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1 Yaroslavl State Pedagogical University named after. K.D. Ushinsky Laboratory work 8 Determination of the parameters of the Rowland diffraction grating Yaroslavl 010
2 Contents 1. Questions to prepare for work Theoretical introduction Diffraction by slits Interference from many slits Grating as a spectral device Description of the installation Procedure for performing the work Assignment Assignment Assignment Assignment Test questions
3 1. Questions to prepare for work Laboratory work 8. Determination of the parameters of the Rowland diffraction grating Purpose of the work: familiarization with the principle of operation and determination of the parameters of the reflective diffraction grating, measurement of the light wavelength using this grating. Instruments and accessories: metal diffraction grating, mercury-quartz lamp, specially designed machine. Literature: 1. Landsberg G.S. Optics, M. Science, 1976. Savelyev I.V. Physics course, vol. 3, 1971 1. Questions for preparation for work 1. Fraunhofer diffraction by a slit. Design, operating principle and parameters of a diffraction grating. Rowland grid. 3. The grid is like a spectral apparatus. Dispersion and resolution of a diffraction grating.. Theoretical introduction A diffraction grating is a collection of a large number of narrow parallel slits, closely spaced at equal distances from each other. The slits can be applied to an opaque screen or, conversely, opaque grooves are applied to a transparent plate (glass). The action of the grating is based on the phenomenon of diffraction by a slit and interference from many slits. Before clarifying the effect of the grating as a whole, let us consider diffraction at a single slit. 3
4.1. Diffraction by a slit Let a plane monochromatic wave be incident on a screen with a narrow infinitely long slit. In Fig. 1 FF 1 is a projection of a screen with a slit AB onto the drawing plane. The slit width (b) is of the order of the wavelength of light. Slit AB cuts out part of the front of the incident light wave. All points of this front oscillate in the same phases and, based on the Huygens-Fresnel principle, are sources of secondary waves. b F A B F 1 L F A ϕ C B F 1 L O 1 O Fig..1 E O 1 Fig.. Secondary waves propagate in all directions from (0) to (± π) to the direction of wave propagation (Fig..1). If you place a lens behind the slit, then all the rays that went parallel to the lens will converge at one point on the focal plane of the lens. At this point, interference of secondary waves is observed. The result of interference depends on the number of half-wavelengths that fits into the path difference between the corresponding beams. Let us consider rays that travel at a certain angle ϕ to the direction of the incident light wave (Fig..). BC = δ path difference between the outer rays. Let us divide AB into Fresnel zones (Fresnel zones in this case are a system of parallel planes perpendicular to the plane of the drawing and constructed so that the distance from the edges of each zone to point O 1 differs by). If δ contains an even number of half-wavelengths, then at point O 1 there will be attenuation of light min. If odd, then the light gain is 4 E
5 . Theoretical introduction max. Therefore, with δ = ±m min with δ = ±(m + 1) max where m = 0; 1; ;... Since δ = b sin ϕ (see figure..), these conditions can be written in the following form: b sin ϕ = ±m b sin ϕ = ±(m + 1) min (.1) max (. ) Figure 3 shows the distribution of light intensity during diffraction by a slit depending on the angle. It can be calculated using the formula: I ϕ = I o sin (π b sin ϕ) (π b sin ϕ) where I o is the intensity in the middle of the diffraction pattern; I ϕ intensity at the point defined by the value. I ϕ 3 b b b 0 b b 3 b sin ϕ Fig..3.. Interference from many slits Consider several parallel slits of the same width (b), located at a distance (a) from each other (diffraction grating) (see Fig..4 ). 5
6 a d b δ 1 ϕ L O Fig. 4 The diffraction pattern from slits, as in the previous case, will be observed in the focal plane of the lens (L). But the phenomenon is complicated by the fact that in addition to diffraction from each slit, addition of light vibrations also occurs in beams arriving at the focal plane of the lens from individual slits, i.e. interference of many beams occurs. If the total number of slits is N, then N beams interfere with each other. The path difference from two adjacent slits is equal to δ 1 = (b+a) sin ϕ or δ 1 = d sin ϕ, where d = a + b is called the lattice constant. This path difference corresponds to the same phase difference ψ = π δ1 between adjacent beams. As a result of interference in the focal plane of the lens, resulting oscillations are obtained with a certain amplitude, which depends on the phase difference. If ψ = mπ (which corresponds to the path difference δ 1 = m), then the oscillation amplitudes add up and the light intensity reaches a maximum. These maxima are called main ones because they have significant intensity and their position does not depend on the total number of slits. If ψ = m () π N (or δ1 = m N), then minima of light are formed in these directions. Therefore, with interference N 6 E
7. The theoretical introduction of beams of the same amplitude gives rise to a number of main maxima, determined by the condition: d sinϕ = ±m (.3) where m = 0;1;;... and additional minima, determined by the condition: d sinϕ = ±m N (.4) where m = 1;;3;... except m = 0;N;N;..., because in this case, condition (.4) turns into condition (.3) of the main maxima. From conditions (.4) and (.3) it is clear that between the two main maxima there are (N 1) additional minima, between which there are, respectively, (N) secondary maxima, defined by the condition: d sinϕ = ±(m + 1) N ( .5) I ϕ N = sinϕ N = 3 sinϕ N = 4 sinϕ Fig..5. (without taking into account diffraction at one slit) As the number of slits increases, the number of additional minima increases, and the main maxima become narrower and brighter. In Fig.5 it is given 7
