Optics, light, and lasers  the practical approach to modern aspects of photonics and laser physics

We will see in the chapter on wave optics that light penetrates into the thinner medium for a dis­tance of about one wavelength with the so-called "evanescent” wave and that the point o

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Optics, Light and Lasers

The Practical Approach to Modern Aspects

of Photonics and Laser Physics Third, Revised and Enlarged Edition

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Optics, Light, and Lasers

Optics, Light, and Lasers

The Practical Approach to Modern Aspects of Photonics and Laser Physics

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1.4.2 Total Internal Reflection 4

Problems 25

2.1.8 Energy Density, Intensity, and the Poynting Vector of

2.3.1.1 Rayleigh Zone, Confocal Parameter b 42

2.3.1.2 Radius of Wave Fronts R(z) 42

2.3.1.3 Beam Waist 2w0 42

2.3.1.4 Beam Radius w(z) 43

2.3.1.5 Divergence 0 div 43

2.3.1.6 Gouy Phase rj(z) 43

2.8 Fresnel Diffraction 71

3.2.1.1 High Frequencies: copr cot » 1 90

3.2.1.2 Low Frequencies: wr « 1 « cop T 90

3.2.3.1 Surface Plasmon Polaritons (SPPs) 92

3.2.3.2 Properties of Surface Plasmon Polaritons (SPPs) 93

3.4.3.1 Lyot Filter 108

3.4.4 Biréfringent Polarizers 109

3.5.6 Optical Isolators and Diodes 118

Problems 119

4 Light Propagation in Structured Matter 121

4.1 O p t i c a l Wave Guides and Fibers 122

4.1.1.2 f — 0: TE and TM Modes 127

4.1.1.3 f > 1: HE and EH Modes 128

4.1.4.4 Photonic Crystal Fibers (PCF) 132

Problems 147

5 Optical Images 149

5.4.1.2 Abbe s Theory of Resolution 156

5.4.1.3 Exploiting the Abbe-Rayleigh Resolution Limit 157

5.6.3 Image Distortions of Telescopes 168

5.6.3.1 Lens Telescopes and Reflector Telescopes 168

5.6.3.2 Atmospheric Turbulence 169

5.7.1.2 Biconvex Lenses and Doublets 171

5.7.1.3 Meniscus Lenses 171

5.7.2.1 Ray Propagation in First Order 172

5.7.2.2 Ray Propagation in Third Order 172

5.7.2.3 Aperture Aberration or Spherical Aberration 173

5.7.2.4 Astigmatism 174

5.7.2.5 Coma and Distortion 175

Problems 177

6 Coherence and Interferometry 181

6.6.3 Resonance Frequencies of Optical Cavities 204

6.6.5 Optical Cavities: Important Special Cases 205

6.6.5.1 Plane Parallel Cavity: f /R = 0 205

6.6.5.2 Confocal Cavity: t / R = 1 206

6.6.5.3 Concentric Cavity: f /R = 2 207

6.7.1.1 Minimal Reflection: AR Coating, AR Layer, and A/4 Film

6.7.1.2 Reflection: Highly Reflective Films 209

2 0 9

Problems 216

7 Light and Matter 219

7.1.2.1 Linear Polarization and Macroscopic Refractive Index 225 7.1.2.2 Absorption and Dispersion in Optically Thin Media 226 7.1.2.3 Dense Dielectric Media and Near Fields 227

