Control of spectral aberrations in a monochromator using a plane holographic chirped grating

Control of spectral aberrations in a monochromatorusing a plane holographic chirped grating

Alexandre April and Nathalie McCarthy*Centre d’optique, photonique et laser (COPL), Département de physique, de génie physique et d’optique,

Université Laval, Quebec City, Qc, Canada, G1K 7P4

*Corresponding author: [email protected]

Received 12 October 2007; revised 8 April 2008; accepted 10 April 2008;posted 15 April 2008 (Doc. ID 88582); published 12 May 2008

Amethod aimed to minimize the impact of spectral aberrations in a monochromator is proposed in whichthe spectrum of the source of radiation under study is scanned by the rectilinear translation of a planechirped grating. The chirped grating, which has a spatially variable groove spacing, is used to diffractand to spectrally focus the radiation. Imaging properties of the chirped grating were analyzed in orderto develop the expression of the aberration coefficients of the system and the expression of the widthof the instrument line shape due to aberrations. The optimal rectilinear trajectory required to operatethemonochromator without significant spectral aberrations inmeasurements has been obtained numeri-cally and tested in the laboratory. Experimental measurements of the emission spectrum of a seven-wavelength helium–neon laser are presented, as well as the sensitivity of the monochromatorperformance to different geometrical parameters. © 2008 Optical Society of America

OCIS codes: 050.0050, 120.0120, 050.1590, 120.4140, 050.1950, 120.6200.

1. Introduction

Amonochromator is an instrument basically made ofanentranceslit, adispersiveelementsuchasadiffrac-tion grating, and an exit slit. While a source of radia-tion illuminates the entrance slit, themonochromatorselects a given wavelength, or a narrow wavelengthband, of the spectrum of the source by allowing onlythat wavelength or that portion of the spectrum topass through the exit slit. The spectrum is scannedby the relative motion of the diffraction grating withrespect to the exit slit [1,2]. In all kinds of monochro-mators, there is always the challenge of controllingaberrationspresent in the instrumentduring thedataacquisition. In fact, efforts have to be made to elimi-nate the spectral aberrations (in the plane of disper-sion of the grating) as well as the spatial ones (inthedirectionparallel to theexit slit).A comprehensivediscussion about aberrations in spectrometers andmonochromators can be found in Ref. [1].

Many conventional monochromator mounts, suchas the Czerny–Turner, the Ebert–Fastie, and theMonk–Gilliesonmonochromators,makeuse of a stan-dard plane diffraction grating and a focusing concavemirror [2]. The grating (whose surface is flat andwhose grooves are equally spaced) is required to sepa-rate the spectral components of the radiation presentin the monochromator, while the focusing concavemirror is necessary to image the entrance slit onthe exit slit. The wavelength tuning is usuallyachieved bya rotation of the gratingabout anaxis par-allel to the grooves [3,4]. As a consequence, two dis-tinct optical elements are employed to provide thediffracting and the focusing properties needed to op-erate themonochromator properly.However, the com-binationof thediffracting and the focusing componentinto a single optic is often desirable.

The design of optical elements both dispersive andimaging has been studied for decades. In fact,Rowland suggested as early as 1882 to employ a con-cave reflection grating, thus combining the principleof the standard plane diffraction grating (with con-stant groove separation) with the focusing propertiesof a concave mirror [5]. Such a concave grating can be

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2750 APPLIED OPTICS / Vol. 47, No. 15 / 20 May 2008

used in a monochromator, replacing both the planegrating and the focusing concave mirror: if the en-trance slit is placed on the circumference of a circlewhose diameter equals the radius of curvature ofthe concave grating, then the spectrum is focusedon the same circle (the so-called Rowland circle) [1].Many concave grating mounts are now routinelyused: apart from the Rowland circle mounting, thereare among other things the Wadsworth mount, theflat-field spectrograph and the Seya–Namioka con-cave grating monochromator [2].It is also known that a diffraction grating can focus

a diffracted beam, even if the surface of the grating isflat, when the groove spacing varies appropriatelywith the groove position; such gratings are hereincalled chirped gratings (CGs). In literature, CGsare also called varied line-space, varied-space, vari-able spacing, variable line-space, varied-line-spacing,or varied pitch gratings. As opposed to gratings withconstant groove separation, CGs provide the imagingproperties to the grating that otherwise must be sup-plied by separate optical elements. For that reason, aplane CG can be incorporated in a monochromator toact as both the dispersive and the spectrally focusingcomponents needed in the instrument [6–10]. Theuti-lization of a CG in a monochromator gives the oppor-tunity to get rid of auxiliary optics (such as focusingmirrors or lenses) in the design of the instrument; itmay indeed simplify the structure of the monochro-mator, reducing the number of optical elementsneeded for its fabrication, minimizing the numberof reflections of the radiation, and possibly reducingthe size of the overall instrument. The idea of usinga CG to spectrally focus the radiation in a monochro-mator has beenalready exploited; in fact,manymono-chromators using CG are regularly produced andsold. They are usually classified into two categories:the normal-incidence configuration [11,12] and thegrazing-incidence configuration, the latter beingespecially suitable for spectral measurements withsoft X-ray radiation [13–15]. Many efforts have beendedicated to correct spectral and spatial aberrationssimultaneously. In normal incidence, a single opticalcomponent corrects for the spectral and the spatialaberrations while, in grazing incidence, a supplemen-tary mirror is usually added to correct for spatialaberrations.The aim of this paper is to present a method to sys-

