HOLOGRAPHY. New-generation diffraction gratings. E. A. Sokolova

HOLOGRAPHY New-generation diffraction gratings E. A. Sokolova S. I. Vavilov State Optical Institute, St. Petersburg, Russia; Universidade da Beira Interior, Portugal Submitted February 28, 2001 Opticheski Zhurnal 68, 53 60 August 2001 This paper shows how holographic diffraction gratings fabricated on spherical substrates by means of wavefront interference can be used to create aspheric wavefronts and thereby to fabricate concave diffraction gratings that have virtually no aberration limitations in high-resolution spectral devices. 2001 The Optical Society of America. INTRODUCTION N. A. Rowland showed in 1882 that a grating formed on a concave spherical substrate could directly serve the function of both diffracting light and of focusing it. 1 This is especially important if a significant part of the light is lost in the collimating and focusing objectives. Rowland cut the gratings on concave spherical substrates, using the same ruling engine used for fabricating flat gratings. Under these conditions, the distribution of the lines is not constant over the grating surface, but is constant over a chord. The blaze angle of such a grating is also constant relative to the chord, and not relative to the grating surface. Rowland showed that, for a grating fabricated in such a way and illuminated by a point source located on a circle tangent to the pole of the grating but with a radius of curvature equal to half the radius of curvature of the substrate, the diffraction image in the first approximation will also be on this circle. This circle is known as the Rowland circle. The aberrations present in the spectral image are a consequence of the shape of the grating and can be varied and in some cases eliminated if the shape of the grating is changed. There are three methods by which we can affect the aberrations. We can vary the shape of the substrate or the lines, or we can vary the distribution of lines over the grating surface. The possibility of controlling the focal properties of gratings by the distribution of their lines has been known for a long time. This was considered, for example, by Cornu in 1893, when he predicted that changing the distribution of the lines would affect the focal properties of both flat and concave gratings. However, because of the significant technical difficulties of producing even ordinary gratings, this work had only theoretical significance until recently. Because of the difficulty in fabricating and testing an aspheric surface, we must use a spherical substrate in ordinary practice and then must form the lines so as to optimize the correction of the aberrations for a particular case. It is possible in principle to completely control the shape and position of the rulings by using interfering wavefronts of the appropriate shape, but in practice the wavefronts are usually either spherical or flat. The process of designing a holographic grating then consists of determining the expression for the differential function of the optical path in the same way as for an ordinary concave grating. However, the placement and shape of the lines is not constant in this case but depends on the positions of the point sources used to record the grating. The coordinates of these points are present in the expansion in series of the optical path function and affect the focal properties and various aberrations of the grating. 2 The particular aberrations of interest for the application in which the grating must be used can be optimized or even eliminated by a suitable choice of the coordinates of the point sources. Methods of eliminating the first-order astigmatism inherent to concave gratings while simultaneously correcting other aberrations in spectrographs and polychromators have been systematically studied, 3 the relationships between the parameters of ruled gratings and the conditions for recording holographic gratings equivalent to them in terms of three aberrations have been established, 4 and a number of original optical systems of spectrographs, monochromators, and polychromators have been created. 5 7 Both conventional methods of fabricating diffraction gratings mechanical ruling and holographic recording using two point sources make it possible to minimize only three aberrations, and this is not always sufficient to ensure high quality of the spectral image. In order to increase the number of optimization parameters and consequently the number of controllable aberrations, recording in aspheric wavefronts obtained from two ellipsoidal mirrors has been used. 8 However, aspheric optics are expensive, and their production and testing are a complex problem, and therefore such a solution is not suitable for use in industry. Recording in which spherical diffraction gratings are used has been proposed for installation on the Rowland circle. 9 Since only recording in copropagating beams was considered, the method can be used only in certain specific optical systems. It has been proposed to record concave gratings in counterpropagating beams, using a complex system consisting of several lenses. 10 Gratings recorded in counterpropagating beams possess a triangular line profile, which favorably affects their energy characteristics. However, the use of refracting optics is not recommended for any holographic recording. Apparatus of this type produces much scattered light, which reduces the quality of the final grating. A solution of the problem free from the disadvantages mentioned above 584 J. Opt. Technol. 68 (8), August 2001 1070-9762/2001/080584-06$18.00 2001 The Optical Society of America 584