7.2.4 Pseudo-spin, Precession, and Rabi Nutation 234

7.2.7.1 Steady-State Inversion and Saturation Intensity 236

7.2.7.2 Steady-State Polarization 238

Contents xi

8.1.1.2 Operating Conditions 252

8.1.1.3 The Laser Resonator 253

8.1.2.1 Laser Line Selection 254

8.1.3 Gain Profile, Laser Frequency, and Spectral Holes 255

8.1.7.2 The Heterodyne Method 259

8.2.3.4 The Excimer Laser 267

8.4.1.2 Configuration and Operation 272

8.4.2.2 The Monolithically Integrated Laser (Miser) 274

8.4.4.2 Fiber Bragg Gratings 277

xii I Contents

9.1.3.2 Mode Pulling 289

9.1.3.3 Field Strength and Number of Photons in the Resonator 290 9.1.3.4 Laser Threshold 290

9.1.3.5 Laser Power and Outcoupling 291

9.4.1.1 Amplitude Fluctuations 298

9.4.1.2 Phase Fluctuations 299

9.4.3.2 Relative Intensity Noise (RIN) 303

10.2.3 Inversion in the Laser Diode 325

10.3.1 Construction and Operation 330

10.3.1.1 Laser Crystal 330

10.3.1.2 Laser Operation 331

10.3.2.1 Emission Wavelength and Mode Profile 332

10.3.2.2 Electronic Wavelength Control 333

10.3.3.1 Inversion in the Quantum Film 334

10.3.3.2 Multiple Quantum Well (MQW) Lasers 336

10.3.3.3 Quantum Wires and Quantum Dots 337

10.5 Laser Diodes, Diode Lasers, and Laser Systems 343

Problems 350

11 Sensors for Light 353

11.2.2 Intrinsic Amplifier Noise 358

11.3.1 Photon Statistics of Coherent Light Fields 360

11.3.2 Photon Statistics in Thermal Light Fields 361

11.4.3 Pyroelectric Detectors 366

11.5.1 The Photoelectric Effect 366

xiv Contents

11.5.2.1 Amplification 368

11.5.2.2 Counting Mode and Current Mode 368

11.5.2.3 Noise Properties of PMTs 369

11.5.2.4 MicroChannel Plates and Channeltrons 370

11.6.1.1 Sensitivity 371

11.6.1.2 Noise Properties 372

12 Laser Spectroscopy and Laser Cooling 379

Problems 404

13 Coherent Light-Matter Interaction 407

13.2.4 Quantum Beats 414

14 Photons: An Introduction to Quantum Optics 417

Fluorescence 432

14.4.4.1 The Mollow Triplet 433

14.5.1 Fluctuating Light Fields 435

14.5.1.1 First-Order Coherence 435

14.5.1.2 Second-Order Coherence 436

14.5.1.3 Hanbury Brown and Twiss Experiment 437

14.5.2.1 Fock States or Number States 439

14.5.2.2 Coherent Light Fields and Laser Light 439

14.5.2.3 Thermal Light Fields 441

14.7.1.1 The Einstein-Podolsky-Rosen (EPR) Paradox 448

14.7.2 Bells Inequality 450

14.7.4 Polarization-Entangled Photon Pairs 452

14.7.5 A Simple Bell Experiment 453

Problems 455

15 Nonlinear Optics I: Optical Mixing Processes 457

xvi Contents

15.2.2.1 Intrinsic Permutation Symmetry 461

15.2.2.2 Real Electromagnetic Fields 461

15.2.5 Effective Value of the Nonlinear d Coefficient 463

15.4.3.1 Angle or Critical Phase Matching 471

15.4.3.2 Noncritical or 90° Phase Matching 471

15.4.5.1 Passive Resonators 474

Problems 482

16 Nonlinear Optics II: Four-Wave Mixing 485

A Mathematics for Optics 497

469

487

497

B Supplements in Quantum Mechanics 503

Bibliography 507

Contents | xvii

Index 519

Preface

The field of optics - in modern times perhaps equivalent to photonics - has ecome an important branch of the physical sciences and technologies The say­ing that the twenty-first century will be the century of the photon, following the era of the electron, is gaining more and more credibility

This textbook has attempted - and continues to do so - to link the central topics ot optics that were established 200 years ago to the most recent research topics Since the very first German language edition published already in 1999,

an impressive evolution of entire new directions has emerged

Optical and photonics research is no longer conceivable without a wide range

o structured materials shaping the propagation of light fields Optical fibers,

p otonic materials, and metamaterials offer an unprecedented degree of control over propagating optical fields Inspiring concepts such as the “perfect lens” and optical angular momentum” are posing new conceptual and experimen­tal challenges and promise new and wide applications Optical imaging has overcome the classic and seemingly unsurmountable Rayleigh—Abbe limit of resolution Femtosecond laser frequency combs allow us today to perfectly count every cycle of an optical frequency, that is, with ultimate precision The next eve opment is looming: optical sciences are going to marry with information technology - perhaps a natural consequence of the photon following the footsteps of the electron

While a single book can impossibly be big enough to discuss all these new topics

at epth, this new edition picks them up selectively and links them to the funda­mentals of optical science It can serve as a reference for teaching the foundations

° mo<^ern optics: classical optics, laser physics, laser spectroscopy, concepts of quantum optics, and nonlinear optics - those remain the building blocks of all experimental efforts and applications, but are frequently taken as "given” and no onger scrutinized for a deeper understanding

As for the former edition, graphical educational material is available through the following website: http://tiny.iap.uni-bonn.de/oll/OLLWS/index.html

Light Rays

1.1 Light Rays in Human Experience

The formation of an image is one

of our most fascinating emotional

experiences (Figure 1.1) Even in

ancient times it was realized that

our “vision” is the result of rec-

tilinearly propagating light rays,

because everybody was aware of

the sharp shadows of illuminated

objects Indeed, rectilinear propa­

gation may be influenced by certain

optical instruments, for example, by

mirrors or lenses Following the suc­

cesses of Tycho Brahe (1546-1601),

knowledge about geom etrical optics

led to the consequential design

and construction of magnifiers,

microscopes, and telescopes All

these instruments serve as aids to

vision Through their assistance,

insights” have been gained that

added to our world picture of natural science, because they enabled observations

of the world of both micro- and macro-cosmos

Thus it is not surprising that the terms and concepts of optics had tremendous impact on many areas of natural science Even such a giant instrument as the Large Hadron Collider (LHC) particle accelerator in Geneva is basically nothing other than an admittedly very elaborate microscope, with which we are able to observe the world of elementary particles on a subnuclear length scale Perhaps as important for the humanities is the wave theoretical description of optics, which spun oft the development of quantum mechanics

Figure 1.1 Light rays

Optics, Light and Lasers: Lhe Practical Approach to Modern Aspects of Photonics and Laser Physics,

Third Edition Dicier Meschede.

