to lens design

for fiber optics

Andrei Tsiboulia

Beam profile approach

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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

zR

q

zR(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 ) = Ro2[ 1 + (z / nm Ro

2)2]

(nm - media refractive index)

z

znq

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+ z

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) = Ro2[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.

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.

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) = Ro2[1+ (z / zn )

2] ,

where zn = Ro nm / NA is the length of the beam near zone.

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`

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.

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:

fNANA zz wn

GI

22

'

zz wn

GI

fRR

220'

zz wn

wwGI

ffzz

22

.' '

Substituting zn = 0 gives the expressions for SI MMF

beam image waist, including Newton formula:

NA`IM NA zwf; R`IM =Ro fzw; zw`IM zw = f` f .

Substituting zw = 0 gives the expressions for SI MMF beam focal waist:

0

0 0

0

00

0

Hyperbolic profile

Conical profile

Numerical aperture: Beam waist size: Waist position:

MMF Beam Focusing by a Lens – Math

NA`FC - Rof ; R`FC = - NA f ; zw`FC = 0 .

Formulas for GI MMF beam focusing are derived above by ray tracing:

fNANA zz wn

GI

22

'

zz wn

GI

fRR

220'

zz wn

wwGI

ffzz

22

.' '

If we regard zn as a Gaussian beam Rayleigh range: zR=nmRo2/, 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

ZRZR

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.

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)

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.

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

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

     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.