Near field diffraction of cylindrical convexgratings

Francisco Jose Torcal-Milla, Luis Miguel Sanchez-Brea andEusebio Bernabeu

Applied Optics Complutense Group, Optics Department, Universidad Complutense de Madrid, Facultad deCiencias Físicas, Ciudad Universitaria s.n., 28040, Madrid, Spain

E-mail: fjtorcal@ucm.es

Received 7 October 2014, revised 12 December 2014Accepted for publication 31 December 2014Published DD MM 2014

AbstractWe analyze the field produced by a cylindrical convex diffraction grating at the Fresnel regimefor several kinds of light sources, including a monochromatic quasipunctual source, finite size,and polychromatic sources. These results can help one understand the functioning of rotaryoptical encoder technology. A decrease in the self-image contrast is produced for finitenonpunctual sources. In addition, the polychromaticity of the source affects the smoothness ofthe self-images, making them quasicontinuous from a certain distance from the grating forward.Finally, we experimentally validate the obtained analytical predictions.

Keywords: diffraction gratings, self-imaging, polychromatic lightPACS numbers: 42.25.Fx, 42.79.Dj

(SQ1 Some figures may appear in colour only in the online journal)

1. Introduction

Diffraction gratings are optical elements that periodicallymodulate one or more properties of light passing throughthem or reflecting from them [1, 2]. The most common dif-fraction gratings are those that modulate the amplitude or thephase of the incident field [2]. The applicability of diffractiongratings is quite extensive. One can find them as fundamentalparts of many different kinds of devices, such as telescopes,spectrometers, and optical encoders [1, 3–5]. For example, theself-imaging phenomenon produced at the near field [3, 6] isused in diffraction-based optical encoders to measure therelative position and/or displacement between two differentparts. Due to their robustness and precision, optical encodersare used in machine tools, robotics, motion control, and more.They are formed by an illumination source, a diffractiongrating—commonly called scale—a mask, and some photo-detectors. The relative displacement between the scale and thereading head produces a variation in the photodetector’ssignals and gives the measurement of the linear (linearencoders) or angular (rotary encoders) movement. In parti-cular, rotary encoders use a diffraction grating engraved on acylindrical substrate. Usually, gratings for rotary opticalencoders are engraved by lithographic methods on the plane

facet of a cylinder [7, 8]. The behavior of this kind of gratinghas been already analyzed in previous works [9, 10]. On theother hand, rotary encoders with the grating engraved by laserablation [11, 12] on the external curved side of a steelcylinder have recently been developed [13–15]. It has beendemonstrated that gratings engraved by nanosecond laserablation on steel substrates act like amplitude gratings due totheir surficial roughness [16, 17].

For plane diffraction gratings, a deep analysis of the near-field and far-field diffraction has been performed in recentdecades, including the near-field pattern when the grating isilluminated with polychromatic and finite extension sources[18, 19]. The diffraction pattern and image formation pro-duced by concave curved gratings—such as torus, spherical,etc—has also been analyzed in several works [20–23]. Sincecurved gratings have generated interest in recent years, a morecomplete analysis needs to be performed to optimize thebehavior of optical encoders and other near-field applications.For example, the near-field diffraction of convex gratingsengraved on the external side of a cylinder and illuminated bya monochromatic plane wave has recently been analyzed [24].

In the present work, we extend the research to curved,convex diffraction gratings illuminated by a punctual or finitesource that also may be monochromatic or polychromatic. We

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present analytical equations for the self-imaging process forthe considered cases, and we experimentally verify the mostgeneral case. The obtained results are directly applicable torotary optical encoders.

2. Theoretical approach

To analyze the general case of a polychromatic and finiteextension source, we first consider the simple case of amonochromatic and punctual source placed at a distance, z ,0

from a cylindrical convex diffraction grating and at a distance,x ,0 from the z-axis. The scheme of the setup is shown infigure 1. Light propagates from right to left until it impingesthe diffraction grating and is diffracted back. In this figure, Pis a plane perpendicular to the optical axis (z-axis) at theorigin of coordinates, δ θ( ) is the distance between each pointof the grating and the plane P, R is the curvature radius of thegrating, ξ z( , ) are the coordinates centered at the point wherethe grating crosses the z-axis, a is the size of the source, and Lis the area of the grating illuminated by the source over the ξ-axis. The radius of curvature is defined to be positive whenthe center of the curvature is placed at the left side of thegrating. In addition, black and white regions of the gratingrepresent nonreflective and reflective areas, respectively.

