2.2 Basic Optical Laws and Definitions - bohr.wlu.ca course Note3.pdf · 2.2 Basic Optical Laws and...

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2.2 Basic Optical Laws and Definitions 2.2 Basic Optical Laws and Definitions Total Internal Reflection Total Internal Reflection - - Critical angle: Critical angle: v = c / n λ = v / f Fundamental parameters: Fundamental parameters: - - Refractive index : Refractive index : n n - - Frequency : Frequency : f f - - Wavelength : Wavelength : λ λ - - Speed of light in materials : Speed of light in materials : v v - - Speed of light in free space (vacuum) : Speed of light in free space (vacuum) : c =3*10 c =3*10 8 8 m/s m/s Snell’ law of reflection: θ 1 = θ 2 , φ 2 = φ 2 , Snell’ law of refraction: n 1 sinφ 1 = n 2 sinφ 2 2 1 sin / c n n φ = Note: definition of Note: definition of θ θ , , φ φ in this book in this book

Transcript of 2.2 Basic Optical Laws and Definitions - bohr.wlu.ca course Note3.pdf · 2.2 Basic Optical Laws and...

2.2 Basic Optical Laws and Definitions2.2 Basic Optical Laws and Definitions

Total Internal ReflectionTotal Internal Reflection-- Critical angle: Critical angle:

v = c / nλ = v / f

Fundamental parameters:Fundamental parameters:-- Refractive index : Refractive index : nn-- Frequency : Frequency : ff-- Wavelength :Wavelength :λλ-- Speed of light in materials : Speed of light in materials : vv-- Speed of light in free space (vacuum) :Speed of light in free space (vacuum) :

c =3*10c =3*108 8 m/sm/s

Snell’ law of reflection: θ 1 = θ 2 , φ 2 = φ 2 , Snell’ law of refraction: n1 sinφ 1 = n2 sinφ 2

2 1sin /c n nφ =

Note: definition of Note: definition of θ θ , , φφ in this bookin this book

2.2 Basic Optical Laws and Definitions2.2 Basic Optical Laws and Definitions

Note :Note : in page 35, book uses N, p for s and p polarizationin page 35, book uses N, p for s and p polarization

A few concepts:A few concepts:-- Plane of incidence (plane including incident ray and normal liPlane of incidence (plane including incident ray and normal line)ne)-- p polarization (E parallel to plane of incidence)p polarization (E parallel to plane of incidence)-- s polarization (E perpendicular to plane of incidence)s polarization (E perpendicular to plane of incidence)-- TE (s polarization, E TE (s polarization, E zz = 0 ) = 0 ) Transverse electric waveTransverse electric wave-- TM (p polarization, HTM (p polarization, Hzz = 0 ) = 0 ) Transverse magnetic wave Transverse magnetic wave

2.2 Basic Optical Laws and Definitions2.2 Basic Optical Laws and Definitions

,s pir r e δ=

Phase shifts:Phase shifts: When light is totally internal reflection, a phase shift When light is totally internal reflection, a phase shift δδ occurs: occurs: -- FresnelFresnel’’ Equation gives reflective coefficient rEquation gives reflective coefficient rpp, r, rss for s and p polarization:for s and p polarization:

2 2 2 22 1 2 1

2 2 2 22 1 2 1

2 2 21 2 1

2 2 21 2 1

sin cos

sin cos

sin cos

sin cos

p

s

n n n nr

n n n n

n n nr

n n n

θ θ

θ θ

θ θ

θ θ

− −=

+ −

− −=

+ −

1 2n n>WithWith

(Note(Note: : φφ is the incident angle and is the incident angle and θ θ = = ππ/2 /2 –– φ φ ))

Question: what happen if incident angle is greater than critical angle ?

2

1

sin cnn

φ =

Complex number

2 2 2 22 1 1 2

2 2 2 22 1 1 2

2 2 21 1 2

2 2 21 1 2

sin cos

sin cos

sin cos

sin cos

p

s

n jn n nr

n jn n n

n j n nr

n j n n

θ θ

θ θ

θ θ

θ θ

− −=

+ −

− −=

+ −

WhenWhen cφ φ>

cos sinie iθ θ θ= +Euler’ identity:

Phase Shifts (or call phase change) :Phase Shifts (or call phase change) :

:sδ,pδ

Expression of Phase Change for reflection coefficients :

The phase shifts between Er and Ei for s and p polarizations

2.2 Basic Optical Laws and Definitions2.2 Basic Optical Laws and Definitions

,s pir r e δ=

2 2

2 2

cos 1tan2 sin

cos 1tan2 sin

p

s

n n

nn

δ θθ

δ θθ

−= −

−= −

1 2/n n n=WithWith(Eq.2(Eq.2--19a need a negative sign)19a need a negative sign)

