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Engineering

Diffraction Gratings: Part 2

BY MASUD MANSURIPUR, LIFENG LI, AND WEI-HUNG YEH

Last month we began in this col­umn a discussion of diffraction

gratings. We continue this month by covering reciprocity, the resolving power, Littrow mount, blazed grat­ings, and dielectric-coated transmis­sion gratings. Please note that some of the figures and references discussed in July are cited here. You will need to refer back to the July issue, pages 42-46, as the figure and reference numbering scheme continues here.

Reciprocity theorem There exists a powerful and quite unexpected reciprocity relation be­tween the beam incident on a grat­ing and any one of the resulting dif­fracted orders. Suppose the incident beam arrives at the grating at an an­gle θ, and the mth diffracted order emerges at an angle θ ( m ) , having dif­fraction efficiency p(m) or, in the case of a transmitted order, τ ( m ) . If the direction of incidence is now changed so that the incident beam is a long the path o f the mth order beam (in the reverse direction, of course), there emerges a - m t h dif­fracted order along the path of the original incident beam (again in the reverse direction). The reciprocity theorem states that the efficiency of this particular diffracted order will be exactly equal to p ( m ) (or τ ( m ) ) . T h i s t h e o r e m can be r i g o r o u s l y proved under general condit ions. 2

In Figure 6 (see Part 1, page 45) the ± 1st order efficiency curves in the c lassica l m o u n t , i.e., p s

( ± 1 ) a n d pp

( ± 1 ) , show several manifestations of the reciprocity theorem. A few more consequences of reciprocity

will be pointed out in the examples that follow.

Resolving power Consider a grating of period p hav­ing a total of N grooves. The width of the mth order diffracted beam that covers the entire grat ing is Np cos θ ( m ) . If this beam is brought to diffraction-limited focus by a lens of focal length f, the focused spot di­ameter D will be 1

Spectroscopists are interested in the focused spots formed by two nearby wavelengths, λ 0 and λ 0 + Δ λ . A c ­cording to E q . (1), the diffraction angle θ(m) in the classical mount is given by s i n θ ( m ) = sinθ + m λ 0 / p , in which case for a small change of wavelength Δ λ we have

Therefore, in the focal plane of the lens, a shift of the wavelength from λ 0 to λ 0 + Δ λ causes a shift of the fo­cused spot by the following amount:

Figure 10. A normally incident b e a m of light is specular ly ref lected from the incl ined facet of a metal l ic pr ism (inclina­t ion angle = α ) . For a given integer m, imagine cut t ing the prism along the d a s h e d l ines, wh ich are parallel to the direc­t ion of inc idence and have lengths that are mul t ip les of m λ 0 / 2 . The var ious s e c t i o n s are then rearranged to form the eche le t te grat ing shown in the lower part of the f igure. If the grat ing is similarly i l luminated at θ = α , the di f f racted order that re t races the inc idence path in the reverse di rect ion will be quite s t rong. Th is is the reason why this kind of grat ing has c o m e to be known as a blazed grat ing.

Figure 11. C o m p u t e d plots of diffraction eff iciency versus sin θ , where θ is the angle of inc idence on the eche le t te grat ing of Figure 10 [λ0 = 0.633 μm, α = 30°, p = 2λ0, (n, k) = (2, 7)]. The inc idence is from the side of the large facet of the triangular g rooves when θ > 0, while θ < 0 cor responds to inc idence from the small - facet s ide. The displayed e f f ic iences are for p- and s- polarized incident light in the c lass ica l mount. (a) Oth order, (b) +1st order, (c) -1st order, (d) ±2nd order, (e) ±3rd order. The arrows at the top of e a c h frame indicate the locat ions of Rayle igh anomal ies .

44 Optics & Photonics News/August 1999

The two wavelengths are resolved when the above shift equals the spot diameter D in Eq. (4), that is, when f Δ θ ( m ) = D, which leads to the follow­ing expression for the resolving power

It is thus seen that the resolving power of a grating is directly pro­portional to N, its total number of grooves, and to m, the order of dif­fraction. The resolving power is completely independent of such seemingly relevant factors as the groove period, the groove geometry, and the incidence angle.

