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## Corrections for the Case of Graded-Index Optical Fibers

Closing the section about optical fibers is a brief point about the so-called ‘graded-index’ fibers (which will be relevant for the experimental work presented in following chapters, where such a fiber was used). The definition and analysis relies on references [10] and [11].

Graded-index (GRIN) fibers are fabricated with a profile of refractive index that gradually changes across the core, as a function of the radial distance from the center. The maximum index, which we shall denote , is found at the center of the guiding core; moving away towards the cladding the refractive index falls according to the following power law, commonly referred to as an ‘alpha-power’ profile: ( ) { √ ( ⁄ ) (1.10) √0.

Where, as before, denotes the radius of the core. The parameter dictates the smoothness of the profile as a function of the radial coordinate : for , the fiber will approach the step-index profile which has been studied so far in this section, while for the profile will constitute a parabolic one. Usually, graded-index fibers are fabricated with alpha-powers of a little more than 2, depending on the desired modal behaviour. Finally, the parameter scales the index profile in the vertical axis, or in other words determines the total index change between the center and the uniform value found throughout the cladding. Similarly to the relation eq. (1.1) expressed for the step-index case, it may be shown that the GRIN fiber’s numerical aperture is √ .

Although the analytic treatment of the modes’ behavior (characteristic equation, field distribution, dispersion etc.) is more complicated for the step-index case, it should be understood that the essential features remain the same: a set of discrete LP modes may be found, defined by two parameters (angular and radial, respectively), and it serves as an orthogonal base for all lightfields guided in the fiber. For our purposes in this work, it shall suffice to state that the total number of solutions to the characteristic equation may be approximated by: { 0} ( ) (1.11).

Again, due to the degeneracies of mode and polarization orientation (same as for eq. (1.5)), the number of modes is nearly four times larger than the number of solutions. From eq. (1.11) we can see that as the alpha parameter decreases (i.e. the fiber’s profile becomes more graded), less modes are supported; the parabolic GRIN fiber ( ) guides half the number of modes as the equivalent stepindex fiber.

Lastly, we mention that as the alpha parameter decreases and the refractive index profile departs from the step-index form, the modes’ field distributions scale in the radial coordinate to become more confined within the core. Generally, exact solutions require numerical approaches. An approximation for the fundamental mode was derived, by a trial and error approach, in reference [11]. A formula (depending on a large number of heuristically-found constants) for the factor by which the waist of the mode is reduced as a function of may be found there.

### The Gain Medium and the Rate Equation of an Amplifier

The subject of this section is the interaction of light with the electronic states of an atom that is laser-active, i.e. capable of acting as an optical gain system. The practical example kept in mind is that of rare-earth elements incorporated as dopants into an optically transparent environment such as silica, as was the case in our work with doped fibers. Nonetheless, the theory covered is general and may equally describe other types of amplifying media, such as the noble gases found in common gas lasers.

The theory presented follows the fundamental textbook of Davis [17], as well as the papers of Barnard [12] and Paschotta [13]. Again, this is a well-known and established subject, for which many sources may be found; the following section therefore covers it in a concise and abridged manner, so as to quickly arrive at the important definitions and results.

#### The Two-Level Gain System and the Rate Equations

A photon may interact with a pair of energy levels in the medium by either being absorbed so that an electron is excited from the lower level to the higher one, or by being created (i.e. emitted) when an electron makes the opposite transition. The latter might take place as a spontaneous emission process, or by the stimulated emission interaction, wherein an existing photon induces the electron to decay from the upper level by emitting a new photon, identical in its properties to the first. To each of the three interactions is assigned a probability (the well-known Einstein coefficient), as shown in the following illustration: The medium’s ability to emit photons raises the prospect of optical gain; however, the important notion of population inversion must first be introduced. The third process depicted in figure 1.6 is the basis of optical amplification – an existing photon may be duplicated by the downwards transition of an electron; but it should be understood that the photon’s annihilation by the opposite transition upwards (i.e., the first process depicted in the same figure) – is equally permitted. The physical principle named the Einstein coefficient relation demonstrates from that in fact, for a pair of (non-degenerate) energy levels in thermodynamic equilibrium, the probability of absorption and of stimulated emission must be identical: The result is that if an equal number of electrons is found in the upper level of the relevant transition as in its bottom level – the rate at which signal photons are created (by stimulated emission) and annihilated (by absorption) would be the same, with a zero net gain. Therefore, in order for light amplification to take place, the population at the higher energy level must be larger than the one at the lower level.