was found by more fully using holographic resources to fabricate diffraction gratings. A diffraction grating in this case is regarded not only as a periodic structure used for diffracting light but in a wider sense as a carrier of information in optics. 11,12 In this context, any hologram can be defined as a modulated grating, and a ruled diffraction grating can be regarded as a synthesized hologram. A concave grating is the hologram of a point. It follows that there is no fundamental difference between a hologram and a diffraction grating, and an optical element that possesses a modulated periodic structure in one degree or another is classified as a hologram or as a diffraction grating only according to how it is used. The use of a conventional holographic i.e., interference diffraction grating fabricated on a spherical substrate coated with a photoresist layer can be broadened to the formation of wavefronts of the necessary shape by photographing the interference pattern of two spherical waves, followed by development and aluminization. Such wavefronts, which possess in general a complex aspheric shape but are formed only by spherical optical elements, can in turn be used both for recording holographic diffraction gratings with improved aberrational characteristics and in other applications that use monochromatic radiation. If such a grating must form, for example, a wave incident on the substrate of a diffraction grating to be fabricated at a subsequent stage from its back side, which must converge to a point after passing through this substrate, such a grating objective can be fabricated by means of a spherical wave coming from this point and passing through the said substrate. At the second stage, with the appropriate illumination, such a holographic objective reconstructs a wavefront of the necessary shape. Thus, instead of a complex system of lenses or mirrors that has to be calculated and fabricated for each case, we obtain a single reflecting optical element that completely solves the problem of forming a wavefront of the necessary shape. In especially complex cases, when it is not possible to form a wavefront of the necessary shape by means of a grating objective recorded in homocentric beams, this grating-objective can also be fabricated in two stages. The problem of forming a wavefront of any given shape reduces to the successive fabrication of a series of holographic gratings. Prospectively, any illuminator system for monochromatic radiation can be reduced to one diffraction grating. This paper discusses how this idea can be used to calculate and fabricate diffraction gratings for spectral devices. FIG. 1. Optical setup for two-stage recording. General case. The most general case of recording a diffraction grating by means of aspheric concave wavefronts is two-step recording in counterpropagating beams in spatially coherent light. 13 Instead of the three optimization parameters that we have available in an ordinary recording setup, we have in the general case six parameters, three for each step. Such an increase in the number of optimization parameters allows us to compensate all five aberrations under consideration: residual defocusing, meridional and sagittal coma, first-order astigmatism, and spherical aberration. In this recording system Fig. 1, a ray of wavelength 0 coming from point A* diffracts at point P* in the direction of point P on substrate G. Ray DP of wavelength 0 is directed to point P from point D. Grating G* is recorded by means of rays of wavelength 0 incident on the substrate from points C* and D*. The optical path function for a ray coming from point A and diffracted at point P(x,y,z) at the kth line in direction PB is defined as F AP PB km. 1 The interference fringes formed on the surface of grating substrate G, or the grating lines, are determined by k 0 n AP PP km 0 DP DO, 2 where k* is the line number corresponding to point P* on the surface of grating G*; n is the refractive index of the material from which the substrate of grating G is made, which can be chosen to be equal to the refractive index of the photoresist; and m* is the order of diffraction at which grating G* is used. The line number k* is formed in accordance with k 0 CP C*O* D P D*O*. 3 Substituting Eqs. 2 and 3 into Eq. 1, weget TWO-STEP RECORDING OF DIFFRACTION GRATINGS IN COUNTERPROPAGATING BEAMS F AP PB m n AP PP m CP CO 0 [DP DO DP DO }. 4 Since we introduce no changes into AP PB by comparison with the classical case, in what follows we can consider the function f n AP PP m CP CO [DP DO ] DP DO, 5 585 J. Opt. Technol. 68 (8), August 2001 E. A. Sokolova 585