tematically minimize the impact of spectral aberra-tions in a monochromator in which the wavelengthtuning is achieved by the rectilinear translation ofa given flat CG. Themethod proposed has been testedexperimentally with a prototype CG monochromatorthat we built in our laboratories. Since we deal withplane gratings with parallel grooves, we restrict thediscussion to spectral aberrations and we do not con-sider the spatial ones. Astigmatism, for example, canin fact be tolerated in a monochromator since only fo-cusing in the plane of dispersion of the grating is re-quired to separate the variouswavelengths present inthe spectrum of a source of radiation [5]. To minimize

the existence of spectral aberrations in our CGmono-chromator, we have developed amodel for calculatingthe width of the instrument line shape (ILS) due toaberrations. It allows the determination of the opti-mal rectilinear trajectory of the grating, for a givenCG and for an initial monochromator geometry, tobe used to scan the spectrumof the source of radiationunder study. It was then possible to experimentallyobtain spectra without significant defocus and spec-tral aberrations. Experimental results were obtainedwith a visible source of radiation, but themethod pre-sented in this paper can be applied to other regions ofthe electromagnetic spectrum.

2. Holographic Chirped Grating Fabrication andCharacterization

The CG used in the monochromator is made by a ho-lographic technique [16]. A uniform thin layer (thick-ness ∼1:3 μm) of photoresist (Olin-ArchmicronHiPR6512) is deposited upon a flat substrate of glassin a clean room (class 1000) with a spin coater. Thenthe substrate is placed in the recording setup schema-tically represented in Fig. 1. The optical source is anargon-ion laser (Spectra-Physics, model 2020-05) ofwavelength λw ¼ 457:9nm. The single-mode (Gaus-sian) laser beam is divided into two Gaussian beamswith a beam splitter. Each beam is then magnified bythe use of a microscope objective and is incident ontothe thin layer of photoresist, with an angle of inci-dence θ with respect to the normal to the plane ofthe substrate. The sign convention for the angles ofincidence is such that the angle associated with a gi-ven beamhas the same sign as the x-component of thewave vector of the beam. Consequently, in Fig. 1, θ1 isa positive angle whereas θ2 is a negative one. There isno angle between the propagation axe of the beamsand the plane xz. The distance along each optical axisbetween themicroscope objective and the center of thesubstrate is zo (Fig. 1). The superposition of beams 1and 2 in the plane of the substrate gives rise to an in-terference pattern that is recorded in the photoresist.The values of the parameters represented in Fig. 1and used to fabricate the CG are listed in Table 1.The photoresist is exposed to the interference fringesduring a time that is adjusted according to the inten-sity of the light incident to the substrate; power at theexit of laser cavity was 196:8mW and exposure timewas then 75:0 s. The development of the photoresist is

Fig. 1. Schematic representation of the experimental setup usedto write the CG.

20 May 2008 / Vol. 47, No. 15 / APPLIED OPTICS 2751

donewith adiluted developer (Shipley 303AMicropo-sit Developer). The last step is the deposition of a120nm-thick layer of aluminum onto the developedsubstrate in a vacuum system at a rate of 0:3nm=s.This process results in a 7 cm wide reflective gratingwhose depth of surface relief is approximately propor-tional to the local intensity of the interference patternused to produce the grating.The local groove spacing of the holographic grating,

identical to the period of the interference pattern, isgiven by

Λðx; yÞ ¼ λw��G1 −G2

�� ; ð1aÞ

with

Gj ¼ sin θj þxcos2θjRjðxÞ

−sin θj2R2

j ðxÞðx2cos2θj þ y2Þ; ð1bÞ

RjðxÞ ¼ zoj þ x sin θj; ð1cÞ

where j ¼ 1; 2 is the beam label. Equations (1a)–(1c)are valid as long as the Rayleigh range of Gaussianbeams 1 and 2, beyond the microscope objectives, ismuch smaller than the distances zo1 and zo2, respec-tively. This requirement is satisfied when the beamsare strongly focused. As Eqs. (1a)–(1c) show, the chirpin the groove spacing is introduced by the difference ofthe radii of curvature R1ðxÞ and R2ðxÞ of the twobeams, that is, mainly by the difference of zo1 andzo2 in the recording setup. Experimentally, we prefer-ably keep the widths of beams 1 and 2 as similar aspossible by choosing objectives with different but ap-propriate numerical apertures. Angles θ1 and −θ2 aremeasured with the help of a precise motorized goni-ometer (Micro-Control, model RT-120PP). It shouldbe mentioned that the value of the central groovespacing (at x ¼ y ¼ 0) of the CG is given byΛð0; 0Þ ¼ λw=j sin θ1 − sin θ2j; the smallest theoreticalvalue of the central groove spacing is λw=2 and istherefore limited by the wavelength λw of the laserwriting beams.The local groovespacingof theCGisexperimentally

