Phys 322 Lecture 15 Chapter 5

Geometrical Optics Fiber optics Fiberoptics: first lightguide

1870: water as a light guide

John Tyndall 1820-1893 Fiberoptics: first optical communication

Alexander Graham Bell 1847-1922

1880: photophone 4 years after inventing a telephone! Fiber optics: communications 1960: First laser 1966: coupling with fibers for communication 1970: 1% of light transmitted over 1 km (losses 20 dB/km) Today: over 96% of power transmitted over 1 km Why light? - frequencies ~1015 Hz Theoretical bandwidth limit: each oscillation is 1 bit, bandwidth is ~1014 bytes/second (100,000 GB/s) Speech ~3 kB/s: can support ~10 billion phone connections over one fiber simultaneously!

DVD quality videophone: ~10 million channels! Note: Modern fiber systems record 100 Tb/s http://www.newscientist.com/article/mg21028095.500-ultrafast-fibre-optics-set-new-speed- record.html What is an Optical Fiber? An optical fiber is a waveguide for light consists of : core inner part where wave propagates cladding outer part used to keep wave in core buffer protective coating jacket outer protective shield Design of optical fibers

Core: Thin glass center of the fiber that carries the light

Cladding: Surrounds the core and reflects the light back into the core

Buffer coating: Plastic protective coating

ncore > ncladding Microstructure fiber

Air holes In microstructure fiber, air holes act as the cladding surrounding a glass core. Such fibers have different dispersion properties.

Core

Such fiber has many applications, from medical imaging to optical clocks. Propagation of light in an optical fiber

Light travels through the core bouncing from the reflective walls. The walls absorb very little light from the core allowing the light wave to travel large distances.

Some signal degradation occurs due to imperfectly constructed glass used in the cable. The best optical fibers show very little light loss -- less than 10%/km at 1,550 nm.

Maximum light loss occurs at the points of maximum curvature. Fiberoptics: single core fiber losses Consider large fiber: diameter D >>   can use geometric optics

Path length traveled by ray:

l  L / cost Number of reflections: l N  1 D /sint

Example: Using Snell’s Law for t: L = 1 km, D = 50 m, n =1.6,  = 30o f i Lsin N  i 1 N = 6,580,000 2 2 D n f  sin i Note: frustrated internal reflection, irregularities  losses! Step Index Fiber: TIR

escapes core

escapes core cladding nt

core ni stuck in core i

i i

nt critical angle sinc   nti (pg 121) For total internal ni reflection need i c for TIR nc

max must be > critical angle

NA  noutside sinmax 2 2 n f  nc NAstep  ni

2 2 NA in air NAstep  n f  nc Fiber and f/#

2 2 n f  nc sinmax  ni

Angle max defines the light gathering efficiency of the fiber, or numerical aperture NA:

2 2 NA  ni sinmax  n f  nc 1 And f/# is: f /#  Largest NA=1 2NA Typical NA = 0.2 … 1 Bundles

Bundles can collect light from larger area and be still flexible

Flexible light carriers Coherent bundles: flexible image carriers

Fibers are arranged in a coherent fashion Data transfer limitations

1. Distance is limited by losses in a fiber. Losses  are measured in decibels (dB) per km of fiber (dB/km), i.e. in logarithmic scale:

10  P  P L /10 P - output power  o  o o    log  10 P - input power L  P  P i  i  i L - fiber length

Example:  Po/Pi over 1 km 10 dB 1:10 20 dB 1:100 30 dB 1:1000

Workaround: use light amplifiers to boost and relay the signal

2. Bandwidth is limited by pulse broadening in fiber and processing electronics Attenuation

page 297 IR absorption Rayleigh Scattering

absorption and scattering in fiber in the IR: “low-OH” versus “high-OH” Pulse broadening Dispersion: The Basics Light propagates at a finite speed

fastest ray

slowest ray fastest ray: one traveling down middle (“axial mode”) slowest ray: one entering at highest angle (“high order” mode)

will be a difference in time for these two rays Types of Dispersion in Fibers modal time delay from path length differences usually the biggest culprit in step-index material n() : different times to cross fiber (note: smallest effect ~ 1.3 m) waveguide changes in field distribution (important for SM) non-linear n can become intensity-dependent

NOTE: GRIN fibers tend to have less modal dispersion because the ray paths are shorter Effect of Dispersion initial pulse farther down farther still

time time time

modal example: step index ~ 24 ns km -1 GRIN ~ 122 ps km-1 Fibers carry modes of light

2  D  number of modes   NA  0 

a mode is : •a solution to the wave equation • a given path/distribution of light (pg 196)

higher # modes gives more light, which is not always desirable Example of # of Modes @ 850nm

Silica step-index fiber has nf = 1.452, nc = 1.442 (NA = 0.205) SELFOC graded index fiber with same NA

diameter 2.5 50 200 400 1000 (microns)

# step-index 1.5 580 9.3 E3 37 E3 230 E3 modes

# GRIN 1.8 716 11 E3 46 E3 1150 E3 modes

high # modes implies classical optics NA and # of Modes killed ray propagated ray

large NA

small NA Pulse broadening

Multimode fiber: there are many rays (modes) with different OPLs and initially short pulses will be broadened (intermodal dispersion)

For ray along axis: tmin  L v f  Ln f c

2 For ray entering at max: tmax  l v f  Ln f cnc  The initially short pulse will be broadened by:

Ln f  n f  Making nc close to nf   t  tmax  tmin   1 reduces the effect! c  nc  Pulse broadening: example

nf = 1.5 nc=1.489 Estimate the bandwidth limit for 1000 km transmission.