In a parabolic approximation, the field produced by apunctual source at a distance, z ,0 is given by

ξ=

−⎡⎣⎢⎢

⎤⎦⎥⎥

( ) ( )U A

ikz

z

ik x

z

expexp

2, (1)0 0

0

0

02

0

where A0 is the amplitude, x0 is the position of the sourcealong the x-axis, π λ=k 2 , and λ is the illumination wave-length. To calculate the diffraction pattern produced by thegrating, we consider the three-dimensional topography bycomputing the optical path. Each point of the wavefront alongthe ξ-axis covers a different distance from the plane, P, to thesurface of the grating. The optical path for each point of thewavefront at ξ θ θ= R( ) sin is δ θ θ= −R( ) (1 cos ). Then,each point Q2is affected by the propagation coefficient,

θ θ= −t ikR( ) exp[ (1 cos )], which can be expressed inCartesian coordinates as

ξ ξ= − − ⎜ ⎟⎧⎨⎪⎩⎪

⎡⎣⎢⎢

⎛⎝

⎞⎠

⎤⎦⎥⎥

⎫⎬⎪⎭⎪

t ikRR

( ) exp 1 1 , (2)2

where ξt ( ) represents the propagation coefficient due to theoptical path of each point of the wavefront from the plane, P,to the surface of the grating. The diffraction grating isangularly periodic, not linearly periodic. Then, its reflectioncoefficient can be expressed as a Fourier series expansiondepending on the angle, θ,

∑θ θ= θ( )r a i q n( ) exp , (3)n

n

where an are the Fourier coefficients with n integer,π=θ θq p2 , and θp is the angular period whose relationship

with the linear period, p, along the perimeter is =θp p R.Changing to Cartesian coordinates, equation (3) results in

∑ξ ξ= θ ⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥r a iq n

R( ) exp arcsin . (4)

n

n

This corresponds to the Fourier series expansion of thereflectance produced by the grating expressed in Cartesiancoordinates. From equations (1), (2), and (4), the field at theplane, P, after being diffracted by the grating is

∑

ξ ξ ξ ξ ξ

ξ

ξ

ξ

= = =

∝−

× − −

× θ

⎜ ⎟

⎜ ⎟

⎡⎣⎢⎢

⎤⎦⎥⎥

⎧⎨⎪⎩⎪

⎡⎣⎢⎢

⎛⎝

⎞⎠

⎤⎦⎥⎥

⎫⎬⎪⎭⎪

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

[ ] ( )

U z U z t r t

Aikz

z

ik x

z

ikRR

a iq nR

( , 0) ( , 0) ( ) ( ) ( )

expexp

2

exp 2 1 1

exp arcsin . (5)n

n

1 0

00

0

02

0

2

The field is obtained by the product of the incident field,the reflectance of the grating, and the phase difference cor-responding to each point of the wavefront under thin elementapproximation. Since light covers twice the distance from the

Figure 1. Scheme of the analyzed configuration, where ξ z( , ) are thecoordinates centered at the point where the grating crosses the z-axis,P is a plane perpendicular to the optical axis (z-axis) at the origin ofcoordinates, δ θ( ) is the distance between each point of the gratingand the plane P, R is the curvature radius of the grating, L is the areaof the grating illuminated by the source over the ξ-axis, and a is thesize of the source placed at the point x z( , ).0 0 The black and whiteregions of the grating represent nonreflective and reflective areas,respectively.

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J. Opt. 00 (2015) 000000 F J Torcal-Milla et al

plane, P, to the grating due to the reflective configuration, theoptical path, δ θ( ), doubles its value.

