Basic angles in microelectromechanical systemscanning grating spectrometers
Tino Puegner,1,* Jens Knobbe,1 and Hubert Lakner1,2
1Fraunhofer Institute for Photonic Microsystems (IPMS), Maria-Reiche-Strasse 2, 01109 Dresden, Germany2Technical University Dresden, Faculty of Electrical Engineering and Information Technology, Institute for Semiconductor
and Microsystem Technology (IHM), Nöthnitzer. Strasse 64, 01187 Dresden, Germany
Spectroscopic instruments like spectrometers andspectrographs are well-established tools for the mea-surement of spectral properties of electromagneticradiation. Among the various types of spectroscopicinstruments, the scanning grating spectrometer(SGS) is one of the most frequently used systems.Interference phenomena and diffraction theory on agrating as the key component in these instrumentsare, therefore, well understood and extensively de-scribed in literature [1–3]. Over the years, continuousdevelopment has led to numerous improvements inthe design of spectroscopic instrumentation. Espe-cially optical design aspects, e.g., for better resolution,have been addressed [4–9].
The vast majority of the SGSs available today havebeen produced using discrete optical components,such as mirrors, gratings, and slits. These instru-
ments are comparatively large, very often quite ex-pensive, and limited to only a few applications, e.g.,in a laboratory environment. As new applicationscome into focus, the need for miniaturized systemsincreases, since they have several advantages, suchas portability and energy efficiency, over traditionalsystems. However, an ordinary SGS cannot be downsized without further effort. One reason is the diffi-culty in miniaturizing the grating and its drivingmechanism. The development of microelectromecha-nical systems (MEMS) in the past decades offers newpossibilities to build small scanning gratings with in-tegrated driving mechanisms [10]. The typical man-ufacturing and assembly processes associated withMEMS technologies allow for very precise definitionof components or details thereof but, on the otherhand, can severely limit the degrees of freedom foralignment and assembly of the whole system. Thisis especially the case for planar mounting technolo-gies or wafer stacking, where it is nearly impossibleor impractical to tilt a component.
Furthermore, these modern MEMS are often sym-metrically moving devices and some of them are re-sonantly driven. These characteristics, which are notcommon to traditional systems, lead to some con-straints that have not so far been considered. In par-ticular, relations among geometrical quantities, suchas the deflection amplitude of the grating, as well asthe application and performance related quantities,such as the attainable wavelength range, have notbeen examined in great detail.
These constraints initiated our analysis of theaforementioned relations among geometry and per-formance issues, as well as the search for an appro-priate analytical description. Special attention hasbeen paid to relations between the orientation ofthe grating in rest position, its deflection amplitude,the symmetry of the motion, and application-specificparameters, such as the spectral range, as indicatedin Fig. 1. Even though this work originates from thedevelopment of miniaturized SGSs using MEMStechnology, the results can, of course, be applied toall types of SGS systems.
The paper is organized as follows. The analysisstarts with the basic grating equation complementedby the typical constraints of orientation and motionof a MEMS scanning grating in an SGS. Importantequations that display the interrelations amongthe geometry of the system and performance relatedquantities are then derived. Finally, some detailedexamples for the application of the equations arepresented.
2. SGSs with Boundary Conditions
In a SGS, very often realized as a Czerny–Turnerconfiguration, the radiation entering through an en-trance slit is collimated and subsequently impingeson a scanning diffraction grating. The radiation isspectrally dispersed by the grating and then refo-cused on an exit slit. The exit slit cuts out a narrow
part of the spectrum. When the grating is rotated,the spectrum is scanned across the slit [11–14]. Fol-lowing Chupp and Grantz, a modified arrangementof the original Czerny–Turner configuration is pre-sented in Fig. 2, featuring a common orientation ofthe slits and the grating in rest position [15]. Thisarrangement is favorable when miniaturizing aSGS by applying MEMS technology with its planarsubstrates and mounting characteristics.
Interference and dispersion of light at the planegrating are governed by the basic grating equation:
mλg
¼ sinðαÞ þ sinðβÞ: ð1Þ
It gives the directions of constructive interference forlight with the wavelength λ in terms of the angle ofincidence α and diffraction β on a grating with thegrating constant g in the diffraction order m. Thegrating constant g is the periodical distance betweentwo adjacent repeating grating structures, referredto as grating lines. The diffraction order is an integerfor the optical path difference that is necessary forconstructive interference on the grating. In accor-dance with Figs. 1–3, the angle α is spanned betweenthe incident beam and the grating normal and theangle β is spanned between the diffracted beamand the grating normal.
