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Figure 6. Plane Grating Configuration

2.3 Czerny-Turner Configuration

The Czerny-Turner (CZ) monochromator consists of two concave mirrors and one plano diffraction grating (see Figure 7).

Although the two mirrors function in the same separate capacities as the single spherical mirror of the Fastie-Ebert configuration, i.e., first collimating the light source (mirror 1), and second, focusing the dispersed light from the grating (mirror 2), the geometry of the mirrors in the Czerny-Turner configuration is flexible.

By using an asymmetrical geometry, a Czerny-Turner configuration may be designed to produce a flattened spectral field and good coma correction at one wavelength. Spherical aberration and astigmatism will remain at all wavelengths.

It is also possible to design a system that may accommodate very large optics.

Figure 7. Czerny-Turner Configuration

2.4 Czerny-Turner/Fastie-Ebert PGS Aberrations

PGS spectrometers exhibit certain aberrations that degrade spectral resolution, spatial resolution, or signal­to­noise ratio. The most significant are astigmatism, coma, spherical aberration and defocusing. PGS systems are used off­axis, so the aberrations will be different in each plane. It is not within the scope of this document to review the concepts and details of these aberrations, (reference 4) however, it is useful to understand the concept of Optical Path Difference (OPD) when considering the effects of aberrations.

Basically, an OPD is the difference between an actual wavefront produced and a "reference wavefront" that would be obtained if there were no aberrations. This reference wavefront is a sphere centered at the image or a plane if the image is at infinity. For example:

Defocusing results in rays finding a focus outside the detector surface producing a blurred image that will degrade bandpass, spatial resolution, and optical signal-to-noise ratio. A good example could be the spherical wavefront illuminating mirror M1 in Figure 7. Defocusing should not be a problem in a PGS monochromator used with a single exit slit and a PMT detector. However, in an uncorrected PGS there is field curvature that would display defocusing towards the ends of a planar linear diode array. Geometrically corrected CZ configurations such as that shown in Figure 7 nearly eliminate the problem. The OPD due to defocusing varies as the square of the numerical aperture.

Coma is the result of the off-axis geometry of a PGS and is seen as a skewing of rays in the dispersion plane enlarging the base on one side of a spectral line as shown in Figure 8. Coma may be responsible for both degraded bandpass and optical signal-to-noise ratio. The OPD due to coma varies as the cube of the numerical aperture. Coma may be corrected at one wavelength in a CZ by calculating an appropriate operating geometry as shown in Figure 7.

Figure 8. The Effect of Coma

Spherical aberration is the result of rays emanating away from the centre of an optical surface failing to find the same focal point as those from the centre (See Figure 9). The OPD due to spherical aberration varies with the fourth power of the numerical aperture and cannot be corrected without the use of aspheric optics.

Figure 9. The Effect of Spherical Aberration

Astigmatism is characteristic of an off-axis geometry. In this case a spherical mirror illuminated by a plane wave incident at an angle to the normal (such as mirror M2 in Figure 7) will present two foci: the tangential focus, F_t, and the sagittal focus, F_S. Astigmatism has the effect of taking a point at the entrance slit and imaging it as a line perpendicular to the dispersion plane at the exit (see Figure 10), thereby preventing spatial resolution and increasing slit height with subsequent degradation of optical signal­to­noise ratio. The OPD due to astigmatism varies with the square of numerical aperture and the square of the off­axis angle and cannot be corrected without employing aspheric optics.

Figure 10. Effects of Astigmatism in a Concave Mirror used "Off-Axis"

2.4.1 Aberration Correcting Plane Gratings

Recent advances in holographic grating technology now permits complete correction of ALL aberrations present in a spherical mirror based CZ spectrometer at one wavelength, with excellent mitigation over a wide wavelength range (Reference 12).


2.5 Concave Aberration Corrected Holographic Gratings

Both the monochromators and spectrographs of this type use a single holographic grating with no ancillary optics.

In these systems, the grating both focuses and diffracts the incident light.

With only one optic in their design, these devices are inexpensive and compact. Figure 11a illustrates an ACHG monochromator. Figure 11b illustrates an ACHG spectrograph in which the location of the focal plane is established by:

b_H - Angle between perpendicular to spectral plane and grating normal.

L_H - Perpendicular distance from spectral plane to grating.

Figure 11a. An ACHG Monochromator

Figure 11b. An ACHG Spectrograph

2.6 Calculating alpha and beta in a Monochromator Configuration

From Equation (1-2),
(remains constant)
Taking this equation and Equation (1-3),
(2-1)

Use Equations (2-1) and (1-2) to determine a and b, respectively. See Table 3 for worked examples.

