1.
to lens design
for fiber optics
Andrei Tsiboulia
Beam profile approach
The slide show goes automatically

2.
Optical Fiber
Multimode (MMF)
local area network
Singlemode(SMF)
wide area network
Graded Index(GI)Step Index (SI)
Traditionally MMF
lens system has been
designed to form an
image of the emitting
fiber end-face upon the
receiving one.
However, image is not essential for coupling. Coupling can be
performed by non-imaging elements (tapered fibers, axicons).
The beam profile shaped by a coupling system should match
the profile of the beam transmitted by the receiving fiber.
We will talk about beam profiles of different fiber
types and transformation of these profiles by lenses.
Introduction

3.
zR
q
z
R(z)
Ro
Divergence
q =/nmRo
Hyperbolic profile;
SMF
Mode field radius Ro
Rayleigh range
zR= nmRo
2/
Singlemode fiber beam profile (Gaussian beam)
R2(z ) = Ro
2[ 1 + (z / nm Ro
2)2]
(nm - media refractive index)

4.
z
zn
q
R(z)
Truncated cone profile:
Divergence
q =NA/ nm
Near zone length
zn=Ro /q
Fiber core radius Ro
SI MMF
Each point on the fiber end-face is a point source,
emitting a local ray cone, which defines a local NA.
The axes of these cones are parallel to the fiber axis.
Step Index Multimode Fiber Beam Profile
Local NAL=const over the fiber cross-section.
R(z) = Ro+ q z

5.
z
zn q
R(z)
Hyperbolic profile:
Divergence q =NA/ nm
(nm - media refr. index)
The paraxial ray trace in a GI MMF is sinusoidal.
Fiber core radius Ro
Near zone length:
zn=Ro/ q
The same profile, as for SMF beam.
GI MMF
R2(z) = Ro
2[1+ (z / zn )2]
Gradient-index Multimode Fiber Beam Profile
Local NAL = NA [1 – (RL/ Ro )2]1/2
Calculation shows that the envelope
of the sinusoidal rays bundle at the
exit of a GI MMF is hyperbolic.

6.
MMF Beam Profile
The result of non-
paraxial computer ray
tracing at the exit of
GI and SI MMF
having the same core
radius Ro and NA
(NA=0.2).
Hyperbolic profile
Ray-tracing (OPTICAD)
Conical profile
The difference between
hyperbolic (GI MMF)
and conical (SI
MMF) beam profile is
clearly seen.

7.
The paraxial ray trace in a GI MMF is sinusoidal:
r (z) = ro cos (z) + (o / ) sin(z),
ro and o - initial ray radial position and slope,  - index gradient constant,
z - axial coordinate, nf - fiber core refractive index.
At any fiber cross section ray radial position r and slope  are the cosine
and the sine of the same argument  ( changes between 0 and 2):
r = Ro cos  and  = (NA/nf ) sin 
The ray equation after refraction: r (z) = Ro cos  + z ( NA/nm ) sin .
Solving together this equation and the same equation,
differentiated with respect to , gives:
Hyperbolic profile
GI MMF Beam Profile – Mathematics
Output light beam hyperbolic envelope: R2(z) = Ro
2[1+ (z / zn )2] ,
where zn = Ro nm / NA is the length of the beam near zone.

8.
MMF ray bundles are traced paraxially through a lens, modeling beam profile.
2. Gradient index MMF ray bundle:
O
The transformed GI MMF beam profile is hyperbolic again.
-f-zw
Source
fiber
SIGI
1. Step index MMF ray bundle:
Hyperbolic profile
Conical profile
MMF Beam Profile Transformation by a Lens
I
z`wIM
z`wGI
W
The waist size and position are different for GI MMF and SI MMF beams.
Focused SI MMF beam envelope consists of three conical sections.
SI MMF beam has two waists: at the rear focal and at the image plane.
F`
f`

9.
Here is the result of non-paraxial computer ray tracing
of two MMF ray bundles through the lens.
Hyperbolic profile
Conical profile
Ray-tracing (OPTICAD)
MMF Beam Profile Transformation by a Lens
We see the difference in profile, waists’ size and
position for GI & SI MMF beams after the lens.

