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Figure 1:
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Figure 2:
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 ![]() ![]() |
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Figure 3:
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Figure 4:
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Figure 5:
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 ![]() |
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Figure 6:
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Figure 7:
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Figure 8:
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 ![]() ![]() |
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Figure 9:
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 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 10:
Illustration of an arm cavity of the Virgo gravitational-wave detector [56]: the macroscopic length ![]() ![]() ![]() ![]() |
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Figure 11:
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Figure 12:
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Figure 13:
Finesse example: Mirror reflectivity and transmittance. |
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Figure 14:
Finesse example: Length and tunings. |
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Figure 15:
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 ![]() ![]() |
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Figure 16:
Some of the lowest-order Bessel functions ![]() ![]() ![]() |
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Figure 17:
Electric field vector ![]() ![]() ![]() ![]() |
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Figure 18:
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 ( ![]() |
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Figure 19:
A sinusoidal signal with amplitude ![]() ![]() ![]() ![]() ![]() |
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Figure 20:
Finesse example: Modulation index. |
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Figure 21:
Finesse example: Mirror modulation. |
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Figure 22:
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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Figure 23:
Finesse example: Optical beat. |
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Figure 24:
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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Figure 25:
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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Figure 26:
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 ![]() |
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Figure 27:
Power transmitted and circulating in a two mirror cavity with input power 1 W. The mirror transmissions are set such that ![]() ![]() ![]() ![]() ![]() |
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Figure 28:
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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Figure 29:
Power in the South port of a symmetric Michelson interferometer as a function of the arm length difference ![]() |
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Figure 30:
Finesse example: Michelson power. |
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Figure 31:
Finesse example: Michelson modulation. |
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Figure 32:
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 ![]() ![]() |
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Figure 33:
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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Figure 34:
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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Figure 35:
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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Figure 36:
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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Figure 37:
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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Figure 38:
Finesse example: Cavity power and slope. |
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Figure 39:
Finesse example: Michelson with Schnupp modulation. |
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Figure 40:
One dimensional cross-section of a Gaussian beam. The width of the beam is given by the radius ![]() ![]() |
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Figure 41:
Gaussian beam profile along z: this cross section along the x-z-plane illustrates how the beam size ![]() ![]() ![]() ![]() ![]() |
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Figure 42:
This plot shows the intensity distribution of Hermite–Gauss modes ![]() |
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Figure 43:
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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Figure 44:
These plots show the amplitude distribution and wave front (phase distribution) of Hermite–Gaussian modes ![]() ![]() |
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Figure 45:
These plots show the amplitude distribution and wave front (phase distribution) of helical Laguerre–Gauss modes ![]() ![]() |
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Figure 46:
Intensity profiles for helical Laguerre–Gauss modes ![]() ![]() ![]() |
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Figure 47:
Intensity profiles for sinusoidal Laguerre–Gauss modes ![]() ![]() ![]() ![]() ![]() |
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Figure 48:
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Figure 49:
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Figure 50:
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Figure 51:
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Figure 52:
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Figure 53:
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Figure 54:
Coupling coefficients for Hermite–Gauss modes: for each optical element and each direction of propagation complex coefficients ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Figure 55:
Finesse example: Beam parameter |
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Figure 56:
Finesse example: Mode cleaner |
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Figure 57:
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’. |
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