A wavefront is a surface of constant phase function. A plane wave has planar wavefronts; variations in refractive index distort them.
The optical path length along a ray is the integral of its refractive index over geometric length. A difference in optical path length produces a phase difference , where is the vacuum wavelength.
A wavefront error is the difference in optical path length from a reference wavefront, often an ideal plane or sphere. Its root mean square should specify the illuminated pupil and which modes, such as constant phase or image displacement, have been removed.
Wave interference occurs when amplitudes add coherently. Equal phases give constructive interference; opposite phases can cancel.
A diffraction grating is a periodic optical structure that directs different wavelengths into different angles through wave interference.
A diffraction order labels a constructive interference maximum by the integer number of wavelengths in the path difference between adjacent grating elements.
For a reflection diffraction grating with both ray directions measured on the same side of its normal, adjacent grooves have path difference . Constructive interference requires this to be an integer multiple . Signed-angle conventions can turn the sum into a difference; the physical path difference is unchanged.
At fixed incidence angle, differentiating the grating equation gives . A camera with focal length therefore has local focal-plane dispersion near its optical axis. With coordinate , the full derivative is .
At fixed wavelength, the grating equation gives . Thus an incident angular width is magnified by in the dispersion direction. Including the ratio of camera and collimator focal lengths gives the projected slit width. The ESO B&C operating manual, Appendix A explicitly includes this anamorphic factor.
A slit of width at the focus of a collimator subtends . Its projected width is . Dividing this width by the grating dispersion gives . Thus the spectral resolving power is . Omitting the anamorphic factor gives a denominator instead; these agree in the Littrow configuration.
In the Littrow configuration the selected diffracted ray retraces the incident ray, so and the grating equation becomes .
An echelle grating works in high diffraction orders. A separate cross-disperser separates their overlapping optical spectra on a two-dimensional detector.
The free spectral range is the wavelength or frequency spacing between adjacent orders or resonances of an optical instrument. It identifies when different orders can overlap.
At fixed incident and central outgoing angles, the grating equation gives . Adjacent central wavelengths are separated byThis is the usual adjacent-order free spectral range. A detector width of approximately , where is the grating dispersion, covers one such interval.
The point spread function is the image of an unresolved point source. It describes blurring from diffraction, optical aberrations, and astronomical seeing.
A diffraction-limited system has imaging performance set primarily by diffraction rather than by optical aberrations.
For a circular illuminated aperture of diameter , the characteristic angular diffraction scale is . The first zero of the ideal intensity pattern lies at approximately .
The Strehl ratio compares the peak of an observed point spread function with the ideal diffraction-limited system peak for the same pupil and total flux. It measures how strongly wavefront errors redistribute light away from the central peak.
For small residual phase errors, , where . The variance is taken after removing irrelevant constant phase. This approximation to the Strehl ratio is most useful near the diffraction limit of a telescope.
A speckle pattern is a granular intensity pattern from coherent wave interference. In stellar imaging, residual wavefront errors produce speckles around the central point spread function.
For a small sinusoidal optical path length error , expand its phase factor to first order. The two exponential harmonics each have field amplitude relative to the unperturbed field, so each speckle has contrast . Its angular displacement is . Here is optical path error, not physical mirror displacement; a reflecting mirror introduces twice its surface displacement in the near-normal limit.
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Physical optics is a branch of optics that focuses on the wave nature of light and its interactions with matter. Unlike geometrical optics, which primarily deals with the propagation of light in terms of rays and prisms, physical optics examines phenomena such as interference, diffraction, and polarization, which cannot be adequately explained by ray optics alone. Key concepts in physical optics include: 1. **Wave Nature of Light**: Light is treated as a wave, which means it is subject to wave phenomena.