Photonics

Applications for Mode-Locked Lasers

- Short pulses allow for time-resolved measurements, e.g. electro-optic sampling measurements on integrated electronic circuits, or pump-probe measurements on semiconductor devices such as SESAMs. - Various methods of imaging, laser microscopy and laser spectroscopy greatly profit from short pulses for various reasons. For example, the high peak powers of femtosecond lasers are useful in two-photon absorption fluorescence microscopes, reaching a very high spatial resolution in all three dimensions. - In the field of optical metrology, mode-locked lasers can be used for distance measurements, but also in frequency metrology (time keeping) and other fields. In the context of frequency metrology, the frequency combs of mode-locked lasers play a particularly important role. - A number of processes for nonlinear frequency conversion are greatly facilitated by the high peak powers of mode-locked lasers, even when the average power remains moderate. - Other fields with a large potential are microwave, millimeter-wave and terahertz optics, and picosecond optoelectronics. Mode-locked lasers are also often combined with ultrafast amplifiers for obtaining higher average powers and in particular higher pulse energies and peak powers - The high pulse intensities are used for applications in material processing, such as micromachining, surface treatment, drilling holes, and three-dimensional laser prototyping. - In the medical domain, mode-locked lasers may again be used for a kind of material processing, e.g. as a laser scalpel or in ophthalmology (e.g. vision correction). There are also photochemical effects used e.g. for certain skin treatments. - High-power laser projection displays may be realized with mode-locked lasers and frequency conversion stages, the latter often being much simpler when working with ultrashort pulses. - High intensity physics relies on amplified systems with very high pulse energies and peak powers, so that extremely high optical intensities are achieved when focusing that laser radiation down to small spots.

Mode-Locking Advancements

-The very shortest pulses with durations below 10 fs (few-cycle pulses) are usually achieved with Kerr lens mode locking of a Ti:sapphire laser -High average utput powers of well over 200 W in sub-picosecond pulses and pulse energies above 10 μJ have been obtained in pulses from passively mode-locked thin-disk lasers, even 80 μJ in picosecond pulses -Very high pulse repetition rates have been obtained with passively mode-locked miniature bulk lasers and also with harmonically mode-locked fiber lasers. Even higher values of > 1 THz are possible with small laser diodes -Various kinds of lasers (normally with high pulse repetition rates) have reached quantum-limited timing jitter performance, thus outperforming many high-quality electronic oscillators.

Q factor of a resonator is related to various other quantities

3 Important relations The Q factor equals 2π times the exponential decay time of the stored energy times the optical frequency The Q factor equals 2π times the number of oscillation periods required for the stored energy to decay to 1/e (≈ 37%) of its initial value The Q factor of an optical resonator equals the finesse times the optical frequency divided by the free spectral range.

Fabry-Pérot interferometer Operation Principle

A substantial circulating optical power in the resonator is possible only if the input wave has an optical frequency close to one of the resonance frequencies of the resonator In resonance, the contribution of the input wave leaking through the input mirror adds constructively to the circulating wave. And, there is destructive interference for the reflected field: the input field reflected at the input mirror is canceled by the field leaking out of the resonator. Therefore, there is effectively no reflection in resonance In anti-resonance, the circulating field is quite weak, and most radiation is reflected at the input mirror. The weak field leaking out of the resonator towards the input source adds constructively to the reflected field Resonance frequencies can be tuned by changing the cavity length with a piezo actuator.

Gaussian beam

Also known as transverse-electromagnetic modes (TEM nm). TEM 00 (where n and m are both equal to 0) is an ideal Gaussian beam. fundamental transverse mode of a resonator, the transverse profile of the optical intensity of the beam with a power P can be described with a Gaussian function the beam radius w(z) is the distance from the beam axis where the intensity drops to 1/e2 (≈ 13.5%) of the maximum value. A hard aperture with radius w can transmit ≈ 86.5% of the optical power. For an aperture radius of 1.5 w or 2 w, this fraction is increased to 98.9% and 99.97%, respectively. The larger the radii, the larger the beam profile

Resonator Mode

As light is confined within an optical cavity, light will continue to reflect back and forth between the mirrors, and due to the effects of interference, only certain patterns and frequencies will be sustained, while others will be suppressed by destructive interference. Two types: longitudinal modes which differ in frequency, and transverse which may differ in both frequency and and the intensity pattern of the light Resonator cavities are stable if the reflected light remains within the cavity. Stable resonators have powers up to 2kW to achieve high gain and improve directionality. Unstable resonators consist of reflected light that continuously diverges as the number of reflections approaches infinity. The beam size will grow untile it is larger than the reflectors and escapes the system. These modes are typically used with high power lasers to reduce the chance of damaging reflectors.