Littrow mount and blazed gratings To build compact spectrometers, it is desirable to have one of the dif­fracted orders return along (or al­most along) the direction of inci­dence. In the so-called Littrow mount, the nth order beam, where n is negative, returns along the direc­tion of incidence. For instance, in the -1st order Littrow mount, we find from Eq. (1)

Under this condition, if p < 1.5λ0,

the only possible diffracted orders are the 0th and the -1st. Further­more, if the efficiency for the 0th or­der can be reduced to zero, all the available power that is not absorbed by the grating will return along the -1st reflected order, thus maximiz­ing the sensitivity of the spectrome­ter. Gratings that direct all or most of the incident optical power into a single diffracted order are known as blazed gratings. Although in the ear­ly days ruled gratings having a trian­gular groove profile satisfied the blaze condition, a triangular cross-section is no longer a prerequisite to the blazing property. Gratings with triangular cross-section and a 90° apex angle are now more appropri­ately referred to as "echelette" grat­ings.

Figure 10 shows a metallic prism with an inclination angle α. When a plane wave is normally incident on the inclined facet of this prism, the specularly reflected light returns along the direction of incidence. Let the lengths of the equidistant lines drawn on the prism parallel to the direction of incidence be integer-multiples of mλ 0 /2 , where m is an arbitrary (but fixed) integer. If the metal prism is cut along these lines and its segments rearranged, one obtains an echelette grating with pe­

riod p = mλ 0 /(2 sin α) , as shown in the lower part of the figure. With an incidence angle θ = α on this grat­ing, Littrow's condition for the -mth diffracted order will be satisfied. In the geometric-optical approxima­tion, this grating should be equiva­lent to the original prism, because the various reflected rays from its individual facets suffer phase delays in multiples of 2π only, making the grating's reflected wavefront indis­tinguishable from that of the prism. In reality, however, the electromag­netic field "feels" the groove struc­ture, and the actual diffraction effi­ciency of the beam returning along the direction of incidence will not always be the same as the specular reflectance of the polished metal prism, although they are usually close.

Figure 11 shows computed effi­ciency curves in the classical mount for the echelette grating of Figure 10 having α = 30°, p = 2λ 0 , and (n, k) = (2, 7) at λ 0 = 0.633 μm. 1 2 The hori­zontal axis depicts sin θ, with the in­cidence angle θ being positive (nega­tive) when incidence is from the side of the large (small) facet of the trian­gular grooves. The arrows at the top of each frame indicate the locations of Rayleigh anomalies, in the neigh­borhood of which resonance features

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and slope discontinuities are seen to occur . T h e 0th order e f f ic iency curves for p- and s-polarized light are shown in Figure 11(a). Despite the asymmetrical groove geometry, the plots of p p

( 0 ) and ps

(0) are per­fectly s y m m e t r i c a r o u n d θ = 0, which is a manifestation of the reci­procity theorem mentioned earlier. The + 1st order efficiency curves in Figure 11(b) show the same kind of symmetry around θ = - 1 4 . 4 8 ° (i.e., sin θ = -0.25), which is the angle of incidence for the + 1st order Littrow m o u n t . Similar ly, the -1s t order curves in Figure 11(c) show the reci­procity theorem at work around θ = 14.48°, the angle of incidence for the - 1 s t order L i t t row m o u n t . T h e Rayleigh anomalies at θ = ± 3 0 ° (i.e., sin θ = ± 0 . 5 ) mark the disappear­ance of the + ±1st order beams beyond these angles, as may be seen clearly in Figures 11(b) and 11(c).

The ± 2 n d order efficiency curves are shown in Figure 11(d). These curves peak at, and are symmetrical around, θ = ±30° , where the Littrow condition for the ± 2 n d order beams is satisfied. Reciprocity between the incident beam and the ± 2 n d order reflected beams is evident in the symmetr ica l values of eff ic iency around θ = ±30° . Note in the case of p - p o l a r i z e d b e a m i n c i d e n t at θ = 30°, where the - 2 n d order effi­ciency reaches 80% while that of all other orders essentially vanishes,

that the remaining 20% of the inci­dent power must have been ab­sorbed by the grat ing. A simi lar consideration applies to both p p

( + 2 )

and p s

( + 2 ) at θ = - 3 0 ° . The ± 3 r d or­der beams exist only at large an­gles of incidence, as may be in­ferred from Figure 11(e). Again note the s y m m e t r y o f these curves (due to rec ip roc i ty ) a r o u n d sin θ = ± 0 . 7 5 , wh ich c o r r e s p o n d to the L i t t r o w mount in the ± 3 r d order.