To calculate the field propagation from the plane, P,forward, we use the Fresnel approach λ≫p( ),

∫ ξ

ξ ξ

= =

× −

−

⎡⎣⎢

⎤⎦⎥

U x zikz

ikzU z

ik x

zd

( , )exp( )

( , 0)

exp( )

2, (6)

L

L

2/2

/2

1

2

where x is the coordinate perpendicular to the propagationaxis at the observation plane. To solve equation (6), weconsider that the radius is much longer than the illuminatedarea of the grating, ≫R L. Then, we can perform a Taylorseries expansion in the exponents of equation (5). In addition,we extended the integral limits of equation (6) to infinity sinceareas of the grating far from the optical axis do not contributeto the field around the optical axis at near distances. Solvingthe integral, the intensity calculated as

= ⁎I x z U x z U x z( , ) ( , ) ( , )2 2 2 results in

∑∝+ +

×− ′ +

+ +

× −− ′

+ +

θ

θ

′′

⁎

⎡⎣⎢

⎤⎦⎥

⎧⎨⎪⎩⎪

⎫⎬⎪⎭⎪

( )[ ]

I x zA R

R z z zza a

iq n n zx xz

R z z zz

iq n n zz

kR R z z zz

( , )( ) 2

exp( )( )

( ) 2

exp2 ( ) 2

, (7)

MPS

n n

n n2,0

2

0 0 ,

0 0

0 0

2 2 20

0 0

where * denotes complex-conjugate, n and ′n are integers, thesuperindex PS denotes punctual source, and the subindex, M,indicates monochromatic. Both effects—the curvature of thegrating and the spherical wave coming from the punctualsource—are present in the intensity. The effects are basicallythe same, affecting the period of the fringes and also thedistance between the self-images along the z-axis. In figure 2,we show two examples of the diffracted intensity consideringa Ronchi convex diffraction grating [1, 2]. We observed howdivergent and convergent incoming light affects the diffractedintensity at the near field. The period of the self-images andthe distance between them vary. From equation (7), we canextract the expressions for the period of the self-images andthe period of the self-imaging phenomenon along the z-axis.From the first exponential factor of equation (7), the period ofthe self-images, p̂ , depends on the radius and the distancebetween the source and the grating as follows

= + +⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥p p z

z Rˆ 1

1 2. (8)

0

The period increases linearly in terms of z for fixedvalues of z0 and R. On the other hand, the period of the self-images phenomenon along the z-axis also depends on the

radius of the grating and the source distance as

= + +⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥z z z

z Rˆ 1

1 2. (9)T T

0

2

It responds to a quadratic form for fixed values of z0 andR. The parameter, λ=z p2 ,T

2 is the Talbot distance for alinear diffraction grating with period, p, illuminated by aplane wave [3].

An interesting example occurs when the light source isplaced on an axis, =x 0,0 at a distance of = −z R/20 fromthe grating, which corresponds to a virtual convergent lightsource. Then, the diffracted intensity exactly correspondswith what would be produced by a plane wave impinging on a

Figure 2. Analytical self-images for a Ronchi cylindrical convexdiffraction grating with period μ=p 20 m, =R 50 mm, illuminationwavelength λ μ= 0.6328 m, and taking the first three diffractionorders. The punctual monochromatic source is placed at μ=x 0 m,0

(a) =z 10 mm,0 (b) = −z 10 mm0 .

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J. Opt. 00 (2015) 000000 F J Torcal-Milla et al

plane diffraction grating,

∑ π

π

∝ − ′

× −− ′

′′

⁎⎡⎣⎢

⎤⎦⎥

⎧⎨⎪⎩⎪

⎫⎬⎪⎭⎪

( )

I x z A a ai n n x

p

i n n z

z

( , ) exp2 ( )

exp2

, (10)

MPW

n n

n n

T

2, 02

,

2 2

where we have used the relationships, =θq qR, π=q p2 ,and superindex PW denotes the plane wave output.

