## Optics Tutorial

## Section 2: Monochromators & Spectrographs

## 2.1 Basic Designs

Monochromator and spectrograph systems form an image of the entrance slit in the exit plane at the wavelengths present in the light source. There are numerous configurations by which this may be achieved - only the most common are discussed in this document and includes Plane Grating Systems (PGS) and Aberration Corrected Holographic Grating (ACHG) systems.

**Definitions**

L_{A} - entrance arm length

L_{B} - exit arm length

h - height of entrance slit

h' - height of image of the entrance slit

α - angle of incidence

β - angle of diffraction

w - width of entrance slit

w' - width of entrance slit image

D_{g} - diameter of a circular grating

W_{g} - width of a rectangular grating

H_{g} - height of a rectangular grating

## 2.2 Fastie-Ebert Configuration

A Fastie-Ebert instrument consists of one large spherical mirror and one plane diffraction grating (see Figure 6).

A portion of the mirror first collimates the light which will fall upon the plane grating. A separate portion of the mirror then focuses the dispersed light from the grating into images of the entrance slit in the exit plane.

It is an inexpensive and commonly used design, but exhibits limited ability to maintain image quality offaxis due to system aberrations such as spherical aberration, coma, astigmatism, and a curved focal field.

## 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.

## 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, Ft, and the sagittal focus, FS. 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.

## 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:

β_{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 α and β 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 α and β, 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)

Ω - 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

f/value | f/2 | f/3 | f/5 | f/7 | f/10 | f/15 |
---|---|---|---|---|---|---|

n (degrees) | 14.48 | 9.6 | 5.7 | 4.0 | 2.9 | 1.9 |

NA | 0.25 | 0.16 | 0.10 | 0.07 | 0.05 | 0.03 |

## 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 ω is ~ tan ω 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 β (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.

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 for a Czerny-Turner configuration with 68 x 68 mm, 1800 g/mm grating and LA = LB = F = 320 nm. Dv = 24°.

λ(nm) | α | β | f/value_{in} |
f/value_{out} |
---|---|---|---|---|

200 | 1.40 | 22.60 | 4.17 | 4.34 |

320 | 5.12 | 29.12 | 4.18 | 4.46 |

500 | 15.39 | 9.39 | 4.25 | 4.74 |

680 | 26.73 | 50.73 | 4.41 | 5.24 |

800 | 35.40 | 59.40 | 4.62 | 5.84 |

## 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).

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 2-16). Magnification may change substantially with wavelength (see Table 4).

(2-16)

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

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

Wavelength (nm) | α (degrees) | β (degrees) | dispersion (nm/mm) | horizontal magnification | bandpass* (nm) |
---|---|---|---|---|---|

200 | -1.4 | 22.60 | 1.60 | 1.08 | 1.74 |

260 | 1.84 | 25.84 | 1.56 | 1.11 | 1.74 |

320 | 5.12 | 29.12 | 1.46 | 1.14 | 1.73 |

380 | 8.47 | 32.47 | 1.41 | 1.17 | 1.72 |

440 | 11.88 | 35.88 | 1.34 | 1.21 | 1.70 |

500 | 15.39 | 39.39 | 1.27 | 1.25 | 1.67 |

560 | 19.01 | 43.01 | 1.19 | 1.29 | 1.64 |

620 | 22.78 | 46.78 | 1.10 | 1.35 | 1.60 |

680 | 26.73 | 50.73 | 1.00 | 1.41 | 1.55 |

740 | 30.91 | 54.91 | 0.88 | 1.49 | 1.49 |

800 | 35.40 | 59.40 | 1.60 | 1.60 | 1.42 |

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 λ_{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 λ_{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(λ) be the real spectrum of the source to be analysed.

Let F(λ) be the recorded spectrum through the spectrometer.

Let P(λ) be the instrumental line profile.

(2-18)

The recorded function F(λ) 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

Each of these factors may be characterized by a special function P_{i}(λ), each obtained by neglecting the other parameters. The overall instrumental line profile P(λ) is related to the convolution of the individual terms:

(2-19)

## 2.12.1 Influence of the Slits (P1(λ))

If the slits are of finite width and there are no other contributing effects to broaden the line, and if:

W_{ent} = width of the image of the entrance slit

W_{ex} = width of the exit slit or of one pixel in the case of a multichannel detector

Δλ1 = linear dispersion x W_{ent}

Δλ2 = linear dispersion x W_{ex}

then the slit's contribution to the instrumental line profile is the convolution of the two slit functions (see Figure 18).

## 2.12.2 Influence of Diffraction (P2(λ))

If the two slits are infinitely narrow and aberrations negligible, then the instrumental line profile is that of a classic diffraction pattern. In this case, the resolution of the system is the wavelength, λ, divided by the theoretical resolving power of the grating, R (Equation 1-11).

## 2.12.3 Influence of Aberrations (P3(λ))

If the two slits are infinitely narrow and broadening of the line due to aberrations is large compared to the size due to diffraction, then the instrumental line profile due to diffraction is enlarged.

