Moller K.D. Optics. Learning by Computing, with Examples Using Mathcad, Matlab, Mathematica, and Maple
Подождите немного. Документ загружается.
11.9. CHROMATIC ABERRATION AND THE ACHROMATIC DOUBLET WITH SEPARATED LENSES 433
FileFig 11.10 (A10ACHRTWOS)
Calculation of the distance of separation for two lenses with different radii of
curvature in an achromatic doublet.
A10ACHRTWOS is only on the CD.
Application 11.10.
1. Choose other values of n
1
and n
2
and give the distance of the two lenses for
no chromatic aberration.
2. Assume two lenses of different focal length and different materials and find
the separation for a chosen focal length of the achromatic system.
See also on the CD
PA1. Calculation of LSA = x
i1
−x
i1sal
for a single Spherical Surface and fixed
Value of the object Distance. (see p. 408)
PA2. Calculation of LSA = x
i1
-x
i1sal
for a single spherical Surface for a Range
of object Distances. (see p. 408)
PA3. Spherical Aberration of a thin Lens. (see p. 412)
PA4. Spherical Aberration and Coma depending on the Shape Factor. (see p.
412)
PA5. Coma of a thin Lens. (see p. 415)
PA6. Calculation of Coma for the aplanatic Lens. (see p. 417)
PA7. Astigmatism of a thin Lens. (see p. 419)
PA8. The achromatic Doublet. Determination of Radii of Curvature and Focal
Length. (see p. 421)
PA9. The achromatic Doublet. Determination of Materials and Focal Length.
(see p. 422)
A
A
APPENDIX
About
Graphs and Matrices
in Mathcad
CHANGING NUMBERS IN A FILE AND PLOTTING A
GRAPH
x : 1, 2 ...10 a : 3 b : 4 f (x): a · x +b
2D GRAPH
λ : 0.5 A : 1 T ≡ 1 δ1 ≡ 1 t 1 ≡ 0.1
Specification of the number of x and t1 values
N : 15 i : 0 ...N j : 0 ...N
Specification of the range
x
i
:−.4 + .025 · it1
j
:−.4+, 025 · j
435
436 APPENDIX A. ABOUT GRAPHS AND MATRICES IN MATHCAD
In the specification of the function only x and t1 are used
uc(x, t1)
:
2 ·A · cos
2 ·π
δ1
2 ·λ
·
cos
2 ·π ·
x
λ
−
t1
T
− 2 ·π
δ1
2 ·λ
2
.
In the plotting function one needs the i and j notation
M
i,j
: uc(x
i
,t1
j
).
Call on “Surface plot” and type at the place holder just M and push “F9.”
Go with the mouse on the graph and change the angle of the “point of view.”
Click twice on the graph and get “3D-Plot Format” for “graph options.” Switch
to contour plot.
MATRICES
Go to “Insert” and “Matrix” and select 2 by 2
Type M
Indicate the matrix and insert
M :
to get M :
APPENDIX A. ABOUT GRAPHS AND MATRICES IN MATHCAD 437
The manipulation of matrices can easily be seen from files containing matrices.
Here we give an example of a matrix composed of functions and how to access
the matrix elements after a multiplication has been done.
Fill in functions of x directly and call M now M(x) x : 0,.1 ...5
M(x):
cos(x) −sin(x)
+sin(x) cos(x)
One can access the matrix elements separately. Note that in Mathcad one starts
with 0. For the 0, 1 and 1, 1 elements one has
M(x)
0,1
M(x)
1,1
0
1
-0.1 0.995
-0.199 0.98
-0.296 0.955
-0.389 0.921
-0.479 0.878
Consider the matrix product M1(x) M(x)
3
.After multiplication one can again
access the matrix elements
M1(x): M(x)
3
one gets for the 0, 1 element
M1(x)
0,1
0
-0.296
-0.565
-0.783
-0.932
-0.997
B
B
APPENDIX
Formulas
CONSTANTS
100µm ⇒ 3000 GHz
100µm ⇒ 100 cm
−1
10µm ⇒ 1000 cm
−1
1 meV 10
3
eV 1.6 ×10
−16
joule ⇒ 8.07 cm
−1
100 nm 1000
˚
A
10000
˚
A 1µ
1
˚
A 10
−8
cm 10
−10
m
FORMULAS
√
−1 ii
2
−1
z a + ib r(cos φ + i sin φ) re
iφ
cos x
e
ix+e
ix
2
sin x
e
ix
− e
ix
2i
e
x
1 + x +
x
2
2!
