Chapter 2 Theoretical Analysis and Simulation
2.2 Theoretical Analysis of HOE
In physics, ray-tracking method[35] can be used to calculate the beam conditions of propagating in medium. Any combination of four things might happen with this light ray:
absorption, reflection, refraction and fluorescence. The idealized beam is calculated to explain the real optical device. In order to comprehend the properties of the holographic pickup head, the ray-tracking method is applied. Figure 2.6 illustrates the schematic diagram of the holographic pickup head.
Figure 2.6 Schematic ray-tacking of holographic pickup head
In the simplified optical system, the laser beam is transmitted through the labeled points O, P, Q, R, S, T, U, V and W on the holographic plate as and creates some skew rays respectively. Each labeled point is denoted as a known point (X0, Y0, Z0) on each skew ray, and its direction cosines vector are defined as K, L, and M
X K D X 0 (2.3) Y L D Y0 (2.4)
( 0 )
Z M D Z d (2.5) where D is the distance from the object point (X0, Y0, Z0) to the point of incidence (X, Y, Z), and d is the distance between the collimator and the objective lens. For a sphere of the radius r, the equation of the next refracting surface is X2+Y2+Z2-2rZ=0. Substituting Eqs.
(2.3-2.5) in the formula gives the equation to be solved for D as
2 2 0 calculate the coordinates of the point of incidence. The property of direction cosines that the angle between two intersecting lines is given by
cosI Kk'Ll'Mm' (2.10) here, I is the incidence angle and k’, l’, m’ are the direction cosines vectors of the
normal at the point of incidence. For a spherical surface
' X
k r (2.11)
' Y
l (2.12)
' 1 Z
m r (2.13) In order to derive the refraction equations, Figure 2.7 displays the refraction of a skew ray. OA is a vector of magnitude n in the direction of the incident ray, OB is a vector of magnitude n’ in the direction of the refracted ray, while AB is a vector of magnitude n’cosI’-ncosI in the direction of the normal. Substituting Egs.(2.11-2.13) in Eq.(2.10), the value of incidence angle I and refraction angle I’ are obtained.
Figure 2.7 Refraction of skew ray [35]
For the holographic optics, the light beam is diffracted by the surface grating. Figure 2.8 shows transmission grating diffracted orders and blazed transmission grating. For transmission gratings at normal incidence, the diffracted orders governed by the following equation
(sin sin )
g i m
md (2.18)
where m is the order of the diffracted wave, λ is the diffracted wavelength (650 nm), dg is the groove spacing (54.982 µm), θi is the incidence angle, φm is the angle of diffraction of the m order measured from the grating normal.
Figure 2.8 Transmission grating diffraction
The gratings are commonly used with oblique incident angles. Figure 2.9 illustrates the diffraction process on a holographic grating plate, and a unit sphere is applied to describe the relation between the incident ray and the diffracted beam[36]. Every point on the unit sphere may correspond to a point on the α-β plane. The incident ray has an incident point of (αi, βi), and the diffracted ray has an outcoming point of (αm, βm). The diffraction order m is decided by the outcoming point (αm, βm), which can be mapped on one of the orthogonal planes parallel to the grating grooves.
The direction cosine diagram provides a simple and intuitive means of determining the diffraction grating behavior even for these general cases. The general grating equation for arbitrarily oriented lines is given by
m i sin
where is the angle between the direction of the grating grooves and the -axis.
Therefore from each diffracted ray, the projected spot coordinate (X, Y) on the photodiode can be calibrated by the direction cosine space. Through these mentioned equations by the ray-tracking method, the approximate spot shape variation on the photodiode is derived. Figure 2.10 demonstrates the projected spot on the photodiode coordinate.
Figure 2.10 Projected spot on photodiode coordinate
2.2.2 Spot size and depth of focus
In DVD system, the focus laser spot is actually an area where the intensity of light varies as an airy pattern function. The spot size is determined by the light wavelength λ and the numerical aperture NA of the lens. Therefore, a simple formula is given to calibrate the radius of the focus laser beam by
0.61 Spot size
NA
(2.21)
A commercial DVD pickup head uses the red laser (λ=650 nm), the NA of the objective lens is 0.6. The diameter of spot size is 1.32 μm. However, in optics, the laser beam is a Gaussian beam. The aperture diameter is defined as 1/e2 (=0.135) of its maximum intensity value on the optical axis. For a Gaussian beam propagating in free space, a laser beam converges to a point where the beam is the smallest, known as the beam waist W0. The beam waist can be regarded as the minimum spot size. Besides, the depth of focus (DOF) is the distance range between the nearest and the farthest objects in a scene that appears acceptably sharp in an image. It can be defined from Rayleigh length ZR[37], which is the distance from the waist to the place where the cross section area is double.
Figure 2.11 illustrates the characteristics of a laser beam through a focusing lens. The beam waist and depth of field can be derived by the following equations, where f is the focal length and D0 is the diameter of the entrance pupil.
0
Figure 2.11 Focusing path of laser beam
The numerical aperture of objective lens brings significant influence on lateral resolution. Figure 2.12 illustrates the effect of numerical aperture on the desirable lateral resolution[38]. An objective lens with low numerical aperture NA generates a bigger beam waist, resulting in relatively long DOF. A high numerical aperture NA objective lens induces a smaller beam waist, leading to relatively shorter DOF. Besides, the distance from focal plane to de-focal plane is directly proportional to the DOF.
Figure 2.12 Effect of numerical aperture on lateral resolution
2.2.3 Optical analysis in liquid
Figure 2.13 shows how the optical path changes when the optical system applies in liquid environment. The objective lens doesn’t contact the liquid directly. A cover glass is placed between air and liquid to avoid scattering by a cambered liquid surface and create a clear window. And the relationship between the angles of incidence and refraction can be formulated by Snell's law.
Figure 2.13 Focusing path of laser beam in liquid
In order to know the focus position in liquid, the thickness from the focal point to the surface of the liquid is defined as hliquid, the thickness of the glass is hglass, and the length form the airy focal point to the top surface of the glass is hair. In optics, the numerical aperture of an optical system such as an objective lens is defined by NA=nsinθ. Where n is the index of refraction of the medium and θ is the half-angle of the maximum cone of light that can enter or exit the lens. Due to Snell's law, the numerical aperture remains the same:
sin sin sin
air air glass glass liq liq
NA n n n (2.25) where nair is the refractive index of the air (=1), nglass is the refractive index of the glass (=1.52), nliq is the refractive index of the liquid (e.g. 1.33 for pure water). The θair, θglass, and θliq are the half-angles of the medium. The incident spot which is on the top surface of the glass is the same both in air and in liquid, the equation should be satisfied:
tan tan tan
air air glass glass liquid liq
h h h (2.26)