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Fiber Amplifiers

Tutorial at the University of Neuchâtel, 2016-04-20

Dr. Rüdiger Paschotta

RP Photonics Consulting GmbHBad Dürrheim, Germanywww.rp-photonics.com

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The Presenter: Dr. Rüdiger Paschotta

Originally: career as a researcher at various places including the

Optoelectronics Research Centre (ORC) Southampton and ETH Zürich;

research on laser physics, ultrashort pulse generation, nonlinear optics

2004: founded RP Photonics Consulting GmbH

in Zürich (moved to Bad Dürrheim, Germany).

Independent technical consulting services:

product design, calculations and simulations,

feasibility studies, independent advice, staff training.

Also: simulation software for designing fiber lasers

and amplifiers, mode-locked lasers, laser resonators, etc.

Encyclopedia of Laser Physics and Technology:

available online and as two-volume book.

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More Informationhttps://www.rp-photonics.com/tutorial_fiber_amplifiers.html

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Topics of this CourseI Active Fibers for Amplifiers and Lasers

rare-earth-doped fibers

amplified spontaneous emission

core-pumped and cladding-pumped fibers

II Continuous-wave Fiber Amplifiers

pump absorption

laser gain

amplified spontaneous emission

high-power operation of double-clad fibers

different laser-active dopants: Yb, Nd, Er

methods for calculating the power conversion

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Topics of this CourseIII Light Pulses in Fibers

simulating pulse propagation

nanosecond vs. picosecond and femtosecond pulses

soliton formation, higher-order solitons

supercontinuum generation

dispersive and nonlinear effects

chirped-pulse amplification

parabolic pulses, self-similar pulse propagation

multi-stage fiber amplifiers

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Part IIActive Fibers for Amplifiers and Lasers

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Setup of a Fiber Amplifier

Active medium: some length of single-mode erbium-doped fiber

Pumping: copropagating, counterpropagating, or bidirectional

Dichroic couplers for injecting pump light

One or two fiber-coupled optical isolators suppress reflections

Amplifier module can contain pump diodes, electronics, etc.

coupler

LD

980 nm

LD

980 nm

Er3+-doped fiber

signal input

signal outputcoupler

isolator

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Aspects of Amplification in Fibers

Long gain medium and small mode area:

can reach high gain even for low doping densities,

“difficult” laser transitions, etc.

high gain efficiency (in dB/mW or dB/J)

strong saturation effects

strong optical nonlinearities

Waveguiding:

single-mode fibers: fixed output beam shape, high beam quality

(insensitive to thermal effects), but limited mode size

multimode fibers: larger beam areas, reduced beam quality,

intermodal dispersion

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Aspects of Amplification in Fibers

Basic idea: dope the fiber core with some laser-active ions:

Additional dopants are usually used for refractive index control,

sometimes for improving the chemical homogeneity.

The cladding usually remains undoped.

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Aspects of Amplification in Fibers

Gain characteristics of the doped core:

strong broadening of absorption and emission transitions

(although not necessarily inhomogeneous saturation behavior!)

large gain bandwidth, large acceptance range of pump wavelength

low absorption and emission cross sections

(compared e.g. to Nd:YAG)

high saturation fluence and saturation intensity

(but small saturation energy and saturation power: small mode area!)

low solubility of laser ions in silica: enforces low doping density,

but can use a long length of fiber

strong multi-phonon non-radiative transitions for small energy gaps

(depending on maximum phonon energy of host glass)

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Some Laser-active Ions for Fibers

Neodymium (Nd3+) for 1.03–1.1 μm, 0.9–0.95 μm, 1.32–1.35 μm:

mostly used with silicate glasses; four-level transition (except for 0.9–0.95

μm)

Ytterbium (Yb3+) for 1.0–1.1 μm:

mostly with silicates; can be highly efficient – more than Nd3+

Erbium (Er3+) for 1.5–1.6 μm, 2.7 μm, 0.55 μm:

mostly used for telecom amplifiers; moderate efficiency;

low solubility of Er3+ in silica, better in phosphate glasses

Thulium (Tm3+) for 1.7–2.1 μm, 1.45–1.53 μm, 0.48 μm, 0.8 μm:

mostly used for 2-μm amplifiers; fluoride glasses for other transitions

Praseodymium (Pr3+) for 1.3 μm, 0.635 μm, 0.6 μm, 0.52 μm, 0.49 μm;

typically with fluoride glasses

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Effective Transition Cross-sections

General idea: consider whole Stark level manifolds

rather than all the sublevels.