8 intensity distribution during the interference of several beams (slits). Thus, under the action of many slits we have in directions determined by the conditions: b sinϕ = ±m min from each slit, b sinϕ = ±(m + 1) max from each slit, d sinϕ = ±m main maxima result d sinϕ = ± m N d sinϕ = ±(m + 1) N interference of many beams, additional minima, secondary maxima. When observing the picture given by a diffraction grating, we clearly see only the main maxima, separated by almost dark intervals, because the secondary maxima are very weak, the intensity of the strongest of them is no more than 5% of the main one. The intensity distribution between the individual main maxima is not the same. It depends on the slit diffraction intensity distribution and the ratio between (b) and (d). In the case where (b) and (d) are commensurate, some main maxima are missing, because These directions correspond to diffraction minima. Thus, at d = b, all even maxima disappear, which leads to an increase in odd ones. At d = 3b, every third maximum disappears. The described phenomenon is illustrated in Fig. 6. The intensity distribution depending on the angle can be calculated using the formula: I ϕ solve. = I o sin (πbsin ϕ) sin (Nπdsin ϕ) (πbsin ϕ) sin (πbsin ϕ) where I o is the intensity created by one slit in the center of the picture. 8
9 . Theoretical introduction I 1 (ϕ) Diffraction pattern at one slit, N = 1 b b sinϕ I (ϕ x) Interference pattern, N = 4 ()()() 3 d d d d d 3 d sinϕ I(ϕ) Total intensity distribution pattern for the grating N = 5 and d b = 4 d Fig.6 sinϕ 9
10 3. The grating as a spectral device As the number of slits increases, the intensity of the main maxima increases, because the amount of light transmitted by the grating increases. But the most significant change caused by a large number of gaps is the transformation of the diffuse main maxima into sharp, narrow maxima. The sharpness of the maxima makes it possible to distinguish close wavelengths, which are depicted as separate, bright stripes and will not overlap each other, as is the case with vague maxima obtained with one or a small number of slits. A diffraction grating, like any spectral device, is characterized by dispersion and resolution. The angular distance between two lines differing in wavelength by 1 Å is taken as a measure of dispersion. If two lines differing in length by δ correspond to a difference in angles equal to δϕ, then the measure of dispersion will be the expression: D = δϕ δ = m dcos ϕ (3.6) The resolution of the grating is characterized by the ability to distinguish the presence of two close waves (resolve two wavelengths) . Let us denote by the minimum interval between two waves that can be resolved by a given diffraction grating. The measure of grating resolution is usually taken to be the ratio of the wavelength around which the measurement is performed to the specified minimum interval, i.e. A =. The calculation gives that: A = = mn, (3.7) where m is the order of the spectrum, N is the total number of grating slits. High resolution and dispersion of diffraction gratings is achieved due to large values of N and small d (grating periods). Rowland lattices have these parameters. The Rowland grating is a concave metal mirror on which grooves (strokes) are applied. It can simultaneously act as a grating and a collecting lens, allowing 10
11 4. Description of the installation to obtain a diffraction pattern directly on the screen. 4. Description of the installation A D 1 ϕ R 4 3 B l E C Fig. 4.1 Measurement setup in Fig. 4.1 consists of rigidly fixed rails (AB and BC), along which the rail DE can slide freely. A Rowland grid (1) is attached to one end of the rail. The grille is fixed so that its plane is perpendicular to the DE rail. The light source is a slit (4), illuminated by a mercury-quartz lamp (3). When the grating is illuminated along the AB direction, spectra of different orders can be observed. The distance from the slit to the lines under study in the spectrum of mercury is recorded on a scale marked on the BC staff using a telescope (). 5. Work order Task 1. Familiarize yourself with the description of the work and the optical design of the device. eleven
12 Task. Determine the Rowland lattice constant. The lattice constant is determined from the condition of the main maximum: d = m sin ϕ. From the installation diagram Fig. 4.1: sinϕ = l R, where l is the distance from the slit to the position of the spectral line on the bench (BC), R is the length of the staff (DE). The final working formula is: d = m R l (5.8) The constant is determined for three lines in the spectrum of mercury: Line Brightness Å Violet-blue Green Yellow 1 (closest to green) Wavelengths are indicated with greater accuracy than other members of the formula (5.8 ), so we can assume that = const. Rail length (DE) R = (150 ± 5) mm. Take the reliability coefficient α = 3. 1 The task should be performed in the following sequence: 1) turn on the mercury-quartz lamp and warm it up for 5 minutes, and then check whether the gap is well illuminated;) moving the DE rail along the rails, find it using a spotting scope green line in the first order spectrum, m = 1 (left side of bench BC), if the line is wide, then reduce the slit width and take reading (l). The tube is then transferred to the violet-blue line (to the left of the green line along the BC bench);
13 5. Work order 3) carry out the same measurements for the same lines in the second-order spectrum, m = (right side of the bench BC); measurements for m > are not carried out because The BC rail is not long enough for this. In this work, we can limit ourselves to single measurements, because the relative error in determining (R) significantly exceeds the relative error in determining l (δ l = 0.5 mm at α = 3). The final result is thus determined for all lines with approximately the same accuracy, so it can finally be averaged over all measured lines. The error in determining the Rowland lattice constant is determined by the formula: δd = d R δ R, (5.9) δ R = 5 mm standard error in determining the length of the staff (DE). It is convenient to enter the experimental data into a table of the following form: Table 1 m, Å l (mm) d(mm) d avg Yellow Yellow. Task 3. Determine the wavelength of one of the yellow lines. Using the results obtained in the task, determine the wavelength of the second yellow line: Жii = d Жi l Жii mr (5.10) 13