2.3 Optical Fiber Modes and Configurations2.3 Optical Fiber Modes and Configurations

An optical fiber is a dielectric waveguide (WG) that operates aAn optical fiber is a dielectric waveguide (WG) that operates at optical t optical frequencyfrequency

It confines electromagnetic energy in the form of light to withiIt confines electromagnetic energy in the form of light to within its surfaces n its surfaces and guides the light in a direction parallel to its axisand guides the light in a direction parallel to its axis

A set of guided EM waves called the A set of guided EM waves called the modesmodes of the WG are used to of the WG are used to describe the propagation of light along WGdescribe the propagation of light along WG

Only a certain discrete number of modes are capable of propagatiOnly a certain discrete number of modes are capable of propagating along ng along fiber. These modes are EM waves that satisfy Maxwellfiber. These modes are EM waves that satisfy Maxwell’’ homogeneous homogeneous equation in fiber and boundary condition at the WG surface.equation in fiber and boundary condition at the WG surface.

Schematic of singleSchematic of single--fiber structure:fiber structure:

2.3.1 Fiber Types2.3.1 Fiber Types

Fiber type according to index profile: Fiber type according to index profile: stepstep--index; gradedindex; graded--index fiber (GRIN)index fiber (GRIN)

2.3 Optical Fiber Modes and Configurations2.3 Optical Fiber Modes and Configurations

2.3.1 Fiber Types2.3.1 Fiber Types

Fiber type according to modes: Fiber type according to modes: singlesingle--mode fiber ; multimode fiber ; multi--mode fiber mode fiber

Advantage / disadvantage of multiAdvantage / disadvantage of multi--mode fiber :mode fiber :

2.3 Optical Fiber Modes and Configurations2.3 Optical Fiber Modes and Configurations2.3.2 Rays and Modes2.3.2 Rays and Modes What does SingleWhat does Single--mode fiber mean? mode fiber mean?

What are modes ? What are modes ?

A set of guided EM waves called the A set of guided EM waves called the modes of the WG are used to describe modes of the WG are used to describe the propagation of light along WGthe propagation of light along WG

The stable field distribution in the x The stable field distribution in the x direction with only periodic z direction with only periodic z dependence is known as a modedependence is known as a mode

2.3 Optical Fiber Modes and Configurations2.3 Optical Fiber Modes and Configurations

EM light field guided along fiber can be represented by a superpEM light field guided along fiber can be represented by a superposition of osition of bound or trapped modes.bound or trapped modes.

Each of these guided modes consists of a set of simple EM field Each of these guided modes consists of a set of simple EM field configurations. configurations.

( )0

i t zE e ω β−EM light field: EM light field:

ββ : : z z component of wave propagation component of wave propagation constant constant kk = 2= 2π π //λ.λ.

ββ is main parameter of interest in is main parameter of interest in describing fiber modesdescribing fiber modes

2.3.2 Rays and Modes2.3.2 Rays and Modes

2.3 Optical Fiber Modes and Configurations2.3 Optical Fiber Modes and Configurations

2.3.2 Rays and Modes2.3.2 Rays and Modes

Two methods to theoretically study the propagation characteristiTwo methods to theoretically study the propagation characteristics of light in cs of light in an optical fiber : an optical fiber : RayRay--tracing approach; Mode Theory (Maxwelltracing approach; Mode Theory (Maxwell’’ equations)equations)

RayRay--tracing approach provides a good approximation to light acceptantracing approach provides a good approximation to light acceptance and ce and guiding properties of optical fiber when the ratio of fiber radiguiding properties of optical fiber when the ratio of fiber radius to the us to the wavelength is large, which is known as smallwavelength is large, which is known as small--wavelength limit.wavelength limit.

RayRay--tracing approach could give a more direct physical interpretatiotracing approach could give a more direct physical interpretation of light n of light propagation characteristics in an optical fiber, and provide an propagation characteristics in an optical fiber, and provide an intuitive picture intuitive picture of propagation mechanism in optical fiberof propagation mechanism in optical fiber

Mode theory could provide accurate solution, give field distribuMode theory could provide accurate solution, give field distribution, and tion, and analyze coupling efficiency.analyze coupling efficiency.