**Spectral Dependence of the Gain and Line Broadening**

All the characteristics discussed above, which determine the behavior of the gain process, are wavelength dependent; Both the pump and the signal wavelengths must lie in regions of the spectrum for which (in the specific gain medium chosen) their absorption and emission cross-sections, respectively, are significant enough to enable efficient population excitation and amplification. Considering the fact that both processes are resonant interactions of a photon with an electronic transition, it is indeed expected that all relevant cross-sections will attain a maximum at a central wavelength which corresponds to the energy difference of said transition through , and fall off away from this frequency with some finite spectral width.

For the gain media discussed in our work, the widths of the spectral curves around the absorption and emission peaks are at least a few nm, and more typically a few tens of nm. This is several orders of magnitude larger than the width one may naively expect to find when viewing these interactions as with a classical electromagnetic resonance, that is – taking the inverse of the lifetime of the upper energy levels (the so-called ‘natural linewidth’). This is because significant broadening mechanisms, occurring mainly because of interactions between the gain atoms and their surrounding environment, ‘smear’ the energy of the electronic levels over a wide range of possible values, thereby enlarging the range of optical frequencies over which a photon may match the transition’s energy difference.

An important distinction should now be noted between ‘homogenous’ broadening mechanisms and ‘inhomogeneous’ones. It is true, that when considering an amplifier, the details of this distinction are of no crucial importance; Once the specific pump and signal wavelengths are chosen, they remain constant, and each of them “sees”, where they happen to fall within the spectral gain curve, some fixed absorption and emission coefficients, which may be either directly measured or taken from tables in the literature. However, for the case of laser system – as will be discussed further in a later section – the inner details of the line broadening phenomena are of great interest.

When considering the ensemble of all gain atoms incorporated in some gain medium, then: A mechanism of Homogenous Broadening is one that equally affects (broadens the transitions of) all atoms in the ensemble. One example is the thermal exchange all atoms perform with their environment; since the temperature is homogenous over the system, all lines are identically broadened. Another example is the Stark broadening – the fact that the energy levels split into a manifold of sub levels in the presence of an electric field. Conversely, inhomogeneous Broadening mechanisms are perturbations to the transition energy difference, which vary between different sub-groups of atoms within the overall ensemble. An example is local variations within the host material in which the gain atoms are embedded; different sites induce different shifts of the central frequency of the transition, resulting in a total gain curve which is a broadened, ‘smeared’ overlay of the gain curves of sub-group of atoms (sometimes dubbed spectral packets), corresponding to different sites throughout the gain medium. Each one of these ‘packets’ will have the same linewidth, namely the one caused by the homogeneous broadening.

**Control of Light Propagation in Complex Media by Wavefront Shaping**

Diffractive optical elements (DOE) able to efficiently convert a singlemode beam into one with a spatially-complex wavefront – such as diffractive beam shapers/beam splitters – have been around for a long time, practically since the invention of holography. These elements are traditionally not reconfigurable, meaning the modulation they apply upon the wavefront is fixed. Since the 1990’s, with the advent of digitally-programmable diffractive devices – such as deformable mirrors, micro-electro-mechanical (MEMS) mirror arrays, and LCOS (liquid crystal on silicon) spatial light modulators – an intriguing prospect opened up: the use of spatially-complex wavefront shaping for the control of light in disordered media.

It is true that reconfigurable wavefront shaping already existed since the 1970’s as a topic of research, notably in the field of astronomy – where optical devices had been developed for compensating the low-order aberrations created by earth’s atmosphere, resulting in significant improvement of the sharpness of telescope images [1]. However, the aberrated wavefronts found in these applications may be considered as having undergone only weak scattering. The first use of wavefront shaping for the control of light in a truly strongly scattering medium was demonstrated in 2007, in the seminal work of the complex photonic group at Twente [2-3]. Using a liquid-crystal modulator, the local phases across the wavefront of an initially-plane-wave laser beam were optimized so as to create a strong, tight focus point beyond layers of strongly-scattering materials such as TiO2, ZnO, and biological tissue.