14 where d and lattice constant obtained in the task. The values of zii for both orders (m = 1 and m =) are equally accurate, i.e. are determined by the standard deviations δ d and δ R, so they can be averaged. The error is determined by the formula: Жii = (жii d avg. The final result is written in the form:) () δd + Жii δr R. (5.11) Жii = (жiiср ± Жii)Å, with α = 3. Task 4. Determine the angular dispersion of the lattice Rowland. To determine the angular dispersion of a diffraction grating, you need to measure the angular distance between two close spectral lines. It is convenient to use yellow mercury lines for this. is given in the text of the task. zhii taken from task 3. D = δ ϕ δ ϕ zhi ϕ zhii zhi zii. (5.1) The angular dispersion for both orders (m = 1 and m =) should be determined. Compare the obtained values with each other and with the values obtained using the formula: D = m d av cos ϕ (5.13) As instructed by the teacher, evaluate the errors for expressions (5.1) and (5.13). Task 5. Calculate the theoretical value of the resolution of the Rowland diffraction grating. where N is the number of grating lines. A = mn (5.14) 14
15 6. Test questions The value of N is determined based on the length of the grating (L = 9 ± 0.1 mm) at α = 3 and the value of the grating constant (see task). Perform calculations for both orders (m = 1 and m =). Estimate the magnitude of the error for expression (5.14). 6. Test questions 1. Why should the size of the slit be commensurate with the wavelength? Why is the zeroth order maximum when the grating is illuminated with white light white, and the rest are iridescent? 3. How does the grating period affect the diffraction pattern? 4. Show that when determining the period, random error can be neglected. 15
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DIFFRACTION GRATING- optical an element that is a collection of a large number of regularly spaced strokes (grooves, slots, protrusions) applied in one way or another to a flat or concave optical lens. surface. D. r. used in spectral instruments as a dispersing system for spatial decomposition of el-magn. into the spectrum. The front of a light wave incident on a laser is broken up by its streaks into separate beams, which, having passed through the streaks, interfere (see Fig. Interference of light), forming the resulting spatial distribution of light intensity - the emission spectrum.
There are reflective and transparent D. r. In the first, the strokes are applied to a mirror (metallic) surface, and the resulting interference pattern is formed in the light reflected from the grating. On the second, the strokes are applied to a transparent (glass) surface, and. the picture is formed in transmitted light.
If the strokes are applied to a flat surface, then such D. r. called flat, if concave - concave. Modern spectral instruments use both flat and concave D. r., Ch. arr. reflective.
Flat reflective D. R., manufactured using special dividing machines with a diamond cutter have straight, strictly parallel and equidistant strokes of the same shape, the edges are determined by the profile of the cutting edge of the diamond cutter. Such a D. r. represents a periodic structure with post. distance d between the strokes (Fig. 1), so-called. period D. r. There are amplitude and phase D. r. For the former, the coefficient changes periodically. reflection or transmission, which causes a change in the amplitude of the incident light wave (such as a lattice of slits in an opaque screen). In phase D. r. special touches are given. a form that periodically changes the phase of the light wave.
Rice. 1. Scheme of a one-dimensional periodic structure of a flat diffraction grating (highly enlarged): d - grating period; W is the length of the threaded part of the grating.
Rice. 2. Diagram illustrating the principle of operation of a diffraction grating: a- phase reflective, b- amplitude slot.
Rice. 3. Interference functions of a diffraction grating.
If on a flat D.r. a parallel beam of light falls, the axis of which lies in the plane perpendicular to the lines of the grating, then, as calculations show, the result is the result of the interference of coherent beams from all N grating strokes, the spatial (in the corners) distribution of light intensity (in the same plane) can be represented as a product of two functions: . Function Jg determined by the diffraction of light on the part. stroke, function J N caused by interference N coherent beams coming from the grating strokes, and is connected with the periodic. structure of D. r. Function J N for a given wavelength is determined by the grating period d, the total number of grating lines N and the angles formed by the incident (angle) and diffracted (angle) beams with the normal to the grating (Fig. 2), but does not depend on the shape of the lines. It has the form , where , - between coherent parallel beams going at an angle from adjacent strokes of the D.R.: =AB+AC(see Fig. 2, A- for phase reflective D. r., 2, b- for an amplitude slot grid). Function J N- periodic function with sharp intense hl. maxima and small secondary maxima (Fig. 3, A). Between neighboring ch. located at the maxima N-2 secondary maxima and N-1 minima, where the intensity is zero. Provision of Ch. maxima is determined from the condition or 
, Where m=0, 1, 2, ... - integer. Where
i.e. Ch. maxima are formed in directions when the path difference between adjacent coherent beams is equal to an integer number of wavelengths. The intensity of all main maxima is the same and equal 
, the intensity of the secondary maxima is small and does not exceed from .
The relationship, called the grating equation, shows that for a given angle of incidence, the directions to the main maximum depend on the wavelength, i.e. 
; therefore, D. r. spatially (in the corners) decomposes the radiation. wavelengths. If diffracting. When the radiation coming from the grating is directed into the lens, a spectrum is formed in its focal plane. In this case, several are formed at the same time. spectra at each value of the number , and the value T determines the order of the spectrum. At m=0 (zero order of the spectrum), the spectrum is not formed, since the condition is satisfied for all wavelengths (the main maxima for all wavelengths coincide). From the last condition at t=0 it also follows that
, i.e. that the direction to the zero-order maximum is determined by the specular reflection from the grating plane (Fig. 4); the incident and diffracted beams of zero order are located symmetrically relative to the normal to the grating. On both sides of the direction to the zero-order maximum there are maxima and spectra m=1,
m=2 and so on. orders.