For the sake of completeness we present in Figure 12 comput­ed efficiency curves in the case of con ica l m o u n t for the same echelette grating as discussed a b o v e . 1 2 Here the grooves are parallel to the plane of incidence, and symmetry with respect to θ = 0 obviates the need for dis­playing the results for negative values of θ. In this conical mount only the 0th and ±1st diffracted orders are allowed; even then, the ± 1st order beams disappear be­yond θ = 60°. Note that, because o f the asymmet r ica l groove shape, the +1st order efficiency curves are quite different from those of the -1st order. Also note that, beyond θ = 60°, where the 0th order beam is the only beam reflected from the grating, the relatively small values of p'p

(0) and p's

(0) indicate substantial absorp­tion within the grating medium.

Transmission grating Consider a grooved glass plate such as that depicted in Figure 13(a). When a plane wave incident at θ ar­rives on this grating, the directions

Figure 12. Computed plots of diffraction efficiency versus the angle of incidence on the echelette grating of Figure 10 [λ 0 = 0.633 μm. α = 30°, p = 2λ 0, (n, k) = (2, 7)]. The displayed efficiencies are for p- and s-polarized incident light in the conical mount. (a) Oth order, (b) +1st order, (c) -1st order. The arrows at the bottom of each frame indicate the locations of Rayleigh anomalies.

Figure 13. A simple transmission grating may be obtained by ruling or etching a glass substrate, or by a holographic method. The substrate's refractive index being greater than unity, the diffraction angles inside the substrate are smaller than those observed upon reflection from the same grating into the air. (a) When the substrate bottom is flat, Snell's law of refraction reorients the beams as they emerge into the air, making the diffraction angles equal to those observed in reflection. However, one or more diffracted orders may be missing due to total inter­nal reflection at the substrate bottom. (b) If the grating i made on the flat surface of a glass hemisphere, the transmitted orders emerge into the air undisturbed.

46 Optics & Photonics News/August 1999

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of reflected orders may be found from Eqs. (1) and (2), but the trans­mitted orders inside the glass plate obey different equations. In the clas­sical mount the transmitted orders emerge at angles θ ( m ) , where

Here n0 is the refractive index of the substrate. The number of diffracted orders in the substrate could, there­fore, be greater than the number re­flected into the air. However, when the transmitted orders attempt to exit the bot tom of the substrate, those at higher than the critical an­gle for total internal reflection will be fully reflected. The beams that exit the substrate emerge at a greater angle than θ ( m ) , in accordance with Snell's law, thus effectively canceling the coefficient n0 on the left-hand-side of Eq . (9). Consequently, the beams emerging from the bottom of the substrate have exactly the same number and (aside from being mir­ror images) the same directions as those reflected from the top of the grating. The transmitted diffracted orders may be observed in their na­tive form by using a hemispherical substrate, as shown in Figure 13(b).

In the case of conical mount sim­ilar arguments apply, so that the mth order beam inside the substrate wil l have a propagation direction given by the unit vector σ ( m ), where

Again, σz is determined from the relation σx

2 + σy

2 + σz

2 = 1. As be­fore, when this beam exits into the air from the bottom of a flat sub­strate, the Snell's law multiplies σχ

and σy by the refractive index n0, en­sur ing that the emergent beams (aside from being mirror images) have the same propagation direc­tions as the corresponding beams reflected from the top of the grating.

Figure 14 shows the location of the transmitted diffracted orders from a

glass g r a t i n g . 1 2 T h e as­sumed grating in this case is similar to that of Figure 1, except that the metal layer is absent. T h e observat ion system is also similar to that in Figure 3, except for the position of the collimating lens, which is moved to the opposite side of the grating to collect the transmitted orders. The incident beam, arriving at θ = 30° in the conical mount, is p-polar­ized. The pictures on the left-hand-side of Figure 14 represent the component of po la r i za t ion in the X Z -plane (E ‌ ) , while those on the right correspond to po­larization along the Y-axis (E ± ) . The top row shows the intensity distribution at the exit pupil of the collimating lens when the substrate bot­tom is flat; the bottom row corresponds to the case of a hemispherical substrate. As expected, in the latter case there are more diffracted orders, the orders are more closely spaced, and the indi­vidual beam diameters are smaller. For the flat sub­strate the peak-intensity-ratio E ±

2 : E 2 = 0.21 X 1 0 - 4 , while for the hemi ­spherical substrate E 2 : E 2 = 0.89 X 10 - 4 .