Total transition rate from manifold j to k:

where jk = effective transition cross section,

I = optical intensity at wavelength ,

nj = population of starting manifold.

Remark: nj may be a fractional excitation level (0..1).

Characterize all stimulated transitions between the two manifolds of Yb3+

ions simply with two wavelength-dependent transition cross-sections:

12() and 21(), also called abs() and em().

(Yb:YAG)

( )j k jk j

IR n

h

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Calculating the Excitation Level

Assumptions:

narrow-band pump and signal waves

no quenching processes, energy transfers, etc.

homogeneous system: all ions behave in the same way

Local dynamical equation (rate equation) in simple case:

p ps s2abs,p abs,s 1 em,p em,s 2 2

p s p s 2

1I II Inn n n

t h h h h

absorption stimulatedemission

spontaneousemission

Can generalize this for interaction with more wavelengths, energy transfer

processes, etc.

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How Much Gain?

Power amplification factor in some length of fiber: exp(G), with

2 em 1 abs0 0

( ) ( , ) d ( ) ( ) ( ) ( ) dL L

G g z z N z N z z

where if all laser ions either in level 1 or level 2; 1 2 dop( ) ( ) ( )N z N z N z

em abs, are effective transition cross sections (wavelength-dependent!);

is an overlap factor for the signal mode(s):

e.g. 1 if the signal field stays entirely within the fiber core.

For a homogeneously doped core:

core

0

0

( ) d

( ) d

r

I r r r

I r r r

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Alternative form:

How Much Gain?

dop 2 em 1 abs0

dop 2 em abs abs0

( ) ( ) d

( ) d

L

L

G N n z n z z

N n z z

where and we assume for the last step,

if we have only one metastable energy level.

dop/j jn N N 1 2 1n n

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Remarks on the Transverse Variation

We have simplified the treatment of the transverse distribution of optical

intensities and doping densities. For example, ions in the wings of the

signal beam profile contribute less to the gain and are less strongly

saturated for a given signal power.

Consequences: somewhat reduced gain and modified saturation behavior.

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Remarks on the Longitudinal Variation

As long as we neglect the transverse variation:

The overall gain depends only on the average excitation level, not on the

longitudinal distribution of excitation.

The same holds for the pump absorption.

(If the fiber has excess losses, the fraction of useful pump absorption does

depend on the longitudinal distribution of excitation. However, excess loss

is typically low.)

Therefore, the resulting gain typically does not depend on the pump

direction. (This may change if there is strong ASE.)

The calculation of the longitudinal distribution in the steady state requires a

self-consistent solution (discussed later on).

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Gain and Loss in the Ytterbium Systemfor different excitation levels

100% excitation

0% excitation

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Gain and Loss in the Ytterbium Systemfor different excitation levels

some gain already for moderate

excitation levels

huge gain, but only at high excitation levels

strong absorption at low

excitation levels

nearly four-levelbehavior

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Influence of Pump Wavelength in Yb:Glass

975-nm pumping: smaller quantum defect, very high pump absorption (in narrow bandwidth), reach only 50% excitation

short pump wavelengths: large quantum defect, moderate pump absorption, can reach high excitation level

typical lasing wavelengths

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Amplified Spontaneous Emission

Excited ions emit fluorescent light in all directions.

A tiny part of the fluorescence goes into the guided mode(s).

That part may be further amplified amplified spontaneous emission

(ASE)

Consequences:

For sufficiently high gain (order of 40 dB), ASE along the fiber may

acquire a power level comparable to the pump power.

ASE can then saturate the gain, i.e., reduce the signal output power.

ASE also acts as a noise addition to the signal.

Note: shape of ASE spectrum can differ from the shape of the fluorescence

spectrum, because the gain is wavelength-dependent.

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Double-clad Fibers

Problem with ordinary (single-clad) active fiber designs:

single-mode fibers: need to launch pump light into single-mode core

excludes high-power laser diodes, strongly limits the power

multimode fibers: sacrifice the beam quality

Solution: double-clad designs:

signal light propagates in single-mode (or few-mode) core

good signal beam quality

pump light propagates in larger undoped inner cladding,

still has some overlap with the doped core

can utilize multimode pump sources;

increased absorption length

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Double-clad Fibers for Higher PowersStandard solid-core design:

Single-mode core in inner

cladding

Outer cladding made with

low-index polymer

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Double-clad Photonic Crystal Fibers

Single-mode core defined by

array of air holes

Pump cladding limited by

longer air holes;

achieve very high numerical

aperture;

all-glass solution possible

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Pump Absorption in Double-clad Fibers

Simple model: pump absorption (in dB/m) in the pump cladding

is smaller than that in the fiber core by the ratio of cladding and core areas.

