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Powerful simulation software for fiber lasers and amplifiers, resonator design, pulse propagation and multilayer coating design.

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# Paraxial Approximation

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Definition: a frequently used approximation, essentially assuming small angular deviations of the propagation directions from some beam axis

German: paraxiale Näherung

Many calculations in optics can be greatly simplified by making the paraxial approximation, i.e. by assuming that the propagation direction of light (e.g. in some laser beam) deviates only slightly from some beam axis.

## Paraxial Approximation in Geometric Optics

Geometric optics (ray optics) describes light propagation in the form of geometric rays. Here, the paraxial approximation means that the angle θ between such rays and some reference axis of the optical system always remains small, i.e. < < 1 rad. Within that approximation, it can be assumed that tan θ ≈ sin θ ≈ θ. The evolution of beam offset (distance from the reference axis) and beam angle in some optical system can then be described with simple ABCD matrices, because there are linear relations between offset and angle of beams before and after some optical component or system.

## Paraxial Approximation in Wave Optics

When describing light as a wave phenomenon, the local propagation direction of the energy can be identified with a direction normal to the wavefronts (except in situations with spatial walk-off). If the paraxial approximation holds, i.e. these propagation directions are all close to some reference axis, a second-order differential equation (as obtained from Maxwell's equations) can be replaced with a simple first-order equation. Based on this equation, the formalism of Gaussian beams can be derived, which gives a much simplified understanding of beam propagation and of fundamental limitations such as the minimum beam parameter product. Essentially, the paraxial approximation remains valid as long as divergence angles remain well below 1 rad. This also implies that the beam radius at a beam waist must be much larger than the wavelength.

The propagation modes of waveguides, particularly of optical fibers, are also often investigated based on the paraxial approximation. The validity of the analysis is then restricted to cases with a sufficiently large effective mode area and sufficiently small divergence of any beams exiting such a waveguide.

The paraxial approximation is very well fulfilled in a wide range of phenomena of laser physics and fiber optics, but it is clearly violated in cases with very strong focusing, where commonly used equations such as θ = λ / (πw0) for the divergence angle break down. In that regime, polarization issues also demand special care. In particular, polarization components in the propagation direction can occur. For such reasons, the simulation of beam propagation then requires significantly more sophisticated methods. For example, beam propagation methods (propagating a two-dimensional array of complex field amplitudes) can be used which do not need that approximation.

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# RP Fiber Power – the versatile Fiber Optics Software

## An Amazing Tool

This amazing tool is extremely helpful for the development of passive and active fiber devices.

Watch our quick video tour!

## Single-mode and Multi­mode Fibers

Calculate mode properties such as

• amplitude distributions (near field and far field)
• effective mode area
• effective index
• group delay and chromatic dispersion

Also calculate fiber coupling efficiencies; simulate effects of bending, nonlinear self-focusing or gain guiding on beam propagation, higher-order soliton propagation, etc.

## Arbitrary Index Profiles

A fiber's index profile may be more complicated than just a circle:

Here, we "printed" some letters, translated this into an index profile and initial optical field, propagated the light over some distance and plotted the output field – all automated with a little script code.

## Fiber Couplers, Double-clad Fibers, Multicore Fibers, …

Simulate pump absorption in double-clad fibers, study beam propagation in fiber couplers, light propagation in tapered fibers, analyze the impact of bending, cross-saturation effects in amplifiers, leaky modes, etc.

## Fiber Amplifiers

For example, calculate

• gain and saturation characteristics (for continuous or pulsed operation)
• energy transfers in erbium-ytterbium-doped amplifier fibers
• influence of quenching effects, amplified spontaneous emission etc.

in single amplifier stages or in multi-stage amplifier systems, with double-clad fibers, etc.

## Fiber-optic Telecom Systems

For example,

• analyze dispersive and nonlinear signal distortions
• investigate the impact of amplifier noise
• optimize nonlinear management and the placement of amplifiers

Find out in detail what is going on in such a system!

## Fiber Lasers

For example, analyze and optimize the

• power conversion efficiency
• wavelength tuning range
• Q switching dynamics
• femtosecond pulse generation with mode locking

for lasers based on double-clad fiber, with linear or ring resonator, etc.

## Ultrafast Fiber Lasers and Amplifiers

For example, study

• pulse formation mechanisms
• impact of nonlinearities and chromatic dispersion
• parabolic pulse amplification
• feedback sensitivity
• supercontinuum generation

Apply any sequence of elements to your pulses!

## … and even Bulk Devices

For example, study

• Q switching dynamics
• mode-locking behavior
• impact of nonlinearities and chromatic dispersion
• influence of a saturable absorber
• chirped-pulse amplification
• regenerative amplification

RP Fiber Power is an extremely versatile tool!

## Mode Solver

For example, calculate

• amplitude and intensity profiles
• effective mode areas
• cut-off wavelengths
• propagation constants
• group velocities
• chromatic dispersion

All this is calculated with high efficiency!

## Beam Propagation

Propagate optical field with arbitrary wavefronts through fibers. These may be asymmetric, bent, tapered, exhibit random disturbances, etc.

## Laser-active Ions

Work with the standard gain model, or define your own level scheme!

Can include different ions, energy transfers, upconversion and quenching effects, complicated pumping schemes, etc.

## Multiple Pump and Signal Waves, ASE

Define multiple pump and signal waves and many ASE channels – each one with its own transverse intensity profile, loss coefficient etc.

The power calculations are highly efficient and reliable.

## Simple Use and High Flexibility Combined

For simpler tasks, use convenient forms:

Script code is automatically generated and can then be modified by the user. A powerful script language gives you an unparalleled flexibility!

## High-quality Documentation and Competent Support

The carefully prepared comprehensive documentation includes a PDF manual and an interactive online help system.

Competent technical support is provided: the developer himself will help you and make sure that any problem is solved!

Our support is like included technical consulting.

## Boost your competence, efficiency and creativity!

• Stop fishing in the dark! Develop a clear quantitative understanding of your devices.
• Explore the effects of possible design changes on your desk.
• That way, get most efficient in the lab.
• Find optimized solutions efficiently, minimizing time to market.
• Get new ideas by playing with your models.

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