Dielectric-coated grating Figure 15 is a diagram of a dielectric-coated transmis­s ion grat ing on a h e m i ­spherical glass substrate. In the example that follows it is assumed that λ 0 = 0.633 μ m , the grating peri­od p = λ 0 , groove depth d = λ 0 / 8 , side-wall inclination angle α = 60°, and duty cy­cle c = 60%. The coatings are conformal to the grating surface, both dielec­tric layers are 100-nm-thick, and their refractive indices are 2.1 and 1.5, as indicated. Because there are no metallic layers in this case, there will be no surface plasmon excitations, but there is the possibility of guided mode cou­pling to the dielectric waveguide formed by the coating layers. The hemi­spherical substrate allows all transmitted orders to exit and be measured in

Figure 14. Computed plots of intensity distribution at the exit pupil of the collimating lens of Figure 3, when the system is rearranged to allow observation of transmitted orders from the grating of Figure 1, from which the metal layer has been removed ( λ 0 = 0.633 μm, p = 4 λ 0 , d = λ 0 / 8 ) . In this case of conical mount at θ = 30° incidence the grooves are parallel to the plane of incidence, as in Figure 2(b), and the incident beam is p-polarized. The pictures on the left correspond to the com­ponent of polarization in the XZ-plane, while those on the right represent the polarization component along the Y-axis. In (a) and (b), the substrate bottom is flat, as in Figure 13(a), where­as in (c) and (d) it is hemispherical, as in Figure 13(b). The ratio of the peak intensity in (b) to that in (a) is 0.21 x 10-4. Similarly, the peak-intensity-ratio of (d) to (c) is 0.89 x 10-4. These results are based on full vector diffraction calculations.

Figure 15. Cross-section of a dielectric-coated diffraction grat­ing. The side-wall angle α = 60° , and the duty cycle c, which is the ratio of the land width to the grating period, is 60%. Both coating layers are 100-nm-thick and (at λ0 = 0.633 μm) their refractive indices are n1 = 2.1 and n 2 = 1.5. For the substrate, which is also transparent, n 0 = 1.5.

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air. The bottom of the hemisphere is anti-reflection coated to avoid losses upon exiting the substrate.

Figure 16 shows computed plots of diffraction efficiency versus θ for the grating of Figure 15.12 The case of conical mount does not show in­teresting phenomena, as evidenced by the featureless plots of ρ and τ for the various orders. This is not surprising, considering that no guid­ed modes can be launched in the di­electric layers in this case. However, in the classical mount, pp, ps, τp , and τs show peaks and valleys that are in­dicative of resonant behavior. Figure 16(b) shows plots of pp and ps for the -1st order reflected beam, which

carries as much as 8% of the inci­dent beam into this particular direc­tion at several angles of incidence. Reciprocity between the incident beam and the -1st order reflected beam is evident in Figure 16(b) in the symmetrical values of efficiency before and after θ = 30°. Note that, unlike surface plasmon excitations in metals, which occur in p-polariza­tion only, the waveguide modes of dielectric layers can be excited by both p- and s-polarized light. In the classical mount, Figure 16(d) shows that the +1st order transmitted beam is cut off beyond θ = 30°. In its place the -2nd order transmitted beam shown in Figure 16(f) appears

and shows fairly high efficiency for p-polarized light in a narrow range of angles around θ = 33°.

It is impossible to describe in a brief survey the entire range of physical phenomena that occur in diffraction gratings and their poten­tial applications. We hope, however, to have brought to the reader's at­tention the richness and complexity of the physics of gratings, and to have encouraged further exploration of this fascinating subject.

OPN Contributing Editor Masud Mansuripur (masud@u.arizona.edu) is a professor, Lifeng Li a re­search professor, and Wei-Hung Yeh a graduate stu­dent at the Optical Sciences Center of the University of Arizona in Tucson.

Figure 16. C o m p u t e d dif fract ion e f f ic ienc ies v e r s u s θ for the d ie lec t r i c -coa ted grat ing of Figure 15 at λ0 = 0.633 μ m (p = λ0, d = λ0/8). For the c l a s s i c a l mount the e f f ic ienc ies are d e n o t e d by ρp, ρs, and for the c o n i c a l mount by ρp, ρs. (a) re f lected Oth order, (b) ref lected -1st order, (c) t ransmi t ted Oth order, (d) t ransmi t ted +1st order, (e) t ransmi t ted -1st order, (f) t ransmi t ted -2nd order. The arrows at the top or the bot tom of each f rame indicate the loca t ions of Rayle igh a n o m a l i e s in the classical mount .

4 8 Optics & Photonics News/August 1999