Underlying assumption: uniform pump intensity over the whole cross

section, including the core.

Problem: cladding has many different propagation modes, which have

different overlap with the doped core and thus experience different losses.

Intensity profile may change such that the effective absorption becomes

smaller. Some modes have nearly no core overlap at all!

Part of the pump power can not be efficiently used.

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Pump Absorption in Double-clad Fibers

Example: simulated intensity profiles of a double-clad fiber with moderate

area ratio and radially symmetric design:

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Pump Absorption in Double-clad Fibers

Solution: breaking the circular symmetry in some way:

Practical problem: fusion splicing may become more difficult.

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Typical Trade-off with Double-clad Fibers

For poor pump beam quality: need larger area of pump cladding.

Consequences:

lower pump absorption

need longer length

effectively higher nonlinearity

lower pump intensity

tentatively lower excitation level

modified gain spectrum

possibly reduced efficiency

Make the core as large as possible in order to keep the area ratio low!

However, core area for single-mode operation is limited, and the pump

intensity remains low.

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Part IIIContinuous-wave Fiber Amplifiers

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Pumping at 975 nm

The pump absorption cross section is highest at 975 nm:

Strong pump absorption for low excitation levels.

Low saturation power: only 4 mW for core with 5 μm radius!

Example: pumping 1.5 m of

fiber with 100 mW at 975 nm:

nearly linear decay of pump

power; excitation initially 50%.

(Yb concentration: 21025 m3)

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Pumping at 920 nm

The pump absorption cross section is 3.4 times smaller than at 975 nm,

and the pump emission cross section 80 times smaller.

Weaker pump absorption for low excitation levels,

but weaker saturation of absorption.

Example: pumping with

100 mW at 920 nm:

faster initial absorption;

excitation initially >80%

A 0.6 m long fiber would

absorb nearly all power

at 920 nm – more than

for 975 nm!

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Signal Gain vs. Pump Power

Small-signal gain at 1030 nm and 1080 nm for pumping at 975 nm:

Positive gain at 1030 nm only for >35 mW pump,

while nearly no reabsorption effect for 1080 nm.

Higher gain at 1030 nm for

strong pump because of

higher em.

Gain levels off at 0.25 W

– why?

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Internal Powers and Excitation Densities

For > 0.25 W pump power at 975 nm, essentially 50% Yb excitation are

reached throughout the whole fiber.

Consequence: no further increase of gain for stronger pumping;

pump absorption is saturated.

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Signal Gain vs. Pump Power

Small-signal gain at 1030 nm and 1080 nm for pumping at 920 nm:

Slightly slower initial growth due to higher photon energy (lower photon

flux).

Gain levels off more early

and softly – why?

1030 nm

1080 nm

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Internal Powers and Excitation Densities

For > 0.2 W pump power at 920 nm, the curves looks quite strange:

initially faster decay of pump

and lower Yb excitation!

What’s going on here?

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Power Distribution in the Amplifier

Check the distribution of powers for pumping with 500 mW at 920 nm:

Backward ASE extracts substantial power.

Forward ASE is substantial around the middle of the fiber, but becomes

reabsorbed.

Excitation is reduced by ASE

at the left end.

Obviously, numerical modeling

is vital for understanding such

devices – even when nothing

more than pump light is injected!

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ASE Spectra for 920-nm Pumping

ASE spectra for pumping with 500 mW at 920 nm:

Backward ASE is much stronger than forward ASE.

Reason: backward ASE is enhanced by fluorescence from the weakly

pumped right part.

Power around 975 nm is

dominating in backward

ASE (log scale!),

but totally suppressed in

forward ASE.

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ASE Spectra for 975-nm Pumping

ASE spectra for pumping with 500 mW at 975 nm:

Power around 1030 nm is dominating.

Forward and backward ASE about equally strong.

Reason: quite uniform

distribution of excitation.

Total ASE power is small.

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Amplification of a Signal

Pumping with 500 mW at 920 nm, amplifying a 10-mW forward signal

at 1030 nm:

Signal saturates the gain, thus strongly suppresses ASE.

Fiber is over-long:

superfluous part reabsorbs

signal power and slightly

increases backward ASE.

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Influences of Signal Direction

Forward-propagating signal:

Excitation is high at the signal input end, where the pump is strong.

Backward ASE is relatively strong, but forward ASE is weaker.