2.3.3 Step2.3.3 Step--Index Fiber StructureIndex Fiber Structure2.3.4 Ray Optics Representation2.3.4 Ray Optics Representation

CoreCore--cladding index difference cladding index difference Δ Δ : : also call fractional refractive index changealso call fractional refractive index change

1 2

1

n nn−

Δ =

Meridional rays; skew raysMeridional rays; skew rays

Using Meridional rays to examine the propagation will suffice toUsing Meridional rays to examine the propagation will suffice to obtain a obtain a general picture of rays propagating in a fibergeneral picture of rays propagating in a fiber

2.3 Optical Fiber Modes and Configurations2.3 Optical Fiber Modes and Configurations

Minimum angle that support total internal reflection :Minimum angle that support total internal reflection :

Numerical aperture is related to the Maximum acceptance angle ofNumerical aperture is related to the Maximum acceptance angle of a fiber a fiber

2min

1

sin nn

φ =

2 20 ,max 1 2 1sin 2NA n n n nθ= = − ≈ Δ

2.3 Optical Fiber Modes and Configurations2.3 Optical Fiber Modes and Configurations

2.3.5 Wave representation in a dielectric slab waveguide2.3.5 Wave representation in a dielectric slab waveguide

Condition of wave propagation : All points on the same phase froCondition of wave propagation : All points on the same phase front nt (wave front) of a plane wave must be in phase(wave front) of a plane wave must be in phase

That means: Phase change between the two different tracings withThat means: Phase change between the two different tracings withsame phase front must be an integer multiple of 2same phase front must be an integer multiple of 2π π

Wavefront : The surfaces joining all points of Wavefront : The surfaces joining all points of equal phase are known as wavefronts.equal phase are known as wavefronts.

HuygensHuygens’’s Principle : s Principle : every point on a every point on a propagating wavefront serves as the propagating wavefront serves as the source of spherical secondary wavelets, source of spherical secondary wavelets, such that the wavefront at some later time such that the wavefront at some later time is the envelope of these wavelets.is the envelope of these wavelets.

2.3 Optical Fiber Modes and Configurations2.3 Optical Fiber Modes and Configurations

2.3.5 Wave representation in a dielectric slab waveguide2.3.5 Wave representation in a dielectric slab waveguideCondition of wave propagation : All points on the same phase froCondition of wave propagation : All points on the same phase front of a nt of a plane wave must be in phaseplane wave must be in phase

That means: Phase change between the two different tracings withThat means: Phase change between the two different tracings withsame phase front must be an integer multiple of 2same phase front must be an integer multiple of 2π π

k sΔ = ⋅Phase shift :Phase shift : When light wave travels through materials, it undergoes a When light wave travels through materials, it undergoes a phase shift given by: phase shift given by:

---- propagation constant k :propagation constant k :

02 /k n k nπ λ= =

11 2

2 ( ) 2 2n s s mπ δ πλ

− + =

---- δ δ : phase shift due to total internal reflection: phase shift due to total internal reflection

14 sin 2 2n d mπ θ δ πλ

+ =

E F

2.3.5 Wave representation in a dielectric slab waveguide2.3.5 Wave representation in a dielectric slab waveguide

Another method to find mode condition is from Another method to find mode condition is from Resonance conditionResonance condition : : Phase change along complete round trip is integer multiple of 2Phase change along complete round trip is integer multiple of 2ππ

14sin 2 sinn dk d πθ θλ

Δ = ⋅ =

k

β

h

θ

14 sin 2 2n d mπ θ δ πλ

+ =

p polarization : p polarization : δ δ pps polarization : s polarization : δ δ ss

2 2

2 2

cos 1tan2 sin

cos 1tan2 sin

p

s

n n

nn

δ θθ

δ θθ

−= −

−= −

2.3 Optical Fiber Modes and Configurations2.3 Optical Fiber Modes and Configurations

2.3.5 Wave representation in a dielectric slab waveguide2.3.5 Wave representation in a dielectric slab waveguide

2 21

2 21

sin cos 1tan2 sin

sin cos 1tan2 sin

n d m n n

n d m nn

π θ π θλ θ

π θ π θλ θ

−⎛ ⎞− =⎜ ⎟⎝ ⎠

−⎛ ⎞− =⎜ ⎟⎝ ⎠

p polarization : p polarization :

s polarization : s polarization :

m = 0, 1, 2, 3,m = 0, 1, 2, 3,…………Corresponding to different Corresponding to different guide modesguide modes

2.3 Optical Fiber Modes and Configurations2.3 Optical Fiber Modes and Configurations

Modes: TEModes: TE0 0 , TE, TE1 1 , TE, TE2 2 …………TMTM0 0 , TM, TM11 , TM, TM22…………