+
x
3
3!
+···, sin x x −
x
3
3!
+···, cos x 1 −
x
2
2!
+···
s
n
a + aq + aq
2
+······+aq
n−1
a
q
n
− 1
q − 1
if |q| < 1,N →∞ s
∞
a
1 − q
439
440 APPENDIX B. FORMULAS
x
2
+ ax + b 0 x
1,2
−
a
2
±
"
a
2
2
− b
(1 ± x)
n
∼
1 ± nx |x| << 1
a
1
b
1
a
2
b
2
×
c
1
d
1
c
2
d
2
a
1
c
1
+ b
1
c
2
a
1
d
1
+ b
1
d
2
a
2
c
1
+ b
2
c
2
a
2
d
1
+ b
2
d
2
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
a
1
b
2
c
2
b
3
c
3
− a
2
b
1
c
1
b
3
c
3
+ a
3
b
1
c
1
b
2
c
2
b
1
c
1
b
2
c
2
b
1
c
2
− b
2
c
1
TRIGONOMETRIC FORMULAS
030
◦
45
◦
60
◦
90
◦
180
◦
270
◦
360
◦
sin 0
1
2
1
2
√
2
1
2
√
31 0 −10
cos 1
1
2
√
3
1
2
√
2
1
2
0 −10 1
tan 0
1
3
√
31
√
3 ∞ 0 ∞ 0
cot ∞
√
31
1
3
√
30 ∞ 0 ∞
sin
2
α + cos
2
α 1
sin α
cos α
tan α
cos α
sin α
cot α
tan α
1
cos α
1
cos
2
α
1 + tan
2
α sin α
tan α
√
1 + tan
2
α
cos α
1
√
1 + tan
2
α
sin(90
◦
± α) +cos α sin(180
◦
± α) ∓sin α
cos(90
◦
± α) ∓sin α cos(180
◦
± α) −cos α
tan(90
◦
± α) ∓cot α tan(180
◦
± α) ±tan α
cot(90
◦
± α) ∓tan α cot(180
◦
± α) ±cot α
sin(−α) −sin α sin(α ± β) sin α cos β ± cos α sin β
cos(−α) +cos α cos(α ±β) cos α cos β ∓ sin α sin β
tan(−α) −tan α tan(+α ± β)
tan α ± tan β
1 ∓ tan α · tan β
cot(−α) −cot α cot(α ±β)
cot α · β ∓1
cot β ± cot α
APPENDIX B. FORMULAS 441
sin 2α 2 sin α cos α
cos 2α cos
2
α − sin
2
α 1 −2 sin
2
α 2 cos
2
α − 1
sin 2α
2 tan α
1 + tan
2
α
cos 2α
1 − tan
2
α
1 + tan
2
α
tan
2
α
2 tan α
1 − tan
2
α
2
cot α − tan α
, cot 2α
cot
2
α − 1
2 cot α
1
2
(cot α − tan α)
1 + cos α 2 cos
2
α
2
, 1 − cos α 2 sin
2
α
2
tan α
"
1 − cos 2α
1 + cos 2α
sin 2
1 + cos 2α
1 − cos 2α
sin 2α
2 tan
α
2
1 − tan
2
α
2
sin α + sin β 2 sin
α + β
2
· cos
α − β
2
sin α + sin β
cos α +cos β
tan
α + β
2
sin α − sin β 2 cos
α + β
2
· sin
α − β
2
cos α +cos β 2 cos
α + β
2
· cos
α − β
2
sin α − sin β
cos α +cos β
tan
a − β
2
cos α −cos β −2 sin
α + β
2
· sin
α − β
2
tan α + tan β
cot α + cot β
tan α ·tan β
tan α + tan β
sin(α ±β)
cos α cos β
1 + tan α
1 − tan α
(tan 45
◦
+ α)
cot α ± cot β
±sin(α ± β)
sin α sin β
cot α + 1
cot α − 1
cot(45
◦
− α)
cos α +sin α
√
2 sin(45
◦
+ α)
√
2 cos(45
◦
− α)
cos α −sin α
√
2 cos(45
◦
+ α
√
2 sin(45
◦
− α)
cot α + tan α
2
sin 2α
cot α − tan α 2 cot 2α
442 APPENDIX B. FORMULAS
DIFFERENTIATION
(u · ν)
uν
+ u
ν
u
ν
u
ν −ν
u
ν
2
(sin x)
cos x (e
x
)
e
x
(cos x)
−sin x (ln x)
1
x
(tan x)
1
cos
2
x
(arcsin x)
1
√
1 − x
2
(cot x)
−1
sin
2
x
(arccos x)
−
1
√
1 − x
2
INTEGRATION
udν uν −
νdu
dx x
x
dx
x
n+1
n + 1
(n −1)
dx
x
ln x
sin xdx −cos x
cos xdx sin x
cot xdx ln sin x
dx
sin
2
x
−cot x
dx
cos
2
x
tan x
dx
1 − x
2
ln
"
1 + x
1 − x
dx
√
1 − x
2
arcsin x
dx
1 + x
2
+arctan x
e
x
dx e
x
a
x
dx
a
x
ln a
dx
x ±a
ln(x ±a)
References
