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Optics Learning by Computing with Examples using MATLAB_Dieter Moller_2007

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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 n1 and n2 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 = xi1 − xi1sal for a single Spherical Surface and ﬁxed Value of the object Distance. (see p. 408)

PA2. Calculation of LSA = xi1- xi1sal 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 P P E N D I X

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 t1 ≡ 0.1
Speciﬁcation of the number of x and t1 values
N : 15	i : 0 . . . N j : 0 . . . N

Speciﬁcation of the range

xi : −.4 + .025 · i t1j : −.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)

δ1

: 2 · A · cos 2 · π

· cos 2 · π ·

−

− 2 · π

In the plotting function one needs the i and j notation

Mi,j : uc(xi , t1j ).

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.				x : 0, .1 . . . 5
Fill in functions of x directly and call M now M(x)
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

A P P E N D I X

Formulas

CONSTANTS

100µm 3000 GHz

100µm 100 cm−1

10µm 1000 cm−1

1 meV 103 eV 1.6 × 10−16joule 8.07 cm−1

100 nm 1000 A˚

10000 A˚ 1µ

1 A˚ 10−8 cm 10−10 m

FORMULAS

√

i2 −1

−1

z a + ib r(cos φ + i sin φ) reiφ

cos x

eix+eix

sin x

eix − eix

ex 1 + x +

+ · · · ,

sin x x −

+ · · · ,

cos x 1 −

+ · · ·

aq2

+ · · · · · · +

aqn−1

qn − 1

−

if |q| < 1, N → ∞ s∞

−

439

440 APPENDIX B. FORMULAS

	x2 + ax + b 0						x1,2
			(1	±	x)n	1	±	nx
a1				±			±
a1	b1	×	c1		d1
a2	b2	×	c2		d2

−	a	±	a

	2		2

|x| << 1

a1c1 + b1c2 a2c1 + b2c2

− b

a1d1 + b1d2 a2d1 + b2d2

a1b1c1	b2c2	b1c1		b1c1
a2b2c2	a1 b3c3	− a2 b3c3	+ a3	b2c2
a3b3c3
	b1c1	b1c2 − b2c1
	b2c2	b1c2 − b2c1

TRIGONOMETRIC FORMULAS

30◦

45◦

60◦

90◦

180◦

270◦

360◦

√

sin

−1

cos

√

−1

√

tan

∞

√

cot

∞

sin2 α + cos2

sin α

cos α

tan α

cot α

cos α

sin α

1 + tan2 α

tan α

sin α

√

cos α

cos2 α

1 + tan2 α

cos α

√

1 + tan2 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(

β)

tan α ± tan β

tan α · tan β

cot(

α)

cot α

cot(α

β)

cot α · β 1

− −

cot

cot α

APPENDIX B. FORMULAS

441

sin 2α 2 sin α cos α

cos 2α cos2 α − sin2 α 1 − 2 sin2 α 2 cos2 α − 1

sin 2α

2 tan α

cos 2α

1 − tan2 α

tan2

tan2 α

2 tan α

cot 2α

cot2 α − 1

(cot α

−

tan α)

tan2

α cot α

tan α

−

2 cot α

2 α

1 + cos α 2 cos

1 − cos α

2 sin

sin 2

1 − cos 2α

2 tan α2

tan α

1 − cos 2α

cos 2α

tan2 α

cos 2α

sin 2α

−

sin α

sin β

2 sin

α + β

cos

α − β

sin α + sin β

tan

α + β

cos α

cos β

sin α

−

sin β

2 cos

α + β

sin

α − β

cos α cos β

2 cos

α + β

cos

α − β

sin α − sin β

cos α

cos β

cos α

−

cos β

−

2 sin

α + β

sin

α − β

tan α + tan β

tan α

tan β

cot α

cot β

tan α

tan β

sin(α ± β)

1 + tan α

(tan 45◦

cos α cos β

−

tan α

a − β

tan

+ α)

cot α cot β

± sin(α ± β)

cot α +

cot(45◦ α)

sin α sin β

cot 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 cot 2α

cot α + tan α

sin 2α

442 APPENDIX B. FORMULAS

DIFFERENTIATION

(u · ν) uν + u ν

	u		u ν − ν u
	ν		ν2
(sin x)		cos x

(cos x) − sin x

(tan x)	1

		cos2 x
(cot x)		−1
	sin2 x

(ex ) ex

(ln x) 1 x

(arcsin x) √ 1

1 − x2

(arccos x) − √ 1

1 − x2

INTEGRATION

udν uν −

ν du

x dx

xn+1

−

sin xdx − cos x

cot xdx ln sin x

tan x

cos2 x

1 + x

−

√

arcsin x

1 − x2

ex dx ex

ax dx

ln a

dx ln x x

cos xdx sin x

sin2 x

− cot x

1 + x2

+ arctan x

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, SpringerVerlag, 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 PhotoOptical 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, Einfuhrung¨ 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

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