According to Fig. 1, the angles α, β, and ω dependon the position of the grating. They can be can be ex-pressed as a superposition of mean values αd, βd, andωd, as well as the angle of deflection of the grating φ:
α ¼ αd þ φ; β ¼ βd þ φ; ω ¼ ωd þ φ: ð2ÞSince the motion of the grating is assumed to be
symmetrical about the rest position of the grating,the mean values αd, βd, and ωd occur when the
Fig. 1. SGS with its basic design parameters illustrated for therest position of a symmetrically oscillating grating.
Fig. 2. SGS with its basic design parameters illustrated for therest position of a symmetrically oscillating grating in an arrange-ment with a common orientation of the slits and the grating in restposition.
grating is in rest position at φ ¼ 0. To avoid ambigu-ities, we adopt a sign convention that relates the signof the diffraction order signðmÞ to the sign of theangle of incidence signðmÞðαd þ φÞ and the angle ofdiffraction signðmÞðβd þ φÞ in Eq. (1):
mλg
¼ sinðsignðmÞðαd þ φÞÞ þ sinðsignðmÞðβd þ φÞÞ:ð3Þ
The sign convention in Eq. (3) ensures that the signof the diffraction order has no influence on the direc-tion of rotation of the grating and λ increases by anincrease of φ:
jmjλg
¼ sinðαd þ φÞ þ sinðβd þ φÞ: ð4Þ
The symmetrical motion of the grating by the an-gle φ is limited by the deflection amplitude � φ̂.Furthermore, the boundaries of the wavelengthrange Δλ ¼ λmax − λmin are assumed to coincidewith the boundaries of the range of the motion− φ̂ ≤ φ ≤ þφ̂:
jmjλmin
g¼ sinðαd − φ̂Þ þ sinðβd − φ̂Þ; ð5Þ
jmjλmax
g¼ sinðαd þ φ̂Þ þ sinðβd þ φ̂Þ: ð6Þ
For optical design purposes, it proves to be usefulto introduce another characteristic wavelength with-in the spectral range, namely, a design wavelength λdthat strikes the exit slit when the grating is in restposition at φ ¼ 0:
jmjλdg
¼ sinðαdÞ þ sinðβdÞ: ð7Þ
A grating in a SGS at positions determined byEqs. (5)–(7) along with the associated angles of inci-dence and diffraction is illustrated in Fig. 3.
Adding or subtracting the equation containing thelower boundaries [Eq. (5)] to or from the equationcontaining the upper boundaries [Eq. (6)] leads to
jmjðλmax þ λminÞg
¼ sinðαd þ φ̂Þ þ sinðαd − φ̂Þ
þ sinðβd þ φ̂Þ þ sinðβd − φ̂Þ; ð8Þ
jmjðλmax − λminÞg
¼ sinðαd þ φ̂Þ − sinðαd − φ̂Þ
þ sinðβd þ φ̂Þ − sinðβd − φ̂Þ: ð9Þ
By utilizing trigonometric identities, Eqs. (8) and (9)can be rearranged:
jmjðλmax þ λminÞg
¼ 2 cosðφ̂ÞðsinðαdÞ þ sinðβdÞÞ; ð10Þ
jmjðλmax − λminÞg
¼ 2 sinðφ̂ÞðcosðαdÞ þ cosðβdÞÞ: ð11Þ
The sums on the right-hand sides of Eqs. (10) and(11) can be further modified:
jmjðλmax þ λminÞ2g cosðφ̂Þ ¼ 2 cos
�αd − βd2
�sin
�αd þ βd2
�;
ð12Þ
jmjðλmax − λminÞ2g sinðφ̂Þ ¼ 2 cos
�αd − βd2
�cos
�αd þ βd2
�:
ð13ÞA basic characteristic of a SGS is that the angle
spanned between α and β, referred to as the angleof deviation δ ∈ R; ½−π; π�, has to be constant, sincethe positions of the two mirrors and the rotation axisof the grating in spectrometers as shown in Fig. 1 arefixed:
Fig. 3. The grating in a SGS in three different positions. In (a),λmax hits the exit slit, in (b), λd hits the exit slit, and in (c), λmin hitsthe exit slit. A rotation about − φ̂ ≤ φ ≤ þφ̂ results in a spectralrange Δλ ¼ λmax − λmin.
