UNIVERSITY OF CRETE

INTER-INSTITUTIONALPOSTGRADUATE STUDY PROGRAMME

«OPTICS & VISION»

Vibrational and Angular Stability

of Optical Systems for Space Applications

Thesis submitted in partial fulfillment of the requirements for the

Master of Science in

Inter-Institutional Postgraduate Study Programme

“Optics and Vision“

IOANNIS DROUGKAKIS

Supervising Committee

Dimitris G. Papazoglou

Assistant Professor, Materials Science and Technology Department, UoC.

Wolf Von Klitzing

Cretan Matter Waves research group leader, IESL-FORTH

Michael I. Taroudakis

Professor, Mathematics and Applied Mathematics Department, UoC

2

ABSTRACT

Stability of optical systems is a critical issue for optical space applications since long

and short term fluctuations in temperature and flight induced vibrations can

dramatically affect their performance. The technical requirements in respect to the

angular positioning, optical path length, positional and beam alignment are at the

limits of current mounting and optical component fabrication technology. A number

of ongoing and proposed space missions, such as STE-QUEST, use fiber-free space-

fiber schemes in order to manipulate laser beams.

In this thesis we have focused on the analysis and the optimization of the

performance of a typical a fiber-free space-fiber optical communication link. The

receiver and transmitter subsystems were composed by a fiber connected to a fiber

coupler. Using a rigorous approach on a simplified optical system we have deduced

analytical formulas for the power coupling as a function of all system parameters and

possible misalignments of the optical components. This allowed us to optimize the

transmitter/receiver optics so that the optical system is less sensitive to

misalignments. The optimized system, composed by commercial optical components,

was numerically tested using a commercial ray tracing analysis software. The

sensitivity to misalignments was thus numerically estimated. The results were in

excellent agreement with our analytical formulation.

This thesis is a part of the “Optical Breadboard Technologies for Complex Space

Missions” project financed by the European Space Agency (ESA) and took place in

IESL-FORTH facilities.

3

ΠΕΡΙΛΗΨΗ

Η σταθερότητα των οπτικών συστημάτων είναι υψίστης σημασίας για τις σχετικές

διαστημικές αποστολές , μιας και οι διακυμάνσεις στην θερμοκρασία και οι δονήσεις

που προκαλούνται κατά τη διάρκεια της πτήσης, μπορούν να επηρεάσουν σημαντικά

την απόδοση τους. Οι τεχνικές προδιαγραφές σχετικά με την ακρίβεια της

τοποθέτησης των υποσυστημάτων, τον οπτικό δρόμο και την ευθυγράμμιση την

δέσμης είναι στα όρια των σύγχρονων τεχνολογιών τοποθέτησης και κατασκευής

οπτικών. Ένας μεγάλος αριθμός διαστημικών αποστολών που είναι ήδη σε εξέλιξη ή

έχουν προταθεί, όπως το STE-QUEST , χρησιμοποιούν διατάξεις στις οποίες δέσμες

λέιζερ μεταδίδονται από οπτική ίνα σε οπτική ίνα με ελεύθερη διάδοση.

Στα πλαίσια αυτής της διπλωματικής εστιάσαμε στην ανάλυση και βελτιστοποίηση

ενός τυπικού συστήματος οπτικής σύνδεσης ίνας – ελεύθερης διάδοσης - ίνας. Τα

υποσυστήματα του πομπού και του δέκτη αποτελούνταν από μια οπτική ίνα

συνδεδεμένη με ένα οπτικό ζεύκτη ινών. Ακολουθώντας μια αναλυτική προσέγγιση

σε ένα απλοποιημένο οπτικό σύστημα αναπτύξαμε αναλυτικές σχέσεις για την

διαπερατότητα ισχύος ως συνάρτηση όλων των οπτο-μηχανικών παραμέτρων και

των πιθανών απευθυγραμμίσεων των οπτικών στοιχείων. Αυτό μας επέτρεψε να

βελτιστοποιήσουμε τα οπτικά του πομπού και του δέκτη έτσι ώστε το συνολικό

οπτικό σύστημα να είναι λιγότερο ευαίσθητο σε απευθυγραμμίσεις. Στην συνέχεια

η οπτική συμπεριφορά του βελτιστοποιημένου οπτικού συστήματος, αποτελούμενο

από εμπορικά διαθέσιμα οπτικά, εξετάστηκε αριθμητικά χρησιμοποιώντας

κατάλληλο λογισμικό ανάλυσης διάδοσης ακτινών. Με αυτό τον τρόπο μετρήθηκε

αριθμητικά η ευαισθησία στις απευθυγραμμίσεις. Τα αποτελέσματα βρέθηκαν σε

απόλυτη συμφωνία με τις αναλυτικές σχέσεις.

Η παραπάνω εργασία αποτελεί μέρος του ερευνητικού προγράμματος “Optical

Breadboard Technologies for Complex Space Missions” που χρηματοδοτήθηκε από

τον Ευρωπαϊκό Οργανισμό Διαστήματος (ESA) πραγματοποιήθηκε στις

εγκαταστάσεις του Ινστιτούτου Ηλεκτρονικής Δομής και Λέιζερ (ΙΗΔΛ) του Ιδρύματος

Τεχνολογίας και Έρευνας (Ι.Τ.Ε)

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Acknowledgements

I would like to thank my supervisor Prof. Dimitris Papazoglou for all the time he has

dedicated to my work over the past one year and for his thorough and constructive

review of this thesis. I am grateful for allowing me to work on a very interesting thesis

topic and for the patience he showed to me. I would also like to thank Dr. Wolf von

Klitzing for giving me the opportunity to be involved in a research group and for the

fruitful exchange of knowledge we had. It is a great learning experience for me to be

part of all the steps needed for constructing an optical setup, from the design phase,

which is part of my diploma thesis, to the implementation of the theoretical findings

in an optical breadboard. Finally, thanks to my family for their enduring

encouragement and understanding throughout this process.

5

TABLE OF CONTENTS

Chapter 1 ....................................................................................................... 7

Introduction................................................................................................... 7

Chapter 2 ....................................................................................................... 9

Optical communication links .......................................................................... 9

2.1 Optical beam propagation ............................................................................................... 9

2.1.1 Gaussian beam modes ............................................................................................... 9

2.1.2 ABCD Matrices ......................................................................................................... 17

2.2 Optical Fibers .................................................................................................................. 22

2.2.1 Operation Principle .................................................................................................. 22

2.2.2 Fiber modes ............................................................................................................. 25

2.2.3 Single-Mode Optical Fibers...................................................................................... 28

2.2.4 APC fibers ................................................................................................................. 29

2.2.5 Coupling of Gaussian beams ................................................................................... 30

2.3Technology Review ......................................................................................................... 33

2.3.1 Fiber Couplers .......................................................................................................... 33

2.3.2 Anti-Reflection Coatings .......................................................................................... 34

Chapter 3 ..................................................................................................... 37

Theoretical analysis of power transmission in communication links ............ 37

3.1 Fiber to fiber link ............................................................................................................ 37

3.1.1 Generic fiber coupling analysis ................................................................................ 37

3.1.2 Fiber to fiber coupling ............................................................................................. 43

3.1.2 Common Misalignments .......................................................................................... 45

3.1.2.1 Longitudinal Displacement ............................................................................... 45

3.1.2.2 Lateral Displacement ........................................................................................ 46

6

3.1.2.3 Tilt of the fibers ................................................................................................. 47

3.2 Fiber free-space to fiber ................................................................................................. 48

3.2.1 Generic fiber coupling analysis ................................................................................ 48

3.2.2 Commonly used configurations, perfectly aligned link ........................................... 51

3.2.2.1 Symmetric 4f system (Distance L equal to two focal lengths) .......................... 54

3.2.2.2 Distance L equal to the Rayleigh range of the collimated beam ...................... 55

3.2.2.3 Distance L equal to two times the Rayleigh range of the collimated beam ..... 56

Chapter 4 ..................................................................................................... 59

Stability Analysis .......................................................................................... 59

4.1 Rigorous Analysis ............................................................................................................ 59

4.1.1Misaligned optical system ........................................................................................ 59

4.1.2 Sensitivity analysis ................................................................................................... 59

4.1.3 Optimization ............................................................................................................ 61

4.1.4 Lens Selection .......................................................................................................... 64

4.1.5 Conclusions .............................................................................................................. 65

4.2 Optical System Analysis using optical raytracing sofware ............................................. 67

4.2.1 Optical simulation software .................................................................................... 67

4.2.2 ZEMAX introduction ................................................................................................ 68

4.2.3 Confirmation, using ZEMAX, of the validity of analytical equations ....................... 72

4.2.4 Key system performances ....................................................................................... 75

4.2.4.1 System sub-components ................................................................................... 75

4.2.4.2 Stability analysis ................................................................................................ 75

Conclusions .................................................................................................. 84

Bibliography ................................................................................................ 85

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Chapter 1

Introduction

Many optical devices used in space applications such as interferometers, telescopes and

microscopes, are based on stable optical benches, which use optical materials such as fused

quartz [1], silicon carbide, and glass ceramics like Zerodur [2, 3, 4]. One of the most important

element of these benches, is fiber-to fiber coupling. A typical fiber to fiber coupling scheme

consists of a transmitter and a receiver coupler and the optical design can be seen in Fig.1.2.