deduced from themeasurements of the diffraction an-gleθm for agivenorderm. Forknownwavelength λandangle of incidence θi, the local groove spacingΛðxÞ at aposition x on the grating surface is evaluated by theuse of the well-known diffraction grating equation

sin θm ¼ sin θi þmλΛðxÞ : ð2Þ

The sign convention for the order m used to expressEq. (2) is the following: the angle θ0 (which is equalto θi) for the specular order being positive, positive or-ders are those for which θm > θ0 and negative ordersare those for which θm < θ0. The CG has been opti-cally characterized with a helium–neon laser(λ ¼ 632:8nm) in the −1 order with θi ¼ 20:00°. Ex-perimental resultsandthe theoreticalprediction fromEqs. (1a)–(1c) are shown in Fig. 2. As expected, the lo-cal groove spacing of the CG varies monotonicallyfrom Λmax ∼ 2400nm to Λmin ∼ 1100nm along thex-axis. Based on experimental data, a polynomial in-terpolation of degree 3 of the function ΛðxÞ was done:

ΛðxÞ ¼ Λð0Þð1þΛ1xþΛ2x2 þΛ3x3Þ; ð3Þwhere Λð0Þ ¼ 1468nm, Λ1 ¼ −0:110 cm−1,Λ2 ¼ þ0:0165 cm−2, andΛ3 ¼ −0:00202 cm−3. Theori-gin of the x-axis is assumed to be at the center of thegrating. These coefficients are valid at least in therange −4 cm < x < 4 cm (Fig. 2), which covers the en-tire width of the holographic grating.

3. Imaging Equation in the Dispersion Plane

In this Section, we present the expression of the ima-ging equation in the dispersion plane of a reflectiveflat CG whose groove spacing is given by ΛðxÞ. Thez-axis is coincident with the normal to the gratingand the y-axis is parallel to the grooves (that are sup-posed to be straight). As a result, the xz-plane is thedispersion plane (Fig. 3). A beam of wavelength λ ori-ginating from point A (position of the entrance slit ofthemonochromator) is incident on the reflective grat-ing with an angle of incidence θi > 0. The optical axisof the incident beam meets the surface of the gratingat pointOof coordinates ðxo; yo; 0Þ; without loss of gen-

Table 1. Parameters of Fabrication of the Holographic Grating

Beam 1 Beam 2

Angles of Incidence of Writing Beams θ1 ¼ 9:01° −θ2 ¼ 9:05°Distances Between Objective and Substrate zo1 ¼ 19:1 cm zo2 ¼ 64:0 cmFocal Lengths of the Objectives f 1 ¼ 2:9mm f 2 ¼ 9:0mm

Fig. 2. Local groove spacingΛðxÞ of the holographic CG as a func-tion of position x. The theoretical curve is calculated withEqs. (1a)–(1c) and data of Table 1.


erality, we set yo ¼ 0 at this point. The beam in themth order, diffracted with a diffraction angle θm,forms the image of point A at point B (position ofthe exit slit). The distance AO is ri and the distanceOB is rm, as shown in Fig. 3. The path AOB definesthe optical axis of the beam in the monochromator.Appendix A presents the derivation of the aberra-

tion coefficients Fn. The equation F2 ¼ 0 is of greatimportance and is related to the spectral focusingcondition. It enables us to find the distance rm alongthe optical axis of the paraxial image position of theentrance slit with respect to the grating. FromEq. (A6a), we find the imaging equation in the disper-sion plane of the CG [17]:

cos2θiri

þ cos2θmrm

¼ mλΛðxoÞ

Λ1ðxoÞ; ð4Þ

whereΛðxoÞ andΛ1ðxoÞ are defined by Eqs. (A4a) and(A4b), respectively. The beam in themth order is con-verging only if the product mΛ1ðxoÞ is positive. Forexample, the normalized period gradient Λ1ðxoÞ ofthe CG must be negative to produce a convergingbeam in the −1 order.

4. ILS Width Due to Spectral Aberrations

The ILS of a spectrometer is the spectral response ofthe instrument to amonochromatic input. In paraxialoptics and in a diffraction-free situation, the image ofan infinitesimally thin entrance slit illuminated witha monochromatic light should also be infinitesimallythin, leading to a infinitely narrow ILS (Dirac deltafunction in the spectral domain). However, such anILS is never observed, mainly due to the finitenessof the width of the slits and to the presence of aberra-tions in the monochromator. Even when the slits areconceptually infinitesimally thin, aberrations tend tospatially spread out the paraxial line image, leadingto a finite ILS width. In this Section, we evaluate theILS width due to spectral aberrations. We limit theanalysis to spectral aberrations introduced by theCG only.Ideally, the optical path differenceWðxÞ vanishes in

the image plane. In practice, it is not the case and thederivative ofWðxÞ is related to the image deviation inthe dispersion plane. The image deviationΔxm repre-sents the distance, in the image plane, between theparaxial line image of the infinitesimally thin en-trance slit and the actual line image of the slit gener-

ated by the light coming frompoint P. It is given by [7]

Δxm ¼ rmcos θm

dWdx

¼ rmcos θm

Σ∞

n¼2nFnðx − xoÞn−1; ð5Þ

where Eq. (A5a) has been used. The quantity ðx − xoÞin Eq. (5) obviously satisfies the inequality jx − xoj≤ L=2, where L is the width of the illuminated partof the grating.