As Eq. (14) is valid for any combination of α and β, itis valid for the combination of angles in rest position,αd and βd, too. The half-angle of deviation δ=2 thenamounts to
δ2¼ βd − αd
2: ð15Þ
Furthermore, αd and βd can be obtained by adding orsubtracting ωd and δ=2, as illustrated in Fig. 1:
αd ¼ ωd −δ2; βd ¼ ωd þ
δ2: ð16Þ
Thus, the tilt angle ωd can be calculated by Eq. (16)based on αd and βd:
ωd ¼ βd þ αd2
: ð17Þ
The expressions for δ=2 [Eq. (15)] and ωd [Eq. (17)]can be applied as substitutions in Eqs. (12) and (13):
jmjðλmax þ λminÞ2g cosðφ̂Þ ¼ 2 sinðωdÞ cos
�δ2
�; ð18Þ
jmjðλmax − λminÞ2g sinðφ̂Þ ¼ 2 cosðωdÞ cos
�δ2
�: ð19Þ
Squaring and adding Eqs. (18) and (19) leads to
�jmjðλmax þ λminÞ2g cosðφ̂Þ
�2þ�jmjðλmax − λminÞ
2g sinðφ̂Þ�
2
¼ 4cos2�δ2
�: ð20Þ
The resulting equation can be solved for the half-angle of deviation δ=2, which denotes the symmetryof the incoming and the diffracted beam regardingthe spectrometer axis as illustrated in Fig. 1:
ð21ÞA first important result is represented by Eq. (21).
It connects the half-angle of deviation, the wave-length range, the deflection amplitude of the grating,the grating constant, and the diffraction order. Thismeans that, for a given MEMS device, characterizedby the properties of the grating and its deflectionamplitude, the attainable spectral range can becalculated by choosing an appropriate value for thehalf-angle of deviation δ=2 and evaluating Eq. (21).Or alternatively, it can be used to determine the
minimum amplitude φ̂ that is necessary for arequired spectral range.
However, Eq. (21) does not contain explicit infor-mation on the positions of the grating with respectto the spectrometer axis. The values that α and β takeare not determined by Eq. (21), since they are mea-sured relative to the grating normal. For a completedescription of the system, a second equation that in-cludes the angle between the spectrometer axis andthe grating normal, as indicated in Fig. 1, is needed.To derive this additional equation, we can solveEq. (18) for the angle ωd:
ωd ¼ arcsin
0@jmj
2gλmaxþλmin2 cosðφ̂Þ
cos�δ2
�1A: ð22Þ
When the expression for δ=2 [Eq. (21)] is applied toEq. (22), the tilt angle ωd amounts to
The tilt angle of the grating in rest position, thespectral range, and the deflection amplitude are con-nected by Eq. (23). It is important to note that thegrating constant and the diffraction order do nothave any influence on the tilt angle. The expressionfor ωd complements Eq. (21). With Eq. (16), the an-gles of incidence αd and diffraction βd in rest positioncan be calculated from ωd and δ=2 as functions of g,m, λmin, λmax, and φ̂:
ð25ÞThus all important geometric quantities in terms
of angles on the grating are given as functions of thegrating properties and the spectral range by meansof Eqs. (21) and (23)–(25). They can be used for the
optimization of various spectrometer configurationswith symmetrically oscillating diffraction gratings.
3. Exemplary Solution Set
A. General Conditions
For a discussion of the equations derived in Section 2,it is worth noting that only a part of the possiblesolutions is physically significant. Some of the pa-rameters are affected by technical aspects, whileothers are determined by a specific application, likethe spectral range Δλ, which is characterized by λminand λmax.
The choice of an appropriate grating constant g isdetermined on the one hand by Δλ and on the otherhand by the available values of g in case the gratingis not customized. For miniaturized MEMS spectro-meters, preferentially deployed in the visible (VIS)and near-infrared (NIR) range of the electromagneticspectrum, a reasonable choice for the grating con-stant is somewhere between 380 and 780nm forthe VIS or 780 and 2500nm for the NIR. As the grat-ings in MEMS devices are manufactured by litho-graphic, etching, and nanoimprint techniques, thegrating constant can theoretically be chosen out ofthe specified ranges without limitations. Nonethe-less, it is common practice to fabricate grating con-stants that are orientated at values generated byfrequently used groove densities n ¼ 1=g of ruledgratings.