2 I 7 Light Rays

In our human experience, rectilinear propagation of light rays - in a homoge­neous medium - stands in the foreground But it is a rather newer understanding that our ability to see pictures is caused by an optical image in the eye Neverthe­less, we can understand the formation of an image with the fundamentals of ray optics That is why this textbook starts with a chapter on ray optics

1.2 Ray Optics

When light rays spread spherically into all regions of a homogeneous medium,

in general we think of an idealized, point-like, and isotropic luminous source at their origin Usually light sources do not fulfill any of these criteria Not until we reach a large distance from the observer, may we cut out a nearly parallel beam of rays with an aperture Therefore, with an ordinary light source, we have to make

a compromise between intensity and parallelism to achieve a beam with small divergence Nowadays optical demonstration experiments are nearly always per­formed with laser light sources, which offer a nearly perfectly parallel, intense optical beam to the experimenter

When the rays of a beam are confined within only a small angle with a common optical axis, then the mathematical treatment of the propagation of the beam of rays may be greatly simplified by linearization within the so-called paraxial approximation This situation is met so often in optics that proper­ties such as those of a thin lens, which go beyond that situation, are called

“aberrations.”

The direction of propagation of light rays is changed by refraction and reflec­tion These are caused by metallic and dielectric interfaces Ray optics describes their effect through simple phenomenological rules

1.3 Reflection

We observe reflection of, or mirroring of, light rays not only on smooth metallic surfaces but also on glass plates and other dielectric interfaces Modern mirrors may have many designs In everyday life they mostly consist of a glass plate coated with a thin layer of evaporated aluminum But if the application involves laser light, more often dielectric multilayer mirrors are used; we will discuss these in more detail in the chapter on interferometry (Chapter 6) For ray optics, the type

of design does not play any role

1.3.1 Planar Mirrors

We know intuitively that at a planar mirror like in Figure 1.2, the angle o f incidence 6X is identical with the angle o f reflection 02 of the reflected beam,

1.4 Refraction 3

and that incident and reflected beams lie within

a plane together with the surface normal Wave

optics finally gives us a more rigid reason for

the laws of reflection Therefore also details, for

example, the intensity ratios for dielectric reflec­

tion (Figure 1.3), are explained, which cannot be

derived by means of ray optics

1.4 Refraction

At a planar dielectric surface, for example, a

glass plate, reflection and transmission occur

concurrently Therefore the transmitted part of

the incident beam is “refracted.” Its change of

direction can be described by a single physical

quantity, the “index of refraction” (also refractive

index) It is higher in an optically “dense” medium

than in a “thinner” one

In ray optics a general description in terms

of these quantities is sufficient to understand

the action of important optical components But

the refractive index plays a key role in the con­

text of the macroscopic physical properties of

dielectric matter and their influence on the prop­

agation of macroscopic optical waves as well

This interaction is discussed in more detail in

where 0 { and 02 are called the angle of incidence and angle of emergence at the

interface It is a bit artificial to define two absolute, material-specific refractive

indices, because according to Eq (1.2) only their ratio n n = nxjn 2 is determined

at first But considering the transition from medium “1” into a third material “3'

with nVi, we realize that, since w23 = «21wn , we also know the properties of refrac­

tion at the transition from “2” to “3.” We can prove this relation, for example, by inserting a thin sheet of material “3” between “1” and “2.” Finally, fixing the refrac­tive index of vacuum to wvac = 1 - which is argued within the context of wave optics - the specific and absolute values for all dielectric media are determined

Heavy flint glass SF11 LaSF N9 25.76 32.17

Barium crown BaK 1 57.55

Flint glass

F 2 36.37

Refractive index n fo r selected wavelengths

1.4.2 Total Internal Reflection

According to Snell s law, at the interface between a dense medium “1” and a thin­

ner medium 2 (nx > n2), the condition (1.2) can only be fulfilled for angles smaller than the critical angle 0C:

For 0 > 6 C the incident intensity is totally reflected at the interface We will see in

the chapter on wave optics that light penetrates into the thinner medium for a dis­tance of about one wavelength with the so-called "evanescent” wave and that the point of reflection does not lie exactly at the interface (Figure 1.4) The existence

of the evanescent wave enables the application of the so-called "frustrated” total internal reflection, for example, for the design of polarizers (Section 3.4.4)

7.5 Fermat's Principle: The Optical Path Length

Figure 1.4 Total internal reflection at a dielectric surface

occurs for angles 9 > 9C.The point of reflection of the

rays does not lie exactly within the interface, but slightly

beyond (the Goos-Hänchen effect [1, 2])

1.5 Fermat's Principle: The Optical Path Length

As long as light rays propagate in a homogeneous medium, they seem to follow the shortest geometric path irom the source to a point, making their way in the shortest possible time If refraction occurs along this route, then the light ray obviously no longer moves on the geometrically shortest path

The French mathematician Pierre de Fermat (1601-1665) postulated in 1658 that in this case the light ray should obey a minimum principle, moving from the source to another point along the path that is shortest in time.