**Linearity of the Transmission and the Transmission Matrix**

The surprising ability of spatial modulation to form well-controlled intensity patterns on the other side of a disordered, non-absorbing but strongly scattering medium (such as was defined in section 1.1), which is essentially opaque to ‘traditional’ imaging schemes, was explained as follows: The relation between the input optical field (specifically, the one impinging upon the modulator) and the field after the scattering medium (specifically, the one formed upon the camera sensor) is not only deterministic, but linear in the spatial coordinates. Stated more explicitly – if we segment the modulator plane into N sub-parts (namely, pixels), and likewise identify M pixels of interest at the camera plane, then the field at each of these output pixels (which we shall denote ) is a linear combination of the contributions of each of the input pixels: ∑0 (2.1).

and the particular way the light waves have propagated to, within, and from the medium – including the scattering processes by its disorder – are ‘hidden’ in the complex coefficients of transmission .

**Measurement of the Transmission Matrix**

An important issue concerning the measurement of the transmission matrix should be mentioned, and that is the need to measure the output field, meaning both the amplitude and the phase of the light. The common method of doing so by using a sensor sensitive only to intensity (as is almost always the case in optics) is based upon the interference effect, which, indeed, is the effect responsible for the appearance of the speckle pattern in the first place. The method requires however that a constant reference field – i.e., one that is guaranteed not to be modulated throughout the measurement procedure – be superposed, in the output plane, together with the field we wish to measure. We denote the former , and the latter , because it is the field controlled by our modulator. Then, once these two fields are summed up together at the camera sensor, the intensity detected is the well-known expression that appears in any homodyne detection scheme: || | | | | | | { } (2.3).

The first two terms are the stand-alone intensities of each of the two patterns. The third term is the interesting one, because it is sensitive to the phase of the field we wish to measure; suggesting the possibility of extracting the phase from intensity-only measurements. However, this is not so straightforward – the phase modulates this term only through the interference with the reference field. At this point, the distinction between two different popular techniques of applying and using the reference field should be discussed.

The first possible approach is the off-axis holography method, which employs an external reference field – meaning, one that has not gone through the scattering medium. This field is always split-off from the light source as a separate arm, so that it shares a common phase with the main arm, regardless of the ever-present noise and drift of the source’s phase. The reference arm is led around the disordered medium to arrive at the output detection plane (namely, the camera sensor) as a plane wave with a slight misalignment of its beam with respect to the main one (say by some angle ). Combining the two beams at the sensor gives rise to interference fringes whose spatial frequency is determined by the misalignment angle thus: { } { } ( ) (2.4).

A Fourier transform of the image then readily gives access (by looking at the 1st Fourier order) to both the amplitude and the phase of . An example of the use of this method, from ref. [8], is shown in the left panel of figure 2.3.