The transmitter part consists of an optical fiber coupled to a collimating lens. The collimated

beam is transmitted to the receiver coupler, which is a symmetric optical system, consisting

of a collecting lens and a fiber. A typical opto-mechanical design for a system like, commonly

used in laboratory environments, can be seen in Fig.1.3. This setup allows the manipulation

of light, such as frequency or intensity modulation, overlapping and splitting using free-space

components with low losses. A number of space missions, such as STE-QUEST, based on laser-

cooled atomic clock PHARAO use this setup for controlling many different laser beams in

frequency and amplitude, in order to cool and manipulate atomic clouds [5, 6, 7, 8]. The need

for accurate and stable breadboards comes from the fact that the beam must be coupled into

single mode optical fiber after having traversed a number of passive optical elements. The

technical requirements in respect to the angular positioning, optical path length, positional

and beam alignment, in order to achieve the high power transmission required, are at the

limits of current mounting and optical component fabrication technology. The demanding

application environment that includes long and short term fluctuations in temperature and

flight induced vibrations occur, imposes additional constrains. Bounding techniques are

crucial in order to achieve stability and reliability of the optical bench and needs to provide a

stable bond, allow precise alignment and cure reasonably fast. Examples of bonding

techniques are hydroxide-catalysis bonding [9,10,11] ,which has been successfully used in the

Gravity Probe-B telescope [12], and adhesive bonding using two component epoxy [13]. The

optical stability of such systems will be studied, using both analytical descriptions and

simulated ray-tracing.

8

Chapter 2 introduces the reader to the technologies used in fiber-to-fiber coupling schemes.

A basic introduction to the component properties and techniques used in application where

high coupling efficiencies are crucial is made.

Chapter 3 provides in-depth theoretical treatment and analysis of our scheme and calculation

of values of interest and merit functions that are used to derive conditions for the stability of

a specific breadboard. A generic fiber coupling analysis, using thin lenses, has been made

describing power transmission of a perturbed Gaussian beam reaching a receiver fiber. This

analysis was applied in a simple transmitter-receiver link by correlating the input beam waist

and possible misalignments with the receiver and transmitter characteristics, obtaining

analytical equations.

Chapter 4 includes ray-tracing simulations for potential misalignments of a breadboard with

the specifications obtained by Chapter 3, using optical design software ZEMAX. We also

evaluate the generic analysis results using the Physical Optics Propagation (POP) function. The

numerical analysis made at this chapter, takes into account optical aberrations induced by

the optical elements and optical materials/surface losses which were not included in the

analysis at the previous chapter.

Fig. 1.2 Simple transmitter-receiver link optical interface

Fig. 1.3 Simple transmitter-receiver link opto-mechanical interface

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Chapter 2

Optical communication links

2.1 Optical beam propagation

2.1.1 Gaussian beam modes

A Gaussian beam is a monochromatic electromagnetic radiation with the characteristic that

the transverse magnetic and electric field amplitudes and intensity profiles are given by the

Gaussian function. This is called the fundamental mode and is the intended output of most

lasers because of its property to be focused into the most concentrated spot. The electric field

amplitude of such a beam is given by equation 2.1.

2 2

2exp( )

x y

w (2.1)

The spot size w used at the above equation is position dependent and in particular its

variations with distance is hyperbolic. The minimum value of w is called the beam waist radius

w0. For such a beam the Gaussian profile is maintained during propagation and is determined

by a single parameter, which is w0. The Gaussian beam is only one solution of the paraxial

Helmholtz equation.

Paraxial approximation

The propagation of harmonic optical waves is described by the Helmholtz equation:

2 2( ) ( , , ) 0, k x y z (2.2)

Where k = 2π/λ is the wavenumber, is the wavelength, and Ψ(x, y, z) is the complex field

amplitude. Let’s assume that the wave is propagating along z axis and describe it as a carrier

wave with a slow changing amplitude u(x,y,z) :

( , , ) ( , , )exp( ),x y z u x y z ikz (2.3)

Substituting this into the wave equation yields the reduced wave equation

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2 2 2 2

2 2 22 0

u u u uik

x y z z

(2.4)

In the paraxial approximation

2 2 2 2 2

2 2 2 2 22 , ,

u u u u u uik

z z z x z y

(2.5)

And we obtain the paraxial equation which is:

2 2

2 2( 2 ) 0ik u

x y z

(2.6)

The general solution to the exact wave equation - which corresponds to a uniform spherical

wave diverging from a source point r0(x0, y0, z0) is given by:

exp[ ( , )]

( ; )( , )

oo

o

ik r ru r r

r r

(2.7)

Where ( ; )ou r r is the field at point r and 2 2 2( , ) ( ) ( ) ( ) o o o or r x x y y z z , and when

we are in the Fresnel approximation, all terms of the power series of ρ higher that the

quadratic are dropped, yielding a “paraxial-spherical wave” solution [14]

2 2

2 2

[( ) ( ) ]1( , , ) exp{ }

2( )

[( ) ( ) ]1exp{ },

( ) 2 ( )

o o

o o

o o

ik x x y yu x y z

z z z z

ik x x y y

R z R z

(2.8)

Where R(z) is the radius of curvature and (xo,yo ,zo) are coordinates of the origin. The problem

with this approximation is that the beam extents to infinity in the transverse direction.

In order to avoid this we can use complex source coordinates and a paraxial beam propagating

in the z direction can be written as:

2 21

( , , ) exp( ( ),( ) 2 ( )

x yu x y z ik

q z q z (2.9)

Where ( ) o oq z q z z is the complex radius

11

We can now expand the term of 2.9 in imaginary and real parts and obtain:

1 1 1

( ) ( ) ( )real imi

q z q z q z (2.10)

And by replacing it to equation 2.9 we get:

2 2 2 2

2 2 2 2

2

1 ( ) ( )( , , ) exp( )

( ) 2 ( ) 2 ( )

1 ( ) ( )exp( )

( ) 2 ( ) ( )

real im

ik x y ik x yu x y z

q z q z q z

ik x y i x y

q z R z w z

(2.11)

Where R(z) is the radius of curvature and w(z) is the beam radius such that

2

2

1 1

( ) ( ) ( )

Mi

q z R z w z (2.11a)

It is important to notice that not all beams are perfectly Gaussians. A very important

parameter is introduced in the above equation that is the M2. In the case of a Gaussian beam

profile M2 = 1 and its value increases as the beam profile deviates further from a Gaussian.

Fig 2.1 represents the comparison of the beam profile with M2 = 2 with respect to the ideal

case with M2 = 1.

Fig.2.1 Transverse electric field distribution of two Gaussian beam with M2 equal to 1 and 2

This particular solution is widespread as it is an eigenmode of free space, [15] it remains

Gaussian upon propagation through optical systems, [16] it is the lowest order resonant mode

12

in spherical mirror cavities and the output mode of most laser systems and single mode fibers

is Gaussian. This mode is given by the following equation:

2 2 2 2

2 2

2 ( )( , , ) exp( ( ( )))

( ) ( ) 2 ( )

x y ik x yx y z i kz G z

w z w z R z (2.12)

Where,

2

2

( ) [1 ( ) ],

( ) 1 ( ) ,

( ) arctan( )

R

o

R

R

zR z z

z

zw z w

z

zG z

z (2.13)

And 2

R

wz is a characteristic length scale for Gaussian beams

The Gaussian beam can be characterized at all points along its propagation axis by two

parameters – the radius of curvature of its wavefront R(z) and the radius w(z) at which the

beam intensity drops to Imax/e2, often called the spot size. The minimum value of the spot

size, [17] known as the waist, is denoted by w0.

A linear increase in w occurs in the far-field region [18]. The angle of divergence at z = ∞ is

given by:

1 1( )lim[tan ( )] tan ( )

zo o

w z

z w w (2.14)

Fig.2.2 and Fig.2.3 shows the dependency of the spot size and the radius of curvature of the

wavefront from distance z. As we can see at the waist the wavefront is flat (R->infinite) and

at one Rayleigh range away from the waist it achieves its minimum radius of 2zR.For distances

beyond the Rayleigh range the wavefront curvature approximates that of a spherical wave.

We can also notice that for 1 Rayleigh range distance the spot size become √2 the waist.

