Book contents
- Frontmatter
- Contents
- Acknowledgments
- 1 Introduction
- 2 Properties of single-mode optical fibers
- 3 Scalar OPA theory
- 4 Vector OPA theory
- 5 The optical gain spectrum
- 6 The nonlinear Schrödinger equation
- 7 Pulsed-pump OPAs
- 8 OPO theory
- 9 Quantum noise figure of fiber OPAs
- 10 Pump requirements
- 11 Performance results
- 12 Potential applications of fiber OPAs and OPOs
- 13 Nonlinear crosstalk in fiber OPAs
- 14 Distributed parametric amplification
- 15 Prospects for future developments
- Appendices
- A.1 General theorems for solving typical OPA
- A.2 The WKB approximation
- A.3 Jacobian elliptic function solutions
- A.4 Solution of four coupled equations for the six-wave model
- A.5 Summary of useful equations
- Index
- References
A.2 - The WKB approximation
Published online by Cambridge University Press: 23 March 2010
- Frontmatter
- Contents
- Acknowledgments
- 1 Introduction
- 2 Properties of single-mode optical fibers
- 3 Scalar OPA theory
- 4 Vector OPA theory
- 5 The optical gain spectrum
- 6 The nonlinear Schrödinger equation
- 7 Pulsed-pump OPAs
- 8 OPO theory
- 9 Quantum noise figure of fiber OPAs
- 10 Pump requirements
- 11 Performance results
- 12 Potential applications of fiber OPAs and OPOs
- 13 Nonlinear crosstalk in fiber OPAs
- 14 Distributed parametric amplification
- 15 Prospects for future developments
- Appendices
- A.1 General theorems for solving typical OPA
- A.2 The WKB approximation
- A.3 Jacobian elliptic function solutions
- A.4 Solution of four coupled equations for the six-wave model
- A.5 Summary of useful equations
- Index
- References
Summary
In Chapter 3 we considered OPAs in which the parametric gain coefficient g is a constant along the fiber. This allowed us to obtain closed-form solutions for the gain in several important situations. In practice, however, fibers generally have properties that may cause g to vary as a function of z. Examples of these properties are: fiber loss, which causes the pump power to drop exponentially; non-uniform dispersion, which causes Δβ to fluctuate; birefringence (fixed or randomly varying), which introduces a complex evolution of SOPs. Under these circumstances the constant-g solutions are no longer applicable.
In several areas of physics that involve wave propagation in media with slowly varying properties, one often uses approximate solutions derived by making use of the Wentzel–Kramers–Brillouin (WKB) approximation, also referred to as the phase-integral method [1, 2]. This method was originally introduced in quantum mechanics. In optics, it has been used extensively to study propagation in multimode fibers [3]. It can lead to closed-form solutions if the properties vary in a simple fashion, such as linearly. If the variations are not simple, one may still be able to use the method to obtain some useful expressions involving the average of g along the fiber.
Karlsson first applied the WKB method to fiber OPAs, in the context of modulation instability (MI) [4]. Here we present a slightly different version, starting from the basic OPA equations.
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- Publisher: Cambridge University PressPrint publication year: 2007