The plate factor of the grating, which enables us torelate the image width in the spatial domain (in thedispersion plane) to the ILS width in the spectral do-main, is defined by [2]

Pm ≡1rm

∂λ∂θm

¼ ΛðxoÞ cos θmmrm

; ð6Þ

where Eq. (2), evaluated at x ¼ xo, has been used.The ILS width due to aberrations is thus given by

Δλa ¼ 2jPmΔxmj; ð7Þ

whereΔxm is evaluated at ðx − xoÞmax ¼ L=2. The fac-tor 2 in Eq. (7) is due to the fact that the spectral im-age is affected by spectral aberrations on both sidesof the paraxial image. Substituting Eqs. (5) and (6) inEq. (7), the ILS width due to spectral aberrationstakes the final form:

Δλa ¼�� 2ΛðxoÞ

m

X∞n¼2

nFn

�L2

�n−1

��≈

��ΛðxoÞm

�2F2Lþ 3

2F3L2 þ F4L3

��: ð8Þ

Only the first three terms in Eq. (8) are consideredfrom now on. If the grating is not totally illuminated,ambiguity remains about the length L that has to beused in Eq. (8). To find an estimate of the length ofthe illuminated part of the grating, let us examinethe diffraction pattern produced by the entrance slit.The intensity in the far-field region of a uniformly il-luminated slit of width we with light of wavelength λis given by [3]

IðβÞ ¼ Imaxsinc2�we

λ sin β�; ð9Þ

where sincðxÞ≡ sinðπxÞ=ðπxÞ and β is the angular po-sition. If the ratio λ=we is very small, which is thecase in practice, we can use the valid approximationsin β ≈ β. The full-width at half-maximum (FWHM)of the diffraction pattern of Eq. (9) is approximatelygiven by Δβ ≈ λ=we, which gives the angular width ofthe illuminated portion of the grating in the plane ofdispersion. Taking into account that the optical axisof the incident beammeets the grating with an angleθi with respect to the normal of the grating, the linearversion of this width in the plane of the grating isapproximately

Fig. 3. Geometry of the incident rays and diffracted rays on a CGin the dispersion plane (y ¼ 0).


L≡riΔβcos θi

¼ riλwe cos θi

¼ riΛðxoÞmwe

�sin θmcos θi

− tan θi�;

ð10Þ

where Eq. (2) evaluated at x ¼ xo has been used inorder to eliminate λ. Equation (10) defines the lengthof the illuminated part of the grating that will beused in the following analysis. It should be men-tioned that the use of Eq. (10) implies that the grat-ing is wide enough to be not totally illuminated.

5. Design of the CG Monochromator

Many conventional grating monochromators use aconstant-period grating with a collimating lens anda focusing lens, the latter being used in order to imagethe entrance slit on the exit slit [Fig. 4(a)]; lenses areusually replaced by spherical concave mirrors as intheCzerny–Turner configuration [2]. Thewavelengthis tuned by rotating the grating about an axis parallelto the grooves. On the other hand, the CG exploited inthe present monochromator design is used to diffractand to spectrally focus the radiation incident on it;therefore, in principle no lens or concave mirror isneeded [Fig. 4(b)]. It should be pointed out, however,that the spectral image on the exit slit is not necessa-rily stigmatic. In thisparticular setup, thewavelengthtuning is chosen to be accomplished by a rectilineartranslation of the grating along the t-axis shown inFig. 4(b) without any rotation. The main challengein the design of this monochromator is to determinethe appropriate trajectory of the grating requiredfor the acquisition of the spectrum. It turns out thata rectilinear translation of the grating is possible incertainconditions tokeepanILSasnarrowaspossiblefor every wavelength injected in the instrument. Weprivilege such a rectilinear translation because it sim-plifies the technical aspect of the data acquisitionme-chanism (a standard translation table controlled by acomputer can be employed). In this Section, we pre-sent themethod used to find the rectilinear trajectorythat minimizes spectral aberrations in the ILS.We consider a CG placed in a reference position (we

use hereafter the index “ref” for this position). Thebeam of wavelength λref coming from the entrance slitis incident on the CG with an angle θi at the positionxref on the grating, as shown in Fig. 5. The distancebetween the entrance slit and the position xref onthe grating is ri;ref , which can be directly measuredin the setup. In this reference setup, the entrance slitimage in themth order is at a distance rm;ref from thegrating and at a diffraction angle θm;ref . The angleθm;ref and thedistance rm;ref canbe foundwith thehelpof Eqs. (2) and (4) evaluated at x ¼ xref :