**Table of contents :**

**1 Theoretical Background **

1.1 Scattering of Coherent Light by Disorder

1.2 Light Guidance in Ideal Passive Optical Fibers

1.2.1 The Guided Modes

1.2.2 Optical Fibers as Examples of Complex Media

1.2.3 Corrections for the Case of Graded-Index Optical Fibers

1.3 The Gain Medium and the Rate Equation of an Amplifier

1.3.1 The Two-Level Gain System and the Rate Equations

1.3.2 Spectral Dependence of the Gain and Line Broadening

1.4 Lasers

1.4.1 The Passive Fabry-Perot Cavity

1.4.2 The Cavity with Gain

1.4.3 Mode Competition within the Lasing Cavity

**2 Wavefront Shaping in Non-Amplifying Complex Media and Multimode Fibers **

2.1 Control of Light Propagation in Complex Media by Wavefront Shaping

2.1.1 Linearity of the Transmission and the Transmission Matrix

2.1.2 Measurement of the Transmission Matrix

2.1.3 Focusing Using the Transmission Matrix

2.2 Wavefront Shaping in Passive Optical Fibers

2.2.1 Multimode Fibers in Telecommunication

2.2.2 Multimode Fibers in Microscopy and Biomedical Applications

2.2.3 Approaches for Shaping the Input Wavefront and Coupling into the MM Fiber

2.3 Focusing Light through a Passive Multimode Fiber – Numerical Simulations

2.3.1 Focus Spot Size

2.3.2 Focus Spot Contrast and the Confinement Metric

2.4 Focusing Light through a Passive Multimode Fiber – Experimental Results

2.4.1 Motivation and Constraints

2.4.2 Ensuring Full Control in the Modal Space: The Reference Beam Problem

2.4.3 Benchmark Experiment with SLM image-conjugated to the Fiber Facet

2.4.4 Ensuring Full Control in tModal Space: The SLM Physical Positioning Problem

2.5 Summary and Discussion

**3 Model and Numerical Simulations of a Multimode Fiber Amplifier with Configurable Pumping **

3.1 Fiber Amplifiers – Background and Major Developments

3.1.1 General Background

3.1.2 Motivation for Multimode or Potentially-Multimode Fiber Amplifiers

3.1.3 Wavefront Shaping in Multimode Fiber Amplifiers – Literature Survey

3.2 Proposed Model for the Gain-Dependent TM of a Multimode Fiber Amplifier

3.2.1 Transmission Matrix for Realistic Passive Optical Fibers

3.2.2 Incorporation of Gain into the MMFA Model

3.2.3 Qualitative Analysis of Controllability Based on the Model

3.3 Numerical Simulations of a Multimode Fiber Amplifier

3.3.1 Specific Values for the Parameters Determining Amplifier Performance

3.3.2 Effect of MM pumping on the Signal – Statistics

3.3.3 Effect of MM pumping on the Signal – Optimization in the Modal Domain

3.3.4 Control of the Signal in the Spatial Domain by Optimization

3.3.5 Towards the Experimental System

3.4 Summary and Discussion

**4 Experimental Results for Pump Shaping in Amplifier Configuration **

4.1 Set-up Configuration

4.1.1 Signal Branch

4.1.2 Pump Branch

4.1.3 Imaging of the Output

4.2 Yb-doped Fiber Characterization

4.2.1 Fiber Specifications

4.2.2 Signal Coupling and Propagation

4.2.3 Absorption and Saturation Measurements

4.2.4 Pump Coupling and Propagation

4.2.5 Design Considerations – Choice of Fiber Length

4.2.6 Benchmark for Effects of Gain – Passive Fiber ‘Benchmark’ Experiment

4.3 Results of Pump Wavefront Shaping Experiments

4.3.1 Gain Effects on the signal speckle – baseline vs. variations by pump shaping

4.3.2 Results of Pump Shaping – Maximizing Signal Decorrelation

4.4 Discussion and Summary

**5 Wavefront Shaping of the Pump in a Lasing Cavity Configuration **

5.1 Background and Motivations

5.2 Experimental Setup and Some Elementary Results

5.2.1 Experimental Setup and Typed of Fiber-Based Cavities Implemented

5.2.2 Lasing in the fiber-only cavity

5.2.3 Lasing in the cavity based on external optics

5.2.4 Range of Lasing Wavelengths in Both Cavity Types

5.3 Theoretical Discussion of Lasing In Fiber-Based Cavities

5.4 Main Experimental Results

5.4.1 Evidence of Modal Discrimination through the Pump Shaping

5.4.2 Measurement of Slopes in a Cavity With Fully MM Guiding (no SM Filter)

5.4.3 Measurement of Slopes in a Cavity With a Spatial Filter– a Spliced SM Pigtail

5.5 Summary and Conclusions

**General Conclusion and Prospects **

Appendix A: Numerical Simulation Tool for the Computing the MMFA Transmission Matrix

Appendix B: Spatial Light Modulation Masks Corresponding to the Experimental Results in the Amplifier Configuration

Appendix C: Spatial Mode-Mixing in a Fiber-Based Cavity Defined By Free-Space