In general, MEMS gratings are limited towardlarge grating constants due to of the limited size ofthe miniaturized grating plates. The number of lineson a grating is determined by the width of the gratingWg and the grating constant g. Since the resolvingpower is also affected by the number of lines and,thereby, Wg and g, a selection of very large gratingconstants reduces the resolution. So, for a certainresolution, a maximum value of g must not beexceeded:
R ¼ λdλ ¼ mN ¼ mWg
g: ð26Þ
Regarding the diffraction order m, it has to be con-sidered that, by an increase of the modulus of m, thehalf-angle of deviation δ=2 increases as well. Thismight be disadvantageous for miniaturized systems.However, an increase ofm results in a higher separa-tion of adjacent wavelength and, thereby, in a higherresolution. Nevertheless, regarding the fabricationtechnologies associated with miniaturized MEMSSGSs, the angles should not be too large. The diffrac-tion orderm is limited tom ¼ ½−2; 2� for such systemsin most cases.
The rotational range of the grating φ ¼ ½−φ̂;þφ̂�depends on the underlying scanning technology. Forthe integration of a diffraction grating in a MEMSdevice, oscillatory scanning principles have been pro-ven especially suitable. An oscillatory motion can beachieved with both high inertia and low inertia
devices [16]. Single axis rotating MEMS devices havemostly been realized in high inertia, resonantdesigns. In these devices, the scanning plate issuspended by elastic solid-body joints and drivenelectrostatically or galvanometrically. Besides thedriving mechanism, the motion behavior is influ-enced by the resulting spring-mass system, mainlydetermined by the geometry of the torsion springs,the substrate thickness, and the geometry and themass distribution of the oscillating plate. In caseof an SGS, the scanning amplitude is the criticalparameter. It is important that the maximum scan-ning amplitude applicable in the optical system islimited, as well. At first sight, a MEMS device witha larger amplitude φ̂might be considered superior toone that exhibits a smaller amplitude, since onemight think it should cover a broader spectrum.However, as will be shown in Subsection 3.B, thereexists an upper limit for the maximum deflectionof the grating associated with a given spectral rangeΔλ. This limit is determined by the geometry of thesystem and is independent of the MEMS properties.So beyond a certain value, the amplitude becomesless important and other properties, such as the sizeof the grating plate, may be more important. Exemp-lary rotational ranges are listed in Table 1 for differ-ent kinds of MEMS devices. To our knowledge, adiffraction grating has only been implemented inthe first type of MEMS device [17] listed in Table 1,but could also be implemented in the two otherdevices in a similar way.
B. Solution Set
In summary, the space spanned by the parametersλmin, λmax, g, and m is quite limited, either by prede-finitions or further restrictions. The only parameterthat can be freely chosen to a certain extent is thescanning amplitude of the grating φ̂. In order to gainsome insight into the general behavior of the equa-tions, we first drop the limitations for the deflectionamplitude φ̂ mentioned above and examine the an-gles as functions of the variable φ̂. The constraintsfor the rotational range of the grating will be consid-ered subsequently. Furthermore, we include the signconvention of Eq. (3) in our considerations:
δ02¼ signðmÞ δ
2; ω0
d ¼ signðmÞωd;
α0d ¼ signðmÞαd; β0d ¼ signðmÞβd: ð27Þ
Table 1. Oscillatory Optical Scanning Technologies Applied inMEMS Devices and the Associated Maximum Scanning Amplitudes
Since there are many parameters, an analysis isbest done for two typical examples. As examples,we choose a spectrometer for the VIS with the pa-rameters m ¼ 1, g ¼ 700nm, λmin ¼ 380nm, andλmax ¼ 780nm, and a spectrometer for the NIR withthe parameters m ¼ 1, g ¼ 1600nm, λmin ¼ 950nm,and λmax ¼ 1900nm. The plots of the anglesω0d, δ0=2, α0d, and β0d as a function of the deflection
amplitude φ̂ are shown in Fig. 4 for the first systemand Fig. 5 for the second system.
