Ординатура / Офтальмология / Английские материалы / Optics Learning by Computing with Examples using MATLAB_Dieter Moller_2007

FIGURE 5.11 Half-wave plate between crossed polarizers. The horizontal polarized light is represented by the two perpendicular vectors EY and EZ . The half-wave plate turns EZ into −EZ . The resultant of EY and −EZ is polarized in the vertical direction and may pass the analyzer.

and a Y component perpendicular to it. The Z component leaves the plate with a phase shift of π, oscillating in the Z direction, while the Y component remains oscillating in the Y direction. The resultant of the two components leaving the half-wave plate is polarized in the vertical direction, and will pass the vertical polarizer. In this setup all the light passing the horizontal polarizer will also pass the vertical polarizer, sometimes called the analyzer.

5.7.6.2 Quarter-Wave Plate Between Crossed Polarizers

A quarter-wave plate is placed between crossed polarizers with its axis at 45◦ (Figure 5.12). The light passing the first polarizer, incident on the quarter-wave plate, is decomposed into two components. One oscillates parallel and the other perpendicular to the axis of the quarter wave plate. The resultant, leaving the quarter-wave plate, is not stationary. It rotates around the direction of propagation X and light passes the second polarizer, also called the analyzer. Rotation of the quarter wave plate around a range of angles will not affect this. However, the axis of the quarter-wave plate should not be parallel to the linear polarizers.

FIGURE 5.12 A quarter-wave plate between polarizers.

5.7.7 General Phase Shift

5.7.7.1 Halfand Quarter-Wave Plates

We have discussed the generation of phase differences of π and π/2 by a halfwave plate and a quarter-wave plate. Another way to produce a phase difference between the two components of the electrical field vector is by internal total reflection. In the case of n1 > n2, we have complex reflection coefficients in the region of total reflection. The range of the phase shifts is between plus or minus π, different for the p- and s-cases. We also have a change in the magnitude of reflection. If we superimpose the vectors of the total internal reflected light for the p- and s-cases, we would, in general, obtain elliptical polarized light.

5.7.7.2 Linear, Circular, and Elliptical Polarized Light

We now examine elliptically polarized light. We consider the plane X L and the corresponding phase difference of φX, for angles between 0 and 360◦. We refer to Eqs. (5.89) and (5.90):

By only using the real part of the Y and Z components and substituting α (k1L − ωt), we have

In FileFig 5.10 we show that one may write Eq. (5.103) in matrix form and that we have for φ 0, linearly polarized light,

for φ π/4, elliptically polarized light, and have the equation of an ellipse,

FileFig 5.10 (M10POELIPSES)

The general equation of the ellipse is shown in vector-matrix notation, using the eigenvalue method. The equations are obtained for φ 0 (linear polarized

light) for φ π/4, (elliptically polarized light), and φ π/2 (circular polarized light).

M10POELIPSES is only on the CD.

Application 5.10. Derive the equations for φ 3π/4, φ 5π/2, π 3π/2, and φ 7π/4 and compare with results of FileFig 5.11.

In Appendix A5.2 we show that the rotation of the coordinate system may be equivalent to a transformation to principal axes. In FileFig 5.11 we show graphs of one component plotted against the other for

One sees that the two components of linearly polarized light, vibrating along perpendicular directions, result in linear polarized light when the phase difference is φX 0, π, and 2π. The resulting vibration takes place along a line tilted by 45◦ for φX 0, 2π, and tilted by 135◦ for φX π. For φX π/2 we have left circular polarized light and for φX 3π/2, equivalent to −π/2, right circular polarized light. The ellipse is left turning, when the φX values are first larger and then smaller than φX π/2, and “right turning" for φX values first larger and then smaller than φX 3π/2. The large axis of the ellipse is always oriented in the same direction as the axis of the “closest" linear polarized light.

FileFig 5.11 (M11POELIPLIS)

Graphs are shown of the equation of the ellipse, that is, Eq. (5.103) for φ 0,

φπ/4, φ π/2, φ 3π/4, φ π, φ 5π/4, φ 3π/2, φ 7π/4, and

φ2π.

M11POELIPLIS

Elliptical Polarized Light

Similarly to that discussed in FileFig 5.9 we plot cos(−2πx/360) on the z-axis and cos(−2πx/360 + ) on the y-axis.

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With · E 0 of the third equation of (A5.1) we have the vector wave equation for the electrical field vector E,

Using a similar formalism we derive the vector wave equation for the magnetic field vector B,

A5.1.2 The Operations and 2

Cartesian coordinates (x, y, z)

i∂/∂x + j∂/∂y + k∂/∂z

2 ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2

Spherical coordinates (r, θ, φ)

i∂/∂r + j(1/r)∂/∂θ + k(1/r sin φ)∂/∂φ

2 (1/r2)∂/∂r(r2∂/∂r)

+[1/(r2 sin θ)]∂/∂θ(sin θ∂/∂θ) + [1/(r2 sin2 θ)]∂2∂φ2

APPENDIX 5.2

A5.2.1 Rotation of the Coordinate System as a Principal Axis Transformation and Equivalence to the Solution of the Eigenvalue Problem

FileFig A5.12 (MA2ROTMAS)

1.Rotation matrices and their multiplication.

2.Demonstrated that the equation of the ellipse is obtained without cross terms when introducing φ π/4.

a.Introduction of φ π/4 into the general equation of the ellipse

b.Introduction of φ π/4 as rotation angle

If the value of φ π/4 is not known, it may be determined from the transformation making the matrix of the general equation of the ellipse diagonal.

MA2ROTMAS is only on the CD.