Up to this point, we have considered an ideal punctualillumination source. Nevertheless, standard illuminationsources such as LEDs are neither punctual nor monochro-matic, but they have finite extension and a certain emissionspectrum. Therefore, we will compute the intensity distribu-tion diffracted by the considered grating for a polychromaticand finite-size source. To do this, we simply need to integrateequation (7) over λ considering the spectrum λg ( ), and overx0 considering that the size of the source is a

∫ ∫ λ λ λ∝−

( )I x z g I x z x x( , ) ( ) , , , d d . (11)PFS

a

a

MPS

2,/2

/2

2, 0 0

To obtain analytical results, we particularize to an uni-form spectrum, centered at λ0 with a width of Δλ. We alsoconsider that all the points of the source emit in the same way.Then, the wavelength integral runs from λ Δλ− 20 toλ Δλ+ 2,0 and the size of the source integral in x0 runs from−a 2 to a 2. The intensity results in

∫ ∫

∑

λ λ

Δλ

π

=

=− ′

+ +

× −− ′

+ +

×− ′

+ +

×− ′

+ +

λ Δλ

λ Δλ

θ

θ

θ

θ

− −

+

′′

⁎

⎪ ⎪

⎡⎣⎢

⎤⎦⎥

⎧⎨⎪⎩⎪

⎫⎬⎪⎭⎪

⎧⎨⎪⎩⎪

⎫⎬⎪⎭⎪

⎧⎨⎩

⎫⎬⎭

( )

( )[ ]

[ ]

[ ]

( )I x z I x z x x

A a aiq n n xz

R z z zz

iq n n zz

kR R z z zz

n n q zz

R Rz R z z

a n n q z

Rz R z z

( , ) , , , d d

exp( )

( ) 2

exp2 ( ) 2

sinc8 ( 2 )

sinc( )

2 ( 2 ), (12)

PFS

a

a

MPS

n n

n n

2,/2

/2

/2

/2

2, 0 0

02

,

0

0 0

2 2 20

0 0

2 2 20

0

0

0

0

where α α α=sinc sin , FS means finite source, and P meanspolychromatic source. As one can see, the intensity is affectedby two sinc functions due to the polychromaticity and finitesize of the source, respectively. All the performed analysesare still valid for lower periods, provided that λ≫p anddiffractive effects occur.

To continue, let us compare the effects of both sincfunctions by means of their widths, ωΔλ, ω ,a defined as the

value, z, that fulfills α =sinc 0 for each sinc function,

ωΔλ

ω

=− ′ − +

=− ′ − +

Δλ⎧⎨⎪⎪

⎩⎪⎪

( ) ( )

( )

z

n n z p z R

z

n n a p z R

2 1 2,

( ) 1 2.

(13)

a

0

2 20

20

0

0

We may compare both widths by dividing them,

ωω Δλ

=− ′ − +

− ′ − +Δλ

( )( )

( )n n a p z R

n n z p z R

( ) 1 2

2 1 2. (14)

a

0

2 20

20

The sinc function with the lesser width will be pre-dominant. The polychromaticity of the source is predominantwhen ω ω<Δλ ,a resulting in an achromatic self-imagingregime from a certain distance forward; the finite size of thesource is predominant for ω ω>Δλ ,a attenuating the fringesfor long distances from the grating. This result is of interest inrotary optical encoder technology, where the self-imagingphenomenon is an important issue to solve.

In figures 3(a) and (b), we show two examples of the self-images produced by a convex cylindrical Ronchi gratingilluminated by a polychromatic and finite extension source.These examples correspond to the same parameters asfigure 2, but they take into consideration polychromatic lightwith Δλ = 50 nm and lateral size μ=a 50 m. In this parti-cular example, ω ω>Δλ a and the contrast of the fringes decayto zero from a certain distance forward. This effect is betterobserved in figure 3(b). These results show that diffractionfringes only occur up to a certain distance. Therefore, thisconfiguration may be used in optical encoders, provided thatthe reading head is placed closer to the grating than thementioned distance. In addition, in figures 3(c) and (d) weshow another example in which the term, ω ω<Δλ ,a is pre-dominant due to polychromaticity. It corresponds to the sameparameters as figure 2, but takes into consideration poly-chromatic light with Δλ = 50 nm and lateral size μ=a 10 m.

For completeness, other particular cases corresponding tofinite monochromatic sources and punctual polychromaticsources are analyzed and shown in the appendix. They areobtained from equation (12). These cases are of interest inother configurations of optical rotary encoders.