Backward-propagating signal:

Excitation is low at the signal input, where the pump is weak.

Backward ASE is weaker, but interferes with the signal.

Note:

No significant difference in output power, when signal is rather strong

(so that ASE is suppressed),

except if there are substantial background losses.

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Cladding Pumping

Naïve idea: for a double-clad fiber, everything works as for core pumping,

just at higher power levels.

This is not true:

The pump, but not the signal, is much less strongly coupled to the laser

ions.

We need a longer length of fiber, and preferably a pump wavelength with

strong absorption (975 nm!).

The pump intensity is typically lower: despite the higher pump power, the

pump brightness is lower.

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Example for Cladding Pumping

Example case:

Use a core with 10 μm radius and a pump cladding with 125 μm radius,

core doping as before (2 1025 m-3)

Use 40 W pump power at 975 nm.

Increase the fiber length

to 15 m.

Can amplify a 10-mW signal

at 1030 nm to 35 W.

Low excitation, except near

the input end.

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Example for Cladding Pumping

Modified case:

Pump at 920 nm. Have to increase the fiber length to 40 m.

Can amplify a 10-mW signal at 1030 nm to 30.4 W.

Efficiency is reduced,

because quantum defect

is higher and the system

contains more ytterbium.

More excited ions are needed

due to the reabsorption.

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Amplification at 975 nm

Large gain at 975 nm is possible for high degree of excitation (>50%).

Required: short pump wavelength (e.g. 920 nm) and high pump intensity.

No success with the 40 m long double-clad fiber: only produce ASE in the

1030-nm spectral region!

Problem: long-wavelength

ASE becomes strong long

before the degree excitation

of excitation is high enough

for amplifying at 975 nm.

Underlying problem:

too much Yb in the system!

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Amplification at 975 nm

Why would it help to have less Yb in the system?

Gain at 975 nm requires high degree of Yb excitation.

Need a shorter fiber – e.g. 4 m length – to obtain sufficient gain at 975 nm.

The higher the average

degree of excitation, the

larger is the ratio of gain

at 975 nm and 1030 nm.

For a 40 m long fiber,

obtain e.g. 50 dB @ 1030 nm

already for 16% excitation –

too low for gain at 975 nm.

But: shorter fiber has in-

sufficient pump absorption!

4 m long fiber, 21025 m3 Yb

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Amplification at 975 nm

Consequence: need to use a shorter fiber, and pump directly into the core

to obtain sufficient pump absorption.

Diagram shows result for 1 m long core-pumped fiber.

Required pump brightness

is enormous, of course.

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Amplification at 975 nm

Solution for double-clad fiber: ring-doped core, see

J. Nilsson et al, “Ring-doped cladding-pumped single-mode three-level

fiber laser”, Opt. Lett. 23 (5), 355 (1998)

Basic idea: reduce the coupling of the signal and ASE to the core

can have more Yb in the system without getting excessive gain

Note: large amount of Yb required for efficient pump absorption.

Yb-doped ringaround the core

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Amplification in Ring-doped Fiber

Use a ring-doped fiber with 10 μm mode radius, Yb-doped ring at

1112 μm, reduced cladding radius of 100 μm

ASE is suppressed, but efficiency is relatively small.

Reason: strong spontaneous emission.

Improvement only via

operation with less Yb

and higher excitation level,

i.e., smaller pump cladding.

Would require higher pump

brightness.

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Photodarkening in Yb-doped Fiber

Frequently made observation: Yb-doped silica fibers degrade during

operation. The fiber develops strong losses, particularly in the blue spectral

region, but also some losses in the 1-μm region.

The speed of degradation depends very strongly on the density of excited

Yb ions: J. J. Koponen et al., “Measuring photodarkening from single-mode

ytterbium doped silica fibers”, Opt. Express 14 (24), 11539 (2006)

often no problem in high-power double-clad devices,

but particularly in core-pumped amplifiers for 975 nm

The effect may also be related to Yb lifetime quenching: R. Paschotta et al.,

“Lifetime quenching in Yb-doped fibres”, Opt. Commun. 136, 375 (1997)

For more details and literature references, see the Encyclopedia of Laser

Physics and Technology.

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Remarks on Nd-based Fiber Amplifiers

Neodymium-doped silica fibers can also amplify at 1.03–1.1 μm.

Differences to Yb-doped fiber amplifiers are:

lower amplification bandwidth

larger quantum defect

lower power efficiency,

more heat generation

four-level gain system

no reabsorption, lower excess noise

additional amplifier transitions

at 0.9–0.95 μm (quasi-three-level)

and 1.32–1.35 μm

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Amplification with Erbium

Er3+ ions have many Stark level

manifolds.