Resonator mode properties

Cavity path length: determines the longitudinal resonator modes, or electric field distributions which cause a standing wave in the cavity. Mode of the beam: give the beam its shape These modes maintain their amplitude profile and reporduce themselves after completing one closed loop path inside the resonator. In order for a resonant mode to occur, it must also experience a phase shift equal to an integer multiple of 2π over one closed loop path.

Chromatic Aberrations

Color fringing image distortions caused by wavelength-dependent (actually frequency-dependent) optical effects they arise from frequency-dependent refraction at air-glass interfaces of optical lenses can also occur in the context of prisms and diffractive optics reflective optics do not normally exhibit chromatic aberrations (such as dielectric mirrors); these will actually minimize them two types: Longitudinal and Lateral Longitudinal: wavelengths of color do not converge at the same point after passing through the lens Lateral: aka transverse, different wavelengths of color come in at an angle an focus at different points along the same focal plane

Focal Length of a Curved Mirror

Curved mirrors are often used for focusing or defocusing light Example: laser resonators use curved laser mirrors with dielectric coatings rather than lenses, mainly because they introduce lower losses. Curved laser mirrors: curvature radius somewhere between 10 mm and 5 m. The fabrication of dielectric mirror coatings can be more difficult for very strongly curved mirror substrates, but with refined techniques it is possible to reach focal lengths of only a few millimeters, as required for some miniature lasers

Active Mode Locking with Higher Pulse Repetition Frequencies

Due to geometric constraints, it can be difficult to reach very high pulse repetition rates by making the laser resonator very short. A solution can be harmonic mode locking, where multiple pulses circulate in the laser resonator. The modulator frequency is then an integer multiple of the round-trip frequency. A variation of the method is rational harmonic mode locking, where the modulation frequency is the round-trip frequency times the ratio of two integers.

Gaussian function with a Hermite polynomial

E0 is the field maximum x and y are the axes that define a cross-section of the beam z is the axis of propagation w0 is the beam waist w(z) is the beam radius at a given z value Hn(x) and Hm(x) are the Hermite polynomial with the non-negative integer indices n and m k is the wavenumber (k=2π/λ) zR is the Rayleigh range R(z) is the radius of curvature of the wavefront

EBS

Electron Beam Deposition involves the evaporation of material in a crucible by heating with an electron beam, which is generated from a hot filament and focused with a magnetic field. In the vacuum chamber, the evaporated material moves to the substrate, which can be covered with a mechanical shutter as soon as the right amount of material has been deposited. The target substrate is heated to improve the quality. For typical coating materials, the obtained thin films tend to be somewhat porous, leading to a reduced density and subsequently to a reduced refractive index. The optical properties can then exhibit a significantly increased temperature dependence, as water may fill the pores, and may be driven out of the coating at elevated temperatures. This can be a problem for some sensitive steep-edge filter designs, for example. Similar to this uses evaporation by resistive heating of the crucible

GTIs

Gires-Tournois Interferometers linear optical resonators used for introducing chromatic dispersion. Similar to the Fabry-Perot Interferometer. Front mirror is partially reflective and back mirror is highly reflective. If no losses occur, power reflectance is unity at all wavelengths, but the phase of the reflected light is frequency dependent due to the resonance effect, causing chromatic dispersion. The phase change of reflected light and the dispersion change periodically with the optical frequency. Ideally a GTI is operated near a maximum or minimum of the GDD (group delayed dispersion) and the usable bandwidth is some fraction of the free spectral range, which is inversely proportional to the resonator length. An optical bandwidth well below the free spectral range indicates that the pulse duration needs to be well above the round-trip time of the GTI. The maximum magnitude of GDD scales with the square of the resonator length.

Perfect Gaussian Beam

Hermite-Gaussian; TEM00 - both n and m are equal to 0

Focusing of Divergent Beams

If a divergent (rather than collimated) beam hits a focusing lens, the distance b from the lens to the focus becomes larger than f (Figure 2). Lens equation showed a is the distance from the original focus to the lens. This shows that b ≈ f if a >> f, but b > f otherwise. That relation can be intuitively understood: a focusing power 1 / a would be required to collimate the incident beam (i.e. to remove its beam divergence), so that only a focusing power 1 / f − 1 / a is left for focusing. If a ≤ f, the equation cannot be fulfilled: the lens can then not focus the beam.