13

Fig2.2 Spot size as a function of distance from the waist

Fig2.3 Radius of curvature as a function of distance from the waist

The overall Gouy phase shift propagating from far field to far field through a focus is π as

shown in Fig.2.4. For higher-order transverse modes, the Gouy phase shift is stronger. For

TEMnm modes, for example, it is stronger by the factor 1 + n +m.

https://www.rp-photonics.com/higher_order_modes.html

14

Fig2.4 Gouy phase as a function of distance from the waist

The profile of a Gaussian beam does not necessarily have to be a bell-shaped Gaussian

function with only one maximum. There are other solutions for the radial field distribution of

a beam with the same Q parameter. Two commonly used, orthogonal, mode sets are

Gaussian-Hermite functions (which are expressed in rectangular coordinates), and Gaussian-

Laguerre functions (which are in cylindrical coordinates). The 2D Gaussian-Hermite modes are

given by:

1 2 2

, 2 2

2 2

2 2 2( , , , ) ( ) ( )exp( )

! !

( )exp( )exp( )exp( ( 1))

m n

m n o m n

x y x yw x y z H H

w m n w w w

i x yikz iG m n

R (2.15)

Since Hermitian polynomial of the lowest order H0 equals 1, it can be observed that, for the

indices m = 0 and n = 0, the complex envelope turns into a Gaussian distribution. This case is

usually referred to as the fundamental mode, or the TEM00 mode where, TEM stands for

Transverse Electric and Magnetic field. The first term of equation 2.14 normalizes the power

of each mode and the second term gives the form of the modes Hm and Hn. The third term

gives the Gaussian amplitude profile, while the other terms gives the phase due to

propagation by distance z, the phase due to the mismatch of the flat reference plane and the

equiphase front of the beam with radius of curvature R and the phase that is dependent on

mode number, called the “Gouy phase” G. Equation 2.13 can be separated into x and y

components as:

, ( , , , ) ( , , , ) ( , , , ) m n o m o n ow x y z w x y z w x y z (2.16)

15

Where

1

2 22

2

2 2 (2 1)( , , , ) ( )exp( )

2!

n

m o m

x x i x G mw x y z H ikz i

w w Rwn ,

1

2 22

2

2 2 (2 1)( , , , ) ( )exp( )

2!

n

n o n

y y i y G nw x y z H ikz i

w w Rwn

The Gaussian-Hermite functions provide a complete basis set of orthogonal functions, which

obey the orthonormality condition and does depend from z

,( , , , ) ( , , , )

m o n o m nw x y z w x y z dx (2.17)

Any arbitrary paraxial beam E(x,y,z) can be expanded in the form

, ,( , , ) ( , , , ) n m n m o

m n

E x y z A w z y z , (2.18)

Where An,m are the mode coefficients given by:

*, ,( , , ) ( , , , )

n m n m oA E x y z w x y z dxdy (2.19)

Fig 2.5 shows the first 10 Hermite modes in one dimension. The modes are indexed 0, 1, 2,. . . ,

where 0 is the fundamental. The mode number is equal to the number of zero crossings of

the function. The product of the one dimensional field in the x and y directions gives us the

2dimensional mode shown in Fig.2.6.

16

Fig2.5 The first 10 Hermite polynomials

Fig.2.6 Intensity profiles of low order Hermite modes.

Feeds which have circular symmetry can be reconstructed with fewer modes if the modes

have the same circular symmetry. On way to reconstruct them is Gaussian-Laguerre

polynomial which is described by

22 2

2 2

2 ( )2 ! 2

( )exp( )( )!

n

n

mn m n

r

m rn L inm n w w w

(2.20)

where Lm|n| is the Laguerre polynomial of index m and n, and r and θ are the polar

coordinates.

17

Fig.2.7 Intensity profiles of low order sinusoidal Laguerre-Gaussian modes

Many other basis sets to describe a field exist [19] and in order to choose the most

appropriate we must take into account the nature of the field and the surfaces with which it

interacts.

2.1.2 ABCD Matrices

An ABCD matrix [20] is a ray transfer matrix which describes the effect of an optical element

on a laser beam. It can be used both in geometrical optics and for propagating Gaussian

beams. ABCD matrix theory we have to accept the paraxial approximation. Multiplying this

matrix with a vector representing the light ray we can trace the light path through the optical

system:

2 1

2 2 1 1

.y yA B

n nC D

(2.21)

where y and θ refer to transverse displacement and offset angle from the optical axis

respectively and n is the refractive index. The subscripts ‘1’ and ‘2’ denote the coordinates

before and after an optical element.

Transformation matrices for various system elements are the following:

18

Free space propagation 1

0 1

LM ,

where L is the propagation distance

Curved interface 2 1

1 0

1M n n

R

,

where n1 , n2 are the refractive index before and after respectively and R>0 if concave to the

left

Thin lens 1 0

11

M

f

,

where f is the focal length of the lens

Thick lens 2 1

1

2

1 0 1 01

1 10 1

1

1tot

DM

P P

DP D

P DP

,

where L

dD d

n , 1

1

L sn nPR

,

'2

2

s Ln nPR

, 1 2 1 2totP P P DPP , Ln , sn , 'sn are the refractive

indices of the lens material, the material before and after the lens respectively and R1, R2 are

the radiuses of curvature of first and second surface, as shown in the Fig.2.8

Fig.2.8 Thick lens

19

ABCD matrices can be used to calculate w and R as a Gaussian beam propagates through an

optical system and are the same for all modes. If the effects of truncation can be ignored (D >

4w), then a beam can be reconstructed at any plane in the system using the initial mode

coefficients and the appropriate values of w, φ and R. The modification of the q parameter by

an optical element can be expressed in terms of the elements of the ABCD matrix.

121

Aq Bq

Cq D (2.22)

Or in a more convenient way

1

2 1

/1

/

C D q

q A B q (2.23)

where q1 and q2 represent the value of the q parameter before and after the optical element,

respectively.

As we can see, using equation 2.22 and 2.23 we can obtain the spot size and the radius of

curvature of the modified beam exiting the optical system described by the ABCD matix. If

the system has a sequence of optical elements, which are described by matrices M1,M2,…,Mn

this is equivalent to a single optical element of the transmission matrix: M = Mn…M2·M1.

Fig.2.9 Sequence of optical elements

As an example of an optical system we can analyze the symmetrical optical system shown in

Fig.2.10. It is composed from 2 thin lenses of the same focal length f which are place at L

distance on from each other and one focal length away from the input and output plane

respectively. The sub-matrices that combine the total transfer matrix are are shown in Fig

2.10 as M1,M2… and the total transfer matrix is given by equation 2.24:

20

5 4 3 2 1

1 0 1 01 1 1

1 11 10 1 0 1 0 1

M M M M M M

f L f

f f

(2.24)

Fig.2.10 Symmetrical optical system

Equation 2.25 can be further simplified to

2

1 0

21

M f L

f

(2.25)

Using equation 2.22 and replacing in Rinq iz we can obtain the q parameter of the beam at

the output plane which is given by:

2

2 2

in

Rout in in

R R

f zq

if f z L z

(2.26)

The spot size and the radius of curvature at the output plane can be estimated from (2.11a)

from 1/qout

2 21 2

( )

output in

Li

q f f w (2.27)

21

Thus the spot size at the output plane is, independently of the lens distance L, equal to the

input’s beam waist while the output radius of curvature depends on the characteristics of the

system:

2 1

2 2 /

out in

outout

w w

RfR

f L f L f

.

We can see in Fig.2.11 that for zero distance L, the normalized radius of curvature is 0.5, while

it reaches infinity (flat wavefront) for 2L f . When we are moving to larger distances the

radius becomes negative approaching zero for significant large distance ( 610L

f ).

Fig.2.11 Radius of curvature as a function of distance L normalized in focal lengths of the

lens.

22

2.2 Optical Fibers

2.2.1 Operation Principle

Optical fibers are cylindrical waveguides that confine or guide electromagnetic radiation in

the optical frequencies and have been studied intensively in bibliography. The most basic

optical fiber design is the step-index fiber in which the index distribution of the materials that

compose the fiber resembles a step function and can be seen in Fig.2.12. The core and the

cladding are composed of dielectric or insulating materials such that the core has a higher

index of refraction than the cladding. Materials usually used are silica (SiO2) and silica that

has been doped to slightly increase or decrease the index. The jacket strengthens the fiber

and protects it from breaking. Optical fibers have several advantages over traditional copper

wires because they are lighter and smaller, have increased bandwidth, and eliminate

electromagnetic interference. In addition, because of advances in the production of low loss

glass, optical fibers are a viable solution for long distance communications. The major

application for optical fibers is in telecommunications, though use in local networks and short

distance deployments are also common.

Fig.2.12 A standard step-index fiber. An end profile appears to the left and a side view to the

right.