Simplest case: pumping at 1.45 µm,

amplification at 1.55 µm

same situation as for Yb3+!

(negligible excited-state absorption;

assume negligible energy transfers)

Another simple case: pumping at

0.98 µm, amplification at 1.55 µm.

Nonradiative transfer from level 3

= 4I11/2 to level 2 = 4I15/2 is often very

fast.

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Effective Absorption and EmissionCross Sections of Er:glass

Example: Er3+ ions in germano-alumino-silicate glass:

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Erbium in Different Glasses

Erbium can be used in different host

glasses:

Erbium-doped silica fibers only amplify

in the 1.5-μm spectral region.

Higher-lying energy levels are

“quenched” by multi-phonon emission.

Certain fluoride glasses lead to much

longer lifetimes of higher-lying energy

levels and thus allow operation on

various other transitions, e.g. at 2.9 μm

or 0.55 μm.

Here we only discuss silica-based devices.

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Erbium-doped Fiber Amplifiers

Various issues:

Limited solubility of erbium in silica glass:

tendency for clustering at higher doping concentrations.

Effect somewhat mitigated e.g. by aluminum codoping.

typically use low doping concentration and long fiber

Low absorption and emission cross sections,

long upper-state lifetime

good energy storage, high saturation energy

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Pumping Erbium-doped Fiber Amplifiers

Available pumping options:

Option 1: 980-nm pumping:

upper laser level reached via fast multi-phonon decay

non-ideal quantum defect

high gain efficiency

low noise figure

1550-nmsignal

980-nmpump

multi-phonondecay

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Pumping Erbium-doped Fiber Amplifiers

Available pumping options:

Option 2: 0.8-μm pumping:

upper laser level reached via two

fast multi-phonon decays

higher quantum defect

low noise figure

problem with excited-state absorption1550-nm

signal0.8-μmpump

ESA

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Pumping Erbium-doped Fiber Amplifiers

Available pumping options:

Option 3: 1.45-μm pumping:

pumping directly into the upper laser level manifold

low quantum defect

limited degree of excitation

lower gain and gain efficiency

higher noise figure

1550-nmsignal

1450-nmpump

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Pumping Erbium-doped Fiber Amplifiers

Available pumping options:

Option 4: pumping via Yb sensitizer:

Pump radiation mainly absorbed by additional ytterbium ions, which

transfer their excitation energy to erbium ions.

That transfer is efficient if the glass composition

is optimized (sacrificing gain bandwidth).

Concentration of Yb can be higher than for Er,

and Yb absorption cross sections are higher

can realize shorter amplifiers,

do cladding pumping,

or use other pump wavelengths (e.g. 1047 nm)

Note: careful optimization of core composition is vital.

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Thulium for 2-μm Amplifiers

Absorption of pump photons

at 0.8 μm, exciting ions to levels 3F2-4

Efficient cross-relaxation process leads

to two ions in 3H4 for one ion in 3F2-4

(works well in silica fiber)

quantum efficiency >100% possible

Amplification on the 2-μm transition3H4 3H6

Problem: spectroscopic data on such

cross-relaxation tricks are often hard

to obtain, e.g. for commercial fibers.

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Upconversion Scheme with Thulium

Sequential absorption of three pump

photons from a single pump source at

1140 nm, with multi-phonon decay in

between; emission of blue light in a single

transition.

Process can be fairly efficient for

sufficiently high pump intensity and for

sufficiently long metastable state lifetimes

( use Tm3+:ZBLAN glass).

In other upconversion lasers, e.g. based on Pr3+, multi-step excitation with

one pump source is not possible. Have to use two pump sources or

employ an energy transfer (e.g. Yb3+ Pr3+)

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Part IIILight Pulses in Fibers

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Fiber Amplifiers for Pulse Amplification

Beneficial aspects of fiber amplifiers:

Power efficiency and beam quality are typically high.

Gain can be very high: easily 40 dB for one amplifier stage.

Broad amplification bandwidth: sufficient for ultrashort pulses.

Negative aspects:

Long device length and small mode area strong nonlinear effects

serious limitations of pulse energy and particularly of the peak power

Damage issues, particular problems with fiber ends.

Gain can be higher than desirable: causes ASE problems

(including ASE at unwanted wavelengths).

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Nanosecond Pulses in Fibers

Chromatic dispersion is normally not relevant:

low optical bandwidth.