Focal Length of an Extended Optical System

Many possibilities A common (but not universally used) approach for the definition of focal lengths of extended systems is based on geometrical optics. For finding the front focal point, one calculates rays which are horizontal on the back side (see Figure 2), using the paraxial approximation. The optical system is considered as a "black box", where one does not care about the actual ray paths; instead, one works with internal rays which are extrapolated from the outer rays. Based on those extrapolated rays, one can define the front principal plane (or first principal plane). The front focal length is then the distance between the front focal point (in the front focal plane) and the front principal plane

What must happen for a resonant mode to occur?

Must experience a phase shift equal to an integer multiple of 2π over one closed loop path

Chromatic Dispersion: Normal and Anomalous

Normal: group velocity decreases with increasing optical frequency, occurs for most transparent media in the visible spectral region. Anomalous: occurs at longer wavelengths, e.g. in silica (the basis of most optical fibers) for wavelengths longer than the zero-dispersion wavelength of ≈ 1.3 μm.

Waveguide fabrication

Planar waveguides can be fabricated on various crystal and glass materials with epitaxy or with polishing methods. The waveguide may be made on the top of the device (as shown on the left side of Figure 1), but it can also be placed between other solid layers. Channel waveguides on semiconductor, crystal and glass materials can be made with lithographic methods in combination with, e.g., epitaxy, ion exchange, or thermal indiffusion. It is possible to make a buried waveguide by growing an additional layer on top of the waveguide. That may lead to lower propagation losses and a more symmetric mode profile. Optical fibers can be fabricated by drawing from a preform, which is a large glass rod with a built-in refractive index profile. Fibers can again be drawn into waveguides of further reduced dimensions, in the extreme case resulting in nanofibers. Waveguides can be written into transparent media (e.g. glasses or crystals) with focused and pulsed laser beams, exploiting laser-induced breakdown and related phenomena. In glasses, the affected volume often exhibits a somewhat increased refractive index, which can be directly used for guiding light. In crystals, the refractive index may be decreased; one then has to treat some region around the desired waveguide region

Resonator types

Plane-parallel: aka Fabry-Perot cavity; consisting of two opposing flat mirrors; not commonly used in large scale lasers due to difficulties in precise placement (must be aligned parallel within a few seconds of arc). Commonly used in microchip and microcavity lases and semiconductor lasers, or other applications with short cavities with a small mirror separation distance of less than 1 cm. Distance is equal to an integral multiple of one half of the lasing wavelength. Concentric (spherical): when the radii of the two mirrors are both the same and equal to half of the cavity length; produces a diffraction-limited beam waist in the center of the cavity, with large beam diameters at the mirrors. Confocal: mirrors of equal radii to the cavity length; produces smallest possible beam diameter at the cavity mirrors for a given cavity length; used in lasers where the purity of the transverse mode pattern is important Concave-convex: one convex mirror with a negative radius of curvature. Produces no intracavity focus of the beam. Useful very high power lasers where the intensity of the intracavity light might be damaging to the intracavity medium if brought to a focus Ring resonator: ring of more than two reflectors where the total closed loop path of the reflected light is equal to an integral multiple of one half of the lasing wavelength .

Designing Dielectric Mirrors

Properties may include - a combination of reflectivities at different wavelengths - very broadband reflection ranges - anti-reflection properties - certain polarization properties (for non-normal incidence; thin-film polarizers) - a certain chromatic dispersion profile - minimum sensitivity to growth errors

Cross sections of Hermite-Gaussian resonator modes

TEMnm - n and m are not 0 and produce more complicated resonator modes Lowest order Hermite-Gaussian modes are with n and m values ranging from 0 to 3

Chromatic Dispersion from Polarization Effects

Takes place in optical fibers For a certain fiber span and a given wavelength, two input principal polarization states can be determined, which exhibit somewhat different values of the group delay. Besides this differential group delay, there are contributions to the group delay dispersion with opposite signs for the two principal polarization states. Such effects can be relevant for optical fiber communications at very high bit rates.