One important parameter that defines the angular spread of light emerging from a fiber is the

numerical aperture (NA). It also defines the acceptance cone of what incoming light that can

be guided due to total internal reflection in the fiber.

The numerical aperture for an arbitrary optical fiber is defined by

23

sin s aNA n , (2.28)

where ns is the index of refraction for the surrounding medium. For a step-index fiber we can

obtain:

2 2 core claddingNA n n (2.29)

We can notice that a small difference in refractive index between core and cladding results in

a small numerical aperture and also a small acceptance angle.

Fig.2.13 Schematic of the acceptance angle of an optical fiber

An optical fiber generally consists of two coaxial layers in cylindrical form: a core in the central

part of the fiber and a cladding in the peripheral part that completely surrounds the core.

Although the cladding is not required for light propagation in principle, it plays important roles

in practical use, such as protecting the core surface from imperfections and refractive index

changes caused by physical contact or contaminant absorption, and enhancement of the

mechanical strength. The core has a slightly higher refractive index than the cladding.

Therefore, when the incident angle of the light input to the core is greater than the critical

angle determined by Snell’s law, the input light is confined to the core region and propagates

a long distance through the fiber because the light is repeatedly reflected back into the core

region by total internal reflection at the core–cladding interface. The critical angle is given by:

1 1sin ( ) c

o

n

n, (2.30)

where no,n1 are the core and the cladding refractive indices respectively

The propagation of light along the fiber can be described in terms of electromagnetic waves

called modes, which are patterns of electromagnetic field distributions. The fiber can guide a

certain discrete number of modes that must satisfy the electric and magnetic field boundary

24

conditions at the core–cladding interface according to its material and structure and the light

wavelength. Optical fibers can be commonly classified into two types: single mode fibers

(SMFs) and multimode fibers (MMFs) [21]. An SMF allows only one propagating mode,

whereas a MMF can guide a large number of modes. Depending on the construction method

SMFs and MMFs are again divided into two types: step-index (SI) and graded-index (GI) fibers.

The SI fiber has a constant refractive index in the entire core. The refractive index changes

abruptly stepwise at the core–cladding boundary. The GI fiber has a nearly parabolic

refractive index distribution. The refractive index decreases gradually as a function of the

radial distance from the core center. Fig 2.14 conceptually illustrates the refractive index

profiles and ray trajectories in SMF and in SI and GI MMFs. The most significant difference

among these types of fibers is modal dispersion. When an optical pulse is input into an MMF,

the optical power of the pulse is generally distributed to a large number of the modes in the

fiber. Different modes travel at different propagation speeds along the fiber, which means

that different modes launched at the same time reach the output end of the fiber at different

times. Therefore, the input pulse broadens in time as it travels along the MMF. This pulse

broadening effect, well known as modal dispersion, is significantly observed in SI MMFs. As

shown in Fig 2.8, different rays travel along paths with different lengths; here each distinct

ray can be thought of as a mode in a simple interpretation. The rays travel at the same velocity

along their optical paths because of the constant refractive index throughout the core region

in an SI MMF. Consequently, the same velocity and different path lengths result in different

propagation speeds along the fibre, which causes a wide pulse spread in time. The pulse

broadening caused by modal dispersion seriously limits the transmission capacity of MMFs

because overlapping of the broadened pulses induces inter-symbol interference and disrupts

correct signal detection, thereby increasing the bit error rate. The arrival of different

components of the signal at different times distorts the shape of the output beam resulting in power

losses.

25

Fig.2.14 Refractive index profiles and ray trajectories in (a) SI SMF, (b) SI, and (c) GI MMFs

[22].

2.2.2 Fiber modes

Description of optical fibers can be done either by geometrical optics, or more accurately in

the context of guided-wave optics. For waveguides such as optical fibers which exhibit a small

change in refractive index at the boundaries, the electric field can be well described by a scalar

wave equation in cylindrical coordinates:

2 2 2( , , ) ( ) ( , , ) 0 or z k r r r z (2.31

the solutions of which are the modes of the fiber. Ψ(r, θ, z) is generally assumed to be

separable in the variables of the cylindrical coordinate system of the fiber:

( , , ) ( ) ( ) ( ) r z R r Z z (2.32)

Which results in the following equation for the radial part of the scalar field:

26

2 2

2 2 2

2 2

1( ( ) ) 0 o

d R dR mk n r R

dr r dr r (2.33)

in which m denotes the azimuthal mode number, and β is the propagation constant. The

solutions must obey the necessary continuity conditions at the core-cladding boundary. In

addition, guided modes must decay to zero outside the core region. These solutions are

readily found for fibers having uniform, cylindrically symmetric regions but require numerical

methods for fibers lacking cylindrical symmetry or having an arbitrary index gradient. A

common form of the latter is the so-called α-profile in which the refractive index exhibits the

radial gradient[23].

1

1 2

[1 ( ) ],( )

(1 ) ,

arn r am r a

n n r a

(2.34)

Analytic field solutions are possible for the step-index fibre of circular symmetry. For this case,

the radial dependence of the refractive index is the step function:

1

2

,( )

,

n r an r

n r a

(2.35)

In Fig.2.15 we can see the solution to this and are the Bessel functions. The zero order has a

maximum at zero while higher orders have more zero crossings.

Fig.2.15 Bessel functions Jm(ρ) for m = 0, 1, and 2.

27

The solution in the propagation region ra (cladding) are given respectively

by the equations:

2 2 2

1

2 2 2

1

( )o o

o o

rJ n k

aR r

J n k

(2.36)

2 2 2

1

2 2 2

1

( )o o

o o

rK n k

aR r

K n k

(2.37)

The number of modes in an optical fiber is related to the V number of the fiber and is given

by

2a

V NA

(2.38)

Where a is the core radius, is the vacuum wavelength and NA is the numerical aperture of

the fiber. The parameter is just a convenient definition in the mathematical treatment and

does not have a physical meaning, but is useful when determining the number of modes in

an optical fiber. We can see in Fig 2.16 the number of modes for an optical fiber as a function

of this parameter.

Fig.2.16 Total number of modes M as a function of the fiber parameter V [24]

28

2.2.3 Single-Mode Optical Fibers

Using equation 2.36 if V

29

2.2.4 APC fibers

In some cases, it is important to have a cleaved fiber surface just perpendicular to the fiber

axis. For example, this is often the case when a fiber is inserted into a fiber, although some

connectors require angle cleaves. Mechanical splices also work better with perpendicular

ends. Note that due to refraction at the fiber end, a non-normal cleave causes a deviation of

the output beam direction from the fiber axis as we can see in Fig.2.18. Especially for an APC

fiber end the beam has a propagation axis tilt angle 4 degrees in relation to the mechanical

axis, while the mechanical angle of the cleave is 8 degrees. Also, one then requires an

appropriately tilted input beam for efficient launching and this is making the use of angle

cleaves somewhat inconvenient [25].

The cleave angle also has a big advantage in relation to the perpendicular fiber end and this

is related to the back-reflected light. At the case the fiber has no cleave angle, light reflected

at the output surface from Fresnel reflection due to the index difference to air, will essentially

travel backward in the fiber core. The Fresnel reflection is given by equation 2.39 and for a

typical fiber end is about 4% of the total power. For APC fibers, however, the reflected light,

due to the tilt, is not a fiber mode and thus cannot back-propagate to the source so it is

absorbed in the cladding. This means that there is a very large return loss despite a significant

reflection, which for a normal cleave would cause a much smaller return loss.

21 0

1 0

( )n n

Rn n

(2.39)

Fig.2.18 APC fiber end

https://www.rp-photonics.com/fresnel_equations.htmlhttps://www.rp-photonics.com/return_loss.html

30

2.2.5 Coupling of Gaussian beams

A central issue in designing and setting up a Gaussian beam transfer line is the sensitivity of

the system to misalignment [26]. In principle there are three different kinds of misalignment:

Both beams share the same optical axis, but their spot sizes w or radius of curvature

R are mismatched, they are the so–called axially aligned beams.

The optical axis of both beams is laterally displaced.

The optical axis of both beams is tilted with respect to each other.

As described in the previous sections only specific modes, the eigenmodes ( , )i

R x y , can

propagate in a fiber. In the case of a single mode fiber this reduces to only one eigenmode,

typically Gaussian. Thus, the propagation of an arbitrary distribution ( , )A x y is

decomposed to the propagation of the linear combination of the eigenmodes that synthesize

it:

( , ) ( , )iA i R

i

x y a x y

, where i are projection coefficients that can be estimated from overlap integrals:

*( , ) ( , )ii A Ra x y x y dxdy

Another useful metric is the power coupling of the arbitrary ( , )A x y , i.e. the percentage of

power that can actually be transmitted, though such a system. In order to estimate the power

coupling (power transmission) we use the normalized projection coefficients i given by:

*

* *

( , ) ( , )

( , ) ( , ) ( , ) ( , )

i

A Ri

tri i

A A R R

x y x y dxdyC

x y x y dxdy x y x y dxdy

(2.40)

The power coupling (in percentage) in this case is given by:

2

i

tr tr

i

P C (2.41)

31

Let’s see some examples of this projection for simple single mode, one dimensional, systems

with a single Gaussian eigenmode ( )R x . In the following examples the eigenmode is a

Gaussian beam with infinite radius of curvature and waist size w. As shown in in Fig.2.19 the

input beam ( ) ( 2)A Rx x is shifted along the x axis replica of the eigenmode.. The

fraction of power coupled from one beam to the other is the common area of the two curves.