Second function Jg, which affects the resulting intensity distribution in the spectrum, is due to the diffraction of light on the part. stroke; it depends on the quantities 
, and also on the shape of the stroke - its profile. Calculation taking into account Huygens-Fresnel principle, gives for the function Jg expression
where is the amplitude of the incident wave, - ; , , X And at- coordinates of points on the stroke profile. Integration is carried out over the profile of the stroke. For the special case of a flat amplitude D. r., consisting of narrow slits in an opaque screen (Fig. 2, b)or narrow reflective stripes on the plane,, where , A- the width of the slits (or reflective stripes), and represents the diffraction. intensity distribution during Fraunhofer diffraction by a slit width A(cm. Diffraction of light). Its appearance is shown in Fig. 3(b). Direction to the center ch. diffraction maximum function Jg determined from the condition u=0 or , whence, i.e. this direction is determined by the specular reflection from the plane of the d.r., and, consequently, the direction to the center of the diffraction. the maximum coincides with the direction to the zero - achromatic - order of the spectrum. Therefore, max. the value of the product of both functions, and therefore the max. the intensity will be in the zero order spectrum. The intensity in the spectra of other orders ( m 0) will be correspondingly less than the intensity in zero order (which is schematically depicted in Fig. 3, V). This is unprofitable when using amplitude D. r. in spectral instruments, since most of the light energy incident on the laser is directed to the zeroth order of the spectrum, where there is no spectral decomposition, while the intensity of the spectra of other and even first orders is small.
If the strokes of D. r. give a triangular asymmetrical shape, then such a phase grating has the function Jg also has diffraction. distribution, but with argument And, depending on the angle of inclination
edges of the stroke (Fig. 2, A). In this case, the direction to the center of the diffraction The maximum is determined by the specular reflection of the incident beam not from the plane of the d.r., but from the edge of the stroke. By changing the angle of inclination of the stroke edge, you can align the center of the diffraction pattern. maximum function Jg with any interference ch. maximum function J N any order m 0, usually m=1 (Fig. 3, G) or m=2. The condition for such a combination is that the angles and must simultaneously satisfy the relations and . Under these conditions, the spectrum of a given order T 0 will have max. intensity, and the indicated ratios allow us to determine the required value for the given ones. Phase D. r. with a triangular line profile, concentrating most (up to 80%) of the light flux incident on the grating into a non-zero order spectrum, called. echelettes. The angle at which the specified concentration of the incident light flux into the spectrum occurs is called. brightness angle D. r.
Basic spectroscopic characteristics of D. r. - angular dispersion, resolution and dispersion area - are determined only by the properties of the function J N. associated with periodic structure of the D. line, and do not depend on the shape of the stroke.
Angle dispersion, which characterizes the degree of spatial (angular) separation of rays with different wavelengths, for D. r. obtained by differentiating ; then , from which it follows that when working in a given order of the spectrum T magnitude
the larger the smaller the grating period. In addition, the value increases with increasing diffraction angle. However, in the case of an amplitude grating, an increase in the angle leads to a decrease in the intensity of the spectrum. In this case, it is possible to create a line profile such that the concentration of energy in the spectrum will occur at large angles j, and therefore it is possible to create high-aperture spectral devices with a large angle. dispersion.
Theoretical resolution of D. r. , where - min. difference in wavelengths of two monochromatic lines of equal intensity, which can still be distinguished in the spectrum. Like any spectral device, R D. r. determined by spectral width hardware function, cut in the case of D. r. are the main maxima of the function J N. Having determined the spectral width of these maxima, we can obtain expressions for R in the form where W=Nd- the full length of the shaded part of the D. r. (Fig. 1). From the expression for R it follows that at given angles the value R can be increased only by increasing the size of the D. r. - W. Magnitude R increases with increasing diffraction angle, but more slowly than increases. The expression for A can also be represented as 
, Where 
- full width of parallel diffracters. beam coming from D. r. at an angle.
The dispersion region of the D. r. is the value of the spectral interval, for which the spectrum of a given order T does not overlap with the spectra of neighboring orders and, therefore, there is an unambiguous relationship between the diffraction angle. is determined from the condition where . For m=1, i.e. the dispersion region covers an interval of one octave, for example. the entire visible region of the spectrum from 800 to 400 nm. The expression for can also be presented in the form , from which it follows that the smaller the value, the larger d, and depends on the angle, decreasing (unlike and R) with increasing .
From the expressions for and the relation can be obtained. For D. r. the difference between them is very large, because modern D. r. total number of strokes N great ( N~ 10 5 and more).
Concave D. r. In concave D. r. the strokes are applied to a concave (usually spherical) mirror surface. Such gratings serve as both a dispersing and focusing system, i.e., they do not require the use of input and output collimator lenses or mirrors in spectral instruments, unlike flat gratings. In this case, the light source (entrance slit S 1) and the spectrum turns out to be located on a circle tangent to the lattice at its vertex, the diameter of the circle is equal to the radius of curvature R spherical surface D. r. (Fig. 5). This circle is called around Rowland. In the case of a concave D. r. from a light source (slit), a diverging beam of light falls on the grating, and after diffraction on the streaks and interference of coherent beams, resulting light waves are formed, converging on Rowland's circle, where the interference is located. maxima, i.e. spectrum. The angles formed by the axial rays of the incident and diffracted beams with the axis of the sphere are related by the relation. Several are also formed here. spectra differ. orders located on the Rowland circle, which is the line of dispersion. Since the lattice equation for a concave D. r. the same as for flat, then the expressions for spectroscopic. characteristics - ang. dispersion, resolution and dispersion region - turn out to be the same for both types of gratings. The expressions for the linear dispersions of these lattices are different (see. Spectral devices).
