veronique moeyaert optical transmission, vi – storage area networks revision : optical...

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Veronique MoeyaertOptical Transmission, VI – Storage Area Networks

Optical Telecommunications

Chapter 2 - Optics Refresher

Author : Kevin Heggarty

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Lecture plan

Light as an Electromagnetic wave – orders of magnitude . Quantum nature - «photon» Polarisation Interaction with matter, reflection, refraction, TIR. Dioptre, paraxial approximation Simple lenses, imaging, 2 et 4f setups Lens performance, resolution Aberrations, ray tracing GRIN lenses Gaussians beams.

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Bibliography

• Optics, E. Hecht, Addison-Wesley (Second edition

1984).

• Principals of Optics, M.Born and E.Wolf,

Pergamon.

• The practical application of light, Melles Griot

Catalogue: www.mellesgriot.com

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Light as an electro-magnetic wave Light is an electro-magnetic wave A wave solution of Maxwell’s equations

∇ . E=0 ∇ . H=0 ∇∧H= D

t∇∧E= −

B

t

Progressive wave: E x ,t =E0cos t−kx

Speed: c=1/ ~ 3x108 m/s in vacuum

Light polarisation = polarisation of the electrical field

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EM Spectum refresher

Visible spectrum~400nm (violet) à 700nm (red) Ultraviolet (UV) ~50-400nm, Infrared (IR) ~ 700nm-1mm

Optical « Telecoms » : 850nm, 1300nm, 1550nm

Frequencies : c= 1550nm : ~1015 Hz

Huge potential bandwidth !

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Quantum behaviour Wave-particle duality. Simultaneously a EM wave and a particle =

«photon» Photon Energy E=h

In the visible spectrum, photon energy ~10-19 J Photonic behaviour is rarely visible in the IR, more in the l’UV

Rarely need to allow for quantum nature of light in optics. Exceptions:

Light emission by sources, particularly semi-conductor sources,

LED, Diode Laser ... Detection, photosensitive materials, photodiodes ... In the fundamental limit of «statistical» noise in a detector

(“quantum noise”).

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Polarisation Defined by the electrical field (the field that interacts with matter) In general a propagating wave is separable in x and y

The phase and amplitude relationship between these two components determines

the state of polarisation : Linear (equal amplitude, in phase) Circular (equal amplitude, phase difference of /2, 3/2) Elliptical (unequal amplitude and/or unrelated phase)

In general – light is not perfectly polarised. We talk of the “degree of polarisation”

E z , t =Excos kz− t E

ycoskz− t

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Polarisation : birefringence and dichroics

The interaction between a light wave and matter often depends on polarisation:Reflection – dielectrics can reflect some polarisations preferentiallyDichroics – absorb one polarisation preferentially (polarisers)Birefringents – have different refractive indices (speed of light propagation) for

different polarisations (waveplates).

DichroïqueRéflexion Biréfringent

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Geometrical OpticsLight is an EM wave

wave solution

speed

Spherical, planar ... wavefronts

Huygens’ principle (remission of a wavefront)

Light “rays”

Conceptual construction (ex. Laser “beam”)

Corresponds to the direct of energy flow

Orthogonal to the wavefront

Parallel to the wavevector, k

E x , t =E0cos t−kx

c = 1 /o o = 1 / = c /n

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Light-matter interaction The EM field acts on charges (atoms, electrons) in

matter.

Principally interaction with electrical field

Several models can be used:

− EM – continuity of the E field.

− Fermat - « shortest path »

− Huygens (re-emission

− Stokes - conservation of energy

− Quantum mechanics – conservation of k

In general reflected and transmitted components

Exact behaviour depends on type of matter (metal,

dielectric), on polarisation, wavelength ...

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Reflection and refraction

Reflection:

Refraction: Snel/Decartes law

Angles are measured between the ray and te

normal to the surface.

r=

i

n t s int = n i s in i

i

r

t

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Critical angle and total internal reflection Light ray passes into a lower index medium (eg. glass-to-air)

We increase i

When t > 90o no longer a transmitted wave – all light is reflected

Total internal reflection (TIR). Waveguide – optical fibre

A more complete treatment of light guiding leads to propagation “modes”

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Fresnel coefficients

Ray treatment ignores polarisation

and energy considerations of

reflected and transmitted rays

A more complete EM analysis

leads to the Fresnel Coefficients :

Amplitude coefficients, energy coefficients : |t|2

itti

ii

ttii

ii

itti

tiit

ttii

ttii

θn+θn

θn=t

θn+θn

θn=t

θn+θn

θnθn=r

θn+θn

θnθn=r

coscos

cos2

coscos

cos2

coscos

coscos

coscos

coscos

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Some consequences of Fresnel coefficients

Reflection losses

eg. Normal incidence, air/glass interface

(n=1.0/1.5)

R = 0,2

|r|2 = 0,04 ~ 4% reflected

Multiple interfaces (0,96)n

AR coatings, multilayer ...