If we use equation 2.41 in order to calculate power coupling we obtain a ~3% coupling. The

low value is well anticipated because we can clearly see at the plot that the common area is

only a small fraction of the total area of the curves.

Fig2.19 Gaussian beams projection. The input Gaussian beam is displaced along the x-axis.

For the case that the two beams have different waist but again infinite radius of curvature we

obtain Fig.2.20. As shown in in Fig.2.20 the input beam ( ) ( )1.5

A R

xx is a scaled replica

of the eigenmode, with 1.5 times larger waist. As we can see the common area this time is

larger than the previous case and equation 2.41 results in 42% power coupling.

32

Fig2.20 Gaussian beams projection. The input Gaussian beam has different waist

Let’s assume now the case that the two beams have infinite radius of curvature, w spot size

and the optical axis of the input beam is tilted at an angle γ with respect to the reference. The

input beam is then described by ( ) ( )exp[ sin ]A Rx x ikx where k is the wave

number for a wavelength λ and γ is the angle. We can see there is an extra phase term

depending on the tilt of the beam. If we use equation 2.41 for a tilt angle 50 mrad and

wavenumber 8 106 rad/m we obtain 80% power coupling.

33

2.3Technology Review

2.3.1 Fiber Couplers

Fiber couplers, are sub-components at a fiber-to fiber coupling scheme, that consists of a fiber

and lens and are used either for the collimation or the coupling of a beam into a fiber. There

is a large range of applications at laboratory environments that need to either collimate or

gather a beam and use commercial fiber couplers for this purpose. As we can see in Fig.2.21

these coupler are mainly made of metal and are offered with an option of adaptable distance

between the lens and the fiber. They are mainly used with fiber connectors that typical have

more relaxed specification with 2dB insertion and 50 dB return losses in comparison to the

space-qualified which have smaller than 0.2dB and larger than 75dB respectively.

Fig.2.21 Typical commercial fiber coupler

As mentioned before, fiber coupling and collimation requires highly accurate alignment. The

positional and angular tolerances of the fiber and the lens are extremely tight in order to

achieve the high power transmission required at space applications. This strict requirements

lead to an increase to the complexity of the fiber couplers in terms of manufacturing and

assembling their parts. Also they need to withstand harsh environments, including high

dosages of radiation, extreme temperatures and low outgassing in vacuum conditions.

The mechanical design of these devices needs to comply with extremely tight tolerances on

the sub-components level and also addition of moving parts for the proper alignment. Typical

values for the main tolerances of the sub-components are surface polish accuracies smaller

than λ/20, angular tolerances smaller than 2 arcsec and CTE

34

stability and to minimize short and long term driven effects.. The invar screws located at the

side, are used for the proper lateral positioning of the fiber with respect to the lens. The

assembling and bonding technique used for this design are quite complicated imposing

difficulties to the constructing laboratory.

Fig.2.22 Fiber coupler design to be used in space applications [7]

2.3.2 Anti-Reflection Coatings

It is well known from electro-dynamics that at the boundary between two media which have

different refractive indices, reflection is observed. These losses for glass and air, are typically

4% for normal incidence of light. These losses decrease the efficiency and in imaging systems,

such as telescopes, reduce the image contrast [27]. Furthermore, in high power transmission

applications such as fiber-to fiber coupling these reflections need to be minimized. For a

simple transmitter-receiver link there are six glass-air boundaries, including the fiber ends and

the lenses. This leads us to about 22% reflection losses which is not acceptable in space

applications, where power consumption and photon economy is essential. In order to avoid

these reflections anti-reflection (AR) coatings are used and typical values can be seen in Table

2.1.

The simplest anti-reflection coating, is the so called single-layer anti-reflection coating. This

thin-film coating, as shown in Fig.2.22, is designed for normal incidence and consists of a

35

single quarter-wave layer, 0

34

n, of a material whose refractive index is close to the geometric

mean value of the refractive indices of the two adjacent media 3 1 2n n n [24]. The two

reflections of equal magnitude from each of the two interfaces, cancel each other by

destructive interference. Disadvantages of this method are the limited operation bandwidth

and the difficulty in finding coating materials with suitable refractive index [28].

Fig.2.23 Schematic of single-layer anti-reflection coating (AR)

Using multilayer, numerically designed, arrangements anti-reflective coatings are fabricated

for a broad wavelength range. . Multilayer designs trade a low residual reflectivity with a

large bandwidth. V coatings have a high performance only in a narrow bandwidth, whereas

broadband coatings offer moderate performance but in a wide wavelength range [29].

Fig.2.24 shows reflectivity as a function of wavelength for an optimized anti-reflection coating

on BK7 using two layer pairs of TiO2 and SiO2.

Fig.2.24 Reflectivity as a function of wavelength for a numerically optimized anti-reflection

coating for 532nm and 1064nm [29].

https://www.rp-photonics.com/bandwidth.html

36

AR-coating type Reflectivity (%)

Without coating 4

Single Layer

37

Chapter 3

Theoretical analysis of power transmission in communication links

The key parameter of a telecom optical link is the transmissivity of the system, i.e. the fraction

of the power transmitted from fiber coupler to fiber coupler. In the following analysis we are

going to use Gaussian beam optics, in order to evaluate the system behavior based on the

sub-components that are used. Chapter 3 analyzes a generic fiber coupling scenario and

shows the dependency of transmissivity from the characteristics of an input beam. The results

are then applied to a simple fiber to fiber link and to a specific telecommunication showing

how potential misalignments can affect the system performance in terms of losses in power

transmission.

3.1 Fiber to fiber link

3.1.1 Generic fiber coupling analysis

The variations, caused by misalignments, in the transmissivity of an optical link consisting of

a receiver and a transmitter part can be analyzed theoretically in a generic way by

concentrating on the receiver part. Our focus on small misalignments enables us to analyze

the transmissivity on a simplified receiver optical system consisting of a thin collecting lens

and the receiver fiber as shown in Fig. 3.1.

Fig. 3.1 Schematic of a perturbed Gaussian beam reaching a receiver fiber, f is the receiver lens focal length, γ is the misalignment angle.

In this simplified system, the misaligned Gaussian beam is being collected by a thin lens, of

focal length f, and then couples in the receiver fiber at angle γ, with a displacement given by

38

f tanγ. The Gaussian beam that would optimally lead to maximum transmissivity is the primary

optical mode 0( , )x y of the receiver fiber which we refer to as reference mode. It is actually

the mode (Gaussian beam) that the receiver fiber would emit if we reversed the light

propagation direction (i.e. receiver becomes transmitter and vice versa). The amplitude

distribution of the reference Gaussian mode 0( , )x y of the fiber and of the perturbed input

Gaussian mode on the receiver fiber entrance, ( , )input x y , can be described as:

2 2 2 2

0 2

1( , ) [ ] [ i ]

r r r

x y x yx y Exp Exp

w w R

2 2

2

2 2

1 ( tan )( , ) [ ]

( tan )[ i ] [ ik sin ]

input

x f yx y Exp

w w

x f yExp Exp x

R

(3.0a)

where w and wr are the input and reference beam waists respectively, R and Rr are the input

and reference radii of curvature, f is the focal length of the lens and γ is the misalignment

angle.

In order to estimate the power coupling (power transmission) of a perturbed Gaussian beam,

with amplitude distribution Ψinput (x,y), onto the receiver fiber, we use its normalized overlap

integral Ctr to the amplitude distribution of the reference Gaussian mode 0( , )x y of the

fiber. This normalized overlap factor is given by:

*

0

* *

0 0

( , ) ( , )

( , ) ( , ) ( , ) ( , )

input

tr

input input

x y x y dxdyC

x y x y dxdy x y x y dxdy

(3.0b)

The power coupling is then simply given by:

2

tr trP C (3.0c)

39

Normalized parameters

Typical values Description

r

RR

w ±100 →

Radius of curvature of the input beam (receiver fiber plane)

rr

r

RR

w

Radius of curvature of the reference Gaussian mode

r

ww

w 1

Waist size of the input beam (receiver fiber plane)

dd

r

ww

w 230

Waist size of the collimated beam (optical link)

r

LL

w

250111111

2.25

mm

m Distance between the lenses

r

zz

w 1

Longitudinal displacement of the receiver fiber tip from the focal plane

of the receiver lens

rw

0.7800.35

2.25

m

m Wavelength

r

ff

w 2000

Focal length of the transmitter /receiver lenses

γ (μrad) Misalignment angle γ

Table 3.1 Table of normalized parameters (with respect to the reference beam waist)

(schematic reference to these values is given in Figs. 3.1 and 3.6)

In order to obtain generic, scalable analytic results we normalize all length units over the

beam waist wr of the reference Gaussian mode of the receiver fiber. The normalized

parameters are summarized in Table 3.1 (for our specific application the distance L and the

wavelength λ are set respectively to 250 mm and 780 nm).

