Diffractive origin of fractal resonator modes

New, G.H.C.; Yates, M. A.; Woerdman, J. P.; McDonald, GS

doi:10.1016/s0030-4018(01)01237-8

Cited by 28 publications

(18 citation statements)

References 6 publications

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“…(10), the exponent in each oscillatory term is a quadratic function of x, rather than a linear function. This gives rise to complicated substructure in the Fourier transform of ux, as has been noted already [6] and as we will discuss in Section 4. To side step this complexity, we now employ a more natural strategy, based on linearization of the exponent.…”

Section: Scaling Of the Edge Wave Summentioning

confidence: 73%

“…7 shows that for a given mode P k possesses a complicated structure. Some of this has been interpreted [6] in terms of the individual magni®ed Fresnel edge-diraction patterns; the edges of the`bands' can be calculated from Eq. (14) by setting x AE1.…”

Section: Fractal Implicationsmentioning

confidence: 99%

“…(14) by setting x AE1. The slope À2 of the straight-line parts of the spectrum was also noted [6], and interpreted according to Eq. (20) as implying D 3=2 for ux, in contradiction to the variable scaling described in the previous section.…”

Section: Fractal Implicationsmentioning

confidence: 99%

“…Recently, it was suggested [4,5] that the repeated geometrical magni®cations associated with the instability might imply that the modes have a fractal structure, and an attempt has been made [6] to calculate the fractal dimension D of the intensity pro®le by numerical and analytical means. The self-similarity associated with the magni®cation has been studied in Ref.…”

Section: Introductionmentioning

confidence: 99%

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Theory of unstable laser modes: edge waves and fractality

Berry

Storm

Saarloos

2001

Optics Communications

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For large Fresnel numbers N, unstable laser modes are highly irregular and resemble fractals. To explore this, we derive an explicit formula for the lowest-loss mode of a one-dimensional laser (i.e. where the cavity is two dimensional) in terms of edge-diracted waves, and demonstrate its accuracy for large N. Between the size a of the mirror (outer scale), and the inner scale a=N , there is no distinguished scale, and the graph of mode intensity has a fractal dimension close to 2. Near the inner scale, the scaling is scale dependent, and the crossover is described by an explicit formula for à local fractal dimension' DK, describing the mode on scales near Dx a=2pNK. As K increases through the inner scale K 1, DK decreases from 2 when K ( 1 to 1 when K ) 1 (re¯ecting the smoothness of the mode on ®ne scales). Ó

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Section: Scaling Of the Edge Wave Summentioning

confidence: 73%

Section: Fractal Implicationsmentioning

confidence: 99%

Section: Fractal Implicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Theory of unstable laser modes: edge waves and fractality

Berry

Storm

Saarloos

2001

Optics Communications

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“…In the late 1990s, Karman and Woerdman [1] found that the eigenmodes of one-dimensional (1D) confocal resonators are fractals -patterns that exhibit comparable levels of detail spanning many scalelengths. The fractality of these self-reproducing mode profiles was later shown to originate in the interplay between periodic aperturing and diffraction at the outer boundary of the system (i.e., the feedback mirror) [2].…”

Section: Introductionmentioning

confidence: 98%

Kaleidoscope Lasers - Complexity in Simple Optical Systems

Christian

McDonald

Huang

2010

Advanced Photonics &Amp; Renewable Energy

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On the Computation of Geometric Features of Spectra of Linear Operators on Hilbert Spaces

Colbrook

2022

Found Comput Math

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Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences. However, in many applications gaining an approximation of the spectrum is not enough. Often it is vital to determine geometric features of spectra such as Lebesgue measure, capacity or fractal dimensions, different types of spectral radii and numerical ranges, or to detect gaps in essential spectra and the corresponding failure of the finite section method. Despite new results on computing spectra and the substantial interest in these geometric problems, there remain no general methods able to compute such geometric features of spectra of infinite-dimensional operators. We provide the first algorithms for the computation of many of these long-standing problems (including the above). As demonstrated with computational examples, the new algorithms yield a library of new methods. Recent progress in computational spectral problems in infinite dimensions has led to the solvability complexity index (SCI) hierarchy, which classifies the difficulty of computational problems. These results reveal that infinite-dimensional spectral problems yield an intricate infinite classification theory determining which spectral problems can be solved and with which type of algorithm. This is very much related to S. Smale’s comprehensive program on the foundations of computational mathematics initiated in the 1980s. We classify the computation of geometric features of spectra in the SCI hierarchy, allowing us to precisely determine the boundaries of what computers can achieve (in any model of computation) and prove that our algorithms are optimal. We also provide a new universal technique for establishing lower bounds in the SCI hierarchy, which both greatly simplifies previous SCI arguments and allows new, formerly unattainable, classifications.

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Diffractive origin of fractal resonator modes

Cited by 28 publications

References 6 publications

Theory of unstable laser modes: edge waves and fractality

Theory of unstable laser modes: edge waves and fractality

Kaleidoscope Lasers - Complexity in Simple Optical Systems

On the Computation of Geometric Features of Spectra of Linear Operators on Hilbert Spaces

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