sin θm;ref ¼ sin θi þmλrefΛðxref Þ

; ð11Þ

cos2θiri;ref

þ cos2θm;ref

rm;ref¼ mλref

Λðxref ÞΛ1ðxref Þ; ð12aÞ

with

Λ1ðxref Þ≡�

1ΛðxÞ

dΛðxÞdx

�x¼xref

¼ Λð0ÞðΛ1 þ 2Λ2xref þ 3Λ3x2ref ÞΛðxref Þ

; ð12bÞ

whereΛðxref Þ is evaluated fromEq. (3).We define alsoa translation axis, called t-axis, which makes a con-stant angle Θ with the x-axis [Fig. 4(b)], along whichthe grating is moved. The position xo ¼ xref on thex-axis corresponds to the position t ¼ tref on the t-axis.Fromgeometry, the position xo (seeFig. 5) is related tothe coordinate t by

xoðtÞ ¼ xref − ðt − tref ÞcosðΘ − θiÞ

cos θi; ð13Þ

which gives the displacement along the grating sur-face of the impact point of the optical axis of the inci-dent beam when the grating is displaced along thetranslation axis. One can show with laws of sinesand cosines that the parameters in the monochroma-tor as a function of the position t are (see Fig. 5)

Fig. 4. Optical monochromator (a) using a constant-period grat-ing with collimating and focusing components, (b) using a CGwith-out any other component. The translation axis makes an angle Θwith the CG axis.


ri ¼ ri;ref − ðt − tref ÞsinΘcos θi

; ð14Þ

rm ¼�r2m;ref þ

�ðt − tref Þ sinΘcos θi

�2

−2rm;ref ðt − tref Þ sinΘ

cos θicosðθi þ θm;ref Þ

�1=2

ð15Þ

θm ¼ arcsin�rm;ref sin ðθi þ θm;ref Þ

rm

�− θi: ð16Þ

Since the grating does not experience any rotation,the angle of incidence θi remains constant. The wave-length detected at the exit slit is then given by Eq. (2)and depends implicitly on the coordinate t through θmand ΛðxoÞ:

λ ¼ ΛðxoÞm

ðsin θm − sin θiÞ: ð17Þ

6. Spectral Aberrations Minimization in the CGMonochromator

The strategy retained to reduce the impact of spectralaberrations on the ILS width of the CGmonochroma-tor is to choose the angles θi andΘ tominimizeΔλa foreverywavelength accessible to the instrument. UsingEq. (8) with Eqs. (A4a)–(A4d), (A6a)–(A6c), (A7), and(10), we have the expressionΔλaðxoÞ of the ILS widthdue to spectral aberrations as a function of the posi-tion xo on the grating. With the help of Eq. (13), wehave the expression Δλaðθi;Θ; tÞ of that width as afunction of the coordinate t, as well as the angles θiand Θ, which are both still unknown.In the reference setup, spectral defocusing is zero

(F2 ¼ 0 at t ¼ tref ); this assumption is satisfied ifthe alignment of the monochromator is set properlyduring the initial configuration.Wewant tominimizethe function Δλaðθi;Θ; tÞ for all the possible positionstðtmin ≤ t ≤ tmaxÞ. The method employed to attain thisgoal is to minimize the value of the area under the

curve Δλaðθi;Θ; tÞ as a function of t, that is, to mini-mize the following integral [18]:

gðθi;ΘÞ≡Z

tmax

tmin

Δλaðθi;Θ; tÞdt

¼Z

tmax

tmin

��ΛðxoÞm

ð2F2Lþ 32F3L2 þ F4L3Þ

�� dt: ð18Þ

Since no analytical solution for this integral seems tobe available in the general case, one has to minimizenumerically the function gðθi;ΘÞ. As long as numeri-cal evaluation of the integral is concerned,we can sim-ply replace the integral of Eq. (18) by the sum

~gðθi;ΘÞ ¼Xk

Δλaðθi;Θ; tkÞ; ð19Þ

where tmin ≤ tk ≤ tmax. When the minimal value of~gðθi;ΘÞ is known, the corresponding values for the op-timal angles θi;opt and Θopt can then be found.

In the conception of the CG monochromator, wehave included a collimating lens of focal length fplaced at a distance f after the entrance slit. The pre-ceding analysis remains valid in presence of this col-limating lens, but one has to replace ri by f in theeffectivewidthL ofEq. (10) and to replace ri by infinityin the expressions of Fn [Eqs. (A6a)–(A6c) and (4)].Using the collimating lens in our design, we haveri;ref → ∞ in Eq. (12a); in addition, Eq. (14) is thenmeaningless. Among the reasons for which we haveincluded a collimating lens are the following. In gen-eral, it simplifies the expressions of theFn coefficientsand reduces their values because the first terms of Fnvanishasa consequenceof ri → ∞. Furthermore, it en-sures that theentranceslit image is real (asopposed tovirtual) for any position of the grating with respect tothe slit.

