertical, and anisotropic plates are oriented at 0° of the lab. vertical. Under such scenario, the intensity at the exit of the filter is given by, (17)

where E₀ is the amplitude the incident field, S_LP45°, is the Stokes vector of a linear polarization at 45° of the lab vertical, M_R0° is the Mueller matrix of a retarder oriented at 0° of the laboratory vertical and with retardance , n_e and n_o being the extraordinary and ordinary indices of the anisotropic crystal and d its thickness.

Equation (17) leads to,

where E₀ is the amplitude of the incident field and δ(λ) is the retardance of the first plate.

The largest stage sets the bandwidth and the smallest stage sets the Free Spectral Range. If you use two of the second largest crystals, it will increase the contrast of the desired line. If you split the crystals in half and add a 1/2 waveplates in the middle you can increase the field of view of the filter. The separation and narrowness of the transmission peaks depends on the number, thicknesses, and orientation of the plates.

2.3.2

Solc filter

Another birefringent filter, composed of a series of retardation plates between a single pair of polarizers, was described by Solc.¹⁹ Two main variants of this filter exist. In the fan configuration the two polarizers are parallel to one another and the ith plate is inclined to an angle θ_i = (i – 0.5) π/2N to them. By contrast, the polarizers in the folded configuration are crossed, and the ith plate makes an angle of θ_i = (–1)ⁱ π/4N with respect to the first polarizer. Beyond these main Solc schemes, some authors have studied other configurations (different angular functions among plates or number of plates) to analyze their influence with the final filter efficiency.²¹ A sketch of a possible configuration for the Solc filter is shown in Fig. 2(b).

It has been shown that the Solc filter is inferior to the Lyot filter in the suppression of parasitic light in secondary maxima near the primary transmission bands, although its band width is slightly less, and its transparency is very much better than Lyot filters.²⁰ In fact, a greatest advantage of this filter over the Lyot filter is that only two polarizers are required for an efficient performance of the filter, this leading to far less light losses.

The intensity at the exit of the Solc filter shown in Fig. 2(b) can be modeled by using the Stokes-Mueller formalism. Let us suppose that the polarizers at the entrance and exit of the retardation plates are oriented at 0° of the laboratory vertical (fan configuration), and that the orientation of the N anisotropic plates progressively increase starting from 0°. In this case, the intensity at the exit of the filter is given by,

where E₀ is the amplitude of the incident field, θ_i is the orientation angle of each anisotropic plate, and δ (λ) is the retardance selected for the plates.

2.3.3.

Optical filters and human eye perception

The optical filters described in sections 2.3.1 and 2.3.2 lead to certain intensity distributions as a function of the wavelength. However, the final color observed by users of these filters will be affected by the spectral distribution of the source illuminating the filter, and the spectral sensitivity function of the detector.

The three color sensitivities of the human eye (cones sensitivities) must be considered if light exiting from the filter is observed to the naked eye. The Commission Internationale de l’Eclairage (CIE) 1931 color spaces were the first quantitative relations defined to describe the link between electromagnetic wavelength distributions in the visible spectrum, and physiologically perceived colors in human color vision. The CIE 1931 RGB color space and CIE 1931 XYZ (tristimuli) color space were created by the CIE as a result of a collection of studies done by W.D. Wright²² and J. Guild.²³

The tristimulus X, Y, Z that define the color of the transmitted beam are given by the following integrals ^24,25,

where S (λ) is the spectral distribution of the source, G (λ) is the transmission of the considered filter, and are the three sensitivity functions of the human eye (CIE 1931), and λ₁, λ₂ represent the spectrum interval in which the receiver is sensitive, that in the human eye case, evaluates a wavelength range approximately of [400,700] nm.

If the XYZ space is determined, the RGB channels can be calculated through the following linear transformation^26,27,

3.2.1

Intensity vs parameter

By choosing this option, we can study the intensity (S₀ stokes channel) at the exit of the customized cascade as a function of some parameter of interest corresponding to a specific polarizing element in the cascade. In particular, in the box “Changes” we have to choose the specific optical element to be studied (“number #” box), and the parameter to change (orientation or retardance). The output intensity corresponding to the values selected is then displayed (see Fig. 6(a)).

Figure 6.

Intensity vs Parameter (orientation or retardance) value; and (b) output intensity versus wavelength and color image of the output beam displayed in the Spectrum tab.

3.2.2.

Spectral dependence

By selecting the “Spectral dependence” option in the “Options MMC” button, the applet conducts a representation of the observed color at the exit of the optical filter corresponding to the designed polarizing elements cascade. The applet calculations are based on the mathematical description shown in section 2.3.3, but some extra assumptions are made. First, we are assuming that the input light beam is polychromatic with a constant intensity in all the spectral range. The studied spectral range is controlled by the “ini wavelength” and “final wavelength” boxes that can be modified by users, these boxes only appearing if selecting the radiobutton “Spectral Dependence” (this is why they do not appear in Fig. 4(c), where the “Intensity v Parameter” is selected”). The second assumption is that the refraction indices of the retarders in the cascade do not depend on the wavelength, and the only dependence of the retardance with the wavelength is due to the light wavenumber. Therefore, the relation between the retardance corresponding to the first wavelength in the selected spectrum with the retardance for the ith wavelength is obtained by,

where Δn = n_e – n_o is the difference between the extraordinary and ordinary indices of refraction.

As the transmission could be very low, the output intensities are multiplied by a factor that can be selected by the user. According to the description in section 2.3.3, the RGB channels are obtained for a particular cascade configuration customized by users, and the output intensity versus the wavelength and the color of the output beam are displayed in the corresponding figures at the Spectrum tab (see an example in Fig. 6(b)).

3.2.3.

Polarization change

By selecting the “Polarization Change” option in the “Options MMC” button (see Fig. 4(c)), the polarization of the output beam is calculated and displayed in the corresponding figures. Note that you can set to ON mode the button “Continuous Draw”, and then, the history of the polarizations is displayed on the Poincaré sphere.