For an explanation of this principle, one cannot imagine a better one than that given by the American physicist Richard P Feynman (1918-1988), who visualized Fermat s principle with a human example: One may imagine Romeo discovering his great love Juliet at some distance from the shore of a shallow, leisurely flowing river, struggling for her life in the water Without thinking, he runs straight toward his goal — although he might have saved valuable time if he had taken the longer route, running the greater part of the distance on dry land, where he would have achieved a much higher speed than in the water

Considering this more formally, we determine the time required from the point

of observation to the point of the drowning maiden as a function of the geometric path length Therefore we find that the shortest time is achieved exactly when a path is chosen that is refracted at the water—land boundary It fulfills the refrac­

tion law (1.2) exactly if we substitute the indices of refraction n x and n2 by the

inverse velocities in water and on land, that is,

^ = ^

According to Fermats minimum principle, we have to demand the following The

propagation velocity of light in a dielectric ctl is reduced in comparison with the velocity in vacuum c by the refractive index n:

cn = c/n.

6 ^ 1 Light Rays

Now the optical path length along a trajectory C, where the refractive index

depends on the position r, can be defined in general as

With the tangential unit vector et, the path element ds = e t ■ d r along the pat

can be calculated

Example: Eermat’s principle and refraction

As an example of the use of the integral princip '

\

Figure 1.5 Fermat's principle and

refraction at a dielectric surface

^opt — Hiei ■ r AO + n2e2 • rOB, 5Copl = {n\Z\ - n2e2) • Sr'.

In the homogeneous regions, light has to folios

a line; thus variations can only occur along th surface with the normal N, that is, Sr' = N X Si

We use the commutativity of the triple product

'Ibis relation is a vectorial fo

1.5 Fermat's Principle: The Optical Path Length 1

d r is valid in accordance with Eq (1.4), which yields the relation

net = n % = V £ opt and " 2 = (V £ opt)2>

which is known as the eikon al equ ation in optics We get the important ray equ ation of optics by differentiating the eikonal equation after the path1:

A linear equation may be reproduced for homogeneous materials (V n = 0) from

(1.5) without difficulty

Example: Mirage

As a short example, we will treat reflection at a hot film of air near the ground, which induces a decrease in air density and thereby a reduction of the refrac­tive index (Another example is the propagation of light rays in a gradient wave guide; Section 1.7.3.) We may assume in good approximation that for calm air

the index of refraction increases with distance y from the bottom, for example, n(y) = n0( 1 — te~ <xy) Since the effect is small, e 1 is valid in general, while the

scale length a is of the order tr = 1 m_1 We look at Eq (1.5) for r = for

all individual components and find the x coordinate with constant C:

We may use this result as a partial parametric solution for the y coordinate.

where the new parameter k has to be determined from boundary conditions For

arge distances from the point of reflection at x = x0, we find straight propaga­

tion as expected The maximum angle 0 = arctan(2K*/a), where reflection is still possible, is defined by k < n0a ( t /2 )1/2 As in Figure 1.6, the observer registers two images — one of them is upside down and corresponds to a mirror image,

e curvature of the light rays declines quickly with increasing distance from the ottom and therefore may be neglected for the “upper" line of sight At (^0,y0) a virtual” point of reflection may be defined

1 Iherefore we apply d /d s = et • V and

| v c = (et V)V/: = i(VC-V)Vi: = ^ V (V £ )— ^ V >r

1 Light Rays Figure 1.6 Prof\\e of the refractive

index and opt\ca 1 path for a mirage

Figure 1.7 Reflection in 90° prism This prism is used for rectangular umh >

aiso be used for the design of a retroreflector, whereby an optical deiay M •

by simple adjustment

1.6 Prisms

The technically important rectangular reflection is achieved with an angle o f inc1'

dence of 0, = 45° For ordinary glasses (n ~ 1.5), this is above the angle of tota

internal reflection 0C = sin"'1.5 = 42° Glass prisms are therefore often used as simple optical elements, which are applied for beam deflection (Figure 1.7) M ° re complicated prisms are realized in many designs for multiple reflections, where they have advantages over the corresponding mirror combinations due to theirminor losses and more compact and robust designs

r

s

Often used designs are the Porro prism and the “cat’s-eye retroreflector (Figure 1.8) " other names for the latter are corner cube reflector,” 'c a ts eye,” or “triple mirror.” ^ e Porro prism and its variants are applied, for example, telescopes to create upright lrnages "the retroreflector notonly plays an important role in-

but also enables functioning

°f safety reflectors — cast in plastics - | ■ *

one gets an upright image The cat's-eye retroreflector Ihght) throws back every Ught ray

^dependenUv oi its angle oi incidence, causing a plastics - in vehicles

prisms » t e e (Mat lite r a l reflection n te , ■ drt“ 1 ,ods “ »

and again, witlrom changing * pari, ^

7.6 Prisms 9

rods are used, for example, to guide light from a source toward a photodetector In

miniaturized form they are applied as w avegu ides in optical telecommunications

Their properties will be discussed in the section on beam propagation in wave guides (Section 1.7) and later on in the chapter on wave optics (Section 4.1) in more detail

1.6.1 Dispersion

Prisms played a historical role in the spectral decomposition of white light into

its constituents The refractive index and thus also the angle of deflection 8 in Figure 1.9 actually depend on the wavelength, n = n(A)} and therefore light rays

of different colors are deflected with different angles Under n orm al dispersion, blue wavelengths are refracted more strongly than red ones, n(Ahiuc) > ^(^rcd)-

Refractive index and dispersion are very important technical quantities for the application of optical materials The refractive index is tabulated in manufac­turers’ data sheets for various wavelengths, and (numerous different) empirical formulae are used for the wavelength dependence The constants from Table 1.1

are valid for this formula, which is also called the S ellm eier equ ation :