We have experimentally tested the CG monochro-mator with a He–Ne laser source tunable on sevenwavelengths. The value of the parameters of the refer-ence configurationandall othermeaningful constantsof the design are listed in Table 2. The optimal anglesobtainedwith themethoddescribed in thisSectionareθi;opt ¼ 4:8° and Θopt ¼ 89:5°. The initial position ofthe paraxial image of the entrance slit is then theore-tically such as θ−1;ref ¼ −17:3° and r−1;ref ¼18:0 cm; the exit slit has to be placed initially at thisposition. Of course, these optimal and referencevalues are strongly dependant on the CG used, thechosen initial configuration, and the width tmax −

tmin of the interval of t values.To appreciate the validity of the solution, we can

compare the ILS width due to spectral aberrationsΔλa with the spectral image width due to the slits, de-noted by Δλg, for all the wavelengths that the mono-chromator has to explore. The width Δλg, called thegeometrical ILS width, is given by

Fig. 5. Representation of the geometrical parameters in the CGmonochromator for both the reference position and an arbitraryposition.


Δλg ¼

8><>:

½wsΛðxoÞ cos θm�=jmjrm for ws > jw0ej

½weΛðxoÞ cos θi�=jmjf for ws < jw0ej

;

ð20aÞwith

jw0ej ¼

rm cos θif cos θm

we: ð20bÞ

where we and ws are the entrance and the exit slitswidths, respectively, and w0

e is the image width ofthe entrance slit. With the values specified in Table 2,the widthΔλg is about 0:2nm.With the values of θi;optand Θopt found previously, the calculated ratioΔλa=Δλg for all the allowed positions of the gratingin themonochromator is alwaysmuch less than unity,as shown inFig. 6, suggesting that theangles θi;opt andΘopt are theoretically those that make the ILS domi-nated by the geometrical width of the slits and con-firming that the monochromator is then free ofsignificant spectral aberration in this operation con-figuration. In fact, it has been verified by calculationthat arbitrary values for θi andΘ, in general, lead to aΔλa=Δλg ratio much greater than unity for almost allthe explored wavelengths, meaning that spectralaberrations could greatly affect the ILS.Weshould state that diffraction could also affect the

ILS width. However, its effect can be easily neglectedcompared to the effect of the finiteness of slits andaberrations if the grating iswide enough. An estimateof the width of a purely diffraction-limited ILS (with-out the effects of slits and aberrations) is given by

Δλd ¼ λΛðxoÞjmjD ; ð21Þ

whereD is the physical width of the grating (of signif-icant reflectivity).With the typical values in our situa-tion, we obtain for a 7 cm wide grating (a width of theorderΔλd ∼ 0:01nm), which can be neglectedwith re-spect to the geometrical ILS width Δλg.

7. Experimental Results

The theoretical analysis presented in Section 6 hasbeen applied to an experimental setup of a CGmono-

chromator using a He–Ne laser tunable on sevenwavelengths (Research Electro-Optics, modelLSTP-1010). A laser source is used to measure a nar-row-band spectrum comparable to the ILS of themonochromator at the corresponding wavelength.During the spectral acquisition of data, we used thereference wavelength (λref ¼ 632:8nm) and anotherwavelength (λref 0 ¼ 543:4nm) as calibration points.Spectral results for the entire spectrum of the lasersource is presented in Fig. 7 and Table 3. Over the se-ven spectral lines detected, only five of them are spe-cified by themanufacturer (543.4, 593.9, 604.6, 611.8,and 632:8nm) and are studied below. These five indi-vidual spectral lines are presented inFig. 8.Each line,considered here as the ILS of the monochromator forthe corresponding wavelength, are practically free ofspectral defocusing. The residual aberrations presentin these ILS seem dominated by coma because of theasymmetry of the line. Table 4 lists the measuredFWHM Δλ of the lines of Fig. 8 and their theoreticalwidthΔλg [see Eqs. (20a) and (20b)]. As expected, ex-perimental values are always slightly greater thanthe theoretical ones, due to residual aberrations.Nevertheless, the proximity of values in Table 4 con-firms that spectral aberrationshavevery small effectson the width of the lines in Fig. 8.

We mentioned qualitatively in Section 6 that theILS width is sensitive to the value of the angles θiand Θ. We verified that the angle of incidence θi;optis the appropriate one by measuring the experimen-tal ILS width Δλ as a function of the deviation of theangle θi with respect to the optimal angle θi; optFigure 9(a) shows that the ILS spreads out as theabsolute difference jθi − θi; optj increases, except forthe reference wavelength λref ¼ 632:8nm for whichits ILS width is minimized by default. It is thefarthest wavelength of the measured spectrum(λo ¼ 543:4nm) that suffers the most from an error

Table 2. Configuration of the CG Monochromator

Parameter Value

Reference Wavelength λref ¼ 632:8mmDiffraction Order m ¼ −1Reference Position on the Grating xref ¼ −1:0 cmReference Position on the t-axis tref ¼ 5:0 cmMinimum Value on the t-axis tmin ¼ 0:0 cmMaximum Value on the t-axis tmax ¼ 6:0 cmWidth of the Entrance and Exit Slits we ¼ ws ¼ 25 μmFocal Length of the Collimating Lens f ¼ 25:0 cmGroove Spacing and Variation Parameters see Eq. (3)

Fig. 6. Ratio of the ILS width due to aberrations to the geome-trical ILS width as a function of the wavelength in the optimal con-figuration of the CG monochromator.