3. Experimental approach

To corroborate the theoretical results obtained in the previoussection, we carried out the experiment shown in figure 4. Weused a diffraction grating of period μ=p 20 m engraved bylaser ablation on a cylindrical steel substrate. The nature ofthe substrate allows it to bend into a cylindrical shape. In ourpractical case, we engraved a grating approximately twocentimeters long, which is enough to validate the analyticalresults. The radius of curvature of the grating was

=R 50 mm. We used an LED of wavelengthλ = ±655 20 nm as the illumination source. The emitting areawas circular, with the diameter μ=a 57 m. Afterward, lightwas captured by a CMOS camera (μEye by IDS, pixel size

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J. Opt. 00 (2015) 000000 F J Torcal-Milla et al

μ×2.2 2.2 m) coupled to a microscope objective, which wasused to magnify the self-images. We used a microscopeobjective (model: M Plan Apo NIR 50× by Mitutoyo) tocapture images from the grating surface backward through the

beam splitter. In addition, the camera was free to travel alongthe propagation axis. The experiment consisted of separatingthe camera from the grating plane along the propagation axisand acquiring images of the diffracted intensity at differentplanes. Figure 5 shows that the obtained analytical(figure 5(a))—from equation (12)—and experimental(figure 5(b)) results for this particular case have goodagreement.

4. Conclusions

In this work, we analyzed the near-field intensity pattern ofcylindrical convex diffraction gratings illuminated by a gen-eral source that can be punctual or finite, monochromatic orpolychromatic. We analyzed the effects of the source size and

Figure 3. Analytical self-images for a Ronchi cylindrical convex diffraction grating with period μ=p 20 m, =R 50 mm, polychromaticillumination λ μ= ±0.6328 0.025 m, and taking the first three diffraction orders, the finite source of size a placed at μ=x 0 m.0 (a)

μ=a 50 m, =z 10 mm,0 (b) μ=a 50 m, = −z 10 mm,0 (c) μ=a 10 m, =z 10 mm,0 (d) μ=a 10 m, = −z 10 mm0 .

Figure 4. Scheme of the experimental setup. In our practical case, wehave engraved a grating approximately two centimeters long, whichis enough to validate the analytical results.

5

J. Opt. 00 (2015) 000000 F J Torcal-Milla et al

polychromaticity of the source, giving analytical expressionsfor all cases. In addition, we performed experiments to cor-roborate the most general analytical result: the finite poly-chromatic source. These analyses and results can help oneunderstand the behavior of optical rotary encoder behaviorwith this kind of grating.

Figure 5. Self-images for a Ronchi cylindrical convex diffractiongrating with period μ=p 20 m, =R 50 mm, polychromatic illumi-nation λ μ= ±0.655 0.02 m, finite source with width μ=a 57 mplaced at μ=x 0 m0 and =z 5 mm.0 (a) Analytical, taking the firstthree diffraction orders. (b) Experimental.

Figure 6. Analytical self-images for a Ronchi cylindrical convexdiffraction grating with period μ=p 20 m, =R 50 mm, illuminationwavelength λ μ= 0.6328 m, and taking the first three diffractionorders, finite source of size μ=a 50 m placed at μ=x 0 m.0 (a)

=z 10 mm,0 (b) = −z 10 mm,0 (c) intensity along the optical axisfor different sizes of the source κ μ κ= × = …a 5 m ( 0, , 8)

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Acknowledgments

The authors thank Jose Maria Herrera-Fernandez for his helpwith graphics. This work has been supported by the projectDPI2011-27851 of the Ministry of Science and Innovation ofSpain and the project Fagor Automation-UCM Ref. 92/2014.

Appendix

In this appendix, we show the two intermediate cases—thefinite monochromatic source and punctual polychromaticsource—by particularizing equation (12).