By geometrical considerations we find that the angle of deflection S in Figure 1.9 depends not only on the angle o f incidence 0 but also on the aperture angle a of the symmetrical prism and of course on the index of refraction, n:

8 = 0 — a + arcsin

^min ~ 20Symm ~~ a •

sin(aVw 2 — sin2 6) — cos a sin 0 )] ,

theG ^smT^1111 ^e^ec^on anSle £nijfl is achieved for symmetrical transit through

Th ~~ ^synim^ an<^ enahles a precise determination of the refractive index,

na result is expressed straightforwardly by the quantities a and 8min:

M^ sin[(a + Smln)/2 ]

sin (a /2 )

her ° a ^Uant*tat*ve estimation o f the dispersive power K o f glasses, the Abbe num-

g ma^ e use(I* This relates the refractive index at a yellow wavelength (at A =

e *^erence of the refractive indices at a blue wavelength (A = 486.1 nm,

Figure 1.9 Refraction and dispersion at a symmetrical prism The index of refraction n can be calculated from the minimum angle of deflection 8 = 6min in a simple manner.

Fraunhofer line F of hydrogen) and a red wavelength (A = 656.3 nm, Fraunhofer

line C of hydrogen):

10^ 1 Light Rays

According to the aforementioned, a large Abbe number means weak dispersion,

and a small Abbe number means strong dispersion The Abbe number is also

important when correcting chromatic aberrations (see Section 5.7.3).

The index of refraction describes the interaction of light with matter, and we

will come to realize that it is a complex quantity, which describes not only the

properties of dispersion but also those of absorption Furthermore, it is the task

of a microscopic description of matter to determine the dynamic polarizability and thus to establish the connection to a macroscopic description.

1.7 Light Rays in Wave Guides

Figure A AO Histone station Ho 5A of the Ber\\n-Co\ogne-Cob\enz opt\c-mechan\ca\

’’sight" transmission line on the tower of the StPantaieon Church, Coiogne Picture by NNeiger 0340V

light signals is a very convenient method that has a very long history

of application For example, in the nineteenth century, mechanical pointers were mounted onto high towers and were observed with telescopes to realize transmission lines of many hundreds of miles

Am example of a historic relay station from the 400 mile Berlin-*

line is shown in Figure 1.10 Basi** cally, in-air transmission is also performed nowadays, but with laser light But it is always affected

Y its scattering properties even

^ small distances, because tur- ulence, dust, and rain can easily

lase^b ProPaSat*on a ^ree

Ideas for guiding optical wavesIon6 ^een m existence for a very

hon1^ 1168’ for example, at first, hollow tuhpc “-Hues, tor example, a t”

hollow tubes made of copper were applied, but their attenuation is too large for transmission over long

7.7 Light Roys in Wave Guides 11

distances Later on periodical lens systems have been used for the same purpose, but due to high losses and small mechanical flexibility, they also failed

The striking breakthrough happened to “optical telecommunication” through

the development of low-loss w ave guides, which are nothing other than elements

for guiding light rays In 2009 the Nobel prize was awarded to Charles K Kao (1 9 3 3 -) “for groundbreaking achievements concerning the transmission of light

in fibers for optical communication,” which in fact created the new field offib e r optics and is today the backbone of worldwide communication including the Internet

Optical fibers can be distributed like electrical cables, provided that adequate transmitters and receivers are available With overseas cables, significantly shorter signal transit times and thus higher comfort for phone calls can be achieved than via geostationary satellites, where there is always a short but unpleasant and unnatural break between question and answer

Therefore, propagation of light rays in dielectric wave guides is an important chapter in modern optics Some basics may yet be understood by the methods of ray optics

1.7.1 Ray Optics in Wave Guides

Total internal reflection in an optically thick medium provides the fundamental physical phenomenon for guiding light rays within a dielectric medium Owing to this effect, for example, in cylindrical homogeneous glass fibers, rays whose angle

with the cylinder axis stays smaller than the angle of total internal reflection 0C are

guided from one end to the other Guiding of light rays in a homogeneous glass cylinder is affected by any distortion of the surface, and a protective cladding could even suppress total internal reflection

developed, where the optical waves are

guided in the center of a wave guide through

variation of the index of refraction These

wave guides may be surrounded by cladding

and entrenched like electrical cables

We will present the two most impor­

tant types Step-index fibers consist of

two homogeneous cylinders with different

refractive indices (Figure 1.11) To achieve

beam guiding, the higher index of refraction

must be in the core, the lower one in the

cladding Gradient-index (GRIN) fibers with

continuously changing (in good approxima­

tion, parabolic) refractive index are more

sophisticated to manufacture (Figure 1.12),

but they have technical advantages, for

example, a smaller group velocity dispersion

J

J '

-n2

3 2

Figure 1.11 Profiles of the refractive

index and ray path in optical wave

guides Upper: Wave guide with hom ogeneous refractive index

Center: Wave guide with stepped profile of refractive index (step-index fiber) Lower: Wave guide with continuous profile of refractive index (gradient-index fiber)