Together, the curves of α0d and β0d form a continuousclosed loop. The two intersections of the three anglesα0d, β0d, and ω0
d occur at the positions where the angleof deviation δ0=2 becomes zero:
ifδ2¼ 0
( α0d ¼ β0d ¼ ω0d���dα0dd φ̂
���¼ ���dβ0dd φ̂
���¼ ���dδ0d φ̂
���¼ ∞: ð28Þ
As can be seen for some amplitudes φ̂, the angle β0dexceeds 90°, which is unphysical. Therefore, therange of values for φ̂ must be limited by further con-siderations. We now include the limitations for themotion of the grating in terms of a maximum andminimum amplitude φ̂. Note first that the incidentand diffracted beam both have to be located on thefront side of the grating:
jα0; β0j ≤ π2: ð29Þ
If the grating moves within the given boundaries,the angles take their extrema α̂ ¼ α0d∓φ̂ and
β̂ ¼ β0d∓φ̂. As Eq. (29) is valid for α0 and β0 in general,it is valid for α̂ and β̂ in particular:
j α̂; β̂j ≤ π2: ð30Þ
Thus, the values for the angles α0d and β0d are limitedto:
jα0d; β0dj ≤π2− φ̂: ð31Þ
As can be seen from Eq. (31) and the plots of α0d andβ0d illustrated in Figs. 6 and 7, only a certain sectionof the plots of α0d, β0d ¼ f ðg;m; λmin; λmax; φ̂Þ leads tophysically significant SGS configurations. This sec-tion is limited by a lower boundary φ̂min where bothangles α0d and β0d coincide and an upper boundaryφ̂max where the modulus of α0d or β0d becomes greaterthan ðπ=2Þ − φ̂:
Fig. 4. Plot of the angles ω0d, δ0=2, α0d, and β0d ¼ f ðφ̂Þ in an SGS for
m ¼ 1, g ¼ 700nm, λmin ¼ 380nm, and λmax ¼ 780nm. The limita-tions for the deflection amplitude φ̂ have been omitted.
Fig. 5. Plot of the angles ω0d, δ0=2, α0d, and β0d ¼ f ðφ̂Þ in an SGS for
m ¼ 1, g ¼ 1600nm, λmin ¼ 950nm, and λmax ¼ 1900nm. Thelimitations for the deflection amplitude φ̂ have been omitted.
Fig. 6. Solutions for the angles ω0d, δ0=2, α0d, and β0d ¼ f ðφ̂Þ in an
SGS for m ¼ 1, g ¼ 700nm, λmin ¼ 380nm, and λmax ¼ 780nm forpossible SGS configurations.
Fig. 7. Solutions for the angles ω0d, δ0=2, α0d, and β0d ¼ f ðφ̂Þ in an
SGS for m ¼ 1, g ¼ 1600nm, λmin ¼ 950nm, and λmax ¼ 1900nmfor possible SGS configurations.
Considering the maximum scanning amplitudeachievable with a particular MEMS actuator aslisted in Table 1, the range of motion of the gratingcan be additionally restricted. The boundariesjα0d; β0dj ≤ ðπ=2Þ − φ̂ and φ̂min ≤ φ̂ ≤ φ̂max lead to a tra-pezoidal shaped solution set for an SGS, as illu-strated in Fig. 6 for the first system and Fig. 7 forthe second system.
The domains of the angles φ̂, δ0=2, ω0d, α0d, and β0d
that define the solution space are given in Table 2for the VIS and the NIR systems.
Magnified plots of the highlighted portion of Figs. 6and 7 are shown in Figs. 8 and 9, respectively. Bothillustrations demonstrate that reasonable solutionsexist only for a very limited range of the deflectionamplitudes.
In the design process of an SGS, a scanningamplitude has to be specified within the intervalφ̂min ≤ φ̂ ≤ φ̂max. In accordance with Table 2, the bor-ders of φ̂ for the VIS systems amount to φ̂min ¼ 9:06°and φ̂max ¼ 13:48° and the borders of φ̂ for the NIRsystems amount to φ̂min ¼ 9:58° and φ̂max ¼ 13:87°.As an exemplary deflection amplitude of the grating,we chose φ̂ ¼ 10°, which is well within the given in-tervals and, furthermore, represents a value achiev-able by a MEMS diffraction grating [17]. The basicdesign parameters δ0=2, ω0
d, α0d, and β0d for the two as-sociated SGSs are listed in Table 3.