## 2.12.4 Determination of the FWHM of the Instrumental Profile

In practice the FWHM of F(λ) is determined by the convolution of the various causes of line broadening including:

dλ (resolution): the limiting resolution of the spectrometer is governed by the limiting instrumental line profile and includes system aberrations and diffraction effects.

dλ (slits): bandpass determined by finite spectrometer slit widths.

dλ (line): natural line width of the spectral line used to measure the FWHM.

Assuming a gaussian line profile (which is not the case), a reasonable approximation of the FWHM is provided by the relationship:

(2-20)

In general, most spectrometers are not routinely used at the limit of their resolution so the influence of the slits may dominate the line profile. From Figure 18 the FWHM, due to the slits, is determined by either the image of the entrance slit or the exit slit, whichever is greater. If the two slits are perfectly matched and aberrations minimal compared to the effect of the slits, then the FWHM will be half the width at the base of the peak. (Aberrations may, however, still produce broadening of the base). Bandpass (BP) is then given by:

BP = FWHM ~ linear dispersion x (exit slit width or the image of the entrance slit, whichever is greater).

In Section 2-10 image enlargement through the spectrometer was reviewed. The impact on the determination of the system bandpass may be determined by taking Equation 2-16 to calculate the width of the image of the entrance slit and multiplying it by the dispersion (Equation 1-5).

Bandpass is then given by:

(2-21)

The major benefit of optimising the exit slit width is to obtain maximum THROUGHPUT without loss of bandpass.

It is interesting to note from Equations (2-21) and (1-5) that:

Bandpass varies as cos α

Dispersion varies as cos β

## 2.12.5 Image Width and Array Detectors

Because the image in the exit plane changes in width as a function of wavelength, the user of an array type detector must be aware of the number of pixels per bandpass that are illuminated. It is normal to allocate 3-6 pixels to determine one bandpass. If the image increases in size by a factor of 1.5, then clearly photons contained within that bandpass would have to be collected over 4-9 pixels. For a discussion of the relation between wavelength and pixel position see Section 5. The FWHM that determines bandpass is equivalent to the width of the image of the entrance slit containing a typical maximum of 80% of available photons at the wavelength of interest; the remainder is spread out in the base of the peak. Any image magnification, therefore, equally enlarges the base spreading the entire peak over additional pixels.

## 2.12.6 Discussion

**a) Bandpass with Monochromatic Light**

The infinitely narrow natural spectral band width of monochromatic light is, by definition, less than that of the instrumental bandpass determined by Equation 2-20. (A very narrow band width is typically referred to as a "line" because of its appearance in a spectrum).

In this case all the photons present will be at exactly the same wavelength irrespective of how they are spread out in the exit plane. The image of the entrance slit, therefore, will consist exclusively of photons at the same wavelength even though there is a finite FWHM. Consequently, bandpass in this instance cannot be considered as a wavelength spread around the center wavelength. If, for example, monochromatic light at 250 nm is present and the instrumental bandpass is set to produce a FWHM of 5 nm, this does NOT mean 250 nm ± 2.5 nm because no wavelength other than 250 nm is present. It does mean, however, that a spectrum traced out (wavelength vs. intensity) will produce a "peak" with an apparent FWHM of "5 nm" due to instrumental and NOT spectral line broadening.

**b) Bandpass with "Line" Sources of Finite Spectral Width**

Emission lines with finite natural spectral bandwidths are routinely found in almost all forms of spectroscopy including emission, Raman, fluorescence, and absorption.

In these cases spectra may be obtained that seem to consist of line emission (or absorption) bands. If, however, one of these "lines" is analysed with a very high resolution spectrometer, it would be determined that beyond a certain bandpass no further line narrowing would take place indicating that the natural bandwidth had been reached.

Depending on the instrument system the natural bandwidth may or may not be greater than the bandpass determined by Equation 2-20.

If the natural bandwidth is greater than the instrumental bandpass, then the instrument will perform as if the emission "line" is a portion of a continuum. In this case the bandpass may indeed be viewed as a spectral spread of ± 0.5 BP around a center wavelength at FWHM.

**Example 1:**

Figure 19 shows a somewhat contrived spectrum where the first two peaks are separated on the recording by 32 mm. The FWHM of the first peak is the same as the second but is less than the third. This implies that the natural bandwidth of the third peak is greater than the bandpass of the spectrometer and would not demonstrate spectral narrowing of its bandwidth even if evaluated with a very high resolution spectrometer.

The first and second peaks, however, may well possess natural bandwidths less than that shown by the spectrometer. In these two cases, the same instrument operating under higher bandpass conditions (narrower slits) may well reveal either additional "lines" that had previously been incorporated into just one band, or a simple narrowing of the bandwidth until either the limit of the spectrometer or the limiting natural bandpass have been reached.