Figure 1.12 Manufacturing of wave guides The preform is manufactured with appropriate

materials with distinct indices of refraction, which are deposited on the inner walls of a quartz tube by a chemical reaction

Excursion: Manufacturing optical fibers

10 mm diameter, a so-called preform In the last steD a 6 ^ ° * 7 aSS'Ve 9 'aSS r° d ° f ab° j f

The creation of optical fibers is a cons

one of the oldest materials used in our6^ 61106 ° ^ e a<lvantageous properties of complex structured materials were d ^UIC' ^ ass’ more recent years, more propagation in wave guides, for i , t0 COntro1 the properties of light (see Chapter 4.2.6) eXample> the so-called photonic crystal fibers

1.7.2 Step-index Fibers

Principle of , aPplied in st tota internal reflection is

consist of a T lndexf i bers (Figure 1.13), which

with »T refractive index «, and athe inrl.w ,.r r ^ l■ The relative difference in

1.7 Light Rays in Wave Guides 13

since n2/ n x = 1 - A, and thus is set in relation to A, which yields a < 8.2° for

this case

W hen light rays cross the axis of a fiber, propagation takes place in a cut plane,

which is called the m erid ion al p la n e S kew ed rays do not pass the axis and are

guided on a polygon around the circle It can be shown that the rays must confine

an angle a < a G with the z axis to be guided by total internal reflection.

1.7.2.1 Numerical Aperture of an Optical Fiber

To guide a light ray in an optical fiber, the angle of incidence at the incoupler must be chosen small enough The maximum aperture angle 0a of the acceptance

cone can be calculated according to the refraction law, sin 0a = n x sin a G =

n x cos 0C The sine of the aperture angle is called the n u m erical apertu re (NA) According to Eq (1.8) and co s0 c ~ \flKy it can be related with the physical

parameters of the optical fiber:

This yields, for example, NA = 0.21 for the aforementioned quartz glass fiber, which is a useful and typical value for standard wave guides

1.7.2.2 Propagation Velocity

Light within the core of the wave guide propagates along the trajectory with a

velocity v{r{z)) = c/n (r(z)) Along the z axis the beam propagates with a reduced velocity, (uz) = u c o s a , which can be calculated for small angles a to the z axis

according to

In Section 4.1 on the wave theory of light, we will see that the propagation velocity

is related to the phase velocity

1.7.3 Gradient-Index Fibers

Beam guiding can also be performed by means of a GRIN fiber, where the quadratic variation of the index of refraction is important To determine the curvature of a light ray induced by the refractive index, we apply the ray equation

(1.5) This is greatly simplified in the paraxial approximation (ds a dz) and for a

cylindrically symmetric fiber:

141 1 Light Rays

decreases from the m axim um value n x at r = 0 to n2 a t

th e equation of m otion of a harm onic oscillator,

r — a. O n e e n d s u p w ith

£ L + t* r = 0 ,

a n d realizes im m ediately th at the light ray p erform s o sc illa to ry m o tio n a b o u t t e

r{z) - r0 sin(2/rz/A)

The m axim um elongation allowed is Tq = a, b ecau se o th e rw is e th e b e a m lo s e s

its guiding Therefore also the m axim um angle a G = \ / 2 A for c ro s s in g t h e a x is

occurs It is identical w ith the critical angle for to ta l in te r n a l re f le c tio n in a step-index fiber and yields also th e sam e relation to th e N A (Eq (1.9))

B i a d t e n t - i n d « f ib c r s

T = f d t = i d z l Vz&)- J o J 0

m Uz(2) = ^ ) / V T T ? ( ^ ( y ie ld in g

1.8 Lenses and Curved Mirrors 15

1.8 Lenses and Curved Mirrors

T h e f o r m a tio n o f a n im a g e p lay s a m a jo r ro le in o p tic s , a n d le n se s a n d c u r v e d

m i r r o r s a re e s s e n tia l p a r ts in o p tic a l d e v ic e s F irst, w e w ill d is c u s s th e e ffe c t o f

th e s e c o m p o n e n ts o n th e p r o p a g a tio n o f rays; o w in g to its g re a t im p o r ta n c e , w e

To c o m p e n s a te fo r th e q u a d r a tic in c re a s e o f th e o p tic a l p a th le n g th ^ ( r ) , th e delay

by th e p a th w ith in th e le n s g lass - th a t is, th e th ic k n e s s - m u s t also v a ry q u a d ra ti- cally T h is is a c tu a lly th e c o n d itio n fo r s p h e ric a l su rfa c e s, w h ic h h ave b e e n s h o w n

to b e e x tre m e ly su c c e ssfu l fo r c o n v e r g e n t lenses! T h e r e s u lt is th e sa m e w ith m u c h

m o re m a th e m a tic a l e ffo rt, if o n e e x p lo re s th e p r o p e r tie s o f re fra c tio n a t a len s

su rfa c e a s s u m in g th a t a le n s is c o n s tr u c te d o f m a n y th in p ris m s (F ig u re 1.14) In

th e c h a p te r o n le n s a b e r r a tio n s , w e w ill d e a l w ith th e q u e s tio n of w h ic h c rite ria

s h o u ld b e im p o r ta n t fo r th e c h o ic e o f a p la n a r c o n v e x o r b ic o n v e x lens

ay e 9 ured as a combination of several prisms, (b) A parallel beam of rays originating from