M. Born and E. Wolf, Principles of Optics, 2nd ed. Pergamon Press, New York,
1964.
M. Cagnet, M. Francon, and J. C. Thrierr, Atlas of Optical Phenomena, Springer-
Verlag, Heidelberg, 1962.
P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lecture on Physics,
6th ed. Addison-Wesley Publishing Company, Reading, Massachusetts,
1977.
J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, Inc., New York,
1988.
J. D. Jackson, Classical Electrodynamics, 2nd ed. John Wiley & Sons, NewYork,
1975.
F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed. McGraw-Hill
Inc., New York, 1976.
C. Kittel, Introduction to Solid State Physics, 3rd ed. John Wiley & Sons, Inc.,
New York, 1967.
H. Kogelnik and T. Li, Appl. Opt. Vol 5, p. 1550 (1966).
G. P. Parrent and B. J. Thompson, Physical Optics Notebook, Society of Photo-
Optical Engineers, Redondo Beach, California, 1969.
F. K. Kneub¨uhl and M. W. Sigrist, Laser, B. G. Teubner, Stuttgart, 1988.
E. D. Palik, Handbook of Optical Constants of Solids II, Academic Press, Inc.,
New York, 1991.
R. W. Pohl, Einf¨uhrung in die Optik, Springer-Verlag, Heidelberg, 1948.
443
Index
Aberration, 415
achromatic doublet, 432
aplanatic lens, 425
astigmatism of single surface, 428
astigmatism of a thin lens, 429
chromatic aberration, 430
coma, 423
π-6 equation, 420
spherical of single surface, 415
spherical of thin lens, 418
absorption, 319
achromatic doublet, 430
achromatic doublet with separated lenses,
432
active medium, 295
Airy disc, 149
Airy function, 116
alternating high and low refractive indices,
258
amplitude division, 96
amplitude grating, 152, 158, 170
amplitude reflection coefficients, 321
angle of deviation, 7
angular magnification, 40
angular momentum quantum number, 282,
283
angular quantum number, 283
antireflection coating, 252, 256
aperture function α(η), 385
aperture plane, 382
aperture in random arrangement, 169
aplanatic lens, 425, 427
apodization, 366, 367
Arago, 135
array of source points, 121
nonperiodic, 126
periodic, 121
astigmatic difference ASD, 427, 428, 429
astigmatism of a single spherical surface,
427
astigmatism of a single surface, 429
astigmatism of a thin lens, 428, 429
asymmetric Fourier transform, 370
atomic emission, 273
atomic energy states, 284
exited states, 286
occupation rule, 283
atomic polarizability, 317, 318
Babinet’s theorem, 166, 168
background spectrum, 372
bandpass filter, 364
bandwidth, 285, 288
doppler, 291
homogeneous, 288
Lorentzian, 293
mechanical, 291
natural, 291
quantum, 291
beamsplitter, 104
Bessel function, 149, 165, 166, 389, 390
Bessel function J 1(q), 151
Bessel function as transfer function, 394
birefringent, 231