Ординатура / Офтальмология / Английские материалы / Optics Learning by Computing with Examples using MATLAB_Dieter Moller_2007
.pdf
10.6. HOLOGRAPHY |
413 |
See also on the CD
PW1. Fourier Series. (see p. 370) PW2. Two Bars. (see p.381)
PW3. Two round Objects. (see p. 383)
PW4. Demonstration of the use of Spread Function (siny/y)2 as a Transfer Function. (see p. 386)
PW5. Demonstration of the use of Spread Function (J1(y)/y)2 as a Transfer Function. (see p. 386)
PW6. Rayleigh Distance and Resolution with incoherent Light. (see p. 389) PW7. Rayleigh Distance and Resolution with coherent Light. (see p. 392) PW8. Transfer Function for (sinx/x) of the Coherent Case. (see p. 394) PW9. Transfer Function for (Bess/arg) of the Coherent Case. (see p. 394)
PW10. Blocking Function for removing high Frequencies. (see p. 399) PW11. Blocking Function passing a band of Frequencies. (see p. 399) PW12. Blocking Function passing periodic portion of all Frequencies. (see p.
400)
PW13. Size of Hologram and Quality of Image. (see p. 402)
C H A P T E R
Aberration
11.1 INTRODUCTION
In Chapter 1 on Geometrical Optics we discussed geometrical image formation by using paraxial theory. The essential assumption of paraxial theory is that the angles between an emerging ray from the object and the axis of the system are small. In general small means that one could replace sin α by the angle α (in radians). When this assumption can not be made, one obtains a distorted image. There are elaborate computational programs available for lens design, including systematic corrections for the various types of aberrations. To give an introduction to the most commonly known monochromatic aberrations, we discuss spherical aberration of a single refracting surface and a thin lens, and coma, and astigmatism of a spherical surface and a thin lens. At the end we also discuss chromatic aberration.
11.2SPHERICAL ABERRATION OF A SINGLE REFRACTING SURFACE
In Figure 11.1, two rays are shown from the object point P1 to the spherical surface. After refraction one ray is connected to the image point P2, the other to P2. When paraxial theory can not be used, the ray with the larger angle α2 has the image point P2 closer to the refracting surface. The difference between the points P2 and P2 is called the longitudinal spherical aberration. For its derivation, we look at Figure 11.2 and derive the relations for image formation at one spherical surface
(s1 + r)/(sin(180 − θ1) ζ1/ sin β |
(11.1) |
415
416 11. ABERRATION
|
ζ |
ζ |
α |
ε |
ρ |
α |
FIGURE 11.1 Spherical aberration of a single surface. The image point P2 is formed by the paraxial ray P1QP2. The marginal ray P1Q P2 forms the image at P2, a position closer to the lens.
θ
ζ |
θ |
ζ |
ρ
β
FIGURE 11.2 Coordinates for the treatment of spherical aberration of a single surface.
and |
|
|
(s2 |
− r)/ sin θ2 ζ2/ sin(180 − β). |
(11.2) |
Equations (11.1) and (11.2) may be combined to form |
|
|
(s1 |
+ r)/(s2 − r) nζ1/ζ2, |
(11.3) |
where n is the refractive index of the refracting medium. To get expressions for ζ1 and ζ2, we look at the triangle P1QP2 (Figure 11.2) and have
ζ12 |
r2 + (s1 + r)2 |
− 2r(s1 |
+ r) cos β. |
(11.4) |
Expanding cos β 1 − β2/2 and setting β ρ/r one gets |
|
|||
ζ12 |
r2 + (s1 + r)2 |
− 2r(s1 |
+ r)(1 − [ρ2/2r2]), |
(11.5) |
11.2. SPHERICAL ABERRATION OF A SINGLE REFRACTING SURFACE |
417 |
which results in |
|
ζ12 s12 + (s1 + r)[ρ2/r]. |
(11.6) |
Similarly one obtains |
|
ζ22 ss2 − (s2 − r)[ρ2/r]. |
(11.7) |
Expanding the root one has |
|
ζ1 s1 + (1/s1 + 1/r)[ρ2/2] |
(11.8) |
ζ2 s2 + (1/s2 − 1/r)[ρ2/2]. |
(11.9) |
Introduction of Eqs. (11.8) and (11.9) into (11.3) results in |
|
(s1 + r)/(s2 − r) |
|
n{s1 + (1/s1 + 1/r)[ρ2/2]}/{s2 + (1/s2 − 1/r)[ρ2/2]}. |
(11.10) |
This equation may be rewritten as |
|
1/s1 + n/s2 + (1 − n)/r |
|
(1/r + 1/s1)(1/r − 1/s2)(n/s1 + 1/s2)[ρ2/2]. |
(11.11) |
Introduction of s1 −x0 and s2 xi results in
−1/x0 + n/xi + (1 − n)/r
(1/r − 1/x0)(1/r − 1/xi )(1/xi − n/x0)[ρ2/2]. (11.12)
In the limit of ρ → 0 we must get back to the imaging equation of paraxial theory,
−1/x0 + n/xi1 + (1 − n)/r 0, |
(11.13) |
where we have written xi1 for the image distance for the paraxial case. The coefficient of correction on the right side of Eq. (11.12) depends on x0 and xi . To have it depending only on x0, we use Eq. (11.13), eliminating xi and get
n/xi1sal 1/x0
+ (n − 1)/r + ((n − 1)/n2)(1/r − 1/x0)2(1/r − (n + 1)/x0))[ρ2/2], (11.14)
where we have written xi1sal to indicate the image distance for the longitudinal spherical aberration of a surface of radius of curvature ρ.