Achromatic doublet lens

The basic idea is to realize an overall focusing lens by combining a strongly focusing and a less strongly defocusing lens, where the latter exhibits stronger chromatic dispersion. Although the defocusing effect of the second lens is weaker than the focusing effect on the first one, its chromatic aberrations can compensate those of the other lens. The considered doublet lens design has been numerically optimized such that we obtain a focal length close to 100 mm (measured from the middle of the lens at z = 2 mm) in the wavelength range from 400 nm to 800 nm. The curvature radius on the left side has been fixed at 60 mm, and the other two curvature radii were subject to optimization. The inner radii of curvature of the two lenses are required to be the same, so that the two lenses can be contacted (e.g. cemented together), forming a single optical element and minimizing reflection losses. Note that for the given focal length the double lens requires a stronger curvature due to the defocusing effect of the flint lens.

Early Mathematical Description of Chromatic Dispersion

The denominator is also called the principal dispersion. The Abbe number depends on the refractive indices at only three different wavelengths: 486.1 nm (blue Fraunhofer F line from hydrogen) 589.2 nm (orange Fraunhofer D line from sodium) 656.3 nm (red Fraunhofer C line from hydrogen) Large values of the Abbe number indicate low chromatic dispersion and vice versa. Such values can be used for the design of achromatic optical elements.

Measuring Chromatic Dispersion

The pulse delay technique [2] (for fibers) is based on measuring the difference in propagation time (group delay) for pulses with different center wavelengths. This is typically done using hundreds of meters (or even some kilometers) of a fiber. The dispersion is obtained by differentiation of these data. The phase shift technique or "difference method" [4] (also for fibers): a light beam with a sinusoidally modulated intensity is sent through a fiber, and the phases of the oscillations of input and output power are compared. The group delay can be calculated from that phase, and the dispersion can be measured by performing the measurement at different wavelengths. Dispersion in the resonator of a wavelength-tunable passively mode-locked laser can be measured by monitoring changes in the pulse repetition frequency when the laser wavelength is changed, as this reveals the wavelength-dependent group delay. Different types of interferometry [5] (e.g. white-light interferometry [6] or spectral phase interferometry [10]) can be used to measure the phase delay caused by a dispersive component. The dispersion properties can be obtained from this phase by numerical differentiation. The method is normally used for dispersion measurements on dispersive laser mirrors and sometimes for fibers.

Stable Operation in Active Mode Locking

The round-trip time of the resonator must fairly precisely match the period of the modulator signal (or some integer multiple of it), so that a circulating pulse can always pass the modulator at a time with minimum losses. Even a small frequency mismatch between the laser resonator and the drive signal can lead to a strong timing jitter or even to chaotic behavior, since the obtained "pulling force" on the pulse timing is quite weak. Synchronization between the modulator driver and the laser can be achieved either by careful adjustment of a stable laser setup, or by means of a feedback circuit which automatically adjusts either the modulation frequency or the length of the laser resonator (and thus its round-trip time). A frequently used technique is regenerative mode locking (also called mode locking with regenerative feedback). Here, the modulator signal is not generated by a free-running or slightly corrected electronic oscillator, but rather is derived from the detected intensity modulation of the pulse train itself. Such schemes are particularly important for achieving tunable pulse repetition rates, and are often applied to mode-locked fiber lasers and laser diodes.

How a Cesium Atomic clock works

Use Cesium 133: isotope of cesium of whose atomic transitions is used as a scientific time standard of the real world First measure one of its resonant frequencies by locking a crystal oscillator to the principal microwave resonance of the cesium atom (same sort of frequency as direct broadcast satellite signals) Then heat the cesium so the atoms boil off and pass down a tube maintained at a high vacuum. They pass through a magnetic field that selects atoms of the right energy state. These atoms then pass through an intense microwave field whose frequency sweeps back and forth within a narrow range of frequencies. At some point, each cycle crosses the frequency of exactly 9,192,631,770 Hz. When the cesium atom receives energy at exactly the right frequency, it changes its energy state. At the far end of the tube, another magnetic field separates out the atoms that have changed their energy state. A detector at the end of the tube gives an output proportional to the number of Cesium atoms striking it, and will peak in output when the microwave frequency is exactly correct. This peak is then used to make the slight correction necessary to bring the crystal oscillator and hence the microwave field exactly on frequency. This locked frequency is is then divided by 9,192,631,770 to give the familiar one pulse per second required by the real world

Principle Dispersion

a crude measure for chromatic dispersion of a transparent optical material It is simply the difference of refractive indices between two specific standard spectral lines of hydrogen: the F line at 486.1 nm (blue) the C line at 656.3 nm (red) The principal dispersion nF − nC appears in the definition of the Abbe number.