Rice. 5. Scheme of formation of spectra by a concave diffraction grating on a Rowland circle.
Concave radiators, unlike flat ones, have astigmatism, which manifests itself in the fact that each point of the source (slit) is represented by a grating not in the form of a point, but in the form of a segment perpendicular to the Rowland circle (to the dispersion line), i.e., directed along the spectral lines, which leads to . decrease in spectrum intensity. The presence of astigmatism also prevents the use of decomposition. photometric devices. Astigmatism can be eliminated if the strokes are applied to the aspherical, e.g. a toroidal concave surface or cut into a lattice not with equidistant, but with distances between the strokes varying according to a certain law. But the production of such gratings is associated with great difficulties; they have not yet received widespread use.
Topographic D. R. In the 1970s A new, holographic method for manufacturing both flat and concave DRs was developed, and in the latter, astigmatism can be eliminated, which means. spectrum areas. In this method, a flat or concave spherical. substrate coated with a special layer. photosensitive material - photoresist, is illuminated by two beams of coherent laser radiation (with a wavelength) in the area of intersection of which a stationary interference is formed. a pattern with a cosine intensity distribution (see. Interference of light), changing the photoresist material in accordance with the change in intensity in the picture. After appropriate processing of the exposed photoresist layer and application of a reflective coating to it, a holographic image is obtained. phase reflect. a grating with a cosine shape of the line, i.e. it is not an echelette and therefore has a lower aperture ratio. If the illumination was produced by parallel beams forming an angle with each other (Fig. 6), and the substrate is flat, then a flat, equidistant holographic image is obtained. D. r. with period, with spherical substrate - concave holographic. D. r., equivalent in its properties to a conventional rifled concave lattice. When illuminated, spherical. substrate with two diverging beams from sources located on the Rowland circle, a holographic result is obtained. D. r. with curvilinear and non-equidistant strokes, the edges are free from astigmatism, which means. spectrum areas.
Diffraction grating
Very large reflective diffraction grating.
Diffraction grating- an optical device operating on the principle of light diffraction, is a collection of a large number of regularly spaced strokes (slots, protrusions) applied to a certain surface. The first description of the phenomenon was made by James Gregory, who used bird feathers as a lattice.
Types of gratings
- Reflective: Strokes are applied to a mirror (metal) surface, and observation is carried out in reflected light
- Transparent: Strokes are applied to a transparent surface (or cut out in the form of slits on an opaque screen), observation is carried out in transmitted light.
Description of the phenomenon
This is what the light from an incandescent flashlight looks like when it passes through a transparent diffraction grating. Zero maximum ( m=0) corresponds to light passing through the grating without deviation. Due to lattice dispersion in the first ( m=±1) at the maximum, one can observe the decomposition of light into a spectrum. The deflection angle increases with wavelength (from violet to red)
The front of the light wave is divided by the grating bars into separate beams of coherent light. These beams undergo diffraction by the streaks and interfere with each other. Since each wavelength has its own diffraction angle, white light is decomposed into a spectrum.
Formulas
The distance through which the lines on the grating are repeated is called the period of the diffraction grating. Designated by letter d.
If the number of strokes is known ( N), per 1 mm of grating, then the grating period is found using the formula: 0.001 / N
Diffraction grating formula:
d- grating period, α - maximum angle of a given color, k- order of maximum, λ - wavelength.Characteristics
One of the characteristics of a diffraction grating is angular dispersion. Let us assume that a maximum of some order is observed at an angle φ for wavelength λ and at an angle φ+Δφ for wavelength λ+Δλ. The angular dispersion of the grating is called the ratio D=Δφ/Δλ. The expression for D can be obtained by differentiating the diffraction grating formula
Thus, angular dispersion increases with decreasing grating period d and increasing spectrum order k.
Manufacturing
Good gratings require very high manufacturing precision. If at least one of the many slots is placed with an error, the grating will be defective. The machine for making gratings is firmly and deeply built into a special foundation. Before starting the actual production of gratings, the machine runs for 5-20 hours at idle speed to stabilize all its components. Cutting the grating lasts up to 7 days, although the time of applying the stroke is 2-3 seconds.
Application
Diffraction gratings are used in spectral instruments, also as optical sensors of linear and angular displacements (measuring diffraction gratings), polarizers and filters of infrared radiation, beam splitters in interferometers and so-called “anti-glare” glasses.
