What issphericalaberration in Physics

Table 1 shows how α and β vary depending on the deviation angle for a 1200 g/mm grating set to diffract 500 nm in a monochromator geometry, based on Fig. 2.

Unless otherwise indicated, the efficiency of a diffraction grating is measured in the Littrow configuration at a given wavelength.

Note: Just because a grating is “non-blazed” does not necessarily mean that it is less efficient! See Fig. 7 showing the efficiency curve for an 1800 g/mm sinusoidal grooved holographic grating.

i and o are the image and object distances for the lens as in the lens equation. The minimum spherical aberration occurs when Condition for minimum spherical aberration The higher the index of refraction n, the smaller the aberration for the optimum shape. Lens Aberrations

Ghosts are focused and imaged in the dispersion plane of the monochromator.Stray light of a holographic grating is usually up to a factor of ten times less than that of a classically ruled grating, is typically non-focused, and when present, radiates through 2π steradians.

Blaze is defined as the concentration of a limited region of the spectrum into any order other than the zero order. Blazed gratings are manufactured to produce maximum efficiency at designated wavelengths. A grating may, therefore, be described as “blazed at 250 nm” or “blazed at 1 micron,” etc. by appropriate selection of groove geometry.

Relative efficiency measurements require the mirror to be coated with the same material and used in the same angular configuration as the grating.

As a general approximation, for blazed gratings the strength of a signal is reduced by 50% at two-thirds the blaze wavelength, and 1.8 times the blaze wavelength.

Sphericalaberration correction

Although spherical aberration cannot be eliminated for a single lens, it can be minimized by the appropriate bending of the lens into its best form. The degree of bending can be characterized by the Coddington shape factor:

Table 1 shows how the angle of incidence and angle of diffraction vary depending on the deviation angle for a 1200 g/mm grating set to diffract 500 nm in a monochromator geometry.

First order wavelengths between 200 and 380 nm may be monitored without filters because wavelengths below 190 nm are absorbed by air. If, however, the instrument is evacuated or N2 purged, higher order filters would again be required.

Blazed grating groove profiles are calculated for the Littrow condition where the incident and diffracted rays are in auto collimation.

Linear dispersion defines the extent to which a spectral interval is spread out across the focal field of a spectrometer and is expressed in nm/mm, Å/mm, cm-1/mm, etc. For example, consider two spectrometers: one instrument disperses a 0.1 nm spectral segment over 1 mm while the other takes a 10 nm spectral segment and spreads it over 1 mm.

Efficiency in higher orders usually follows the first order efficiency curve. For a grating blazed in first order, the maximum efficiency for each of the subsequent higher orders decreases as the order k increases.

Sphericalaberration photography

Diffraction gratings are manufactured either classically with the use of a ruling engine by burnishing grooves with a diamond stylus, or holographically with the use of interference fringes generated at the intersection of two laser beams1.

Blazed grating groove profiles are calculated for the Littrow condition where the incident and diffracted rays are in auto collimation (i.e., α = β). The input and output rays, therefore, propagate along the same axis. In this case, at the “blaze” wavelength λB.

When the concept of principal focal length is used, the presumption is that all parallel rays focus at the same distance, which is of course true only if there are no aberrations. The use of the lens equation likewise presumes an ideal lens, and that equation is practically true only for the rays close to the optic axis, the so-called paraxial rays. For a lens with spherical aberration, the best approximation to use for the focal length is the distance at which the difference between the paraxial and marginal rays is the smallest. It is not perfect, but the departure from perfect focus forms what is called the "circle of least confusion". Spherical aberration is one of the reasons why a smaller aperture (larger f-number) on a camera lens will give a sharper image and greater depth of field since the difference between the paraxial and marginal rays is less. Lens Aberrations

Monochromatic light has infinitely narrow spectral width. Good sources which approximate such light include single mode lasers and very low pressure, cooled spectral calibration lamps. These are also variously known as “line” or “discrete line” sources.

Sphericalaberration example

Holographic gratings show no ghosts because there are no periodic ruling errors and, therefore, often represent the best solution to ghost problems.

Before introducing the basic equations, a brief explanation on monochromatic light and continuous spectra must first be presented.

λ = the central wavelength of the spectral line to be resolved Wg= the illuminated width of the grating N = the total number of grooves on the grating

Sphericalaberration in mirrors

A grating blazed in first order is equally blazed in the higher orders. Therefore, a grating blazed at 600 nm in first order is also blazed at 300 nm in second order, and so on.

Sphericalaberration formula

In a spectrograph, the linear dispersion for any wavelength other than the wavelength which is normal to the spectral plane will be modified by the cosine of the angle of inclination or tilt angle (γ) at wavelength λn. Fig. 3 shows a “flat field” spectrograph as used with a linear diode array.

R1 and R2 are the surface radii of the spherical lens surfaces. The minimum spherical aberration also depends on the object and image distances, so another factor enters, called the Coddington position factor: Coddington Position Factor i and o are the image and object distances for the lens as in the lens equation. The minimum spherical aberration occurs when Condition for minimum spherical aberration The higher the index of refraction n, the smaller the aberration for the optimum shape. Lens Aberrations

See Fig. 6 and Fig. 7 for typical efficiency curves of a blazed, ruled grating, and a non-blazed, holographic grating, respectively.