1 Light Rays

/ N^ mirror or parabolic mirrors play the m ost im

por-T^ ° 'r very well known from- c \

«It Uic uuciv,cpi

concave mirror For near-axis incident light, spherical mirrors are used planar reflection to curved m irror su rfa c e s.n r

and th e effect o n a parallel b e a m o T Z ' Figure 1.15 raYs Wlt^ m o n e c u t p la n e is v is u a liz e d m^ aXial

points m ust th en lie o n a parabola N e n T fr° m g e o m e trV t h a t t h e r e ile c tlo n

m anufacture In Figure 1.16b th e ^ enCa m ’r r o r s > w h ic h a re m u c h e a s ie r t o

the dependence of the focal length e^cmen*-s are shown, from which

In general we neglect th e quadratic correction, w h ich c a u se s a n a p e r tu r e e

and is investigated in m ore detail in S ection 5.7.2.3

and within the plane perpendicular to th a t one { f ) will differ f r o m / 0 = R / 2;

R cos « and]^cos«, respectively The difference b e tw e e n th e tw o p la n e s o c c u r r in g

7.9 Matrix Optics 17

1.9 Matrix Optics

A s a r e s u lt o f its r e c tilin e a r p ro p a g a tio n , a fre e lig h t ray m a y b e tr e a te d like a

s tr a ig h t lin e In o p tic s , s y s te m s w ith a x ia l s y m m e tr y a re e sp e c ia lly im p o r ta n t,

a n d a n in d iv id u a l lig h t ra y m a y b e d e s c r ib e d su ffic ie n tly w ell by th e d is ta n c e fro m a n d a n g le to th e ax is (F ig u re 1.17) If th e s y s te m is n o t ro ta tio n a lly

s y m m e tr ic , fo r e x a m p le , a f te r p a s s in g th r o u g h a c y lin d ric a l lens, t h e n w e c a n

d e a l w ith tw o in d e p e n d e n t c o n tr ib u tio n s in th e X a n d y d ir e c tio n s w ith th e

s a m e m e th o d

T h e m o d if ic a tio n o f th e b e a m d ir e c tio n by

o p tic a l c o m p o n e n ts - m ir r o r s , le n se s, a n d

d ie le c tric s u rfa c e s - is d e s c r ib e d b y a tr ig o n o ­

m e tr ic a n d th e r e f o r e n o t alw ays sim p le re la tio n

F or n e a r-a x is rays, th e s e f u n c tio n s c a n o f te n b e

lin e a riz e d , a n d th u s th e m a th e m a tic a l t r e a tm e n t

is sim p lifie d e n o rm o u s ly T h is b e c o m e s o b v io u s,

fo r e x a m p le , fo r a lin e a r iz e d f o r m o f th e law o f

re fra c tio n (1.2):

H e re w e h a v e m a d e u se o f th is a p p r o x im a tio n a lre a d y w ith th e a p p lic a tio n o f

F e rm a t's p r in c ip le fo r id e a l le n se s N e a r-a x is rays a llo w th e a p p lic a tio n o f s p h e r i­cal s u rfa c e s fo r le n se s, w h ic h a re m u c h e a s ie r to m a n u f a c tu r e th a n m a th e m a tic a l

id e a l su rfa c e s F u r th e r m o r e , id e a l sy s te m s a re o n ly “id e a l” fo r s e le c te d ray sy s­

te m s; o th e r w is e th e y su ffe r fro m im a g e a b e r r a tio n s like o th e r sy ste m s

W h e n tr e a tin g th e m o d ific a tio n o f a lig h t ray by o p tic a l e le m e n ts in th is

a p p r o x im a tio n by lin e a r tr a n s f o r m a tio n , m a tr ic e s a re a c o n v e n ie n t m a th e m a tic a l

to o l fo r c a lc u la tin g th e f u n d a m e n ta l p r o p e r tie s o f o p tic a l sy ste m s T he d e v e lo p ­

m e n t o f th is m e th o d le d to th e d e n o m in a tio n m a t r i x o p t i c s T he in tr o d u c tio n

o f tr a n s f o r m a tio n m a tr ic e s fo r ray o p tic s m a y b e v isu a liz e d v e ry easily, b u t

th e y a c h ie v e d s trik in g im p o r ta n c e b e c a u s e th e y d o n o t c h a n g e th e ir fo rm w h e n

tr e a tin g n e a r-a x is rays a c c o rd in g to w ave o p tic s (see S e c tio n 2.3.2) F u rth e rm o re

th is fo rm a lis m is also a p p lic a b le fo r o th e r ty p e s o f o p tic s su c h as “e le c tro n o p tic s

o r th e e v e n m o re g e n e ra l “p a rtic le o p tic s ”

1.9.1 Paraxial Approximation

L et u s c o n s id e r th e p r o p a g a tio n o f a lig h t ray a t a sm a ll an g le a to th e z axis The

b e a m is fully d e te r m in e d by th e d is ta n c e r fro m th e z axis a n d th e slo p e 1J — ta n (X