The power coupling, after normalization, of the misaligned optical system shown in Fig. 3.1 is given by:

2 2 2 2 2 22

2

2 2

4 1)cos

cos2 2

4(trr

w w tan wP Ex w

A A

w

B BR

pR

(3.1)

where:

40

2

2 4 2 2 2 2

2

2

,

1 2( ) ,

1 1( ) (1 )

1 ,

( )

r

r

f zB

f

A w wR R

R

z

w

No misalignment (γ = 0)

Equation 3.1 is greatly simplified in the absence of misalignment, i.e. γ = 0. In this case, the

exponential term vanishes and the power coupling can be written as:

2 2

2 2 4 2 2 2

4

1 1( ) (1 )

tr

r

wP

w wR R

(3.1a)

For the case that the input beam enters the fiber at its waist (R-> ) we can further simplify

the above equation obtaining:

2 2

2 2 2

4

(1 )tr

wP

w

(3.1b)

This equation shows the dependency of transmissivity from the input waist entering the

receiver fiber. We must notice that the curve is not symmetric around its optimum position,

which is when the waist size entering the fiber has the same value with the reference beam

waist. At this point power transmission is maximum reaching 1 as it can be seen in Fig.3.2. In

order to have 15% power losses the entering beam’s waist must be 0.7 times the reference

but requires a much bigger raise for the waist, 1.6 the reference, in order to obtain the same

result. At the inset, that shows behaviour for a larger range of waist sizes, we can notice that

a larger input beam by a factor of 2.4 results in lossing half of the trasmitted energy.

41

Fig.3.2. Power Coupling as a function of normalized waist size, assuming infinite radius of

curvature for input-reference (inset shows behavior for a larger range of waists)

In the case that the input beam has the same the spot size to the reference , 1w , but a

radius of curvature R, but equation 3.1a is simplified to equation:

2 2

2 2 2

4

2

tr

RP

R (3.1c)

We can see in Fig.3.3 that our system does not show the same sensitivity to radius of

curvature as to waist size. When radius is bigger than ten times the waist of the reference

mode we can obtain power transmission larger than 85%. Also we can notice that for larger

normalized radius than 100 power transmission become maximum.

Fig.3.3. Power coupling as a function of normalized input beam’s radius of curvature (inset

shows behavior for a larger range of waists)

42

Misaligned beam on the receiver fiber (γ 0)In the case of an input beam that enters the

receiver fiber in an angle γ,as we can see in Fig.3.4, we have only the effect beam tilt without

any displacement. In this case, assuming infinite radius of curvature and same spot sizes for

input-reference, the dependency of power transmission from the angle is described by

equation 3.1d.

2 2

2

sin

=e

trP (3.1d)

Fig. 3.4 Schematic of a perturbed Gaussian beam reaching a receiver fiber without displacement, f is the receiver lens focal length, γ is the

misalignment angle.

As we can see in Fig.3.5 the effect of a tilt of the beam is not as strong compared to when

both tilt and displacement occurs. For angles up to 45 mrad power coupling is above 85%.

Fig.3.5. Power Coupling as a function of input beam angle γ (inset shows behavior for a

larger range of angles)

43

Fig.3.6 shows how sensitive the power transmission is concerning the input beam’s tilt angle

for different focal lengths of the lens. Comparing with the previous figure, we can notice that

an angle of about only 2 mrad is enough to lose all the power transmission. Another point

worth mentioning is that the bigger the focal length of the lens used, the more sensitive the

system is to inducted misalignment. For example a tilt of about 0.8 mrad result in 50% power

coupling for the smaller lens while the other two have respectively reached 30% and 5%. We

can draw the same conclusion from Fig.3.7, where we can clearly see that for the tilt of the

beam power coupling is decreasing increasing the focal length of the lens.

Fig. 3.6 Power coupling as a function of input beam’s tilt angle γ for different focal length of

the lens.

Fig. 3.7 Power coupling as a function of focal length of the lens for different beam tilt angles

γ

3.1.2 Fiber to fiber coupling

44

Single mode fibres are frequently used in telecom and other applications. A common scheme

is the direct fiber to fiber link, where the two fibers are opposite to each other. Perturbations

can occur at a system like that and the generic analysis presented in the previous section is

used to evaluate the effect of misalignments to the system’s power transmission. For the

general case that is shown in Fig.3.8 that a number of misalignments occur the equations

describing the two Gaussian modes and power transmission at the receiver fiber are the

following:

Fig3.8 Schematic of a perturbed fiber to fiber coupling scheme, a and z are the lateral and longitudinal displacement respectively and γ is the tilt of the

fiber.

2 2 2 2

0 2

1( , ) [ ] [ i ]

r r r

x y x yx y Exp Exp

w w R

2 2

2

2 2

1 ( )( , ) [ ]

( ) 2[ i ] [ i sin ]

input

x a yx y Exp

w w

x a yExp Exp x

R

(3.1e)

2 22 2 2 2

2 22

1 1( ) sin ( ) sin

4}r rtr

w wB B

Exp

wR R R Rw

PA A

(3.1f)

2 4 2 2 2 2

2

1 1( ) (1 )

1

,

2 ,

,

r

A w wR R

w

w

B a

45

, where a is the normalized over the waist lateral displacement and is the tilt angle of the

transmitter fiber with respect to the optical axis of the receiver fiber. In the following analysis

the radius of curvature, rR , of the receiver fiber will be considered infinite.

3.1.2 Common Misalignments

3.1.2.1 Longitudinal Displacement

Fig.3.11 Longitudinal displacement

A common misalignment is the longitudinal displacement of the two fibers. As we know when

a Gaussian beam propagates in space the spot size and the radius of curvature are a function

of the distance z from the waist. This misalignment results in a mode mismatch reducing

power transmission. In this case equation 3.1f reduces to:

2 2

2 42 2 2

2

4

(1 )

tr

wP

ww

R (3.1g)

If we replace w and R with their values at a distance z from the waist we obtain the following

equation describing power losses due to longitudinal displacement:

1

11

2

ZLoss

, (3.1h)

Where 2

zZ

46

Fig 3.12 shows the system losses for the parameters given in table 3.1. We can notice that the

system is not as sensitive to this kind of displacement as it was to the lateral. It requires about

22μm to reach losses of about 15% and about 225 μm to lose all energy.

Fig.3.12 Power losses as a function of the longitudinal displacement z

3.1.2.2 Lateral Displacement

Fig.3.9 Lateral displacement

Lateral misalignment is an offset of the two fibers as shown in Figure 3.9.For this case there

is a lateral displacement a, in the x direction. The modes of the receiver and transmitter fibres

are described by the following equations:

2 2 2 2

0 2

1( , ) [ ] [ i ]

r r r

x y x yx y Exp Exp

w w R

(3.1g)

2 2 2 2

2

1 ( ) ( )( , ) [ ] [ i ]

input

r r r

x a y x a yx y Exp Exp

w w R

Since we assume identical fibers their mode waist is the same, while the radius of curvature

goes to infinity.

47

For this case equation 3.1f is greatly simplified and power transmission is given by

2 atrP e (3.1h)

, where a is the normalized (over the waist) lateral displacement of the fiber and hence the

power losses can be given by:2

1 aLoss e

As it can be seen in Fig.3.10 fiber coupling is very sensitive to lateral displacement of the fiber

and a displacement at the order 0.5 wr (for our case 1.12μm) results in above 20% power loss.

It is also worth noticing that for a displacement equal twice the reference mode losses

become 100%.

Fig.3.10 Power losses as a function of the lateral displacement a

3.1.2.3 Tilt of the fibers

Fig.3.13 Tilt of the receiver fiber For this case equation 3.1f reduces to 3.1i which we can notice that is the same case described

at section 3.1.1 by equation 3.1d where a tilted beam reaches a receiver fiber without any

displacement. Losses for this kind of misalignment are plotted in Fig 3.14. As we can see

approximately 50 mrad tilt of the fiber results in 20% losses.