Resonator

a device or system that exhibits resonance or resonant behavior naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. Oscillations in a resonator can be either electromagnetic or mechanical (including acoustic). Used to either generate waves of specific frequencies or to select specific frequencies from a signal.

Astigmatism

a kind of optical aberration; defect of optical lenses or a property of laser beams light hits a spherical optical lens or a spherically curved mirror under a substantial angle against the optical axis. The focal length of the optical element is essentially dependent on the direction: it is reduced in the tangential plane and increased in the sagittal plane. As a result, perfect focusing of a circular laser beam, for example, is not possible: the smallest beam radius for the tangential direction is reached sooner than for the sagittal direction. cannot be reduced simply by using an optical aperture; must be corrected by using an appropriate combination of different lenses

beam divergence

a measure for how fast a laser beam expands far from its focus (beam waist) in the 'far field'. low beam divergence: important for laser pointers and free-space optical communications collimated beams: beams with very small divergence (constant beam radius) over significant distances; generated from strongly divergence beams with beam collimators some divergence is unavoidable; light propagates in a homogenous medium (not a waveguide) poor beam quality: beams with large divergence

Focal length

a measure of how strongly an optical system focuses or defocuses light Example: thin focusing lens If a sufficiently large collimated beam of light is incident on the lens, the beam will be focused, and the focal length is the distance from the lens to that focus (assuming that the lens is surrounded by vacuum or air, not by some dense substance with a significant refractive index) Defocusing lens: focal length is the distance from the lens to the virtual focus (indicated by the dashed lines), taken as a negative value

Q Factor

a measure of the damping of resonator modes Energy storage: 2π times the ratio of the stored energy to the energy dissipated per oscillation cycle, or equivalently the ratio of the stored energy to the energy dissipated per radian of the oscillation. Resonance bandwidth: the ratio of the resonance frequency (ν0) and the full width at half-maximum (FWHM) bandwidth δν of the resonance; Q = ν0/δν High Q Factor: Q > ​1⁄2, underdamped, the oscillation will carry on; pure oscillation systems (such as a bell that rings forever) has an infinite quality factor) Intermediate Q factor: Q = ​1⁄2; critically damped; output will not oscillate. Critical damping results in the fastest response (approach to the final value) possible without overshoot. Low Q Factor: Q < ​1⁄2, overdamped, the oscillation will die out rapidly. As the quality factor decreases the slower decay mode becomes stronger relative to the faster mode and dominates the system's response resulting in a slower system.

M2 factor.

a parameter for quantifying the beam quality of laser beams beam quality factor or beam propagation factor A diffraction-limited beam has an M2 factor of 1, and is a Gaussian beam (smaller values of M2 are physically not possible) Hermite-Gaussian beam: has an M2 factor of (2n + 1) in the x direction, and (2m + 1) in the y direction Limits the degree to which the beam can be focused for a given beam divergence angle With optical power, it determines the brightness of a laser beam predicts the evolution of the beam radius; helps in designing pump optics for diode-pumped lasers

Active Mode Locking

a technique of mode locking, based on active modulation of the intracavity losses or the round-trip phase change This can be achieved e.g. with an acousto-optic or electro-optic modulator, a Mach-Zehnder integrated-optic modulator, or a semiconductor electroabsorption modulator. If the modulation is synchronized with the resonator round trips, this leads to the generation of ultrashort pulses. A pulse with the "correct" timing can pass the modulator at times where the losses are at a minimum (see Figure 2). It is thus favored against any other radiation circulating in the resonator. As the pulse will in the steady state saturate the laser gain such that its round-trip gain is zero, other circulating radiation will have a negative round-trip gain and will thus die out sooner or later. Wings of the pulse experience a little attenuation, which effectively leads to (slight) pulse shortening in each round trip: The round-trip gain is slightly negative for the wings and slightly positive for the pulse center. As a result, the pulses get shorter and shorter, until the pulse shortening is offset by other effects (e.g. gain narrowing or chromatic dispersion) which tend to broaden the pulse.

Frequency Modulation Mode Locking

aka FM mode locking Another appropriate name "phase modulation mode locking" active mode locking also works with a periodic phase modulation (instead of amplitude modulation) e.g. in a Pockels cell, even though this leads to chirped pulses Some FM mode-locked lasers exhibit an instability: they exhibit random hopping between two operation modes, where the pulses pass the modulator at times where either a minimum or a maximum of the phase delay is reached. This kind of bistability is sometimes removed by dispersive and nonlinear effects.