where AP x x A0 2 y y A0 2 z z A0 2, PP x x0 2 y y 0 2 z z 0 2, CP x C0 x 0 2 y C0 y 0 2 z C0 z 0 2, DP x D0 x 0 2 y D0 y 0 2 z D0 z 0 2, 6 DP x xd 2 y y D 2 z z D 2, CO r C0, D O r D0. The coordinates of point P* can be expressed as a function of the coordinates of point P, and they can be found from the intersection of ray CPP* with the surface of grating G*. Using computer program MATHEMATICA, we obtained five derivatives of function f, corresponding to the grating equation ( f / y) and the four main aberrations: defocusing ( 2 f / y 2 ), meridional coma ( 3 f / y 3 ), first-order astigmatism ( 2 f / z 2 ), and sagittal coma ( 3 f / y z 2 ). In order to convince ourselves of the correctness of the resulting formulas, we computed these derivatives for the optical system of a vacuum monochromator operating in the spectral region 100 300 nm and having an 1800-line/mm grating with radius of curvature of 250 mm. Because of the rather large curvature of the lines on this grating, the only way to fabricate it is by holographic recording in counterpropagating beams. The recording parameters of this grating were calculated earlier using numerical methods and were checked experimentally by mathematical modelling. 14 Another grating, having 1800 lines/mm and a radius of curvature of 500 mm, was used as an objective. The resulting grating equations, as well as the coincidence within good accuracy of the three aberration factors, confirm that the resulting formulas are correct. We were able to use these formulas to improve the grating-recording parameters for the monochromator under consideration and thereby to achieve more complete compensation of sagittal coma and spherical aberration. The theory was also checked on the example of a polychromator operating on the Rowland circle. The grating of this polychromator has 3600 line/mm and a radius of curvature of 500 mm. The angle of incidence of the radiation on the grating equals 50.5. The results for defocusing, firstorder astigmatism, meridional and sagittal coma, and spherical aberration coincide to within 10 8 10 10. A grating having 600 line/mm and intended for a spectrograph with a flat field 15 was not only calculated but was also fabricated. 16 This grating was chosen for fabrication since a similar ruled grating was available, so that the spectral characteristics of the two gratings could be compared experimentally. The mechanically ruled grating for this spectrograph has 600 line/mm, the lines being straight and not equally spaced. The radius of curvature of this grating is 125 mm, and the working region is 36 36 mm 2. To record a grating having only 600 line/mm, we used normal incidence of one of the recording beams. The grating objective was recorded through the substrate of the final grating. To check the aberrational properties of the recorded grating, its focal FIG. 2. Spectra of a mercury cadmium lamp, obtained using the hologram grating considered here a and the corresponding ruled grating b. length was measured in an autocollimation apparatus. The grating focused the expanded beam of a helium neon laser at a distance of 118 mm for the first negative order of the spectrum and 128 mm for the first positive order of the spectrum. According to the focusing equation of a diffraction grating with nonuniformly spaced lines, the stepnonuniformity factor of the recorded grating is 0.001 657 4 0.001 704 47 mm 1. The corresponding factor of the ruled grating is 0.001 72 mm 1. The difference corresponds to total error in setting the distances in the recording system no greater than 0.3 mm, which is small for manual adjustment. We also recorded the spectra of a mercury cadmium lamp using the holographic grating considered here and the corresponding ruled grating mounted in the same geometry Fig. 2. The holographic grating gives the same quality of the spectrum as does the ruled grating. It possesses a smaller blaze angle, so that it diffracts more energy in the blue and less in the red than the ruled grating. Using the recording system considered here, a number of diffraction gratings were fabricated by means of the same grating objective, which was recorded through the substrate of one of these gratings. In this case, no difference was detected in the focusing properties of the fabricated gratings. Consequently, for commercial production of gratings by the two-step technique, it is sufficient to fabricate one grating objective for each size and type of grating to be recorded at the second stage. The service life of such a grating objective is the same as for any holographic diffraction grating in a spectral device. RECORDING DIFFRACTION GRATINGS BY MEANS OF INCIDENT AND REFLECTED BEAMS An important factor that affects the quality of the diffraction gratings being recorded is the scattered light level in the holographic apparatus. The rays reflected from the substrate surfaces can interfere with each other, as a result of which stray lines are formed that degrade the grating quality. When 586 J. Opt. Technol. 68 (8), August 2001 E. A. Sokolova 586