Self-phase modulation (SPM) can be relevant for the spectral evolution,

but normally has little impact on the pulse duration.

Stimulated Brillouin scattering (SBS) can be very nasty for narrow

bandwidth (single-frequency sources): strong nonlinear back-reflection.

Stimulated Raman scattering (SRS) can be important for high peak powers.

Can occur in forward and backward direction.

Pulse energies are often quite high – well above the gain saturation

energy. Therefore, get strong gain saturation effects.

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Gain Saturation with Pulses

Typical situation:

pulse duration far below the pumping time and upper-state lifetime

pulse energy above the gain saturation energy

Resulting gain reduction:

→ strong gain reduction and energy extraction if Ep > Esat

Note: saturation energy of a fiber is relatively small due to the small mode

area.

Example: Yb-doped fiber amplifying pulses at 1060 nm,

em = 0.33 pm2, mode area 100 µm2: Esat = 57 µJ. (At 1030 nm, only 30 µJ.)

final 0 p satexp /g g E E

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Modeling of Pulse Propagation in Fibers

Choose type of model depending on the situation:

Nanosecond pulses:

can often neglect chromatic dispersion,

and only monitor the nonlinear phase shift

(limit the peak power to avoid getting into a strongly nonlinear regime)

→ use a simple model, propagating only optical powers,

ignoring spectral properties

Picosecond or femtosecond pulses:

need to consider the full temporal/spectral behavior:

represent a pulse by an array of complex amplitudes in the time or

frequency domain.

In both cases, often ignore the transverse spatial dimension:

fixed intensity profiles defined by fiber modes.

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Numerical Representation of Short Pulses

FFT

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Modeling of Pulse Propagation in Fibers

Numerical representation: array of complex amplitudes in the time or

frequency domain:

Consider amplitudes of envelope, not directly the electric field amplitudes:

with a slowly varying amplitude A(t)

have to treat only a small frequency range around the central frequency

amplitudes intime domain

amplitudes infrequency domain

FFT

0( ) ( ) expE t A t i t

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Ultrashort Pulse Evolution in Fibers

Basic equation for propagating pulse amplitudes A(z, t):

0

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3

2

0

23

2.( , ) ( , )

1 ( , )

..2 6

( ) ( , ) d

A z t A

i

t

A z tz

i A z t R A z

i

t

t t

Need to treat dispersion and wavelength-dependent gain in the frequency

domain, nonlinear effects in the time domain.

Common method: split-step algorithm: switch between time and

frequency domain in such a way that the resulting numerical errors remain

small.

Automatic step size control is often necessary for efficient calculations.

Altogether, it’s not trivial …

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Effect of Chromatic Dispersion in Fibers

Basic equation for chromatic dispersion only:

2 332

2 3( , ) ... ( , )

2 6A z t i A z t

z t t

In the frequency domain:

2 332( , ) ... ( , )

2 6A z A z

z

Temporal broadening of Gaussian pulses

with second-order dispersion only:

Regime of strong broadening:

For non-Gaussian pulses and/or higher-order dispersion, the pulse shape

changes.

2

20 2

0

1 4ln2L

222 0

0

4ln2 if L

L

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Dispersive Pulse Broadening

Temporal broadening effect on Gaussian pulses with different initial

durations:

2

20 2

0

1 4ln2L

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Intermodal Dispersion

Different guided modes (in a multimode fiber) generally have different

group velocities: effect of waveguide dispersion.

This happens even in step-index fibers with high NA,

where the field is confined to an area with constant refractive index.

Generally, higher-order modes have smaller phase constants .

(Larger transverse wave vector components imply smaller longitudinal

components!)

Nevertheless, higher-order modes tend to have lower group velocities.

(Note: GVD involves frequency derivative of phase constant.)

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Intermodal Dispersion

Numerical simulation for short pulse and 50 cm long fiber:

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Intermodal Dispersion

Temporal evolution as animated image:

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Regimes of Pulse Amplification

Different pulse energy regimes:

Output pulse energy well below the saturation energy, high repetition rate:

boosting the average power with moderate pulse energy.

Output pulse energy above the saturation energy, low repetition rate:

extraction of stored energy (often limited by damage or nonlinear issues).

Different pulse duration regimes:

Nanosecond pulses: can reach substantial pulse energy with moderate

peak power.

Picosecond and femtosecond pulses: peak power limits pulse energy,

except if dispersively stretched pulses are amplified

( chirped-pulse amplification)

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Limitations of Average Power

Launched pump power and extracted output power for a double-clad

continuous-wave fiber amplifier can be multiple kilowatts

higher than for most competing technologies

Cooling issues are not severe. Reasons: long fiber length, short distances

for heat to propagate, waveguiding effect.

