List of Figures

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This set of figures introduces an abstract form of illustration, which will be used in this document. The top figure shows a typical example taken from the analysis of an optical system: an incident field Ein is reflected and transmitted by a semi-transparent mirror; there might be the possibility of second incident field E in2. The lower left figure shows the abstract form we choose to represent the same system. The lower right figure depicts how this can be extended to include a beam splitter object, which connects two optical axes.
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Simplified schematic of a two mirror cavity. The two mirrors are defined by the amplitude coefficients for reflection and transmission. Further, the resulting cavity is characterised by its length D. Light field amplitudes are shown and identified by a variable name, where necessary to permit their mutual coupling to be computed.
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This sketch shows a mirror or beam splitter component with dielectric coatings and the photograph shows some typical commercially available examples [45]. Most mirrors and beam splitters used in optical experiments are of this type: a substrate made from glass, quartz or fused silica is coated on both sides. The reflective coating defines the overall reflectivity of the component (anything between R ≈ 1 and R ≈ 0, while the anti-reflective coating is used to reduce the reflection at the second optical surface as much as possible so that this surface does not influence the light. Please note that the drawing is not to scale, the coatings are typically only a few microns thick on a several millimetre to centimetre thick substrate.
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The relation between the phase of the light field amplitudes at a beam splitter can be computed assuming a Michelson interferometer, with arbitrary arm length but perfectly-reflecting mirrors. The incoming field E0 is split into two fields E1 and E2 which are reflected atthe end mirrors and return to the beam splitter, as E3 and E4, to be recombined into two outgoing fields. These outgoing fields E 5 and E 6 are depicted by two arrows to highlight that these are the sum of the transmitted and reflected components of the returning fields. We can derive constraints for the phase of E1 and E2 with respect to the input field E0 from the conservation of energy: |E0|2 = |E5|2 + |E6 |2.
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Illustration of an arm cavity of the Virgo gravitational-wave detector [56]: the macroscopic length L of the cavity is approximately 3 km, while the wavelength of the Nd:YAG laser is λ ≈ 1μm. The resonance condition is only affected by the microscopic position of the wave nodes with respect to the mirror surfaces and not by the macroscopic length, i.e., displacement of one mirror by Δx = λ∕2 re-creates exactly the same condition. However, other parameters of the cavity, such as the finesse, only depend on the macroscopic length L and not on the microscopic tuning.
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Finesse example: Mirror reflectivity and transmittance.
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Finesse example: Length and tunings.
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Example traces for phase and amplitude modulation: the upper plot a) shows a phase-modulated sine wave and the lower plot b) depicts an amplitude-modulated sine wave. Phase modulation is characterised by the fact that it mostly affects the zero crossings of the sine wave. Amplitude modulation affects mostly the maximum amplitude of the wave. The equations show the modulation terms in red with m the modulation index and Ω the modulation frequency.
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Some of the lowest-order Bessel functions Jk(x) of the first kind. For small x the expansion shows a simple xk dependency and higher-order functions can often be neglected.
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Electric field vector E0 exp (iω0t) depicted in the complex plane and in a rotating frame (x′, y′) rotating at ω0 so that the field vector appears stationary.
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Amplitude and phase modulation in the ‘phasor’ picture. The upper plots a) illustrate how a phasor diagram can be used to describe phase modulation, while the lower plots b) do the same for amplitude modulation. In both cases the left hand plot shows the carrier in blue and the modulation sidebands in green as snapshots at certain time intervals. One can see clearly that the upper sideband (ω0 + Ω) rotates faster than the carrier, while the lower sideband rotates slower. The right plot in both cases shows how the total field vector at any given time can be constructed by adding the three field vectors of the carrier and sidebands. [Drawing courtesy of Simon Chelkowski]
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A sinusoidal signal with amplitude as frequency ωs and phase offset φs is applied to a mirror position, or to be precise, to the mirror tuning. The equation given for the tuning ϕ assumes that ωs∕ω0 ≪ 1, see Section 2.5.
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Finesse example: Modulation index.
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Finesse example: Mirror modulation.
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A beam with two frequency components hits the photo diode. Shown in this plot are the field amplitude, the corresponding intensity and the electrical output of the photodiode.
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Finesse example: Optical beat.
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Typical optical layout of a two-mirror cavity, also called a Fabry–Pérot interferometer. Two mirrors form the Fabry–Pérot interferometer, a laser beam is injected through one of the mirrors and the reflected and transmitted light can be detected by photo detectors.
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Power enhancement in a two-mirror cavity as a function of the laser-light frequency. The peaks marks the resonances of the cavity, i.e., modes of operation in which the injected light is resonantly enhanced. The frequency distance between two peaks is called free-spectral range (FSR).
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This figure compares the fields reflected by, transmitted by and circulating in a Fabry–Pérot cavity for the three different cases: over-coupled, under-coupled and impedance matched cavity (in all cases T1 + T2 = 0.2 and the round-trip loss is 1%). The traces show the phase and amplitude of the electric field as a function of laser frequency detuning.
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Power transmitted and circulating in a two mirror cavity with input power 1 W. The mirror transmissions are set such that T1 + T2 = 0.8 and the reflectivities of both mirrors are set as R = 1 − T. The cavity is undercoupled for T1 < 0.4, impedance matched at T1 = T2 = 0.4 and overcoupled for T1 > 0.4. The transmission is maximised in the impedance-matched case and falls similarly for over or undercoupled settings. However, the circulating power (and any resonance performance of the cavity) is much larger in the overcoupled case.