Dielectric mirror

aka laser mirror mirrors consisting of multiple thin layers of different transparent optical materials (dielectric coatings, thin-film coatings, interference coatings) Even if the Fresnel reflection coefficient from a single interface between two materials is small (small difference in the refractive indices) the reflections form many interfaces (in a certain wavelength range) can constructively interfere to result in a very high overall reflectance of the device Ex: Bragg Mirror (all optical layer thickness values are just one-quarter of the design wavelength); highest possible reflectance Ex: resonator mirrors of a laser Purpose: - to create a broader reflection bandwidth - a combination of desirable reflectance values in different wavelength ranges - special polarization properties - edge filters - tailored chromatic dispersion properties amount of layers is determined by the difference in refractive indices between the materials characteristic property: optical properties depend on the angle of incidence Ex: Bragg Mirror reflectance spectra: larger the angle, the more the reflection spectrum is shifted towards shorter wavelengths. This is because the component of the wave vector perpendicular to the layer surfaces become smaller for a given wavelength, which can be compensated by reducing the wavelength. Ex: Optimized bragg mirror; extreme high reflectivities - 99.9999% allowing for the creation of optical resonators (cavities) with extremely high Q factor. High reflection in the smaller part of the visual spectrum - transparent to visible light and shine in colors depending on the angle of view Can be made of both plane and curved surfaces Used for focusing and defocusing

Optical cavity

an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Major components for lasers: surround the gain medium and provide feedback of the laser light Designed to have a large Q factor causing the frequency line width of the beam to be very small compared to the frequency of the laser

Pseudopotential

an attempt to replace the complicated effects of the motion of the core (i.e. non- valence) electrons of an atom and its nucleus with an effective potential an effective potential constructed to replace the atomic all-electron potential (full-potential) such that core states are eliminated and the valence electrons are described by pseudo-wavefunctions with significantly fewer nodes A simplified approximation of the effective potential of electrons in a complex system such as a crystal lattice

diffraction-limited beams

beams with a minimum possible beam divergence for a given waist radius potential to be focused to small spots is as high as possible for the given wavelength (if its beam quality is ideal) beam waist with a given beam radius (generated from a beam by focusing with a curved mirror) is associated with the minimum possible beam divergence.

Effects of chromatic dispersion

causes wavelength-dependent refraction, which is responsible, e.g., for the occurrence of rainbows. Wavelength-dependent diffraction at a diffraction grating allows the spatial separation of different frequency components of light. Important impact on the propagation of pulses, because a pulse always has a finite spectral width (bandwidth), so that dispersion can cause its frequency components to propagate with different velocities. Normal dispersion: leads to a lower group velocity of higher-frequency components, and thus to a positive chirp, whereas anomalous dispersion creates negative chirps.

Fabry-Perot Interferometer applications

check whether a laser operates on a single resonator mode or on multiple modes. High-Finesse Fabry-Perot Interferometers: used as reference cavities and for spectral analysis Optical Spectrum analysis: Fabry-Perot interferometer is often made short enough to achieve a sufficiently large free spectral range; bandwidth of the resonances is then the free spectral range divided by the finesse. Due to the high reflectivities, the finesse can be high (well above 1000, and with supermirrors even much higher). For a given finesse, the wavelength resolution can be improved by increasing the mirror distance, but only at the cost of reducing the free spectral range, i.e., the range within which unique spectral assignment is possible. Gires-Tournois interferometer another variant; used for generating chromatic dispersion

Fabry-Pérot interferometer Variants

commonly defined to consist of 2 planar mirrors, but the term is also frequently used for resonators with curved mirrors plane-plane mirrors experience diffraction losses because their resonator modes extend up the edges of the mirror Fabry-Peot interferometers are usually used with input beams of much smaller diameter, which are not actually matched to the resonator modes. the term is sometimes also used for devices containing in a waveguide Fabry-Pérot lasers are laser diodes containing an active (amplifying) waveguide with some kind of mirrors at the ends

Q factor of a resonator

depends on the optical frequency ν0, the fractional power loss l per round trip, and the round-trip time Trt a resonator consisting of two mirrors with air (or vacuum) - Q factor rises as the resonator length increases (because this decreases the energy loss per optical cycle) High Q values are achieved by strongly reducing the losses per round trip

Waveguide modes

field distribution for a given optical frequency and polarization in a plane perpendicular to the propagation direction Special interest placed on distributions which do not change during propagation, apart from a common phase change. Multimode fiber: each mode has a so-called propagation constant, the imaginary part of which quantifies the phase delay per unit propagation distance. A fiber also has a large number of unguided modes (cladding modes) which are not restricted to the vicinity of the fiber core.