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Limitations of Stored Energy

Spontaneous emission limits the stored energy to the order of Pp 2.

Example: 1 W pump, 1 ms (Yb): 1 mJ

values similar to those of bulk devices; not the limiting factor

Amplified spontaneous emission (ASE) becomes strong when the gain

exceeds roughly 40 dB.

Gain efficiency in dB / μJ is very high.

Example: Yb at 1030 nm: Esat = 30 μJ for mode area 100 μm2

0.15 dB / μJ = 150 dB / mJ

cannot store 1 mJ, except for substantially larger mode area

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Limitations for Energy Extraction

Nonlinear signal distortions can set in when the nonlinear phase shift

reaches the order of .

Example: 1 m length, 1 μm wavelength, mode area 100 μm2,

n2 = 2.6 · 1020 m2 / W, Pp = 1 kW: nl = 1.6 rad

standard single-mode fibers start exhibiting nonlinear effects

at 1 kW over 1 m.

Note: 1 kW over 1 ns is only 1 μJ!

There are some ways to cope with stronger nonlinear phase shift,

for example chirped-pulse amplification and parabolic pulse amplification.

pnl 2

eff

2 Pn L

A

40

79

Limitations for Energy Extraction

Nonlinear self-focusing sets a hard limit to the peak power in a fiber:

Above this power, the beam collapses, and the glass is destroyed.

For silica, the limit is roughly 4 MW – independent of the mode area!

2 2 2

crit2 2

0.610.15

8P

n n n n

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Chirped-pulse Amplification (CPA)

General principle:

dispersively stretch the seed pulses

→ very low peak power

amplify the stretched pulses

→ still moderate peak power

dispersively compress the amplified pulses

→ high peak power only after the compressor,

not in the fiber

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Chirped-pulse Amplification (CPA)in Fiber Amplifiers

High fiber nonlinearity → require strong temporal stretching.

Large compressor dispersion and matching higher-order dispersion of stretcher and compressor can be challenging→ often non-ideal pulse quality.

Difficult to implement in all-fiber form:

Long fiber can often serve as stretcher,but not as compressor: high nonlinearity!

Fiber Bragg gratings (FBGs) provide huge dispersion in short length,but still limited peak power capability.

Therefore, often have to use bulk diffraction gratings.

→ Loose some of the advantages of fibers!

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Limitations ofChirped-pulse Amplification in Fibers

Maximum practical stretched-pulse duration for bulk diffraction gratings:

a few nanoseconds.

Hard peak power limit from self-focusing: a few megawatts.

Example: 3 MW · 3 ns = 9 mJ.

→ pulse energy limited to the order of 10 mJ

Problem for high-bandwidth pulses: limited gain bandwidth of fiber

(compared to Ti:sapphire, for example).

Problem for low-bandwidth pulses: need a huge amount of dispersion.

→ Method is not practical e.g. for >1-ps pulses with <1 nm bandwidth;

it works best in the moderate femtosecond regime.

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83

Parabolic Pulse Amplification

General idea: up-chirped pulses can undergo self-similar propagation

(“similariton pulses”) in a fiber amplifier:

temporal pulse shape stays close to parabolic

pulse bandwidth and duration scale with E1/3

pulses exhibit nearly linear chirp

→ dispersive compression leads to high quality pulses

where the pulse duration scales with E1/3

When beginning e.g. with unchirped pulses,

pulses asymptotically approach the self-similar parabolic regime.

Advantages:

appropriate stretching occurs automatically

maintain high pulse quality

84

Parabolic Pulses in Fiber Amplifiers

Realistic example case: ytterbium-doped pulse amplifier:

2 m long step-index fiber, GVD = 20’000 fs2/m

input pulses with 0.1 nJ, 100 fs, unchirped

150 mW pump at 940 nm, applied for a long time (low repetition rate)

→ amplifier gain 27 dB

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85

Parabolic Pulses in Fiber Amplifiers

Output pulse in time domain:

Note: step edges of temporal profile; up-chirp; parabolic phase.

86

Parabolic Pulses in Fiber Amplifiers

Output pulse in frequency domain:

Note: steep edges of spectral profile; wiggles; parabolic phase.

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87

Fiber Laser Systemsfor Higher Pulse Energies

MOPA approach: combine a low-power seed laser with a high-gain

amplifier system.

Can get millijoule output energies (at lower repetition rates) and/or kilowatt

average powers, but require complex systems with multiple amplifier

stages, stretchers and compressors etc.