48

2 2

2

sin

=e

trP (3.1i)

Fig.3.14 Power losses as a function of the longitudinal displacement z

System Peak value

perfectly aligned

100%

z equals one Rayleigh range of the

reference Gaussian mode

80%

z equals two times the Rayleigh range

of the reference Gaussian mode

50%

Table 3.2 Summary for common misalignments

3.2 Fiber free-space to fiber

3.2.1 Generic fiber coupling analysis

49

The generic result of eq. 3.1 can be now applied in a telecommunication link by correlating

the misalignment and the input beam waist to the receiver and transmitter characteristics.

Such a complete link is shown in Fig.3.15.

Fig. 3.15 Schematic of a complete optical link with symmetric optics in the transmitter and

receiver

In this case the power transmission is now also a function of collimated beam’s waist, dw

(normalized) and transmitter-receiver distance L :

4 4 22

2

4 cos cos2( , , , , ) ( tan )

dt

o o

or d

w BP w L z E

A Axp

(3.2)

where

4 2 3 2 2 20

2 2 2 2 2 2 2 2 3 2 2 4

4 2 2 3 2

4

3 2

3

2 2 2 2 2

,

4 ( ) 4 ( )

2 (4 )] 4

2 (5 2 ) 4 ( 2 5 2 ) (12 5 )

[

2 18 ]

4 (

[

d d d d

d d d d

o d d d d d d

d

A w w w L w z

L Lw z w z w L w z L z

B w w w L z w z w L Lz z w

Lw z 42 23 ) ,3 L z L z

3 3 2 2 2

0

4 4 3 2

2 2 3

3 2 32 2 42 2 2

4( )[2 ( (4 ) ) ]

[ 2 ]

,

(4 4

4

2 4 )

d d d d d

o d d d d d

w z w w L w z L w z L z

w w z w z L w z Lw z L z

50

By estimating the waist size of the collimated beam as a function of the transmitter lens focal

length df

w

we can express the transmissivity as a function of the focal length f:

4 2 22 21 1 1

1 1

cos cos24( , , , , ) ( tan ),

Tr

BfP f L z Exp

A A

(3.3)

where

4 2 2 2 4 2 2 2 4 2 41

2 22 2 2 2

1

[ ] ,( 2 ) 4 2 (2 ) ( 2 )

([2 2 2 ]( ) ( ) ) 2 (2 )(3 2( )) 3 (2 ) ,

A f L f f z f L z f L f z

B f f f f L f L f f L f f L z z f L

4 2

1

2 2 2 22 4

1

2

2 4

4( )[ ( ) ](2 )( )

[2 2 (2 ) (2

,

) ]

f z f L f zf z f L f z

f f z f L z f L f z

Using now Eq. 2.3 we can answer the following question:

“Given a specific receiver-transmitter distance is there an optimal choice

of transmitter-receiver optics so that the optical system is less sensitive

to misalignments?”

Before answering this question let’s review some commonly used configurations, using and

the specified parameters (table 3.1) under the prism of our analytic results.

51

3.2.2 Commonly used configurations, perfectly aligned link

We review 3 commonly used receiver-transmitter configurations. Between these cases the

only variation is the distance L between the lenses. We first assume that the systems are

perfectly aligned (tilt angle = 0) as shown in Fig.3.16.

Fig.3.16 Schematic of a perfectly aligned transmitter-receiver optical system

For this aligned optical system equation 3.3 simplifies to:

4 2 2

4 2 2 2 4 2 2 2 4 2 4

4

( 2 ) 4 2 (2 ) ( 2[ ])

Tr

fP

L f f f z f L z L f f z (3.4)

From equation 3.4 we can obtain a graphical representation showing the dependency of

power coupling from distance L for a 5mm lens. As we can see in Fig.3.17 increasing the

distance L coupling decreases and reaching a level of 85% when the distance is about 95cm

(380000wr). Inset shows behaviour at a larger range of distances and we can notice that for

a distance of 2.25m (1000000wr) half the energy is lost.Of course this result assumes that the

lenses have infinite diameter, so we do not have power losses due to clipping of energy.

52

Fig.3.17 Power coupling as a function of normalized distance L (inset shows behavior for a

larger range of distances)

We must also take into account the finite dimensions of the lenses. As we now from theory

when a Gaussian beam propagates in a media, its diameter increases with distance. When

this increase is comparable,or even bigger, from the diameter of the lenses, we have loss of

energy due to clipping. At Fig.3.18 we can see power transmission against normalized distance

L in Rayleigh ranges for different diameters of a 4.5mm lens assuming zero longitudinal

displacement z. We can notice two things. First of all, when the distance between the lenses

becomes bigger from about 0.6 Rayleigh ranges, transmission begins to decrease. Secondly

the smaller the diammeter of the lens, the greater the effect of the clipping, leading to a

bigger power loss. When the diameter is 1.2mm (500wr) we can see that at about 1 Rayleigh

Range we have power losses, while for bigger diameters such as 2.25 mm (1000wr) this effect

is smaller.

53

Fig.3.18 Power coupling as a function of normalized in Rayleigh Range distance L, for

different diameters of the lens.

By tuning the distance z of the receiver fiber from the focal plane of the receiver lens (Fig.

3.13) we obtain the maximum transmissivity:

2 2 4 2 2 2 8 4max

2 2 4 2 2 2 4 2 2

( 2 ) 4 ( 2 ) 4

( 2 ) [( 2 ) 2 ]opt tr

f L f f L f fz P

L f f L f f

(3.4a)

So for a perfectly aligned system the peak transmissivity is a function of 3 parameters , ,f L

out of which only f is free for optimization. We can find the optimal values for f by using

in eq. (3.4a) the condition:

max0 2 , 1max

trtr

PL f P

f

(3.4b)

The condition 2L f defines the optimal configuration as far as the power transmissivity is

concerned. On the other hand we have to point out that this is not a very strict constrain,

since max 0.85 100trP L f for the range of typical values of our application (Table 3.1) as

shown in Fig. 3.16. It is clear if 2.1f mm then max 0.85trP which is much lower than the

125mm set by the condition 2L f .

54

Fig.3.19 Transmissivity as a function of the focal distance for our application

parameters (Table 3.1)

Let’s now review the characteristics 3 typical cases starting from the optimal in respect of the

transmissivity.

3.2.2.1 Symmetric 4f system (Distance L equal to two focal lengths)

Fig.3.20 Case 1: Symmetric 4f system (L = 2f)

As shown in Fig.3.20, in this case the distance between the lenses L is twice the focal length

(or the distance from fiber to fiber is 4f. In this case:

22

Lf L f (3.5)

55

Eq. 3.4 then reads:

12 2

21

4( , , , ,0)Tr

zP f L z

. For the fully symmetric case

0 1Trz P . Applying our typical parameters (table 3.1) in equation 3.5 we get that

125f mm and 14 mm collimated beam diameter. Unfortunately as we will demonstrate

later on this system it is very sensitive to misalignments. The behavior of a system like that to

longitudinal displacement can be seen in Fig.3.

Fig.3.21 Case 1: power transmissivity as a function of the normalized longitudinal

displacement of the receiver fiber

3.2.2.2 Distance L equal to the Rayleigh range of the collimated beam

Fig.3.22 Case 2: L equals one Rayleigh range of the collimated beam

As shown in Fig. 3.22, in this case distance between the lenses L is equal to the Rayleigh range

of the collimated beam:

2

Rd

f LZ L L f

(3.6)

Eq. 3.4a then reads:

56

max

2

8(2 2 )

(4 3 4 )tr

L L LP

L L

(3.6a)

Applying our typical parameters (table 3.1) in equation 3.6a we get that 2.2f mm and 0.5

mm collimated beam diameter. At this case, as shown in Fig. 3.23 we can reach 89%

transmissivity by slightly adjusting the longitudinal fiber position.

Fig. 3.23 Case 2: power transmissivity as a function of the normalized longitudinal

displacement of the receiver fiber

3.2.2.3 Distance L equal to two times the Rayleigh range of the collimated

beam

Fig.3.24 Case 3: L equals two times the Rayleigh range of the collimated beam

As shown in Fig.3.24, in this case the distance between the lenses L is two times the Rayleigh

range of the collimated beam:

22

2 2 2Rd

L f L LZ f (3.7)

Eq. 3.4a then reads:

57

max

2

(8 5 8 2 )

(4 3 4 2 )tr

L L LP

L L

(3.7a)

Applying our typical parameters (table 3.1) in equation 3.7 we get that f = 1.6 mm which

corresponds to a 0.34 mm diameter beam and a 125 mm Rayleigh range. From equation 3.4

we can then calculate the transmissivity as the receiver fiber is shifted along the optical axis.

As we can see in Fig.3.25 the transmissivity never exceeds of 56% a value that is well below

our typical threshold value.

Fig. 3.25 Case 3: power transmissivity as a function of normalized longitudinal

displacement of the receiver fiber.