6. Finite monochromatic source

We obtain the intensity distribution for a monochromaticfinite light source of lateral size, a, by particularizingequation (12) for Δλ = 0. It results in

∑∝

× −− ′

+ +

×− ′

+ +

×− ′

+ +

θ

θ

θ

′′

⁎

⎪ ⎪

⎧⎨⎪⎩⎪

⎫⎬⎪⎭⎪

⎡⎣⎢

⎤⎦⎥

⎧⎨⎩

⎫⎬⎭

( )[ ]

[ ]

I x z A a a

iq n n zz

kR R z z zz

iq n n xz

R z z zz

a n n q z

Rz R z z

( , )

exp2 ( ) 2

exp( )

( ) 2

sinc( )

2 ( 2 ), (15)

MFS

n n

n n2, 02

,

2 2 20

0 0

0

0 0

0

where the superindex FS denotes finite source, and we haveconsidered the source centered at the optical axis, =( )x 0 .0

The effect of considering a finite source is a loss of intensityand contrast when we separate from the grating. Infigures 6(a) and (b), we show two examples with the sameparameters as figures 2(a) and (b), but considering Δλ = 0.The effect of the sinc function is better observed infigure 6(b), presenting areas with high and low contrast alongthe z-axis. In this case, the periods of the self-images alongthe x-axis and the z-axis are also affected in the same fashionas in equations (8) and (9).

In addition, figure 6(c) shows the intensity dependencealong the optical axis in terms of the size of the source for thefinite monochromatic source. In this case, the fringes dis-appear from a certain distance forward, and the intensitystabilizes at a certain constant value. This occurs because theintensity along the optical axis tends to a constant value.

7. Punctual polychromatic source

We obtain the diffracted intensity produced by a polychro-matic punctual source placed at a distance, z ,0 by

Figure 7. Analytical self-images for a Ronchi cylindrical convexdiffraction grating with period μ=p 20 m, =R 50 mm, polychro-matic illumination λ μ= ±0.6328 0.05 m, and taking the first threediffraction orders, punctual source placed at μ=x 0 m.0 (a)

=z 10 mm,0 (b) = −z 10 mm,0 (c) intensity along the optical axisfor different spectrum widths, Δλ κ μ κ= × =0.2 m ( 0,...,8).

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J. Opt. 00 (2015) 000000 F J Torcal-Milla et al

particularizing equation (12) for =a 0. It results in

∑

Δλ

π

∝− ′ +

+ +

× −− ′

+ +

×− ′

+ +

θ

θ

θ

′′

⁎⎡⎣⎢

⎤⎦⎥

⎧⎨⎪⎩⎪

⎫⎬⎪⎭⎪

⎧⎨⎪⎩⎪

⎫⎬⎪⎭⎪

( )

( )[ ]

[ ]

I x z A a aiq n n zx xz

R z z zz

iq n n zz

kR R z z zz

n n q zz

R Rz R z z

( , ) exp( )( )

( ) 2

exp2 ( ) 2

sinc8 ( 2 )

, (16)

PPS

n n

n n2, 02

,

0 0

0 0

2 2 20

0 0

2 2 20

0

where the subindex P indicates polychromatic, and we haveconsidered the source centered at the optical axis, =( )x 0 .0 Infigures 7(a) and (b), we show two examples of intensity dif-fracted by a curved grating illuminated by a polychromaticpunctual source. These examples correspond to the sameparameters as those in figures 2(a) and (b), but considering

=a 0. The effect is similar to that obtained for a finitemonochromatic source. A decay of the intensity is producedfor long distances from the grating, but the polychromaticityof the beam has an additional effect. From a certain distanceforward, the fringes remain almost equal—referring to thecontrast—without the contrast inversion inherent to the clas-sical self-imaging phenomenon. This effect is better observedin figure 7(b). In addition, the effect of the sinc function isalso observed in figure 7(b), presenting areas with high andlow modulation of the fringe intensity.

Finally, figure 7(c) shows the dependence of the intensityalong the optical axis, considering the punctual polychromaticsource in terms of the amount of polychromaticity. As onecan see, by increasing the amount of polychromaticity, theintensity of the central fringe stabilizes to a constant value. Inthis case, this value is due to the stabilization of the fringeswithout contrast inversion. This fact is of interest for opticalencoder technology, since it would allow placement of thereading head at any distance from the grating without losingor inverting contrast.

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