The longitudinal spherical aberration (LSA) is defined as xi1 − xi1sal , which is the difference of the image positions calculated for the paraxial and the spherical aberration cases. In FileFig 11.1, we study the LSA for an object distance, which corresponds to a real image, and two different refractive indices. In FileFig 11.2, we study the dependence of LSA on object distances, which corresponds to real images, and on the refractive index and the radius of curvature. For real images spherical aberration may not be eliminated.
418 11. ABERRATION
FileFig 11.1 (A1SPHASS)
Calculation of LSA xi1 − xi1sal for a single spherical surface and negative object distance for two different refractive indices.
A1SPHASS is only on the CD.
Application 11.1.
1.Consider positive and negative object distances and choose one refractive index. Show that one can get positive and negative values for the LSA.
2.Decide how small you want to make the LSA and determine the corresponding value of n.
FileFig 11.2 (A2SPASSS)
Calculations for a single spherical surface to demonstrate the dependence of LSA xi1 − xi1sal on the object position, the refractive index, and the radius of curvature. Note that for the choice of parameters used, there is no value of the refractive index for which the LSA is zero.
A2SPASSS is only on the CD.
11.3LONGITUDINAL AND LATERAL SPHERICAL ABERRATION OF A THIN LENS
In Figure 11.3 spherical aberrations are shown for a positive and a negative thin lens. To calculate the LSA xi1 − xsph for a thin lens we use twice the result obtained for a single spherical surface, as discussed in Section 11.2. There we calculated for a spherical surface with refractive index n, the position xi1sal (see Eq. (11.14)). The light entered from the medium of index 1 and traveled into the medium of index n. We obtained
− 1/x0 + n/xi |
(11.15) |
(n − 1)/r1 + ((n − 1)/n2)(1/r1 − 1/x0)2(1/r1 − (n + 1)/x0)[ρ2/2].
This result may be used to get a similar expression for light incident on the second surface traveling from the medium with index n into the medium of index 1. We substitute x0 with → xii and xi with → x00 and obtain
− n/x00 + 1/xii |
(11.16) |
−(n − 1)/r2 − ((n − 1)/n2)(1/r2 − 1/xii )2(1/r2 − (n + 1)/xii )[ρ2/2],
420 11. ABERRATION
spherical aberration case, 1/ff (x0), where ff (x0) is the focal length of the case of spherical aberration. The result is
−1/x0 + 1/xiisph 1/ff (x0) |
(11.21) |
with |
|
ff (x0) 1/{1/f + {(n − 1)/n2}[ρ2/2]{a(x0) − b(x0)c(x0)}} |
(11.22) |
and the abbreviations |
|
a(x0) [(1/r1 − 1/x0)2(1/r1 − (n + 1)/x0)], |
|
b(x0) [1/x0 + (n − 1)/r1 − n/r2]2 |
(11.23) |
c(x0) [n2/r2 − (n + 1)/x0 − (n2 − 1)/r1]. |
|
In Figure 11.4 we define the lateral spherical aberration as LAT LSA times ρ/xiisph. In FileFig 11.3 we study the question of elimination of spherical aberration and calculate numerical values of LSA and LAT for a choice of parameters of n, r1, r2, ρ, and object distance x0. The elimination of spherical aberration is further discussed in the next section using the π–σ equation.
FileFig 11.3 |
(A3SPHTINS) |
|
|
|
|
Calculations of the spherical aberration of a thin lens. Longitudinal spherical aberration and lateral spherical aberration.
A3SPHTINS is only on the CD.
FIGURE 11.4 Longitudinal spherical aberration (LSA). The lateral spherical aberration (LAT) is calculated using tan α.
11.4. THE π–σ EQUATION AND SPHERICAL ABERRATION |
421 |
Application 11.3.
1.Use different values of n and observe that spherical aberration may not be eliminated for real objects and images.
2.Assume x0 values for virtual images and find positive and negative values for the LSA. Therefore, for this case, spherical aberration may be eliminated.
11.4THE π–σ EQUATION AND SPHERICAL
ABERRATION
We now study whether spherical aberration can be removed when using a special choice of parameters. We introduce the parameters π (position factor) and σ (shape factor),
π (xii + x0)/(xii − x0) and |
σ (r2 + r1)/(r2 − r1). |
(11.24) |
|
Using Eqs. (11.18) and (11.19) we may write |
|
||
1/x0 −(π + 1)/2f |
and |
1/xi (1 − π)/2f |
(11.25) |
1/r1 (σ + 1)/{2f (n − 1)} |
and |
1/r2 (σ − 1)/{2f (n − 1)}. (11.26) |
|
Introducing the expressions of Eqs. (11.24) to (11.26) into Eq. (1.17) we get
− 1/x0 + 1/xiisph |
|
|
(n − 1)(1/r1 − 1/r2) + [ρ2/f 3]{Aσ 2 + Bσ π + Cπ2 + D}, |
(11.27) |
|
where we abbreviated |
|
|
A (n + 2)/{8n(n − 1)2} |
B (n + 1)/{2n(n − 1)} (11.28) |
|
C (3n + 2)/(8n) |
D n2/{8(n − 1)2}. |
(11.29) |
We may look at Eq. (11.27) as the thin lens equation plus a correction term. To study whether spherical aberration can be removed we look at the correction term
Y [ρ2/f 3]{Aσ 2 + Bσ π + Cπ2 + D}. |
(11.30) |
When Y is equal to zero, spherical aberration is eliminated.
In FileFig 11.4 a graph is shown for f 10, n 1.5, ρ 4, and x0 4. There are Y values smaller than zero and, using these parameters, spherical aberration may be eliminated. In FileFig 11.4 one may study an example for positive and negative values of x0 and ρ between 1 and 4. When Y shows negative values, spherical aberration may be eliminated.