Summary

System maxtrP Peak value

symmetric 4-f

21

2

2

14

z

100%

L equals one Rayleigh range of the

collimated beam

2

8(2 2 )

(4 3 4 )

L L L

L L

89%

58

L equals two times the Rayleigh range

of the collimated beam

2

(8 5 8 2 )

(4 3 4 2 )

L L L

L L

56%

Table 3.3 Summary of common misalignments

59

Chapter 4

Stability Analysis

4.1 Rigorous Analysis

4.1.1Misaligned optical system

The generic behavior of a misaligned system as described in eq. 3.3 can be further simplified

if we take into account that we are in the paraxial regime so that tan . In this case eq. 3.3

is simplified to:

4 2 221

1 1

4( , , , , ) ( ),Tr

BfP f L z Exp

A A

(4.1)

where

4 2 2 2 4 2 2 2 4 2 41

2 2 2 2 2 2 2 2 2

1

( 2 ) 4 2 (2 ) ( 2 )

2 (2 2 ) (2 2 )

[ ]

[ ]

A L f f f z f L z L f f z

B f f f L L f f f z z

In the range of parameters that we study 1 1 100B A , so even a small misalignment will

have a strong deteriorating effect on the transmissivity.

In order to provide an analytic guideline on the optimization of the optical system

parameters we will furthermore narrow the misalignment range to very small angles

100 rad , which are in the target range for space applications that use fiber-to fiber

coupling. We can then perform a power series expansion for Eq. 4.1 in order to simplify our

analysis.

4.1.2 Sensitivity analysis

Taking this into account that 100 rad and using the parameters range (table 3.1) of our

application we can approximate equation 4.1 (or 3.3) with a power series angle γ:

4 2 2

12

1

2

1(4

)tr AA

fP B

(4.2)

60

The sensitivity over misalignment is then expressed as the variation of the transmisivity over

angle γ and is described by:

2

4 2 2

1

1

( , , , ) ,

( , , , ) 8

trP C f z L

BC

Af z L f

(4.3)

Thus the sensitivity of optical system on misalignment is, for very small angles γ, proportional

to γ with a proportionality factor ( , , , )C f z L that depends on the optical parameters of the

system. Using now eq. 4.2 we can analytically estimate the optimal parameters for the optical

system so that sensitivity on misalignment is as low as possible.

For this purpose we need to calculate the partial derivative f :

3

1

2( , , , ) ,C f z L

Af

B

(4.4)

where

4 2 2 2 4 2 2 2 4 2 4

1

4 5 2 2 2 2 2 2 2 2

2

4 4 2 2 2 4 2 2 2 4 2

2 2

[ ]

[

( 2 ) 4 2 ( 2 ) ( 2 )

2 (16 8 7 3 10 9 4

4 2 2 ( 2 ) ( 2 )

2({4 [4

( ) ( )]

[ ] )

A L f f f z L f z L f f z

B f f fL L f f fz z

f z f f z f L z L f L f

f

3 2 3 4 2 4

4 8 4 2 2

6 6 2 2 2

( 2 ) (3 )] 4 4 ( 2 )}

[ 8 2 2 ])

8 (2 2 )

z L f f z f L f z L f

f f f z z

f f f L L

This partial derivate is simplified if we assume, without loss of generality that the fiber is

positioned on the focal plane of the receiver lens ( 0)z :

5 2

2 2 4 2 3

4 2 2

4 2 2 2 2 8 4

( ,0, , ) 32

[( 2 ) 4

( )(2

]

2

)(8 6 3 )

(12 3 )4 81

C f L f

f L f f

f L f L f fL L

f f f L L f

(4.4a)

61

Furthermore the partial derivative with respect to normalized distance L between the lenses,

for ( 0)z , is obtained by:

6 6 2 4 3 2 2 3 5 2 2

4 2 4 2 2 3

( ,0, , ) 32 ( 4 4 3 ) 4

[ ( 2 ) 4

( , , )

[ ]

]

C f L f f f L fL L f

L f L f

D f L

(4.5)

4.1.3 Optimization

Optimization of the system in a way that high power transmission is achieved, is crucial as

mentioned in the introduction. Another critical issue is the sensitivity to potential

perturbation. Using the analysis made at the previous sub-sections we can optimize the

system in order to obtain transmissivity higher than 95% (or 85%) and minimal sensitivity to

the misalignment. In order to do that some criteria have to be applied to the system. These

are shown in the following section.

Criterion 1: The transmisivity should be higher than 85% (optimally 95%)

During the previous analysis we obtained analytical equations describing maximum power

transmission in the system, Eq. 3.4a. For the range of typical values of our application (Table

3.1) we get the following conditions for the lens focal distance:

2.55 0.95

2.1 0.85

tr

tr

f mm P

f mm P

(C1)

We can see at Table 4.1 the results obtained by C1 for different distances L between the lenses

Distance L (mm) Focal length for 0.85trP

(mm)

Focal length for 0.95trP

(mm)

100 1.3 1.6

500 3 3.6

1000 4.2 5.1

2000 6 7.3

Table 4.1 Conditions for the lens focal length distance

62

Fig.4.1 The effect of finite lens size.

Another factor that can reduce the transmisivity and is the finite diameter of the lenses

though the effect of clipping. As shown in Fig. 4.1 the limiting case is when the input beam

completely fills the receiver lens. Assuming a typical value of transmitter-receiver lens

diameter of 6 dD w where dw is waist of the collimated beam (Fig.4.1) we can obtain that

26 0s f L f f L

.

This, for the range of typical values of our application (Table 3.1) leads us to the following

condition for the lens focal distance: that 0.91f mm .

0.91f mm (C1a)

Since this condition is less strict than (C1) we keep (C1)

The minimum focal length needed in order to avoid clipping as a function of distance L can be

seen in Fig.4.2.

Fig.4.2 Minimum focal length to avoid clipping as a function of L

63

Criterion 2: Sensitivity to misalignment should be minimized

The sensitivity to misalignment is estimated from Eq. 4.3. In order fulfill this criterion, we must

focus on minimization of the proportionality factor ( , , , )C f z L in eq. (4.3). This minimization

over the focal length can be obtained by the partial derivate over the focal distance 0C

f

,

that leads to the condition:

2

3

1

0 0C

fAf

B

(C2)

So the smaller the focal distance of the lens the smaller the sensitivity of misalignments

Criterion 3: Sensitivity to longitudinal positioning should be minimized

The sensitivity to variations of the optical link distance L is estimated from Eq. 4.5. In order

fulfill this criterion, we must on minimize C

L

by setting ( , , ) 0D f L in Eq. 4.5. This leads

to the condition:

0 4.3 ( 1911)C

f mm fL

(C3)

An overview of this behavior can be seen in Fig.4.3 where ( , , )C

D f LL

is plotted as a

function of the distance L.

Fig.4.3 D factor as a function of normalized distance L/wr

64

4.1.4 Lens Selection

All lens selection conditions described in section 3.4.2 are summarized in Table.2.2

Criteria Condition

The transmisivity should be higher than 95%

(85%)

2.55 0.95

2.1 0.85

tr

tr

f mm P

f mm P

Minimal sensitivity to misalignment 0f mm

Minimal sensitivity to variations in L 4.3f mm

Table.4.2 Lens Selection

Since most of the conditions in table 4.2 refer to ranges of values it is useful to plot the

variation of the sensitivity to misalignment, as expressed by the proportionality factor in Eq.

4.4, as a function of the focal lens. As we can see the sensitivity in the range

2.55 4.3mm f mm is in the range of 2 22.5 7rad C rad . This indicates that we can

select a slightly longer focal length lens as long as the misalignment sensitivity although not

optimal is below a threshold which we set to 210C rad .

This relaxed condition enables us to select lenses in the range of the 4.3f mm which

combines enhanced transmisivity (>95%) combined with minimal sensitivity to L variations

and fair sensitivity to misalignment 27C rad .

Fig. 4.4 Sensitivity to misalignment (red curve 0z , black dotted curve optz z eq. 2.1.4a)

65

4.1.5 Conclusions

In the previous section we have selected a focal length of 4.3f mm based, among others,

on small misalignment angle approximations.

Fig. 4.5 Transmissivity as a function of tilt angle γ for possible lenses

In order to finalize our selection we study the transmisivity as a function of angle for a larger

range of angles at a range of 4-2.67 mm in focal length. Our numerical results are shown in

Fig. 4.5. For all three lenses we note that a tilt angle of the beam of 200 μrad reduces the

coupling efficiency by about 10%. Therefore the required angular stability of the beams is not

more than 100 μrad. We can see that if we choose a 4mm lens then we have a 0.5% decrease

in maximum power transmission and a 12.5% decrease in angle sensitivity relatively to the

2.3 mm lens. For the 2.67 mm (2080wr) lens we have a 0.5% increase in maximum sensitivity

and a 18.3% increase in sensi