Brewster Angle: one polarisation transmitted:

photography (remove reflections)

Laser mirrors – lower losses

i

t=90o , tan

i=n

t/n

i, r∥ = 0

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DiopterRefraction at a spherical, dielectric interface

Snel and basic geometry gives usn

1

lo

n

2

li

=1

R n2si

li

−n

2so

lo

So

SiS

o

lo

li

i

r

n2n

1

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Paraxial Approximation

Consider rays close to the optical axis.

Small angles ().

Hence : and we obtain

When

We can define «focalisation »

lo≈s

oet l

i≈ s

i R

nn=

s

n+

s

n

io

1221

R

nn=

s

n,s

ii

121

Rnn

n=fi

12

2

0sin753

sin753

θ!

θ

!

θ+

!

θθ=θ

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Thin lens approximation

Two surfaces (R1 and R

2), at least one spherical

We assume Thin lens (d << f ) In air (n

i = 1)

This gives

and the “imaging”

equation

1

f= n

l−1

1

R1

−1

R2

1

so

1

si

=1

f

(NB Other types of lenses exist, eg. GRIN lenses: flat surfaces, varying index, n.

nl

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Simple imaging

So Si

f f

Objet Image

Graphical technique:Centre ray : undeviatedParallel ray : through focal pointFocal ray : parallel to axis.

Mathematical technique

1

So

1

Si

=1

f⇒ S

i=

f So

So− f

Transverse magnification

Object before f - real image (eg. Photographic lens)

Object at f - image at infinity (ex. collimation of a light

source)

Object after f - virtual image (eg. Magnifying glass)

MT=

hi

ho

=si

so

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Basic lens arrangements

Collimation 1:1 imaging, “2f” setup

Coherent imaging, “4f” setup

2f 2f

f f2f

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Characteristics of simple (singlet) lenses

Focal length, f

Diameter, D

F number (f#) = f/D

Numerical aperture (N.A.) :

N.A. = sin = D/2f

N.A. et f# are measures of the optical « power » or curvature of a lens

(N.A. Also used for other components, eg. fibre).

D

f

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Resolution and diffraction limit

Resolving power

Distinguish two points of an object : Rayleigh criterion

Transfer function – impulse response

Limited by the aperture (diameter), , of the lens

Diffraction theory gives :

“Diffraction limited”

φ

λf=d 2,44

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Resolution – test targets

Evaluated visibility of a test target.

Examples test targets :

Binary amplitude grating (B&W)

USAF test target

Pros :

Quick measurement

Cons : Disadvantages

Qualitative measurement only - difficult to compare lenses

Tends to over-estimate resolving power (aided by 3rd order harmonics)

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Modulation transfer function (MTF) An object with a sinusoidally varying optical density (grey)

Measure the contrast, C, of the image of the grating

Plot contrast against grating spatial frequency

A quantitative criterion – probably best found so far

Can be used for all imaging systems (2f, at infinity, intermediate, on-

axis, off-axis ...)

C=T

max−T

min

TmaxT

min

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Aberrations Paraxial theory assumes small

True close to optical axis, low curvature lenses, long focal lengths ...

However, remains an approximation

Deviations due to this approximation = « aberrations »

Several types of aberration : (monochromatic ...)

1st order aberrations (3rd order theory, Seidel ...)

spherical

coma

astigmatism

field curvature

distortion

sin ≈ , cos≈1

s in =− 3 /3 !

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Aberrations - 2

Spherical Coma

Off-axis object : different rays

image at different points.

External rays focus at different points

(typically closer to the lens)

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Aberrations – 3

Astigmatism Field curvature

Off-axis object : asymmetry, different

focal lengths in different directions.

In reality, image forms on a sphere –

deviation from true position = “distortion”

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Aberrations - 4

Distortion Magnification, M

T varies with off-axis

Chromatic aberrations Refractive index, n, depends on , n() Different colours refracted by different angles Focus (and so image) at different points

Lateral Colour

For similar reasons, MT varies with

Plan

focale

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Lens shape

Aberrations depend strongly on how a lens is used. For each “conjugate” there is a optimal lens shape. “Optimal” when the rays are deviated equally at each surface. Simple rule : “most curved surface toward the infinite conjugate”

2f (1:1) - Symmetrical Biconvex Infinite conjugate - plano-convex

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Doublets, triplets ... acromats Different types of glass, different refractive indices, n

(eg. “crown” and “flint”, n= 1,51 – 1,72) Aberrations depend on n

(ray deviation too strong or weak for = sin

SOLUTION = a “lens” made of different glasses (« doublet, triplet ...)

(errors of one glass correct those of the other)

H1

H2

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Examples of classic “objectives »

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Advantages of « acromats »

Correction of chromatic aberration (b, r)

Correction of spherical aberration

(improved performance even with

monochromatic light)

Improved performance for varying

conjugates.

Crown glass more durable

Large series production : cost reduction.

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Ray tracing Aberrations eliminated or reduces with two glasses Impossible to eliminate all aberrations – use more than 2 glasses ?