The data in Table 3 represent the optimized solu-tions for a VIS and a NIR SGS, for MEMS gratingswith symmetrical scanning amplitude. For a gratingwith the given grating constant and deflectionamplitude, the predefined spectral range is entirelycovered. As pointed out above, the existence of theupper limit for the amplitude φ̂ implies that, in theexample, a MEMS device need not be capable of am-plitudes larger than 13:48° or 13:87°. However, aminimum value of 9:06° or 9:58° is definitely neededfor the given spectral ranges. Therefore, only the firstdevice listed in Table 1 can cover the spectral range.
In addition to the scanning amplitude that is ne-cessary for spectroscopic function within λmin ≤ λ ≤λmax for a specific diffraction order and grating con-stant, the attainable wavelength range as a functionof the scanning amplitude can be calculated, as well.For this purpose, the minimum and maximum wave-lengths are substituted by
Fig. 9. Plots of the angles ω0d, δ0=2, α0d, and β0d ¼ f ðφ̂Þ in an SGS for
m ¼ 1, g ¼ 1600nm, λmin ¼ 950nm, and λmax ¼ 1900nm withinthe solution space 9:58° ≤ φ̂ ≤ 13:87°.
Fig. 8. Plots of the angles ω0d, δ0=2, α0d, and β0d ¼ f ðφ̂Þ in an SGS for
m ¼ 1, g ¼ 700nm, λmin ¼ 380nm, and λmax ¼ 780nm within thesolution space 9:06° ≤ φ̂ ≤ 13:48°.
Table 2. Solution Set of the Angles φ̂, δ0=2, ω0d , α
0d, and β0d for the Exemplary SGS Shown in Figs. 6 and 7
in Eqs. (32) and (33). The wavelength λm ¼ λmaxþλmin2 is
situated in the middle of the wavelength range:
φ̂min ¼ f ðm; g; λm;ΔλÞ: ð35ÞThe condition for φ̂min in Eq. (35) can be convertedinto a condition for Δλmax:
Δλmax ¼ f ðm; g; λm; φ̂Þ: ð36ÞFollowing the exemplary VIS andNIR systems dis-
cussed above, the attainable wavelength ranges for
systems featuring the same diffraction orders andmiddle wavelengths λm in accordance with Eq. (36)but three additional grating constants are presentedin Figs. 10 and 11. The additional diffraction ordersare orientated at values generated by frequentlyused groove densities n ¼ 1=g of ruled gratings.
The attainable maximum wavelength ranges forthe plots of Figs. 10 and 11 are listed in Table 4. Forthe given grating constants or groove densities, amaximum wavelength range of 804 or 1720nm canbe reached with the VIS SGS or the NIR SGS, respec-tively. For both systems, the gratings are assumed tohave maximum scanning amplitude of 15°.
4. Conclusions and Outlook
The miniaturization of an SGS using a MEMS grat-ing leads to further design constraints that are notcommon to classical SGSs. Important effects, suchas a symmetrical grating deflection of limited ampli-tude, are consequences thereof. An analysis of theassociated basic system design that takes theseboundary conditions into account has led to newanalytical expressions. These expressions connectthe basic angles on the grating, the diffraction order,the wavelength range, and the deflection amplitudeof the symmetrically oscillating grating. The basicangles can be calculated based on application specificparameters and technological constraints withoutthe need of further assumptions. An investigationof the influence of input parameters on the analyticalexpressions results in a description for the limitedsolution set regarding these parameters and an in-sight into interrelations among them.
With the equations for the basic angles on the grat-ing and the knowledge of the solution set, the designand optimization of an SGS in an early design stagecan be significantly simplified. Thereby, the designprocess is both application and technology orien-tated. Although the investigations have been in-itiated by the miniaturization of a SGS by the useof MEMS gratings, the analytical expressions and in-sights have proven useful for all SGS types.
References1. E. G. Loewen, Diffraction Gratings and Applications (Marcel
Dekker 1997).
