# Binary gratings with random heights.

**ABSTRACT** We analyze the far-field intensity distribution of binary phase gratings whose strips present certain randomness in their height. A statistical analysis based on the mutual coherence function is done in the plane just after the grating. Then, the mutual coherence function is propagated to the far field and the intensity distribution is obtained. Generally, the intensity of the diffraction orders decreases in comparison to that of the ideal perfect grating. Several important limit cases, such as low- and high-randomness perturbed gratings, are analyzed. In the high-randomness limit, the phase grating is equivalent to an amplitude grating plus a "halo." Although these structures are not purely periodic, they behave approximately as a diffraction grating.

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Page 1

Binary gratings with random heights

José María Rico-García1,2and Luis Miguel Sanchez-Brea1,3

1Universidad Complutense de Madrid, Optics Department, Applied Optics Complutense Group,

Facultad de Ciencias Físicas, Ciudad Universitaria s.n., 28040 Madrid, Spain

2jmrico@fis.ucm.es

3sanchezbrea@fis.ucm.es

Received 27 January 2009; revised 29 April 2009; accepted 3 May 2009;

posted 7 May 2009 (Doc. ID 106838); published 22 May 2009

We analyze the far-field intensity distribution of binary phase gratings whose strips present certain

randomness in their height. A statistical analysis based on the mutual coherence function is done in

the plane just after the grating. Then, the mutual coherence function is propagated to the far field

and the intensity distribution is obtained. Generally, the intensity of the diffraction orders decreases

in comparison to that of the ideal perfect grating. Several important limit cases, such as low- and

high-randomness perturbed gratings, are analyzed. In the high-randomness limit, the phase grating

is equivalent to an amplitude grating plus a “halo.” Although these structures are not purely periodic,

they behave approximately as a diffraction grating.

OCIS codes:

050.0050, 050.2770, 030.0030.

© 2009 Optical Society of America

1.

Diffraction gratings are one of the most important

optical components [1]. They can be defined as

optical elements that produce a periodical modula-

tion on the properties of the incident light beam, thus

generating diffraction orders [2,3]. Their directions

are given by the well-known grating equation

psinθj¼ jλ, where p is the period of the grating, λ

is the wavelength of the incident beam, j is an inte-

ger, and θjare the directions of the diffracted beams.

The intensity of the diffraction orders is obtained

from the square of the Fourier coefficients of the grat-

ing. Amplitude or phase gratings are used in most

applications [4]. Also, other kinds of gratings are pos-

sible, such as polarization gratings [5–7] or gratings

with random microscopic

topography [8–10].

Theoretical approaches normally assume that dif-

fraction gratings present an ideal optical behavior.

However, gratings are not always ideal. Flaws or de-

fects can be produced during the fabrication process

[10–13]. Other nonideal behavior stems from rough-

Introduction

irregularitiesin the

ness on the surface ofthe gratings, suchas in the case

of steel tape gratings [8]. Roughness produces a mod-

ification in the intensity of the diffraction orders

[14–19].

Also, stochastic irregularities in the height of the

strips can be observed [20]. In this work we theore-

tically and numerically analyze the performance of

phase gratings where the height of the strips pre-

sents a certain random variation around an average

value. Because of the stochastic properties of strips, a

statistical approach needs to be used in order to de-

termine the intensity at the far field. We show that

the intensity of the diffraction orders depends on the

parameters defining the degree of randomness in the

heights of the strips in the grating. This analysis is

required to define the maximum degree of random-

ness acceptable for a good performance of the grat-

ing. This paper has been organized as follows.

Section 2 is devoted to the statistical description of

the perturbed grating. Their properties are ex-

plained in terms of a matrix Γ, which contains all

the statistical information of the correlations of the

field just after the plane where the grating lies. In

Section 3, the average far-field intensity distribution

has been calculated through the analysis of the mu-

tual coherence function. Two cases have been studied

0003-6935/09/163062-08$15.00/0

© 2009 Optical Society of America

3062APPLIED OPTICS / Vol. 48, No. 16 / 1 June 2009

Page 2

in detail, namely, the limits for low and high random-

ness in the perturbed grating. In addition, we ac-

count for the dependence of the diffraction orders

with the degree of randomness. In Section 4 we com-

pare the results obtained using this formalism with

those obtained numerically, showing some examples

of the far-field diffraction pattern of the proposed

gratings. Finally, the main conclusions of this paper

are summarized in Section 5.

2.