May also use a gain-switched laser diode as seed laser: cheap source,

high flexibility with electronic control. But: low peak power, require more

amplifier gain.

Generally, high average power is easier to achieve than high pulse energy

→ high repetition rate systems tentatively more competitive

88

Multi-stage Fiber Amplifiers

Use two or more amplifier stages:

each one with its own active fiber and pump sources

with additional optical components in between

For example, may have a

preamplifier

main amplifier

power amplifier

This approach increases the complexity, but extends the possible

performance in various respects.

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89

Large Mode Areas of Power Amplifiers

Typically, the average power is much higher in the last stage (power

amplifier). At least, the pulse energies are much higher.

A larger mode area is then appropriate:

Keep nonlinear effects down: large mode area reduces optical

intensities.

For double-clad fibers, large core reduces the cladding/core area ratio,

thus increases the pump absorption

→ can use a shorter fiber → lower nonlinearity

Avoid excessive gain saturation for high-energy pulses.

Reduced gain is usually no problem here.

For high-energy amplifiers, need to store high energy without obtaining

excessive amplifier gain!

90

Small Mode Areas for Low-power Amplifiers

For a preamplifier, a smaller more area is appropriate:

Often like to have a high gain efficiency.

Achieve more efficient power conversion in low average power regime.

Profit from strict single-mode regime with robust guiding

(allows tight bending, e.g.).

Simple splicing and good compatibility with fiber-coupled

components due to standard fiber diameter and mode area.

No reasons for large mode area:

No nonlinear issues due to low peak powers.

No gain saturation issues.

No pump absorption issues (use single-mode pump source).

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91

Different Pumping Options

Low-power preamplifiers can be conveniently pumped with fiber-coupled

single-mode pump diodes.

High-power amplifiers require double-clad fibers, combined with suitable

pumping arrangements (free space or special pump couplers).

Forward pumping better for preamplifier in terms of amplifier noise,

while backward pumping is better for power amplifier:

lower ASE, lower nonlinear effects.

May want to apply bidirectional pumping for some stages.

92

Suppressing ASE

Amplified spontaneous emission (ASE) can be a big issue for very high

amplifier gains.

This is mitigated,

if the gain in each amplifier stage remains moderate (e.g., < 40 dB), and

no ASE can get from one stage to another.

How to distribute the gain?

Power amplifiers often have moderate gain from lower gain efficiency.

Preamplifiers can easily reach 30 or 40 dB.

If more is needed, use more stages.

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93

Suppressing ASE

How to kill ASE between amplifier stages?

First consider how ASE differs from the wanted signals:

ASE is very broadband, while the signal bandwidth may be smaller.

→ bandpass filters help

if signal bandwidth is far smaller than ASE bandwidth

ASE is temporally spread, while signal may be ultrashort

→ use an optical switch such as an acousto-optic modulator (AOM)

Note: AOM is “slow”: opens for a time much longer than a short pulse.

can strongly reduce loss in average power,

but cannot suppress an ASE pedestal around each pulse.

94

Suppressing ASE

ASE is bidirectional, while signal goes only in one direction

→ use a Faraday isolator to kill backward ASE

(Note: backward ASE is particularly nasty!)

ASE is unpolarized, while signal may be polarized

→ may use a polarizer (but don’t achieve that much)

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95

Minimizing Amplifier Noise

Amplifiers inevitably do not only amplify noise in the input, but also add

some noise (“excess noise”) to the signal.

Excess noise is at the quantum limit in the ideal case of a four-level

amplifier with no extra power losses (e.g. due to splice losses or scattering

in fiber).

The noise figure is then 3 dB for a high-gain amplifier:

output noise power is two times that for a quantum-limited input.

Additional excess noise comes from extra losses, particularly in quasi-

three-level amplifiers (Yb, Er, …). How much, depends on the excitation

level: low extra noise if excitation level is high near the input end.

Therefore, forward pumping of the preamplifier at a short wavelength is

ideal.

96

Designing Multi-stage Fiber Amplifiers

Obviously, a trial & error approach is particularly inappropriate here.

Need to acquire a good quantitative understanding of all relevant effects,

such as

nonlinearities

power conversion

gain saturation

ASE

Numerical simulations are usually needed: too complex situation for using

a couple of simple equations. However, some initial analytical calculations

are often very helpful.

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The End Many scientific and technical issues are non-trivial, sorry!

But hopefully you have learned a lot.

Questions and feedback are welcome.

Take RP Photonics flyers as you like.