Wavelength Dependence of the Focal Length; Using Curved Mirrors

focal length: slightly wavelength-dependent due to the wavelength dependence of the refractive index (-> chromatic dispersion) Leads to chromatic aberrations of imaging systems and similar problems in other applications where an optical system is used for a wide range of optical wavelengths. Chromatic aberrations may be eliminated by using mirrors

Dioptric Power

focusing power of a lens the inverse of the effective focal length (which is the same is the front and back focal length if the median on both sides of the optics is the same). This means that a strongly focusing lens has a small focal length, but a large dioptric power. Example: Prescription lenses specify the dioptric power

Finesse

free spectral range divided by the FWHM (full width at half-maximum) bandwidth of the resonances of an optical resonator Fully determined by the resonator losses and is independent of the resonator length Related to the Q Factor

GTI drawbacks

fundamentally limited bandwidth (proportional to the square root of the given magnitude of GDD) and the limited amount of control of higher-order dispersion. Dispersive mirrors with significantly broader optical bandwidth can be designed as chirped mirrors

Applications of Achromatic optics

imaging systems, as needed for photography, microscopy and video recording beneficial for focusing or collimating the output of supercontinuum sources or in other cases with ultrabroadband ultrashort pulse Curved mirrors and dielectric mirrors are naturally achromatic; reflective optics avoid chromatic aberrations

Collimated beams

laser beams with weak divergence a beam (typically a laser beam) propagating in a homogeneous medium (e.g. in air) with a low beam divergence, so that the beam radius does not undergo significant changes within moderate propagation distances. Example: Gaussian beam - long Rayleigh length compared to the conceived propagation distance; 1064-nm beam with a 1-mm beam radius at its beam waist has a Rayleigh length of ≈ 3 m in air, so that it can be considered as being collimated within a normal laboratory setup Consists of only parallel light rays

Mode-Locked Lasers

lasers which emit ultrashort pulses on the basis of the technique of mode locking (active or passive) Require a gain medium with a large gain bandwidth Features include not too high nonlinearity and chromatic dispersion, and (particularly for passive mode locking) high enough laser cross sections in order to avoid Q-Switching instabilities Types: dye lasers (argon ion), solid-state bulk lasers (based on ion-doped crystals), fiber lasers, semiconductor lasers Vital to select a gain medium in a particular parameter regime (e.g. concerning pulse duration, center wavelength, and pulse repetition rate

Cavity Dumping in Mode-Locked Lasers

mode-locked laser can generate higher pulse energies of e.g. several microjoules at lower pulse repetition rates (e.g. 100 kHz or 1 MHz) by incorporation of a cavity dumper in the laser resonator. The basic principle is to form a high-energy pulse within the resonator while having low resonator losses, and then to couple out of the energy with the cavity dumper.

GTI: Tunable dispersion

obtained with a variable air gap between the mirrors, whihc however must be carefully stabilized to avoid unwatned drifts. More stable but in general not tunable GDD can be generated with monolithic designs based on thin films of dielectric media Examples: TiO2 and SiO2 in femtosecond lasers

Spherical Aberrations

optical aberrations resulting from spherical optical surfaces Example: ball lens Figure 1 demonstrates this for a ball lens with 10 mm diameter and a refractive index of 1.515 (N-BK7 glass at 633 nm), which is used to focus parallel incoming light. The outer incoming rays are crossing the optical axis substantially sooner than the paraxial ones. causes beam distortions when used for focusing or collimating laser beams

Free-space Optical Communications

optical data transmission through free space, usually through air or vacuum normally done through optical fibers (these allow for transmission over large distances without excessive power losses, alignment issues, and disturbing influences of the atmosphere Ex: "photo phone" (Graham Bell's telephone) and the optical telegraph (uses free-space optics - telescope) depends on the accuracy of a well-collimated laser beam

achromatic optical elements

optical devices or setups with minimized chromatic aberrations can be used in a wide range of wavelengths Achromatic optical lenses are often just called achromats. The property of being achromatic (essentially insensitive to wavelength changes) is called achromatism.