Table 4. Maximum Attainable Wavelength Ranges Δλmax for theExemplary SGS in Figs. 10 and 11 with Different Values of g andm � 1, λm � 580 nm, and φ̂ ≤ 15°, orm � 1, λm � 1425 nm, and φ̂ ≤ 15°,
Respectively
m ¼ 1, λm ¼ 580nm
g [nm] 417 556 700 8331g ¼ n [ l
mm] 2400 1800 1428 1200Δλmax [nm] 300 485 655 804
m ¼ 1, λm ¼ 1425nm
g [nm] 1052 1428 1600 18181g ¼ n [ l
mm] 950 700 625 550Δλmax [nm] 777 1266 1470 1720
Fig. 10. Plot of the attainable wavelength range for an SGS withm ¼ 1, g ¼ f417;556; 700;833gnm, and λm ¼ 580nm for a MEMSdevice with a maximum scanning amplitude up to 15°.
Fig. 11. Plot of the attainable wavelength range for an SGS withm ¼ 1, g ¼ f1052;1428;1600; 1818gnm, and λm ¼ 1425nm for aMEMS device with a maximum scanning amplitude up to 15°.
Table 3. Values of the Angles δ0=2, ω0d , α
0d , and β0d for the Exemplary SGS
Shown in Figs. 8 and 9 for a MEMS Grating with φ̂ � 10°
2. M. Born, E. Wolf, and A. B. Bhatia, Principal of Optics:Electromagnetic Theory of Propagation, Interference andDiffraction of Light (CUP Archive, 2000).
3. C. Palmer, Diffraction Grating Handbook, 6th ed.(Newport, 2005).
4. G. R. Roaendahl, “Contributions to the optics of mirrorsystems and gratings with oblique incidence. I. Ray tracingformulas for the meridional plane,” J. Opt. Soc. Am. 51,1–3 (1961).
5. A. B. Shafer, L. R. Megill, and L. Droppleman, “Optimizationof the Czerny-Turner spectrometer,” J. Opt. Soc. Am. 54, 879–886 (1964).
6. K. Kudo, “Optical properties of plane-grating monochro-mator,” J. Opt. Soc. Am. 55, 150–161 (1965).
7. J. K. Pribram and C. M. Penchina, “Stray light inCzerny-Turner and Ebert spectrometers,” Appl. Opt. 7,2005–2014 (1968).
8. R. A. Hill, “A new plane grating monochromator with off-axisparaboloids and curved slits,” Appl. Opt. 8, 575–581 (1969).
9. M. A. Gil and J. M. Simon, “New plane gratingmonochromatorwith off-axis parabolical mirrors,” Appl. Opt. 22, 152–158(1983).
10. H. Grüger, A. Wolter, T. Schuster, H. Schenk, and H. Lakner,“Realization of a spectrometer with micromachined scanninggrating,” Proc. SPIE 4945, 46–53 (2003).
11. H. Ebert, “Zwei Formen von Spectrographen,” Ann. Phys. 274,489–493 (1889).
12. M. Czerny and A. F. Turner, “Ueber den Astigmatimus beiSpiegelspektrometern,” Z. Phys. 61, 792–797 (1930).
13. M. Czerny and V. Plettig, “Ueber den Astigmatimusbei Spiegelspektrometern II,” Z. Phys. 63, 590–595(1930).
14. W. G. Fastie, “Image forming properties of the Ebert mono-chromator,” J. Opt. Soc. Am. 42, 647–650 (1952).
15. V. L. Chupp and P. C. Grantz, “Coma canceling mono-chromator with no slit mismatch,” Appl. Opt. 8, 925–929(1969).
16. L. Beiser, Unified Optical Scanning Technology (Wiley,2003).
17. F. Zimmer, A. Heberer, T. Sandner, H. Grueger, H. Schenk, H.Lakner, A. Kenda, and W. Scherf, “Investigation and charac-terization of high-efficient NIR-scanning gratings used in NIRmicro-spectrometer,” Proc. SPIE 6466, 646605 (2007).
18. S. Kurth, C. Kaufmann, R. Hahn, J. Mehner, W. Dotzel, andT. Gessner, MEMS Scanner for Laser Projection (ChemnitzUniversity of Technology, Center for Microtechnologies,2004).
19. L. O. S. Ferreira and S. Moehlecke, “A silicon micromechanicalgalvanometric scanner,” Sens. Actuators A Phys. 73,252–260 (1999).