Fig. 7. Spectrum of the He–Ne tunable laser measured with theCG monochromator.


of the angle of incidence θi. In a similar way, wechecked the validity of the translation angle Θopt bymeasuring Δλ as a function of Θ −Θopt [Fig. 9(b)].Analogous conclusions can be stated about thedifference jΘ −Θoptj.8. Discussion

The method proposed here can be extended to mini-mize not only the spectral aberrations but also thespatial ones. The analysis presented in Sections 3–6, and in Appendix A, remains essentially the same,provided that the aberration function Wðx; yÞ is gen-eralized so as to include the coordinate x aswell as thecoordinate y. Moreover, the coordinate z has to be ta-ken in account in the optical pathAPB shown in Fig. 3if the surface of the grating is not flat. Specifically, it ispossible to employ a concave CG to correct for spatialaberrations. For a spherical concave gratingwith a ra-dius of curvature R, the coordinate z on the surfaceis related to x and y by the relation z ¼ R−ðR2 − x2 − y2Þ1=2. The procedure to follow to find theaberration coefficients Fij of a concave spherical CGis available in Ref. [17].We mentioned earlier in Section 2 that the smal-

lest theoretical value of the central groove spacingis limited by the writing wavelength λw of the laserbeams. In the present work, we have produced a ho-lographic CG with a central groove spacing equals toΛð0Þ ¼ 1468 nm or, equivalently, 681 lines per mm.In some applications (in extreme-ultraviolet spectro-meters, for instance), highly dense CG with a fewthousand lines per mm are required. If a holographic

technique is expected to produce such a CG, a smallerwriting wavelength λw has to be used with the appro-priate photoresist. If the suitable writing wavelengthor photoresist is not available, the CG can be me-chanically ruled instead. Nevertheless, no matterhow the CG grating is made, the method proposedto systematically minimize the spectral aberrationsis not fundamentally restricted to the order of mag-nitude of the groove spacing of our CG.

9. Conclusion

We have fabricated a holographic chirped grating(CG), which is aimed to be used in the design of a grat-ing monochromator. In this CG monochromator, thewavelength tuning is achieved through a rectilineartranslation of the grating. The angle of incidence ofthe beam on the grating and the angle between theplane of the grating and the translation axis are de-termined in such a way that spectral aberrations inthe entrance slit image are minimized. Experimentalresults obtained with the prototype of the CG mono-chromator yield to the conclusion that themethod em-ployed to choose the appropriate trajectory of thegrating in the instrument is successful.

Appendix A

In thisAppendix,wederive the expression of the aber-ration coefficients for a reflective CG. Referring to thegeometry of Fig. 3, we examine a point P, lying some-whereon thegratingsurfaceat the coordinate ðx; 0; 0Þ.We consider a rayof light coming fromA, incident onP,and diffracted ideally to point B. Assuming that thegrating is in air, the aberration functionW, which cor-responds to the optical path difference between thisray and a ray defined by the optical axis, is given by

W ¼ APB − AOBþNmλ; ðA1Þ

Table 3. Wavelengths of the He–Ne Tunable Laser

Wavelength λo Difference

Theoretical [20,21](nm) Measured (nm) (nm) %

543.37 543.37 (calibration point) —

593.93 593.94 0.01 0.002604.61 604.63 0.02 0.003611.80 611.77 0.03 0.005629.37 629.34 0.03 0.005632.82 632.82 (calibration point) —

640.11 640.10 0.01 0.002

Fig. 8. Normalized spectral lines of the He–Ne measured withthe CG monochromator.

Table 4. Measured and Theoretical FWHM of the FiveWavelengths under Study

λoðnmÞ ΔλðnmÞ ΔλgðnmÞ�0:05nm �0:001nm �0:04nm543.37 0.198 0.18593.93 0.220 0.20604.63 0.218 0.21611.77 0.218 0.21632.82 0.228 0.22


whereAPBandAOBare lengthsof segments shown inFig.3andN is thenumberof groovesbetweenpointsOand P. Since points A, O, P, and B lie in the dispersionplaneandgroovesareparallel to they-axis, the opticalpath difference functionW is a function of x only. Pathlengths in Eq. (A1) are geometrically given by

AOB ¼ ri þ rm; ðA2aÞ

APB ¼ ½ðri sin θi þ xÞ2 þ r2i cos2θi�1=2

þ ½ðrm sin θm − xÞ2 þ r2mcos2θm�1=2: ðA2bÞ

Theexpression of thenumberNðxÞof groovesbetweenpoints O and P is obtained by integrating the function1=ΛðxÞbetweenxo and x; the function1=ΛðxÞ is before-hand expanded in a Taylor series with respect to xabout the point of impact O of the optical axis onthe grating (x ¼ xo):

NðxÞ≡Zx

xo

1Λðx0Þdx

0 ¼ ðx − xoÞΛðxoÞ

þX∞n¼1

bnΛðxoÞ

ðx − xoÞnþ1

ðnþ 1Þ ;