Crown Glass

optical glasses with low chromatic dispersion and tentatively a low refractive index Abbe number above 50 or 55; Normally will have lower contents of heavy metals, and will have higher contents of alkali metals (sodium and potassium), lower mass density larger band gap energy than flint glasses, leading to a shorter-wavelength UV absorption edge. Their parasitic absorption and scattering losses can be fairly low. Examples: lenses, mirror substrates, optical windows and prisms; imaging applications; window glasses

Flint Glass

optical glasses with strong chromatic dispersion (low Abbe number - below 50) and tentatively a high refractive index (larger than 1.55) Large refractive index normally results from heavy metal contents typically exhibit a UV absorption edge at longer wavelengths, which results from a lower band gap energy parasitic absorption and scattering losses are tentatively higher than for crown glasses Example: dispersive prisms that aim to achieve sufficiently high angular dispersion, combined with crown glass for achromatic lenses, correction glasses

Waveguide

spatially inhomogeneous transparent structures for guiding light i.e. for restricting the spatial region in which light can propagate Usually contains a region of increased refractive index, compared with the surrounding medium (cladding). Guidance is possible through the use of reflections at metallic interfaces. Some also involve plasmonic effects at metals. 2D guidance restricts the extension of guided light in two dimensions and permitting propagation essentially only in one dimension. Most important example is an optical fiber

Rayleigh Length

the distance from a beam waist where the mode radius increased by a factor square root of 2 circular beam: mode area is doubled Gaussian beam: the Rayleigh length is determined by the waist radius (w0) and the wavelength (λ - vacuum wavelength divided by the refractive index (n) of a material) Imperfect beam quality: Rayleigh length is decreased by the M2 factor determines the depth of focus Highest laser gain in a laser medium can be achieved when the focused pump beam has a Rayleigh length of the order of the length of the gain medium (weaker focusing reduces the pump intensity whereas stronger focusing leads to a strong divergence which limits the effective interaction length

Astigmatism of a beam

the focal points for the vertical and horizontal direction do not coincide. May be corrected with an anamorphic prism pair, for example, or with cylindrical lenses or with tilted curved mirrors.

Chromatic Dispersion

the frequency dependence of the phase velocity in a transparent medium In an optical medium, it is the phenomenon that the phase velocity and group velocity of light propagating in a transparent medium depend on the optical frequency; results from interaction of light with electrons of the medium. A related quantitative measure is the group velocity dispersion.

Free spectral range of a Fabry-Pérot interferometer

the frequency spacing of its transmission peaks. Limits the optical frequency range. A large free spectral range can be desirable. However, for a given finesse, a larger free spectral range also leads to a larger resonator bandwidth and thus a poor spectral resolution.

Finesse relation to Q factor

the latter is the finesse times the resonance frequency divided by the free spectral range

Q factor of an Oscillator

the ratio of the resonance frequency ν0 and the full width at half-maximum (FWHM) bandwidth δν of the resonance If the oscillator is based on some resonator, the effective Q factor of the oscillator may deviate substantially from the intrinsic Q value of the resonator Measurements on atomic transitions have a limited measurement time, effective linewidth of the reference transition is significantly increased; example: cesium atomic clock Stabilized oscillator: linewidth is a tiny fraction of the linewidth of the underlying frequency standard; example: in cesium atom clocks, the quartz oscillator is often stabilized to a millionth of the linewidth of the signal from the cesium apparatus

Dielectric Coatings

thin-film coatings made of transparent dielectric materials, e.g. for laser mirrors or anti-reflection coatings layers of transparent dielectric materials function to modify the reflective properties of the surface by exploiting the interference of reflections from multiple optical interfaces may be used for highly reflecting laser mirrors or partially transmissive output couplers, dichroitic mirrors, anti-reflection coatings, crystalline materials, semi conductor devices such as edge-emitting laser diodes, vertical cavity surface-emitting lasers, and photodiodes; polymers (plastic optics) Consist of discrete layers with varying refractive indices; however theyre are also gradient-index coatings for rugate filters, where the refractive index is varied continuously. That can be achieved, for example, by gradually varying the chemical composition of the material during growth. May be applied with EBD, IAD, IBS, and APRS

Beam Radius

w symbol a measure of the transverse extension of a light beam. the distance from the beam axis where the optical intensity drops to 1/e2 (≈ 13.5%) of the value on the beam axis beam diameter is twice the beam radius

Free Spectral Range

Δν Related to optical resonators; the frequency spacing of the axial modes of an optical resonator. (Axial modes = Gaussian shaped) When using different transparent media, the free spectral range is the inverse of the round-trip time of an optical pulse