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Typical optical layout of a Michelson interferometer: a laser beam is split into two and sent along two perpendicular interferometer arms. We will label the directions in a Michelson interferometer as North, East, West and South in the following. The end mirrors reflect the beams such that they are recombined at the beam splitter. The South and West ports of the beam splitter are possible output port, however, in many cases, only the South port is used.
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Power in the South port of a symmetric Michelson interferometer as a function of the arm length difference ΔL.
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Finesse example: Michelson power.
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Finesse example: Michelson modulation.
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Example of an error signal: the top graph shows the electronic interferometer output signal as a function of mirror displacement. The operating point is given as the zero crossing, and the error-signal slope is defined as the slope at the operating point. The right graph shows the magnitude of the transfer function mirror displacement → error signal. The slope of the error signal (left graph) is equal to the low frequency limit of the transfer function magnitude (see Equation (102View Equation)).
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The top plot shows the cavity power as a function of the cavity tuning. A tuning of 360° refers to a change in the cavity length by one laser wavelength. The bottom plot shows the differentiation of the upper trace. This illustrates that near resonance the cavity power changes very rapidly when the cavity length changes. However, for most tunings the cavity seems not sensitive at all.
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Typical setup for using the Pound–Drever–Hall scheme for length sensing and with a two-mirror cavity: the laser beam is phase modulated with an electro-optical modulator (EOM). The modulation frequency is often in the radio frequency range. The photodiode signal in reflection is then electrically demodulated at the same frequency.
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This figure shows an example of a Pound–Drever–Hall (PDH) signal of a two-mirror cavity. The plots refer to a setup in which the cavity mirrors are stationary and the frequency of the input laser is tuned linearly. The upper trace shows the light power circulating in the cavity. The three peaks correspond to the frequency tunings for which the carrier (main central peak) or the modulation sidebands (smaller side peaks) are resonant in the cavity. The lower trace shows the PDH signal for the same frequency tuning. Coincident with the peaks in the upper trace are bipolar structures in the lower trace. Each of the bipolar structures would be suitable as a length-sensing signal. In most cases the central structure is used, as experimentally it can be easily identified because its slope has a different sign compared to the sideband structures.
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Power and slope of a Michelson interferometer. The upper plot shows the output power of a Michelson interferometer as detected in the South port (as already shown in Figure 29). The lower plot shows the optical gain of the instrument as given by the slope of the upper plot.
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This length sensing scheme is often referred to as frontal or Schnupp modulation: an EOM is used to phase modulate the laser beam before entering the Michelson interferometer. The signal of the photodiode in the South port is then demodulated at the same frequency used for the modulation.
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Finesse example: Cavity power and slope.
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Finesse example: Michelson with Schnupp modulation.
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One dimensional cross-section of a Gaussian beam. The width of the beam is given by the radius w at which the intensity is 2 1∕e of the maximum intensity.
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Gaussian beam profile along z: this cross section along the x-z-plane illustrates how the beam size w(z) of the Gaussian beam changes along the optical axis. The position of minimum beam size w0 is called beam waist. See text for a description of the parameters Θ, zR and Rc.
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This plot shows the intensity distribution of Hermite–Gauss modes unm. One can see that the intensity distribution becomes wider for larger mode indices and the peak intensity decreases. The mode index defines the number of dark stripes in the respective direction.
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These top plots show a triangular beam shape and phase distribution and the bottom plots the diffraction pattern of this beam after a propagation of z = 5 m. It can be seen that the shape of the triangular beam is not conserved and that the phase front is not spherical.
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These plots show the amplitude distribution and wave front (phase distribution) of Hermite–Gaussian modes unm (labeled as HGnm in the plot). All plots refer to a beam with λ = 1 µm, w = 1 mm and distance to waist z = 1 m. The mode index (in one direction) defines the number of zero crossings (along that axis) in the amplitude distribution. One can also see that the phase distribution is the same spherical distribution, regardless of the mode indices.
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These plots show the amplitude distribution and wave front (phase distribution) of helical Laguerre–Gauss modes upl. All plots refer to a beam with λ = 1 µm, w = 1 mm and distance to waist z = 1 m.
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Intensity profiles for helical Laguerre–Gauss modes upl. The u00 mode is identical to the Hermite–Gauss mode of order 0. Higher-order modes show a widening of the intensity and decreasing peak intensity. The number of concentric dark rings is given by the radial mode index p.
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Intensity profiles for sinusoidal Laguerre–Gauss modes uapllt. The up0 modes are identical to the helical modes. However, for azimuthal mode indices l > 0 the pattern shows l dark radial lines in addition to the p dark concentric rings.
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Coupling coefficients for Hermite–Gauss modes: for each optical element and each direction of propagation complex coefficients k for transmission and reflection have to be computed. In this figure k1, k2, k3, k4 each represent a matrix of coefficients knmn′m′ describing the coupling of un,m into un′,m′.
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Finesse example: Beam parameter
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Finesse example: Mode cleaner
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Finesse example: LG33 mode. The ring structure in the phase plot is due to phase jumps, which could be removed by applying a phase ‘unwrap’.