Classically ruled gratings may be planar or concave and possess grooves, each parallel with the next. Holographic grating grooves may be either parallel or of unequal distribution in order to optimize system performance. Holographic gratings are generated on planar, spherical, toroidal, and many other surfaces.

If the diffraction grating has periodic ruling errors, a ghost, which is not scattered light, will be focused in the dispersion plane. Ghost intensity is given by:

The amount of spherical aberration in a lens made from spherical surfaces depends upon its shape. Bending the lens can also give partial correction of coma aberration.

Because bandpass is also determined by the slit width of the spectrometer and residual system aberrations, an achieved bandpass at this level is only possible in diffraction limited instruments, assuming an unlikely 100% of theoretical.

It is easy to imagine that fine spectral detail would be more easily identified in the first instrument than the second. The second instrument demonstrates “low” dispersion compared to the “higher” dispersion of the first. Linear dispersion is associated with an instrument’s ability to resolve fine spectral detail.

Sphericalaberration in a lens

For lenses made with spherical surfaces, rays which are parallel to the optic axis but at different distances from the optic axis fail to converge to the same point. For a single lens, spherical aberration can be minimized by bending the lens into its best form. For multiple lenses, spherical aberrations can be canceled by overcorrecting some elements. The use of symmetric doublets like the orthoscopic doublet greatly reduces spherical aberration.

Fig. 4 shows a first order spectrum from 200 to 1000 nm spread over a focal field in spectrograph configuration. From Equation (1) with a grating of given groove density and for a given value of α and β:

In most monochromators, the location of the entrance and exit slits are fixed, and the grating rotates around a plane through the center of its face. The angle, DV, is, therefore, a constant determined by:

If, for example, a light source emits a continuum of wavelengths from 20 nm to 1000 nm, then at the physical location of 800 nm in first order (Fig. 4) wavelengths of 400, 266.6, and 200 nm will also be present and available to the same detector. In order to monitor only light at 800 nm, filters must be used to eliminate the higher orders.

A blazed grating is one in which the grooves of the diffraction grating are controlled to form right triangles with a “blaze angle, ω,” as shown in Fig. 5. However, apex angles up to 110° may be present especially in blazed holographic gratings. The selection of the peak angle of the triangular groove offers opportunity to optimize the overall efficiency profile of the grating.

A continuous spectrum has finite spectral width, e.g. “white light.” In principle, all wavelengths are present, but in practice a “continuum” is almost always a segment of a spectrum. Sometimes a continuous spectral segment may be only a few parts of a nanometer wide and resemble a line spectrum.

Theoretically, a 1200 g/mm grating with a width of 110 mm that is used in first order has a numerical resolving power R = 1200 × 110 = 132,000. Therefore, at 500 nm, the bandpass is equal to:

In a monochromator, LB is the arm length from the focusing mirror to the exit slit, or if the grating is concave, from the grating to the exit slit. Linear dispersion, therefore, varies directly with cos β, and inversely with the exit path length, LB, diffraction order (k), and groove density, n.

Light other than the wavelength of interest reaching a detector (often including one or more elements of “scattered light”) is referred to as stray light.

Sphericalaberration and chromatic aberration

Diffraction gratings are manufactured either classically with the use of a ruling engine, or holographically with the use of interference fringes generated at the intersection of two laser beams.

The minimum spherical aberration also depends on the object and image distances, so another factor enters, called the Coddington position factor:

In most monochromators, the location of the entrance and exit slits are fixed, and the grating rotates around a plane through the center of its face. The angle, DV, is, therefore, a constant.

The efficiency also decreases the farther away from Littrow conditions in which the grating is used (e.g. α ≠ β). Holographic gratings may be designed with groove profiles that discriminate against high orders. This may be particularly effective in the UV and VIS using laminar groove profiles created by ion etching.

Holographic gratings show no ghosts because there are no periodic ruling errors and, therefore, often represent the best solution to ghost problems.

where, dλ is the difference in wavelength between two spectral lines of equal intensity. Resolution is then the ability of the instrument to separate adjacent spectral lines. Two peaks are considered resolved if the distance between them is such that the maximum of one falls on the first minimum of the other. This is called the Rayleigh criterion.

Diffraction gratings are manufactured either classically with the use of a ruling engine, or holographically with the use of interference fringes generated at the intersection of two laser beams.

Remember, ghosts and subsequent stray light intensity are proportional to the square of order and groove density (n2 and k2 from Equation (17). Beware of using ruled gratings in high order or with high groove density.

Regardless of the shape of the surface or whether classically ruled or holographic, the text that follows is applicable to each; explanations are provided where there are differences.

LA = Entrance arm length LB = Exit arm length at λn βH = Angle between the perpendicular to the spectral plane and the grating normal LH = Perpendicular distance from the spectral plane to grating.