Literature
- Sivukhin D.V. General physics course. - 3rd edition, stereotypical. - M.: Fizmatlit, MIPT, 2002. - T. IV. Optics. - 792 p. - ISBN 5-9221-0228-1
- Tarasov K.I., Spectral devices, 1968
see also
- Fourier optics
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See what “Diffraction grating” is in other dictionaries:
Optical instrument; a set of a large number of parallel slits in an opaque screen or reflective mirror strips (stripes), equally spaced from each other, on which light diffraction occurs. The diffraction grating decomposes... ... Big Encyclopedic Dictionary
DIFFRACTION GRATING, a plate with parallel lines applied to it at equal distances from each other (up to 1500 per 1 mm), which serves to obtain SPECTRA during DIFFRACTION of light. Transmission grilles are transparent and lined on... ... Scientific and technical encyclopedic dictionary
diffraction grating- A mirror surface with microscopic parallel lines applied to it, a device that separates (like a prism) the light incident on it into the component colors of the visible spectrum. Topics information technology in...
diffraction grating- difrakcinė gardelė statusas T sritis Standartizacija ir metrologija apibrėžtis Optinis periodinės sandaros įtaisas difrakciniams spektrams gauti. atitikmenys: engl. diffraction grating vok. Beugungsgitter, n; Diffraktionsgitter, n rus.… … Penkiakalbis aiškinamasis metrologijos terminų žodynas
An optical device, a collection of a large number of parallel slits in an opaque screen or reflective mirror strokes (strips), equally spaced from each other, on which light diffraction occurs. D.R. decomposes the light falling on it into... ... Astronomical Dictionary
diffraction grating (in optical communication lines)- diffraction grating An optical element with a periodic structure that reflects (or transmits) light at one or more different angles, depending on the wavelength. The basis is made up of periodically repeated changes in the indicator... ... Technical Translator's Guide
concave spectral diffraction grating- Spectral diffraction grating made on a concave optical surface. Note Concave spectral diffraction gratings are available in spherical and aspherical types. [GOST 27176 86] Topics: optics, optical instruments and measurements... Technical Translator's Guide
hologram spectral diffraction grating- Spectral diffraction grating, manufactured by recording an interference pattern from two or more coherent beams on a radiation-sensitive material. [GOST 27176 86] Topics: optics, optical instruments and measurements... Technical Translator's Guide
threaded spectral diffraction grating- Spectral diffraction grating made by applying streaks on a dividing machine. [GOST 27176 86] Topics: optics, optical instruments and measurements... Technical Translator's Guide
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Books
- Set of tables. Geometric and wave optics (18 tables), Study album of 12 sheets. Article - 5-8670-018. Huygens' principle. Wave reflection. Image of an object in a plane mirror. Light refraction. Total internal reflection. Variance… Category:
Basic concepts and characteristicsspectral device. |
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Illumination distribution in the slit image |
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| Diffraction grating Spectral instruments use diffraction gratings to spatially decompose light into a spectrum. A diffraction grating is an optical element consisting of a large number of regularly spaced lines applied to a flat or concave surface. Grilles can be transparent or reflective. In addition, a distinction is made between amplitude and phase diffraction gratings. For the former, the reflection coefficient periodically changes, which causes a change in the amplitude of the incident wave. In phase diffraction gratings, the grooves are given a special shape, which periodically changes the phase of the light wave. The most widely used is a flat reflective phase diffraction grating with a triangular groove profile - echelette. Lattice equation The front of a light wave incident on a diffraction grating is split by its grooves into separate coherent beams. Coherent beams, having undergone diffraction by the streaks, interfere, forming the resulting spatial distribution of light intensity. The intensity distribution is proportional to the product of two functions: interferenceI N
and diffractionI D
. FunctionI N
is caused by the interference of N coherent beams coming from the grating lines. FunctionI D
determined by diffraction on a separate line. The path difference between coherent parallel beams coming at an angle β from adjacent strokes will be Δs=AB+AC or (1), and the corresponding phase difference This relationship is called the lattice equation. It shows that the main maxima are formed in directions when the path difference between adjacent beams is equal to the total number of wavelengths. Between adjacent main maxima there is located N-2 secondary maxima, the intensity of which decreases proportionally 1/N, And N-1 minima, where the intensity is zero. The lattice equation for application to monochromators is used in a more convenient form. Since the difference between the angles α
And β
is constant when the lattice rotates and this difference is known θ
, it is determined by the design of the monochromator, then it depends on two variablesα
And β
move on to one φ
– angle of rotation of the lattice from zero order. |
Fig. 1. Explanation of the principle of operation of a diffraction grating. |
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Glare wavelength The reflectivity of diffraction gratings depends on the angle of inclination of the lines - by changing the angle of inclination of the edge of the line, you can align the center of the diffraction maximum of the function I D with interference main maximum function I N any order. The direction to the center of the diffraction maximum is determined by the specular reflection of the incident beam not from the grating plane, but from the edge of the line. Thus, the condition for such a combination is: angles α
And β max
must simultaneously satisfy the following relations: Under these conditions, the spectrum of a given order will have the greatest intensity. Corner β max
is called the “splash” angle, and the wavelength is called the “splash” wavelength. λ Blaze. If the spectral region for research is known, then λ Blaze can be determined from the relation: Example 1. The range under study is 400…1200 nm, i.e. λ 1=400nm, λ 2=1200nm. Then from formula (8): λ Blaze=600nm. Select a grating with a gloss of 600nm. Example 2. The studied range is 600…1100 nm. Calculation using formula (8) gives, in rounding, 776 nm. There is no grille with such shine on the proposed list. The grating with the gloss closest to the found one is selected, i.e. 750nm. |
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Energy efficiency area diffraction gratings The region where the grating reflectance is at least 0.405 is called the energy efficiency region: |
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Dispersion area The dispersion region is a spectral interval in which the spectrum of a given order does not overlap with the spectra of neighboring orders. Consequently, there is an unambiguous relationship between the diffraction angle and the wavelength. The dispersion area is determined from the condition: Example. If Thus, for a given grating in the range from 450 nm to 900 nm, the dispersion region is combined with the energy efficiency region. |
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Dispersion The degree of spatial separation of beams with different wavelengths is characterized by angular dispersion. We obtain the expression for angular dispersion by differentiating the equation for the lattice: |
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Resolution Theoretical resolution: |
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Spectral area of the grating depending on from the number of strokes For each diffraction grating with a period d there is a maximum wavelength limit |
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Concave diffraction grating A concave diffraction grating plays the role of not only a dispersing, but also a focusing system. The expressions for the spectroscopic characteristics - angular dispersion, resolution and dispersion area - are the same as for a flat grating. Concave gratings, unlike flat gratings, have astigmatism. Astigmatism is eliminated by applying strokes on an aspherical surface or with the distances between the strokes varying according to a certain law. |
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Holographic diffraction grating The quality of the diffraction grating is determined by the intensity of the scattered light, caused by the presence of small defects on the edges of individual strokes, and the intensity of “ghosts” - false lines that arise when equidistance in the arrangement of strokes is violated. The advantage of holographic gratings compared to rifled ones is the absence of “ghosts” and lower intensity of scattered light. However, the holographic phase reflective grating has a sinusoidal line shape, i.e., it is not an echellette, and therefore has lower energy efficiency (Fig. 2). The production of holographic gratings with a triangular groove profile, the so-called “bladed” gratings, leads to the appearance of microstructures on the edges of the bars, which increases the intensity of scattered light. In addition, the correct triangular profile is not achieved, which reduces the energy efficiency of such gratings. |
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(2). FunctionI N
- a periodic function with different intense main maxima. The position of the main maxima is determined from the condition 
, where 
(3), where k- spectrum order. 
