Robert:•Motivation•Principles of Optics•Applications •Optimization

Andy:Materials

Loss vs. amplification

Theoretical problems

Overview

2 + 2 = 4WM

Motivation

1. Internet relies on fiber optics.

2. Amplification needed.

Current technology inadequate:

Limited amplification bandwidth

Limited internet speed

Linear OpticsLow intensity light in transparent media.

•Refraction

•Dispersion

Light slows down in transparent media.

Refractive index is function of frequency.

Propagation constantBeta (propagation constant) very useful.

Expressible as power series.

Coefficient critical to optimizing FWM.

Limitations1. Photons do not interact.

2. No new frequencies are created.

3. Too simple for our purposes.

But nonlinear optics provides uswith great possibilities…

Nonlinear Optics•Kerr effect: refractive index depends on

intensity of light.

•Nonlinearity causes complex behavior.

•Nonlinear Schrödinger Equation

Photons can mix and change their frequencies!

Nonlinear Term

Four-wave Mixing

Signal

Pump Lasers Photons added to signal

Photons added to idler

Idler(createdto conserve energy)

(Amplified through FWM)2 + 2=4WM

Frequency (ω)/100THz

Log

(In

tensi

ty)

Pump photons mix to form signal and idler photons.

Elastic Collision Analogy

Energy Conservation:

MomentumConservation:

Pump energies

Energies of signal and idler

Pump momenta

Momenta of signal and idler

ApplicationsWhat can we use it for?

•Amplification and Frequency conversion.

•Solves world hunger (for internet speed)

Optimization:•How do we turn ideas into high performance technology?

mathematical analysis and approximation.

Amplification Optimization•Amplification depends on only one number.

•Must be close to –γP for maximum gain.

•Complexity of β solved by quartic approximation.

Conditions for Maximum Flat Gain

1. Average pump frequency at zero dispersion point ω00.

Where:

2. β4 4 must be positive.

3. And lastly, regarding the pumps:

Before and After Optimization

Signal Frequency Offset

Gain

Inferior bandwidth

Optimized bandwidth

Frequency Conversion OptimizationIdler photons used as new signal:

Useful since different frequencies needed in fiber.

Problem: pumps: same average frequency as “a” and “b.”

Stuck with bandwidth we’re given…

Dispersion Engineering•Optical fibers:

Total internal reflection

•Light strays into cladding.

•Samples 2 refractive indices.

•We can engineer β22, , β3, 3, β4 etc.4 etc.

n22n11

Frequency Conversion OptimizationIdler photons used as new signal:

Useful since different frequencies needed.

Problem: pumps: same average frequency as “a” and “b.”

Stuck with bandwidth we’re given…

Solution: dispersion engineering: minimize β44 Make β3 3 and β4 4 into “magic ratio.”

Creates greater bandwidth.

Summary

Optical Fiber

Nonlinear effect ∝ γPL

Silica Low loss Low nonlinearity γ

High P and L needed for FWM

Silica V.S. Chalcogenide

Silica Chalcogenide

Made of

SiO2 S, Se, Te+others

γ Low High

Used in Optical Fiber Optical Chip

Loss Low High

Nonlinear Schrodinger Equation (NLS)

uiuut

u

z

ui

2

2

2

)(2

1

Linear loss coefficient

Numerically solve NLS with loss(Split step Fourier method)

How loss affects gains

1 pump case

Signal GainIdler Gain

γ+γ- INPUTOUTPUT

1 pump case

INPUTOUTPUT

Gain curve – 1 pump

Gain curve – 1 pump

α = (dB/m) 0 20 40 60 80 100

Chalcogenide

Peak Gain– 1 pump

loss ∝ e-αL

(dB/m)

2 pump case INPUTOUTPUT

Signal Gain

Idler Gain

2 pump case

Gain curve - 2 pump case

Signal Gain Idler Gain

Asymmetry Problem

Conclusion FWM : nonlinear optical effect

Parametric amplifications

Conditions for greater bandwidth

How loss affects gain curves— unexpected!!

Future Work

Asymmetry Problem

Coping with loss

References

• C. J. McKinstrie, S. Radic and A. R. Chraplyvy. Parametric Amplifiers Driven by Two Pump Waves. IEEE J. Quantum Electron., vol.QE-8, pp. 538–547, 2002.

• G. P. Agrawal (2001). Nonlinear Fiber Optics. Orlando: Academic Press.

• M. R. Lamont, T. T. Kuhlmey and C. M. de Sterke. Multi-order dispersion engineering for optimal four-wave mixing. Optics Express, vol.16, pp. 7551–7563, 2008.

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