Note: In practice the highest wavelength attainable is limited by the mechanical rotation of the grating. This means that doubling the groove density of the grating will halve the spectral range (see Section 2.14).


2.7 Monochromator System Optics

To understand how a complete monochromator system is characterized, it is necessary to start at the transfer optics that brings light from the source to illuminate the entrance slit (see Figure 12).  Here we have "unrolled" the system and drawn it in a linear fashion.

AS - aperture stop
L1 - lens 1
M1 - mirror 1
M2 - mirror 2
G1 - grating
p - object distance to lens L1
q - image distance from lens L1
F - focal length of lens L1 (focus of an object at infinity)
d - the clear aperture of the lens (L1 in diagram)
W - half-angle
s - area of the source
s' - area of the image of the source


2.8 Aperture Stops and Entrance and Exit Pupils

An aperture stop (AS) limits the opening through which a cone of light may pass and is usually located adjacent to an active optic.

A pupil is either an aperture stop or the image of an aperture stop.

The entrance pupil of the entrance (transfer) optics in Figure 12 is the virtual image of AS as seen axially through lens L1 from the source.

The entrance pupil of the spectrometer is the image of the grating (G1) seen axially through mirror M1 from the entrance slit.

The exit pupil of the entrance optics is AS itself seen axially from the entrance slit of the spectrometer.

The exit pupil of the spectrometer is the image of the grating seen axially through M2 from the exit slit.


2.9 Aperture Ratio (f/value, f/Number), and Numerical Aperture (NA)

The light gathering power of an optic is rigorously characterized by Numerical Aperture (NA).

Numerical Aperture is expressed by:

where μ is the refractive index (μ = 1 in air) (2-2)

and f/value by:
(2-3)

Table 2: Relationship between f/value, half-angle, and numerical aperture

2.9.1 f/value of a Lens System

f/value is also given by the ratio of either the image or object distance to the diameter of the pupil. When, for example, a lens is working with finite conjugates such as in Fig. 12, there is an effective f/value from the source to L1 (with diameter AS) given by:

(2-4)

and from L1 to the entrance slit by:

(2-5)

In the sections that follow f/value will always be calculated assuming that the entrance or exit pupils are equivalent to the aperture stop for the lens or grating and the distances are measured to the center of the lens or grating.

When the f/value is calculated in this way for f/2 or greater (e.g. f/3, f/4, etc.), then sin w is ~ tan w and the approximation is good. However, if an active optic is to function at an f/value significantly less than f/2, then the f/value should be determined by first calculating Numerical Aperture from the half-angle.


2.9.2 f/value of a Spectrometer

Because the angle of incidence alpha is always different in either sign or value from the angle of diffraction beta (except in Littrow), the projected size of the grating varies with the wavelength and is different depending on whether it is viewed from the entrance or exit slits. In Figures 13a and 13b, the widths W' and W'' are the projections of the grating width as perceived at the entrance and exit slits, respectively.

Figure 13. Projection of the grating width on (a) the Entrance and (b) the Exit.

To determine the f/value of a spectrometer with a rectangular grating, it is first necessary to calculate the "equivalent diameter", D', as seen from the entrance slit and D" as seen from the exit slit. This is achieved by equating the projected area of the grating to that of a circular disc and then calculating the diameter D' or D".

(2-6)
(2-7)

In a spectrometer, therefore, the f/value_in will not equal the f/value_out.

(2-8)
(2-9)

where, for a rectangular grating, D' and D" are given by:
(2-10)

(2-11)


where, for a circular grating, D' and D" are given by:

(2-12)
(2-13)

Table 3 shows how the f/value changes with wavelength.

Table 3 Calculated values for f/value_in and f/value_out for a Czerny-Turner configuration with 68 x 68 mm, 1800 g/mm grating and L_A = L_B = F = 320 nm. D_v = 24°.

2.9.3 Magnification and Flux Density

In any spectrometer system a light source should be imaged onto an entrance slit (aperture) which is then imaged onto the exit slit and so on to the detector, sample, etc. This process inevitably results in the magnification or demagnification of one or more of the images of the light source. Magnification may be determined by the following expansions, taking as an example the source imaged by lens L1 in Figure 12 onto the entrance slit:
(2-14)


Similarly, flux density is determined by the area that the photons in an image occupy, so changes in magnification are important if a flux density sensitive detector or sample are present. Changes in the flux density in an image may be characterized by the ratio of the area of the object, S, to the area of the image, S', from which the following expressions may be derived:
(2-15)

These relationships show that the area occupied by an image is determined by the ratio of the square of the f/values. Consequently, it is the EXIT f/value that determines the flux density in the image of an object. Those using photographic film as a detector will recognize these relationships in determining the exposure time necessary to obtain a certain signal-to-noise ratio.