. Using (3) we get 
, substituting in (1): (4). 
And 
, after transforming the sum of sines, we obtain the lattice equation in another more convenient form: 
(5), where
(6).
holds for all wavelengths. Besides, β= -α
i.e., the direction to the zero-order maximum is determined by the specular reflection from the grating plane. 
(7).
(8), where where λ 1 And λ 2– boundary wavelengths of the spectrum range. Relationship (8) helps to choose the right lattice.
(9). The value depends on the order of the spectrum: it is maximum in the first order and quickly decreases in spectra of higher orders. For first order: 
. Wavelengths limiting this area: 
And 
.
.
(10). For first order 
, A 
, i.e. the dispersion region covers an interval of one octave. To combine the dispersion region with the energy efficiency region of the diffraction grating, it is necessary that the following condition be satisfied: 
(eleven). In this case, within the dispersion region, the grating reflectance for k=1 will be at least 0.68.
, Then 
, A 
.
(12). From this expression it follows that angular dispersion is determined solely by the angles α
And β
, but not by the number of strokes. When applied to spectral instruments, inverse linear dispersion is used, which is defined as the reciprocal of the product of angular dispersion and focal length: 
.
, where is the resolution. The resolution of a diffraction grating, like any spectral device, is determined by the spectral width of the instrumental function. For a grating, the width of the apparatus function is the width of the main maxima of the interference function: 
. Then: 
(14). The spectral resolution of a diffraction grating is equal to the product of the diffraction order k for the full number of strokes N. Using the lattice equation: 
(15), where the product 
- length of the shaded part of the lattice. From expression (15) it is clear that at given angles α
And β
magnitude R can only be increased by increasing the size of the diffraction grating. The expression for resolution can be presented in another form from (12) and (15): 
(16), where 
- width of the diffracted beam, - angular dispersion. Expression (16) shows that the resolution is directly proportional to the magnitude of the angular dispersion. 
. It is determined from the lattice equation at k=1 And α=β=90° and is equal to 
. Therefore, when working in different regions of the spectrum, gratings with different numbers of lines are used: 
Illumination distribution in the slit image
The distribution of illumination in the slit image depends on the nature of the aberrations of the optical system, as well as on the method of illuminating the slit.
Aberrations
An ideal optical system produces a pinpoint image of a point. In the paraxial region, the optical system is close to ideal. But with a finite beam width and a distance of the source from the optical axis, the rules of paraxial optics are violated and the image is distorted. When designing an optical system, aberrations have to be corrected.
Spherical aberration
The distribution of illumination in the scattering spot with spherical aberration is such that a sharp maximum is obtained in the center with a rapid decrease in illumination towards the edge of the spot. This aberration is the only one that remains even if the object point is located on the main optical axis of the system. Spherical aberration is especially large in high-aperture systems (with a large relative aperture).
Coma
The image of a point in the presence of coma has the form of an asymmetrical spot, the illumination of which is maximum at the top of the scattering figure.
Astigmatism
It is caused by unequal curvature of the optical surface in different section planes and manifests itself in the fact that the wave front is deformed when passing through the optical system, and the focus of the light beam in different sections appears at different points. The scattering figure is a family of ellipses with a uniform distribution of illumination. There are two planes - meridional and sagittal, perpendicular to it, in which ellipses turn into straight segments. The centers of curvature in both sections are called foci, and the distance between them is a measure of astigmatism.
Field curvature
The deviation of the best focusing surface of the focal plane is an aberration called field curvature.
Distortion
Distortion is the distortion of an image due to unequal linear magnification of different parts of the image. This aberration depends on the distance from the point to the optical axis and manifests itself in a violation of the law of similarity.
Chromatic aberration
Due to light dispersion, two types of chromatic aberration appear: focal position chromatism and magnification chromatism. The first is characterized by a shift in the image plane for different wavelengths, the second by a change in transverse magnification. Chromatic aberration occurs in optical systems that include elements made of refractive materials. Chromatic aberrations are not inherent in mirrors. This circumstance makes the use of mirrors in monochromators and other optical systems especially valuable.