Laws of refraction known – calculate deviation at each surface

Ray path through an optical system can be calculated

The calculation is straightforward (matrices) but long

COMPUTER !!

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Ray tracing - 2 Trace several rays through the optical system (often 1D simulation – use

rotational symmetry).

Gradually build up the image/focal spot by summation.

Can allow for : indices, lens shape, aperture stops, chromatics, dispersion ...

Can optimise (computer chooses) lens shape, indices ...

Can use supplier libraries of standard lenses/glasses

Can calculate/compare expected performance : (MTF), throughput, cost ...

Examples de commercial ray-tracing software : Code V, OSLO, ZEMAX....

Tools for experts !!

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Lecture plan


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GRIN lenses

GRadient d'INdex lens Deflection by variation of refractive index Flat surfaces, no curvature Similar operation to gradient index fibres Optimal index profile is parabolic

nr=nO1−Ar2/2

n(r)

r

n(r)

Continuous deviation Snel’s law and differential calculus

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GRIN lenses – fabricationFabrication

Glass doped to exhange its refractive index

Ag+ or Tl+ ions replace Na+ ions in glass

Refractive index depends on ion concentration

Ion concentration is controlled to control index

“Rod” lenses (individual)

- doped preform and “pulling” (as for fibres)

Linear and planar lenslet arrays

- photolithographic masking followed by doping

Masking

UV

Doping

Ag+

Lenses

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Applications of GRIN lensesApplications

Imaging large areas - photocopiers, scanners, fax

Advantages Good optical quality/cost tradeoff Assembly/alignments greatly simplified - groves et gluing to fibre end

Choice of lens Required length depends on and the application Lens “pitch” = P

L=P /4

Collimation

L= P /2

Imaging – inverted

L=P

Imaging - upright

GRIN lens transformation of Gaussian beams

- Yuan, Riza, Applied Optics Vol 38, p3214, 1999

P =2 A

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Lecture plan


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Gaussian beams

Why important ? We meet them often – naturally occurring :

At the output of a single mode optical fibre Too laser beam Many other wavefronts are close to or tend to Gaussian

A propagating Gaussian beam remains Gaussian : stable

TF(Gauss) = Gauss

A Gaussian beam can be modelled well (analytically) It is a physical solution of Maxwell’s equations cf a spherical or planar wave

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Gaussian beams - definitions

Gaussian profile of the electrical field and irradiance (energy/area)

2220

22

0// weI=rIweE=rE rr

The width, 2w, defines the “waist” of the beam when I(r) falls to 1/e2 of the

maximum (or to 1/e for the electrical field E=1/e E0)

The waist is the only parameter required to characterise the beam profile

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Gaussian beams - propagation Gaussian beam remains Gaussian as it propagates. But it diverges : the waist varies with z.

wz = w0[1 z w

0

2 2]

1/2

R z = z [1w0

2

z 2 ]

Divergence results simply from diffraction - unavoidable A flat wavefront (phase) becomes naturally curved on propagating, the

radius of curvature is given by R(z) The “waist”, w

0 , is the radius where the beam is narrowest

Phase is flat at beam waist: z=0, R(z) infinite.

Asymptote : z∞ , w z z

w0

,

≈ wzz= w

0

, R z ∞ flat

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Gaussian beams - transformation

Gaussian beams do not obey the usual imaging formulae One possible approach « Self’s Law

Input waist = objet, output waist = image

Consequences: Maximum and minimal image distances (not infinity or at the lens) Maximum image distance (collimation) à s

o=f+Z

R not s

o=f

Special case : so/f = s

ii/f = 1 (focus to focus imaging)

1

Soz

R

2 /So− f

1

si

=1

fmagnificationm =

wi

wo

=1

[1−So/ f ]2 zR/ f 2

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Gaussian beams - truncation A true (mathematical) Gaussian beam is of infinite extent

Any practical optical system will truncate the beam

Power loss :

Infinite T, Dt ~ 0, so illumination uniform ... Airy disc pattern

T=1 (Dt= “waist”) 14% of energy lost – allow safety margin ~ 1,5*beam waist

T =D

g

Dt

=Diamètre faisceau Gaussien à 1 /e2

Diamètre de troncature de la lentille

P L = e−2

D t

D g

2

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Conclusion – what you need to remember

• Laws of refraction (Snell/Descartes) and reflection

• Critical angle and normal incidence Fresnel coefficients.

• Laws and schema for simple imaging

• Resolution limits

• Know about the existence of simple lenses, acromats

aberrations and ray tracing ... beyond that is “specialist”.

• GRIN lenses – much used in optical telecoms

• Gaussian beams : slightly unusual behaviour, transformation

– care with truncation.

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References

1.Principals of Optics, M.Born and E.Wolf, Pergamon.

2.The practical application of light, Melles Griot

Catalogue: www.mellesgriot.com

3.http://www.electro-optical.com (Octobre 2005) –

educational site on electro-magnetisme

4.Optics, E. Hecht, Addison-Wesley (Second edition 1984).

veronique moeyaert optical transmission, vi – storage area networks revision : optical...

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