Correlation Matrix and Surface Structure

Let us consider a binary phase grating of period p

and refraction index n as shown in Fig. 1. The phase

grating is engraved in a dielectric substrate, such as

a glass, etc. For simplicity, we will assume that the

grating is illuminated with a monochromatic plane

wave (wavelength λ) in normal incidence. The height

of the strips presents a random perturbation, yet

maintaining the periodicity in their position. The

behavior of the grating can be understood in terms

of its perturbed height distribution:

Correlations on a Grating with Random Heights:

hðxÞ ¼ huðxÞ þ ΔhðxÞ;

ð1Þ

where huðxÞ is the ideal unperturbed grating height

and ΔhðxÞ represents the perturbation over huðxÞ,

which is a random process labeled by the transverse

coordinate x of the structure. Under the thin-element

approximation [21], the phase in the unperturbed

grating betweentwo

Δ ¼ kðn − 1Þh0, where h0is the height of the strips

of the unperturbed grating and k ¼ 2π=λ is the wave-

number. The random transmittance that multiplies

the field in a phase step of height h0due to the

perturbation is

consecutivestripsis

ΔTstep¼ eiϕ;

ð2Þ

where ϕðzÞ ¼ kðn − 1Þz and z is a random variable

that accounts for the perturbation in the phase step.

This random process is associated to a statistical dis-

tribution, wðzÞ. Consequently, the transmittance of

the structure can be described, TðxÞ ¼ T0ðxÞΔTðxÞ,

where T0ðxÞ is the unperturbed grating, which can

be written as a Fourier series [22]:

T0ðxÞ ¼

X

j

ajeiqjx;

ð3Þ

where q ¼ 2π=p, ajis the Fourier coefficient, and j is

an integer number. The term ΔTðxÞ is not purely per-

iodic but is stochastic. It can be described as a sum of

strips with different heights:

ΔTðxÞ ¼

X

a

N

m¼0

smΠ

?x − xm

p=2

?

;

ð4Þ

where

xm¼ ð2m þ 1Þp=4. Without loss of generality, we

have assumed that the fill factor of the grating is

equal to 1=2. N þ 1 is the number of strips of the

grating and

ΠðxÞ

isrectangle functionand

sm¼

?1

if mis odd

if mis even:

eiϕ

ð5Þ

In this case, smare random processes on the m-even

steps. In what follows, the statistical distribution of z,

wðzÞ, is assumed to be a Gaussian function with zero

mean and variance σ, that is, wðzÞ ¼ e−z2=2σ2=

When the random process is statistically station-

ary, the correlation between two points x and x0just

after the grating is [23]

ffiffiffiffiffiffi

2π

p

σ.

Jðx;x0Þ ¼ jA0j2hTðxÞT?ðx0Þi;

where A0is the amplitude of the incident plane wave

and the average is taken over the ensemble of the

quantity in brackets. Considering Eqs. (3) and (4),

the mutual coherence function on Eq. (6) can be cast

into

ð6Þ

Jðx;x0Þ ¼ jA0j2T0ðxÞT?0ðx0Þ

X

where Γnm¼ hsns?mi are the elements of the matrix

Γ, which is the stochastic part of the equation, and m,

n run over the slits of the grating. The mutual coher-

ence function and, therefore, the grating perfor-

mance, depends on its correlation characteristics,

which have been subsumed in Γ. The entries Γnmwill

be functions of the average hΔTðxÞi. In those steps

that can be considered a random variable [24]

×

N

m¼0

X

N

n¼0

ΓnmΠ

?x − xm

p=2

?

Π

?x0− x0n

p=2

?

;

ð7Þ

hΔTstepi ¼ heiϕi ¼

Z∞

−∞

wðzÞeiϕðzÞdz:

ð8Þ

As it can be noted from Eq. (8), the average is the

characteristic function of the distribution wðzÞ, but

Fig. 1.

dashed lines represent the grating without randomness and the

solid lines are realizations of the grating where odd strips present

certain randomness. The period of the unperturbed grating is p

and the refraction index is n.

Scheme of the phase grating with random heights. The

1 June 2009 / Vol. 48, No. 16 / APPLIED OPTICS3063

Page 3

for a scale factor [24]. Then, the average results in

hΔTi ¼ e−g=2¼ α, where g ¼ ½kðn − 1Þσ?2. Owing to

the grating structure, the possible values that Γnm

can take are greatly reduced. They can be computed

in a straightforward way if the correlations between

odd and even steps are put forward explicitly:

8

>

which explains the correlations of the field just after

the grating.