10.
The equation of the ray of a GI MMF bundle refracted by a lens:
r`(z) = (z Ro / f’) cos  +[(z-f`)NA / n` - NA(zw+f) z / f`n] sin 
Solving together this equation and the same equation, differentiated with
respect to , gives the hyperbolic output beam envelope. Its parameters:
f
NANA
zz wn
GI



22
'
zz wn
GI
f
RR
220
'



zz wn
wwGI
ff
zz 22
.
' '


Substituting zn = 0 gives the expressions for SI MMF
beam image waist, including Newton formula:
NA`IM  NA zw  f ; R`IM =Ro f  zw ; zw`IM zw = f` f .
Substituting zw = 0 gives the expressions for SI MMF beam focal waist:
0
0 0
0
0
0
0
Hyperbolic profile
Conical profile
Numerical aperture: Beam waist size: Waist position:
MMF Beam Focusing by a Lens – Math
NA`FC  - Ro  f ; R`FC = - NA f ; zw`FC = 0 .

11.
Formulas for GI MMF beam focusing are derived above by ray tracing:
f
NANA
zz wn
GI



22
'
zz wn
GI
f
RR
220
'



zz wn
wwGI
ff
zz 22
.
' '


If we regard zn as a Gaussian beam Rayleigh range: zR=nmRo
2/, these
formulas are consistent with the Kogelnik’s ABCD law for Gaussian beams.
Formulas are derived by geometric optics,
but they are applicable to Gaussian beams.
The formulas are really universal: they explain image (Newton formula),
focal waist, GI MMF beam waist and Gaussian beam (Kogelnik’s formula).
ZR
ZR
ZR
Hyperbolic profile
Similarity in SMF and GI MMF Beams Focusing
The hyperbolic envelope ray bundle is a basis for geometric
optics approach to the Gaussian beam focusing problem.

12.
A ray bundle is generated in the laser resonator.
It models the laser beam and has a
It is similar to a ray bundle
coming out of a GI MMF
This slide presents an example of a ray bundle describing Gaussian beam.
Hyperbolic profile
Tracing a Ray in Laser Resonator (OPTICAD)

13.
The focused waist size and position are
consistent with the Gaussian beam theory.
This model is convenient
for computer simulation
in ray-tracing codes such
as ZEMAX.
The ray bundle radius and NA are from the catalogue data for SMF-28 fiber.
It is a common opinion that geometric optics is invalid for Gaussian beams.
However, the proper approach (profile, not an image) gives correct results.
Hyperbolic profile
It gives beam visualization
and lens optimization
regarding beam profile
aberrational distortion.
SMF-28 Light Beam Ray-tracing (OPTICAD)
Then again, the common opinion that the GI MMF beam waist is in the image
plane is not correct either. Beam profile, not an image defines the waist.
Similar ray bundle is used to model SMF beam in ray-tracing software.

14.
Gaussian Beam Diffraction?
There is no deviation from the geometric optics for Gaussian beams.
The following features are specific for diffraction:
The beam focusing problem is distinctive from imaging or diffraction.
Both terms “Gaussian beam imaging” and “Gaussian beam diffraction”
could be misleading. The more appropriate term is “Beam shaping”.
It is caused by a beam clipping and it causes fringes in the shaded area.
It is irreversible – the diffracted beam cannot be transformed into the original one.
Diffraction is not linear – matrixes and their combinations cannot describe it.
None of these is specific for Gaussian beams, if there is no beam clipping.
The Gaussian beam focusing theory is based on the diffraction theory,
but the geometric optics can be also derived from the diffraction theory.
Hyperbolic profile

15.
Each original Gaussian beam cross-section is imaged through a lens.
Focused beam profile is an envelope of these images.
F F`
Original and focused waists are not the images of one another.
There is no diffraction involved in this description.
This model is consistent with the Gaussian beam focusing theory.
Gaussian beam focused by a lens is an envelope of
images of the original beam cross-sections.
Hyperbolic profile
w`w
Gaussian Beam as an Envelope of Images
yy1Yp'y2y3Yp

16.
Conclusion
 The GI MMF beam profile is hyperbolic, like a SMF beam profile.
 Geometric optics can describe beam profiles of MMF and SMF.
Diffraction is not an issue, if there is no beam clipping.
 The focused SI MMF beam has two waists - at the image and at the focal
planes. The SI MMF beam profile should be corrected for a good coupling.
 Gaussian and GI MMF beam focusing is described by similar formulas.
 The beam focusing problem is distinct from the imaging or diffraction.
 The beam PROFILE is the main factor affecting the beam
propagation and focusing properties.
 The beam profile approach is universal for imaging and non-imaging
optics, as well as for MMF and SMF.