W ith in th e s o -c a lle d p a ra x ia l a p p ro x im a tio n , w e n o w lin e a riz e th e ta n g e n t o f th e

an g le a n d s u b s titu te it by its a r g u m e n t, r' ~ ( X, a n d th e n m e rg e r w ith tJ to e n d

u p w ith a v e c to r r = (r, a ) A t th e s ta r t, a lig h t ray m ay h ave a d is ta n c e to th e axis

a n d a slo p e o f Tj = ( r j , ƠỊ) H a v in g p a s s e d a d is ta n c e d a lo n g th e z axis, th e n

181 1 Light Rays

A b it m ore com plicated is the m odification by a re fra c tin g o p tic a l s u rfa c e F o r

th a t purpose we look at the situation show n in Figure 1.18, w h e r e tw o o p tic a l

m edia w ith refractive indices n x and n2 are separated by a sp h e ric a l in te r f a c e w ith radius R If the radius vector subtends an angle <f> w ith th e z axis, th e n th e lig h t ray

is obviously incident on the surface at an angle d x = a y + 4> a n d is r e la te d to t h e

angle of emergence by the law of refraction In paraxial a p p ro x im a tio n a c c o r d in g

to Eq (1.2), n x6 x ~ n292 and 4> ^ r , / R is valid, and o ne finds

The linearized relations may be described easily by th e re fra c tio n m a tr ix B:

The m ost im portant optical elem ents may b e specified by th e ir tr a n s to r n w - w " * '

also called ABCD matrices M ABcD, tueû Atf CD matrices M AlicD, — y wv 1

h we collect in Table 1.2 lor look-un m»*—

which we collect in Table 1.2 for look-u

According to Figure 1.18, the effect of a 1

B' at the exit Now the m atrix m ethod s h ^ glass-a n d o n e f u r th e r re f r a c tio n allows to calculate the properties of an en? ^ Stren& ^, b e c a u s e th e lin e a r ity

( > l \ (r \

1.9 Matrix Optics 19

Table 1.2 Important ABCD matrices.

5) O b je c t d is ta n c e s a re d e fin e d to b e p o s itiv e (n e g a tiv e ) ab o v e (b e lo w ) th e z axis.

6) R eflectiv e o p tic s is t r e a te d by flip p in g th e ray p a th a fte r e v e ry e le m e n t

A u se fu l p r o p e r ty o f th e A B C D m a tr ic e s is th e ir d e te r m in a n t2:

t lonC ISf f aS1 ^ Seen f r ° m T ab le 1.2 a n d c a n b e u n d e r s to o d th a t fo r all local o p e ra -

I W1 ° Ut tr a n s ^a t l° n , w e m u s t h av e B = 0 w h ile tra n s la tio n s c h a n g e th e slo p e ces i m m n ite ^ m a ^ w a Ys ( C = 0) S in c e th e d e te r m in a n t o f th e p r o d u c t o f m a tri-

e q U a ^ G P roc*u c t ° f th e ir d e te r m in a n ts , Eq (1.18) h o ld s also fo r a n e n tire

m Wlt I sysl — 1 fo r ray s e n te r in g fro m a n d e x itin g in air

1.9.3 Lenses in Air

oper-2 Siegman's convention â i = n ia i yields in general |M| - 1 [3] for actual calculation

decide for one convention

(1.19)

Explicit calculation of L according to Eq (1.16) a n d ta k in g in to a c c o u n t t h e in d e x

of refraction nair = 1 in Eqs (1.14) and (1.15) yields

Thus the ABCD m atrix for thin lenses be

where the sign is chosen such th /

The r e f r a c tiv f p o w e r ^ ^ r ^ “ ident i cal w i t h have a P o sitiv e re fra c tiv e

Example: ABCD matrix of an imaging system

F o r im a g in g by a n a r b itr a r y A B C D s y ste m , w e m u s t c la im th a t a b u n d le o f rays

T h e m a tr ix m e th o d e n a b le s us to e x p lo re th e e ffe c t o f a sy ste m c o n s is tin g o f tw o

le n se s w ith fo cal le n g th s f x a n d ^ a t a d is ta n c e d W e m u ltip ly th e A B C D m a tric e s

a c c o rd in g to Eqs (1.21) a n d (1.14) a n d g e t th e m a tr ix M o f th e system :

T he sy ste m o f tw o le n se s s u b s titu te s a sin g le le n s w ith focal le n g th g iven by

e c o n s id e r th e fo llo w in g th r e e in te r e s tin g sp e c ia l cases:

) d < ^ f l 2 T w o le n se s th a t a re m o u n te d d ire c tly n e x t to e a c h o th e r, w ith n o sp ace

b e tw e e n th e m , a d d th e ir re fra c tiv e p o w e rs, M - L2LP w ith V = D , + V 2 T his

c ir c u m s ta n c e is u se d , fo r e x a m p le , w h e n a d ju s tin g eyeglasses, w h e n refrac-

e p o w e rs a re c o m b in e d u n til th e r e q u ir e d c o r r e c tio n is found O b v io u sly a

co n v e x le n s c a n b e c o n s tr u c te d o u t o f tw o p la n a r c o n v ex len ses, e x p e c tin g

Ji If th e focal p o in ts c o in c id e , a te le s c o p e is realiz e d A p a ra lle l b u n d lerays w ith ra d iu s is w id e n e d o r c o llim a te d in to a n e w b u n d le o f p a ra lle l

(1.24)

(1.25)