FIG. 3. Optical setup for recording a concave hologram grating by means of incident and reflected beams. copropagating beams are used to record the gratings, the problem is usually solved by darkening the back side of the substrate. When counterpropagating beams are used, the substrate must be transparent. When substrates are fabricated that have few lines per millimeter, for example 600, as for the grating considered above, the angle of incidence of the recording beams on the substrate is relatively small. The percent of reflected radiation is consequently also small. The stray interference fringes are so much weaker than the main ones in this case that they have virtually no effect on the quality of the recorded gratings. When gratings are fabricated with a large number of lines per millimeter, for example 3600, large angles of incidence of the radiation on the substrate are used, the percentage of reflected energy increases, and the effect of stray interference has to be taken into account. A possible way to solve this problem is to use spatially incoherent radiation to record the gratings. 17 In spatially incoherent light, we have interference only between the direct ray and its own reflection. The rays reflected inside the substrate will not encounter the appropriate rays for interference, and no stray gratings will appear. Radiation that possesses temporal but not spatial coherence can be obtained, for example, by introducing a rotating scatterer into the laser beam. An optical system for recording a holographic diffraction grating by means of incident and reflected beams is shown in Fig. 3. At the first stage, grating objective 4 is recorded in spatially coherent light through the substrate of future grating 3, using pinhole stops 1 and 2. Convex mirror 5 is not used at this stage. After grating objective 4 is developed and aluminized, we place it in the same position in which the recording was done, we remove pinhole stop 2, we place spherical mirror 5 so that its center of curvature coincides with the initial position of pinhole stop 2, and we replace pinhole stop 2 with a rotating scatterer in order to eliminate the spatial coherence of the recording beam. If convex mirror 5 is mounted correctly, the light coming from point 1 diffracts at grating objective 4 and is reflected from spherical mirror 5, after which it diffracts and is focused by grating objective 4 back to point 1. The resulting grating 3 we call it the final grating is equivalent to a grating recorded by means of two coincident point sources, one real and one virtual. The breakdown of the spatial coherence prevents interference between the rays reflected from the substrate surfaces and prevents stray lines from appearing. The optical system for recording a diffraction grating usually consists of two point sources and a spherical substrate. In the case considered here, source C is not a material source but the point of intersection of the prolongation of the rays coming from point A* to objective 4, reflected from this objective, and transmitted through the substrate of the final grating. Another source is the specular reflection of point C relatively perpendicular to objective 4 at its center. The recording angles for this setup equal and, respectively. Thus, we have interference between two rays, one of which propagates in air, while the other, oppositely directed, propagates in the photoresist. The grating equation for this case is n 1 sin 0 /, 7 where n is the refractive index of the photoresist. The line formed at point P on the surface of the substrate of the final grating is a result of the interference of rays A*P*P**P and A*P*P, where P* and P** are the points where the prolongation of ray CP intersects objectives 1 and 2, respectively. In this recording system, a ray with wavelength 0, coming from A*, is reflected at point P* on objective 4 to point P on substrate 3. Another ray 0 is incident at point P from point P** on convex mirror 5, on which it was incident from the same point A* after reflection at P*. The interference fringes formed on the substrate surface, or the grating lines, correspond to k 0 nap npp np P**P na*p* P*P n 1 PP**. 8 Substituting Eq. 8 into Eq. 1, weget F AP PB m / 0 n 1 PP**. 9 Recalling that PP** CP CP**, and CP** const, we can differentiate Eq. 9 and find the corresponding aberration factors F ijk M ijk m / 0H ijk, H 200 0, H 300 0, 10 H 020 n 1 1/R cos cos/r, H 120 n 1 sin 1/ R cos cos /cos. The incident beam is focused at a point that in this case lies on the Rowland circle. After passing through the substrate of the grating, the beam is reflected back by the spherical convex mirror, whose center of curvature is at the same point on the Rowland circle. A spherical standing wave is then formed whose antinodes intersect the substrate and determine the position and shape of the lines and the blaze angle. In this case, the faces of the lines will always be perpendicular to 587 J. Opt. Technol. 68 (8), August 2001 E. A. Sokolova 587