ΓmnðαÞ ¼

>

>

>

:

<

1

1

α2

α

m ¼ n

m ≠ n;ðm;nÞodd

m ≠ n;ðm;nÞeven

meven;noddormodd;neven

;

ð9Þ

3.

Function

Once the mutual coherence function hasbeen worked

out in Eq. (7), its counterpart in the far field is

determined after propagating Jðx;x0Þ using the

Fraunhofer kernel [23]:

Far-Field Propagation of the Mutual Coherence

Jðx2;x20Þ ¼ e

ik

2zðx22−x022Þ

Z∞

−∞

Z∞

−∞

Jðx1;x10Þeik

zðx02x01−x2x1Þ

× dx1dx01:

ð10Þ

Taking Eq. (7) into Eq. (10), the normalized mutual

coherence results in

Jðθ2;θ02Þ ¼

X

Z∞

−∞

N

m¼0

X

N

n¼0

Γmn

?Z∞

−∞

T?0ðx01ÞΠmðx01Þeik

zx02x01dx01

×

T0ðx1ÞΠnðx1Þe−ik

zx2x1dx1

?

;

ð11Þ

where

θ02¼ x02=z, and Πm¼ rect½ðx − xmÞ=χp?. As a result,

Eq. (11) is split up into two Fourier transforms. They

are formally identical, except for a sign in the phase.

Then, the mutual coherence function in the far field

is

X

×

2ðpθ2=λ − jÞ

Jðθ2;θ02Þ ¼ Jðθ2;θ02Þ=jA0j2,

θ2¼ x2=z,

Jðθ2;θ02Þ ¼

j;w

aja?wMjwðx2;x02Þsinc

?π

?

sinc

?π

2ðpθ02=λ − wÞ

?

; ð12Þ

where sincðxÞ ¼ sinðxÞ=x, and the products aja?w

quantify the original correlations of the unperturbed

grating. The sinc functions are due to the diffraction

phenomena in each slit labeled by the index m. The

quantities Mjwcontain all the information regarding

the correlations between the diffraction orders in the

perturbed grating. Mjw can be understood as a

modulation factor induced by the randomness in

the perturbed grating over ajand aw:

Mjwðθ2;θ02Þ ¼

X

n;m

Γnmei½ϕmjðθ2Þ−ϕnwðθ02Þ?:

ð13Þ

The phase dependence of Mjwis directly derived from

the properties of the Fourier transform, from which

results ϕmjðθ2Þ ¼ ð2m þ 1Þðqj − kθ2Þp=4.

The normalized average intensity can be easily

recovered from the mutual coherence function in

Eq. (12) as

hIðθ2Þi ¼ Jðθ2;θ2Þ ¼

X

j;w

aja?wMjwðθ2Þ

??

× sinc

?π

2

?pθ2

λ

− j

sinc

?π

2

?pθ2

λ

− w

??

: ð14Þ

The coupling between different orders will vary as a

function of α and the structure of Γ. In the scalar do-

main, the period of the grating is much greater than

the wavelength, p ≫ λ. Then Eq. (14) can be simpli-

fied, because the sinc functions are very narrow and

they do not overlap each other. Thus, the sum is

approximately

hIðθ2Þi ≈

X

j

jajj2Mjjðθ2Þsinc2

?π

2

?pθ2

λ

− j

??

:

ð15Þ

The stochastic part of the proposed grating is con-

tained only in the Mjjðθ2Þ quantities. The quantities

Mjjðθ2Þ have a special meaning. They not only em-

body the new values of the unperturbed diffraction

orders in the far-field pattern but also modify the

shape of the diffraction peaks, as can be noted from

Eq. (12).

When the randomness is null, Γnm¼ 1 for each

ðn;mÞ. Consequently,

Mjjðθ2Þ ¼

X

n;m

eiðm−nÞðqj−kθ2Þp=2¼

sin2h

πN

2

?

pθ2

pθ2

λ− j

?i

sin2h

π

2

?

λ− j

?i ;

ð16Þ

and the well-known grating equation is recovered:

hIðθ2Þi ¼

X

j

jajj2sinc2

?πN

2

?pθ

λ− j

??

:

ð17Þ

A.