2.10 Exit Slit Width and Anamorphism

Anamorphic optics are those optics that magnify (or demagnify) a source by different factors in the vertical and horizontal planes (see Figure 14).

Figure 14. (a) Vertical and (b) Horizontal Magnification

In the case of a diffraction grating-based instrument, the image of the entrance slit is NOT imaged 1:1 in the exit plane (except in Littrow and perpendicular to the dispersion plane assuming L_A = L_B).

This means that in virtually all commercial instruments the tradition of maintaining equal entrance and exit slit widths may not always be appropriate.

Geometric horizontal magnification depends on the ratio of the cosines of the angle of incidence, alpha, and the angle of diffraction, beta, and the L_B/L_A ratio (Equation 2­16). Magnification may change substantially with wavelength (see Table 4).

(2-16)

Table 4 illustrates the relationship between alpha, beta, dispersion, horizontal magnification of entrance slit image, and bandpass.

Table 4 Relationship Between Dispersion, Horizontal Magnification, and Bandpass in a Czerny­Turner Monochromator. L_A = 320 mm, L_B = 320 mm, D_v = 24°, n = 1800 g/mm Entrance slit width = 1 mm

Exit slit width matched to image of entrance slit.

*As the inclination of the grating becomes increasingly large, coma in the system will increase. Consequently, in spite of the fact that the bandpass at 800 nm is superior to that at 200 nm, it is unlikely that the full improvement will be seen by the user in systems of less than f/8.


2.11 Slit Height Magnification

Slit height magnification is directly proportional to the ratio of the entrance and exit arm lengths and remains constant with wavelength (exclusive of the effects of aberrations that may be present).

(2-17)

Note: Geometric magnification is not an aberration!


2.12 Bandpass and Resolution

In the most fundamental sense both bandpass and resolution are used as measure of an instrument's ability to separate adjacent spectral lines.

Assuming a continuum light source, the bandpass (BP) of an instrument is the spectral interval that may be isolated. This depends on many factors including the width of the grating, system aberrations, spatial resolution of the detector, and entrance and exit slit widths.

If a light source emits a spectrum which consists of a single monochromatic wavelength l_o (Figure 15) and is analyzed by a perfect spectrometer, the output should be identical to the spectrum of the emission (Fig. 16) which is a perfect line at precisely l_o.

In reality, spectrometers are not perfect and produce an apparent spectral broadening of the purely monochromatic wavelength. The line profile now has finite width and is known as the "instrumental line profile" (instrumental bandpass) (see Figure 17).

The instrumental profile may be determined in a fixed grating spectrograph configuration with the use of a reasonably monochromatic light source such as a single mode dye laser. For a given set of entrance and exit slit parameters, the grating is fixed at the proper orientation for the central wavelength of interest and the laser light source is scanned in wavelength. The output of the detector is recorded and displayed. The resultant trace will show intensity versus wavelength distribution.

For a monochromator the same result would be achieved if a monochromatic light source is introduced into the system and the grating rotated.

The bandpass is then defined as the Full Width at Half Maximum (FWHM) of the trace assuming monochromatic light.

Any spectral structure may be considered to be the sum of an infinity of single monochromatic lines at different wavelengths. Thus, there is a relationship between the instrumental line profile, the real spectrum and the recorded spectrum.

Let B(l) be the real spectrum of the source to be analysed.
Let F(l) be the recorded spectrum through the spectrometer.
Let P(l) be the instrumental line profile.

(2-18)

The recorded function F(l) is the convolution of the real spectrum and the instrumental line profile.

The shape of the instrumental line profile is a function of various parameters:

• the width of the entrance slit
• the width of the exit slit or of one pixel in the case of a multichannel detector
• diffraction phenomena
• aberrations
• quality of the system's components and alignment

• Bandpass varies as cos a
• Dispersion varies as cos b

• Changes in system aberrations as the grating is rotated (e.g., coma)
• Changes in the diffracted wavefront of the grating in higher orders (most serious with classically ruled gratings)
• Residual system aberrations such as spherical aberration, coma, astigmatism, and field curvature swamping grating capabilities (particularly low f/value, e.g., f/3, f/4 systems)

• A system that will allow the largest entrance slit width for the bandpass required.
• The highest dispersion.
• The largest optics affordable.
• Longest focal length affordable.
• Highest groove density that will accommodate the spectral range.
• Optics and coatings appropriate for specific spectral range.
• Entrance optics which will optimize etendue.
• If the instrument is to be used at a single wavelength in a non-scanning mode, then it must be possible to adjust the exit slit to match the size of the entrance slit image.