the propagation direction of the incident rays, so that perpendiculars dropped to the surfaces of the faces of the lines at the points of incidence of the rays converge to the Rowland circle. The faces of the lines are then parts of the surfaces of a series of concentric spheres, and this is a necessary condition to ensure that the wavelength of the blaze is constant over the entire grating surface. From the viewpoint of the holographic reduction of aberrations, the use of only this type of grating in its traditional version is rather limited. However, for the grating considered above, which has 3600 line/mm, the aberration factors corresponding to such a recording setup are close to optimal. 18 A number of 3600-line/mm diffraction gratings were recorded in such a setup, using both spatially coherent and spatially incoherent radiation. An investigation of the spectra obtained from the recorded gratings confirmed the theoretically predicted compensation of the sagittal coma. As expected, stray spectra are absent with gratings recorded in spatially incoherent light, unlike those recorded in spatially coherent light. In order to use this method to fabricate gratings for other types of spectral devices, some nonuniformity of the step can be introduced by displacing the substrate of the final grating from the Rowland circle along the axis of the incident ray. 19 The defocusing factor in this case is A 200 n 1 cos 2 cos d/ 0, 11 where R/OC, R is the radius of curvature of the grating substrate, and OC is the distance between the grating substrate and point 2 Fig. 3. Then, to satisfy the condition of compensation of the defocusing, the grating must be displaced by A 200 n 1 cos. 2 12 The astigmatism and meridional coma factors for this case are determined by the geometry of the recording layout and must be calculated from A 020 n 1 cos d/ 0, A 300 n 1 sin cos cos 1 d/ 0. 13 The constancy of the blaze angle over the surface of the concave grating, ensured by recording in counterpropagating beams, is especially important for gratings intended for spectrographs with a flat field. Since the size of the detector is limited, spectrometers with a flat field are usually lowdispersion devices. To achieve high energy efficiency in the visible region when using a mechanically ruled grating with a small number of lines per millimeter, we need a grating with a very small blaze angle, no more than 3 5. Since the process for mechanically fabricating concave diffraction gratings is such that the blaze angle varies from one end to the other, the diffraction efficiency over the grating surface varies, especially in the case of a high-aperture grating. This problem can be solved by using multisection gratings with a constant blaze angle at the center of each section. The theoretical resolving power of such a grating corresponds to that of a grating whose width equals the width of one of the sections. Moreover, the grating efficiency still varies within each section, and, if such a grating is used with significant defocusing, the contour of the spectral line may split. 20 Gratings recorded in counterpropagating beams are free from these drawbacks. For certain devices with vertical optical layouts, the use of such gratings provides the additional possibility of compensating second-order astigmatism, which cannot be done with mechanically ruled gratings. 19 We can affect the aberration factors of gratings recorded with incident and reflected beams more strongly if we use a three-step recording process instead of a two-step process. An optical setup for three-step recording is shown in Fig. 4. In this case, grating objective 1 is recorded in the same way as in the two-step recording setup. Grating 1* is recorded in the same setup as is grating 1, but without the substrate of final grating 2 on the path of one of the beams. The second step of the recording is to record convex grating 3 using grating objectives 1 and 1*. One of the recording beams passes through grating substrate 2. At the third stage, we record grating 2, using grating objective 1 and convex grating 3. The defocusing factor can be adjusted in this case in the same way as in the preceding case, by changing the distance between grating 2 and point C. Here we can also adjust the astigmatism factor, if points D and D* do not lie at the centers of curvature of gratings 1 and 1*. CONCLUSION FIG. 4. Optical setup for three stage recording. The new-generation diffraction gratings considered here actually correspond to the term holographic, unlike the already widely used interference gratings. First, a grating recorded in counterpropagating beams results from the interaction of light with the photosensitive layer over its entire thickness, and is not merely a surface photograph of an interference pattern. Second, the grating objective recorded through the substrate of the final grating is essentially a ho- 588 J. Opt. Technol. 68 (8), August 2001 E. A. Sokolova 588