ðA3aÞ

with

bn ¼ ð−1Þn

×

��

Λ1ðxoÞ 1 0 ::: 0Λ2ðxoÞ Λ1ðxoÞ 1 ::: 0: : : : :Λn−1ðxoÞ Λn−2ðxoÞ Λn−3ðxoÞ ::: 1ΛnðxoÞ Λn−1ðxoÞ Λn−2ðxoÞ ::: Λ1ðxoÞ

��n×n

;

ðA3bÞ

ΛnðxoÞ≡1

n!ΛðxoÞdnΛðxÞdxn

��x¼xo

; ðA3cÞ

whereΛðxoÞ is the groove spacing at x ¼ xo. Note thatΛðxoÞ, Λ1ðxoÞ, Λ2ðxoÞ, Λ3ðxoÞ are generally differentfrom Λð0Þ, Λ1, Λ2, and Λ3 of Eq. (3), except forxo ≡ 0. Expanding the n × n determinant of Eq. (A3b),we can explicitly calculate the first coefficients bn:b1 ¼ −Λ1ðxoÞ, b2 ¼ Λ2

1ðxoÞ −Λ2ðxoÞ, b3 ¼ −Λ31ðxoÞ þ 2

Λ1ðxoÞΛ2ðxoÞ −Λ3ðxoÞ, etc. Since the form of ΛðxÞ isknown [see Eq. (3)], we can write the coefficientsΛnðxoÞ in terms of Λð0Þ, Λ1, Λ2, Λ3, and xo:

ΛðxoÞ ¼ Λð0Þð1þΛ1xo þΛ2x2o þΛ3x3oÞ; ðA4aÞ

Λ1ðxoÞ ¼�

1ΛðxÞ

dΛðxÞdx

�x¼xo

¼ Λð0ÞðΛ1 þ 2Λ2xo þ 3Λ3x2oÞΛðxoÞ

; ðA4bÞ

Λ2ðxoÞ ¼12

�1

ΛðxÞd2ΛðxÞdx2

�x¼xo

¼ Λð0ÞðΛ2 þ 3Λ3xoÞΛðxoÞ

;

ðA4cÞ

Λ3ðxoÞ ¼13!

�1

ΛðxÞd3ΛðxÞdx3

�x¼xo

¼ Λð0ÞΛ3

ΛðxoÞ: ðA4dÞ

The function WðxÞ is then explicitly obtained by sub-stituting Eqs. (A2a), (A2b), and (A3a)–(A3c) inEq. (A1). The range of the coordinate x is usuallymuchsmaller than the values of distances ri and rm; it istherefore possible to expand the optical path differ-ence function WðxÞ as a power series of ðx − xoÞ[19]. This gives

WðxÞ ¼X∞n¼0

Fnðx − xoÞn; ðA5aÞ

with

Fn ¼ 1n!

dnWdxn

��x¼xo

: ðA5bÞ

Fig. 9. Experimental ILS width as a function of the difference be-tween (a) the angle of incidence and its optimal value and (b) thetranslation angle and its optimal value.


The coefficients Fn are called the aberration coeffi-cients of the CG. Specifically, F2 relates to defocus,F3 to coma, F4 to spherical aberration and other Fn

tohigher-orderaberrations.Werestrict thediscussionto powers in Eq. (A5a) not greater than four. FromEqs. (A5b) and (A1), we find F0 ≡ 0 and F1 ¼sin θi − sin θm þmλ=ΛðxoÞ. Using Eq. (2) at x ¼ xo,we find F1 ≡ 0 as well. The three remaining coeffi-cients considered in this analysis are explicitly

F2 ¼ 12

�cos2θiri

þ cos2θmrm

�−

mλ2ΛðxoÞ

Λ1ðxoÞ; ðA6aÞ

F3 ¼ 12

�−sin θicos2θi

r2iþ sin θmcos2θm

r2m

�

þ mλ3ΛðxoÞ

ðΛ21ðxoÞ −Λ2ðxoÞÞ; ðA6bÞ

F4 ¼ 18

�cos2θið4 − 5cos2θiÞ

r3iþ cos2θmð4 − 5cos2θmÞ

r3m

�

þ mλ4ΛðxoÞ

ð−Λ31ðxoÞ þ 2Λ1ðxoÞΛ2ðxoÞ −Λ3ðxoÞÞ

ðA6cÞ

We may eliminate the wavelength λ in Eqs. (A6a)–(A6c) with the help of the equation F1 ≡ 0:

mλΛðxoÞ

¼ sin θm − sin θi: ðA7Þ

SubstitutingEq. (A7) intoEqs. (A6a)–(A6c), we obtainthe first aberration coefficients (F2,F3, andF4) only interms of the geometrical parameters of the setup(θi,θm, ri, and rm), the parameters of the CG groovespacing (Λð0Þ,Λ1,Λ2, andΛ3 of Eq. (3)), and the posi-tion xo of the optical axis of the incident beam on thegrating.

This work was supported by grants from NaturalSciences and Engineering Research Council ofCanada (NSERC), the Fonds de recherche sur lanature et les technologies (FQRNT), Québec, theCanadian Institute for Photonic Innovations (ICIP/CIPI), and the Centre d’optique, photonique et lasers(COPL), Québec.

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Control of spectral aberrations in a monochromator using a plane holographic chirped grating