Illumination of the entrance slit
Coherent and non-coherent lighting
The nature of the illumination of the entrance slit of the device is of significant importance for the distribution of intensity over the width of the spectral line, i.e. degree of illumination coherence. In practice, illumination of the entrance slit is neither strictly coherent nor incoherent. However, it is possible to come very close to one of these two extreme cases. Coherent illumination can be achieved by illuminating the slit with a point source located at the focus of a large-diameter condenser placed in front of the slit.
Another method is lensless illumination, when a small source is placed at a great distance from the slit. Incoherent illumination can be obtained by using a condenser lens to focus the light source onto the entrance slit of the device. Other lighting methods occupy an intermediate position. The importance of distinguishing them is due to the fact that when illuminated with coherent light, interference phenomena may occur that are not observed when illuminated with incoherent light.
If the main requirement is to achieve maximum resolution, then the aperture of the diffraction grating is filled with coherent light in a plane perpendicular to the slit. If it is necessary to ensure maximum spectrum brightness, then the incoherent illumination method is used, in which the aperture is also filled in a plane parallel to the slit.
Filling the aperture with light.
F/#-Matcher
.
One of the main parameters that characterizes a spectral device is its aperture ratio. Aperture is determined by the maximum angular size of the light beam entering the device and is measured by the ratio of the diameter (dk) to focal length (fk) collimator mirror. In practice, the inverse is often used, called F/# It is preferable to use another characteristic - numerical aperture. Numerical aperture (N.A.) associated with F/# ratio: 
.
Optimal imaging of an extended incoherent light source onto the input slit of the device is achieved when the solid angle of the incident light beam is equal to the input angle of the device.
A– entrance slit area; θ - input solid angle.If the slit and collimator are filled with light, then no additional system of lenses and mirrors will help increase the total flux of radiation passing through the system.
For a specific spectral device, the maximum input solid angle is a constant value determined by the size and focal length of the collimator: 
.
To match the angular apertures of the light source and the spectral device, a special device called F/# Matcher is used. F/# Matcher is used in conjunction with a spectral device, providing its maximum aperture, both with and without a light guide.
Fig.4. F/# Matcher Scheme
The advantages of F/# Matcher are:
- Using the full geometric aperture of the spectral device
- Reducing stray light
- Maintains good spectral and spatial image quality
- Possibility of using filters of unequal thickness without focusing distortion
DEFINITION
Diffraction grating called a spectral device, which is a system of a number of slits separated by opaque spaces.
Very often in practice, a one-dimensional diffraction grating is used, consisting of parallel slits of the same width, located in the same plane, which are separated by opaque intervals of equal width. Such a grating is made using a special dividing machine, which applies parallel strokes to a glass plate. The number of such strokes can be more than a thousand per millimeter.
Reflective diffraction gratings are considered the best. This is a collection of areas that reflect light with areas that reflect light. Such gratings are a polished metal plate on which light-scattering strokes are applied with a cutter.
The diffraction pattern on the grating is the result of mutual interference of waves that come from all the slits. Consequently, with the help of a diffraction grating, multi-beam interference of coherent beams of light that have undergone diffraction and coming from all slits is realized.
Let us assume that the width of the slit on the diffraction grating is a, the width of the opaque section is b, then the value is:
is called the period of the (constant) diffraction grating.
Diffraction pattern on a one-dimensional diffraction grating
Let us imagine that a monochromatic wave is incident normally to the plane of the diffraction grating. Due to the fact that the slits are located at equal distances from each other, the path differences of the rays () that come from a pair of adjacent slits for the chosen direction will be the same for the entire given diffraction grating:
The main intensity minima are observed in the directions determined by the condition:
In addition to the main minima, as a result of the mutual interference of the light rays sent by a pair of slits, in some directions they cancel each other, which means that additional minima appear. They arise in directions where the difference in the path of the rays is an odd number of half-waves. The condition for additional minima is written as:
where N is the number of slits of the diffraction grating; k’ accepts any integer values except 0, . If the lattice has N slits, then between the two main maxima there are an additional minimum that separates the secondary maxima.
The condition for the main maxima for a diffraction grating is the expression:
Since the value of the sine cannot be greater than one, the number of main maxima is:
If white light is passed through the grating, then all maxima (except for the central m = 0) will be decomposed into a spectrum. In this case, the violet region of this spectrum will face the center of the diffraction pattern. This property of a diffraction grating is used to study the composition of the light spectrum. If the grating period is known, then calculating the wavelength of light can be reduced to finding the angle , which corresponds to the direction to the maximum.
Examples of problem solving
EXAMPLE 1
| Exercise | What is the maximum spectral order that can be obtained using a diffraction grating with constant m if a monochromatic beam of light with wavelength m is incident on it perpendicular to the surface? |
| Solution | As a basis for solving the problem, we use the formula, which is the condition for observing the main maxima for the diffraction pattern obtained when light passes through a diffraction grating: The maximum value is one, so: From (1.2) we express , we get: Let's do the calculations:
|
| Answer |
EXAMPLE 2
| Exercise | Monochromatic light of wavelength . is passed through a diffraction grating. A screen is placed at a distance L from the grating. Using a lens located near the grating, a projection of the diffraction pattern is created onto it. In this case, the first diffraction maximum is located at a distance l from the central one. What is the number of lines per unit length of the diffraction grating (N) if light falls on it normally? |
| Solution | Let's make a drawing. |