logram of this substrate or a reflecting holographic optical element. The gratings recorded by this method are the most general case of holographic diffraction gratings. Gratings recorded in nonhomocentric beams by means of spherical and aspheric mirrors can be considered a particular case of the above, since a mirror in this treatment can be regarded as a particular case of a grating with an infinitely large period. Since the two-step recording method possesses six free parameters, making it possible to compensate the five main aberrations with a reserve, the introduction of aspheric optics seems superfluous at the present time. A mechanically ruled diffraction grating can be regarded in this context as a synthesized hologram. Although mechanically ruled gratings theoretically possess greater possibilities for compensating aberrations, in practice they are limited by the design of the ruling engine, and two-step recording currently possesses the greatest practical possibilities. This situation can change if the method of direct recording by a scanning laser or by electron beam currently used in microelectronics is further improved to make it possible to fabricate thinner structures with which to reproduce with sufficient accuracy the necessary distribution law of the lines and their curvature. However, for the time being, concave diffraction gratings recorded by a two-step process can satisfy the requirements imposed on the resolving power of a wide class of spectral devices, including high-resolution devices. The appearance of such a class of diffraction gratings virtually removes the limitations on the use of concave gratings in spectral devices. 1 H. A. Rowland, Preliminary notice of the results accomplished in the manufacture and theory of gratings for optical purposes, Phil. Mag. 13, 469 1882. 2 H. Noda, T. Namioka, and M. Seya, Geometric theory of the grating, J. Opt. Soc. Am. 64, 1031 1974. 3 I. V. Pe sakhson, Modern Trends in Spectroscopic Technique. A Collection of Articles Nauka, Novosibirsk, 1982, pp. 94 125. 4 Yu. V. Bazhanov, Relationship between the parameters of ruled and holographic concave diffraction gratings, Opt. Mekh. Prom. No. 10, 1 1979 Sov. J. Opt. Technol. 46, 571 1979. 5 A. V. Savushkin, E. A. Sokolova, and G. P. Startsev, Optimizing the spectral and energy characteristics of concave diffraction gratings, Opt. Mekh. Prom. No. 6, 51 1987 Sov. J. Opt. Technol. 54, 376 1987. 6 E. A. Sokolova and G. P. Startsev, Optimizing the parameters of systems of fast monochromators with stigmatic concave diffraction gratings, Opt. Mekh. Prom. No. 7, 1 1988 Sov. J. Opt. Technol. 55, 389 1988. 7 E. A. Sokolova and G. P. Startsev, Minimizing sagittal coma in monochromators with concave diffraction gratings, Opt. Mekh. Prom. No. 8, 1 1988 Sov. J. Opt. Technol. 55, 449 1988. 8 T. Namioka and M. Koike, Aspheric wavefront recording optics for holographic gratings, Appl. Opt. 34, 2180 1995. 9 M. Duban, Third-generation Rowland holographic mountings, Appl. Opt. 30, 4019 1991. 10 I. V. Pe sakhson and L. E. Levandovskaya, Optical systems for recording concave hologram diffraction gratings in opposed beams, Opt. Zh. No. 8, 54 1991 Sov. J. Opt. Technol. 58, 498 1991. 11 G. W. Stroke, An Introduction to Coherent Optics and Holography Academic Press, New York, 1969, pp. 8 13. 12 Yu. N. Denisyuk, Mapping the optical properties of an object in the wave field of the radiation scattered by it, Opt. Spektrosk. 15, 522 1963. 13 E. A. Sokolova, Geometric theory of two steps recorded holographic diffraction gratings, Proc. SPIE 3450, 113 1998. 14 E. A. Sokolova and S. Cortes, Calculation and mathematical model computer experiments with optical mountings for recording and using holographic diffraction gratings, Proc. SPIE 2968, 311 1996. 15 E. A. Sokolova and S. Cortes, New optical mountings of the spectral devices with concave diffraction gratings and high entrance slit, Proc SPIE 2863, 197 1996. 16 E. A. Sokolova, S. Mogo, and M. Hutley, Practical recording of holographic diffraction gratings in opposite-directed beams, in Diffractive Optics and Micro-Optics, Technical Digest, Quebec City, Canada, Conference Edition, 2000, pp. 118 119. 17 M. C. Hutley, Improvements in or relating to the formation of photographic record, U.K. Patent No. 1 384 281 1975. 18 E. A. Sokolova, Concave diffraction gratings recorded in counterpropagating beams, Opt. Zh. 66, No. 12, 88 1999 J. Opt. Technol. 66, 1084 1999. 19 E. Sokolova, Holographic diffraction gratings for flat-field spectrometers, J. Mod. Opt. 47, 2377 2000. 20 L. B. Katsnel son and E. A. Sokolova, Determining the spread function of an IR monochromator with a concave diffraction grating, Opt. Mekh. Prom. No. 4, 46 1986 Sov. J. Opt. Technol. 53, 231 1986. 589 J. Opt. Technol. 68 (8), August 2001 E. A. Sokolova 589