Solution: 6 Ln f  n f  10 1.5  1.5  5 t   1  1 s  3.7 10 s  37s   8   c  nc  310 1.489  Even the shortest pulse will become ~37 s long 1 kilobits per second Bandwidth ~ 5  27 kbps 3.7 10 s = ONLY 3.3 kbytes/s Multimode fibers are not used for communication! Graded Index Fiber

nc

nf varies n quadratically

nc

like a “restoring” force ! Types of fibers

nc

nf

nc step-index multimode

nc nf nc step-index singlemode

nc

nf GRIN nc Single mode fiber

To avoid broadening need to have only one path, or mode

Single mode fiber: there is only one path, all other rays escape from the fiber

clad core

jacket

Geometric optics does not work anymore: need wave optics. Single mode fiber core is usually only 2-7 micron in diameter Single mode fiber: broadening

clad Problem: shorter the pulse, broader core the spectrum. refraction index depends on wavelength jacket ‘Transform’ limited pulse product of spectral full width at half maximum (fwhm) by time duration fwhm: ft  0.2 A 10 fs pulse at 800 nm is ~40 nm wide spectrally If second derivative of n is not zero this pulse will broaden in fiber rapidly Solitons: special pulse shapes that do not change while propagating Critical Bend radius An example:

I need a fiber that will conduct NIR light.

I have to keep a tight pulse pattern.

It must couple into an LED.

What do I do? Putting It All together

(a) I needed a fiber that will conduct get a low OH fiber NIR light.

(b) I had to keep a tight pulse pattern. you want low dispersion: SM or a GRIN fiber, low diameter, low NA

(c) It must couple into an LED. The LED has a high divergence angle; better get a bright one. A laser might be better, and use a GRIN lens to couple. These are design considerations, as well as cost! Phys 322 Lecture 15 Chapter 5

Geometrical Optics Optical systems Human eye Human eye Most of the bending n1.376 Iris serves as aperture stop. Diameter changes from ~2 mm in dark to ~8 mm in bright light Note: it also contracts to increase sharpness when doing close work. collagen n1.337 (protein polymer)

blind spot Floating specs (floaters): muscae volitantes

http://en.wikipedia.org/wiki/Floater The cornea, iris, and lens

The cornea is a thin membrane that has an index of refraction of around 1.38. It protects the eye and refracts light (more than the lens does!) as it enters the eye. Some light leaks through the cornea, especially when it’s blue.

The iris controls the size of the pupil, an opening that allows light to enter through.

The lens is jelly-like lens with an index of refraction of 1.386-1.406 (GRIN lens). This lens bends so that the vision process can be fine tuned. When you squint, you are bending this lens and changing its properties so that your vision is clearer.

The ciliary muscles bend and adjust the lens. Accommodation

1 1 1   so si f 1  1 1     nl 1    f  R1 R2 

change focus

closest: young adult ~12 cm middle-aged ~30 cm 60 yrs old ~100 cm (Birds: change curvature of cornea) Crystalline lens of an eye

Lens: 9x4mm, consists of ~22,000 layers of cortical fibers

n = 1.386…1.406

http://www.bartleby.com/107/illus887.html http://www.owlnet.rice.edu/~psyc351/imagelist.htm Correcting vision Nearsighted eye and glasses

Farsighted eye and glasses Physiological optics: dioptric power D

1  1 1  Instead of f use dioptric power D: D     nl 1    f  R1 R2  1 1 1 For 2 thin lenses closely spaced:   D  D1  D 2 f f1 f2 For intact unaccommodated eye D=58.6 D (Diopter) Far point: the object point whose image lies on the retina for unaccommodated eye Normal eye: far point is 

Nearsightedness (myopia) - far point is closer, D > 58.6 D Farsightedness (hyperopia) - far point is behind the lens, D < 58.6 D Near point: the closest object point whose image could be projected on the retina with accommodated eye Example: nearsightedness (myopia)

Far point is closer, D > 58.6 D

Suppose far point = 2 m

The additional lens must make image si=-2 m for so= 2 m (assume lens-eye distance is small, contact lens) 1 1 1 1 1     Lens f = -2 m, or D=-0.5 D f so si   2m

For spectacle lens distance d away from eyes:

D l D  Dl - distant lens power (d from eye) c 1 D d  l Dc - equivalent contact lens power Astigmatism The lens has different radii of curvature in different planes test pattern:

normal eye astigmatic eye http://www.thineyeglasses.com/glossary/astigmatism.htm Correction: cylindrical lenses

More complex: sphero-cylindrical lenses