**Example 2:**

A researcher finds a spectrum in a journal that would be appropriate to reproduce on an inhouse spectrometer. The first task is to determine the bandpass displayed by the spectrum. If this information is not given, then it is necessary to study the spectrum itself. Assuming that the wavelengths of the two peaks are known, then the distance between them must be measured with a ruler as accurately as possible. If the wavelength difference is found to be 1.25 nm and this increment is spread over 32 mm (see Figure 19), the recorded dispersion of the spectrum = 1.25/32 = 0.04 nm/mm. It is now possible to determine the bandpass by measuring the distance in mm at the Full Width at Half Maximum height (FWHM). Let us say that this is 4 mm; the bandpass of the instrument is then 4 mm x 0.04 nm/mm = 0.16 nm.

Also assuming that the spectrometer described in Table 4 is to be used, then from Equation 2-21 and the list of maximum wavelengths described in Table 6, the following options are available to produce a bandpass of 0.16 nm:

## Table 5: Variation of Dispersion and Slit Width to Produce 0.16 nm Bandpass in a 320 mm Focal Length Czerny-Turner

Groove Density (g/mm) | Dispersion (nm/mm) | Entrance Slit Width (microns) |
---|---|---|

300 | 9.2 | 17 |

600 | 4.6 | 35 |

1200 | 2.3 | 70 |

1800 | 1.5 | 107 |

2400 | 1.15 | 139 |

3600 | 0.77 | 208 |

The best choice would be the 3600 g/mm option to provide the largest slit width possible to permit the greatest amount of light to enter the system.

## 2.13 Order and Resolution

If a given wavelength is used in higher orders, for example, from first to second order, it is considered that because the dispersion is doubled, so also is the limiting resolution. In a monochromator in which there are ancillary optics such as plane or concave mirrors, lenses, etc., a linear increase in the limiting resolution may not occur. The reasons for this include:

- 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)

Even if the full width at half maximum is maintained, a degradation in line shape will often occur - the base of the peak usually broadens with consequent degradation of the percentage of available photons in the FWHM.

## 2.14 Dispersion and Maximum Wavelength

The longest possible wavelength (λ_{max1}) an instrument will reach mechanically with a grating of a given groove density is determined by the limit of mechanical rotation of that grating. Consequently, in changing from an original groove density, n_{1}, to a new groove density, n_{2}, the new highest wavelength (λ_{max2}) will be:

(2-22)

## Table 6: Variation in Maximum Wavelength with Groove Denisty in a Typical Monochromator

LA = LB = F = 320 mm, DV = 24°. In this example maximum wavelength at maximum possible mechanical rotation of a 1200 g/mm grating = 1300 nm

Groove Density (g/mm) | Dispersion (nm/mm) | Max Wavelength (nm) |
---|---|---|

150 | 18.4 | 10400 |

300 | 9.2 | 5200 |

600 | 4.6 | 2600 |

1200 | 2.3 | 1300 |

1800 | 1.5 | 867 |

2400 | 1.15 | 650 |

3600 | 0.77 | 433 |

From Table 6 it is clear that if a 3600 g/mm grating is required to diffract light above 433 nm, the system will not permit it. If, however, a dispersion of 0.77 nm/mm is required to produce appropriate resolution at, say, 600 nm, a system should be acquired with 640 mm focal length (Equation (1-5)). This would produce a dispersion of 0.77 nm/mm with a 2400 g/mm grating and also permit mechanical rotation up to 650 nm.

## 2.15 Order and Dispersion

In Example 2, Section 2.12.6, the solution to the dispersion problem could be solved by using a 2400 g/mm grating in a 640 mm focal length system. As dispersion varies with focal length (L_{B}), groove density (n), and order (k); for a fixed L_{B} at a given wavelength, the dispersion equation (Equation 1-5) simplifies to:

kn = constant

Therefore, if first order dispersion = 1.15 nm/mm with a 2400 g/mm grating the same dispersion would be obtained with a 1200 g/mm grating in second order. Keeping in mind that kλ = constant for a given groove density, n, (Equation 1-9), using second order with an 1800 g/mm grating to solve the last problem would not work because to find 600 nm in second order, it would be necessary to operate at 1200 nm in first order, when it may be seen in Table 6 that the maximum attainable first order wavelength is 867 nm.

However, if a dispersion of 0.77 nm/mm is necessary in the W at 250 nm, this wavelength could be monitored at 500 nm in first order with the 1800 g/mm grating and obtain a second order dispersion of 0.75 nm/mm. In this case any first order light at 500 nm would be superimposed on top of the 250 nm light (and vice-versa). Wavelength selective filters may then be used to eliminate the unwanted radiation.

The main disadvantages of this approach are that the grating efficiency would not be as great as an optimized first order grating and order-sorting filters are typically inefficient. If a classically ruled grating is employed, ghosts and stray light will increase as the square of the order.

## 2.16 Choosing a Monochromator/Spectrograph

Select an instrument based on:

- 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.

Remember: f/value is not always the controlling factor of throughput. For example, light may be collected from a source at f/1 and projected onto the entrance slit of an f/6 monochromator so that the entire image is contained within the slit. Then the system will operate on the basis of the photon collection in the f/l cone and not the f/6 cone of the monochromator. See Section 3.