The relation of Eq. (14) encloses all the pertinent

information to study the far-field correlations of

the perturbed grating regardless of the value of

the randomness parameter σ. Nonetheless, there

are two important cases to be analyzed, the high,

σ ≫ λ, and low σ ≪ λ randomness limits. For both

cases, the matrix Γ considerably simplifies its depen-

dence with the fluctuation level and its behavior is

easier to examine. Concerning the limit σ ≫ λ, Γ

can be approximated to a simpler form knowing that

High-Randomness Limit

3064 APPLIED OPTICS / Vol. 48, No. 16 / 1 June 2009

Page 4

α ≪ 1:

Γðα ≪ 1Þ ≃ Λ þ COOð1Þ:

ð18Þ

The matrices Λ and COOð1Þ subsume the features of

the correlation matrix when the randomness is high

(subscript O stands for odd). Their definitions are

Λmn¼

?1

?1

n ¼ mðn;mÞeven

otherwise

ðn;mÞodd

otherwise:

0

;

COOmnð1Þ ¼

0

ð19Þ

This means that the even steps correlate only with

themselvesbecausethe

fm;ng with m ≠ n and m, n even are zero. The fluc-

tuations are so significant that there are not simila-

rities among different even steps. In the same way,

the correlations between odd and even steps are also

zero. COOð1Þ is present since the odd steps do not fluc-

tuate. Figure 2 shows Γ for a high-random finite grat-

ing. Both contributions to Γ are noticeable. As an

example, in Fig. 3(a) the average intensity is shown

for the case of a highly perturbed grating where

Δ ¼ π=2, which is compared to that obtained for

the same grating without perturbation, σ ¼ 0, in

Fig. 3(b).

The most remarkable feature is the existence of a

halo in the far-field pattern. As Fig. 3(a) shows, the

diffraction peaks are situated over a smooth inten-

sity curve not present in the unperturbed case. This

is the halo. It can be understood as the intensity of an

isolated strip multiplied by the number of strips with

randomness, N=2. In the high-randomness limit, as

it will be seen, the perturbed structure performs as

anamplitude grating plus a nonperiodic structure re-

sponsible of the halo. The halo shape is ruled by the

correlationsbetween

diagonal term Λ, which does not work as a grating-

like arrangement, because it is related to the short

range correlations contained in Γ not typical of a

grating structure. In fact, it represents the average

correlations of the perturbed steps at high random-

ness. The other term, COOð1Þ, is responsible for the

intensity peaks, because it represents the correlation

of an amplitude grating just after the plane where it

lies. Half of the power contained in the original peaks

is driven to the halo because the amplitude grating

created in this limit takes the other half, as can be

Fig. 2.

For this example, the unperturbed grating has N ¼ 31 steps, the

refraction index is n ¼ 1:5, the wavelength is λ ¼ 680nm, and

σ ¼ 5λ. The xmand x0naxes are normalized to p=2. For all the fig-

ures and simulations the following parameters have been used:

the wavelength of the incident beam is λ ¼ 0:68μm, the period

of the grating is p ¼ 25μm, the refraction index is n ¼ 1:5, and

the number of periods is N ¼ 31.

Γ matrix corresponding to high-randomness limit σ ≫ λ.

Fig. 3.

(b) Intensity for the same grating but without randomness σ ¼ 0.

(a) Average intensity for high randomness σ ¼ 5λ when Δ ¼ π=2. The period of the grating studied in every figure is p ¼ 25μm.

1 June 2009 / Vol. 48, No. 16 / APPLIED OPTICS3065

Page 5

seen in Fig. 3. This aspect of the perturbed grating

differs from usual gratings, where there exist only

well-defined directions of propagation for the trans-

mitted radiation of the periodic structure. The mod-

ulation factors Mjware responsible for its existence.

Another remarkable feature of the grating behavior

in this limit is its independence from the concrete va-

lue of σ, as it will be shown in Subsection 3.C This

interpretation will be supported by the numerical re-

sults given in Section 4.

B.

The low-randomness limit occurs when g ≪ 1. Then

a linear series expansion α ≃ 1 − g=2 can be per-

formed. This encompass a distinct correlation matrix

Γ as a linear series in g:

Low-Randomness Limit

Γðg ≪ 1Þ ≃ Cð1Þ þ gP;

ð20Þ

where Cð1Þ ¼ COOð1Þ þ CEEð1Þ and P ¼ Λ − CEEð1Þ

−1

trices CEOand CEEare analogous to the matrix COO

used above. The matrix depends on the degree of ran-

domness g in the perturbed structure. An example of

this is exhibited in Fig. 4. The long-range correla-

tions typical from a grating structure are shown.

Therefore, we expect a different shape in the far-field

pattern because of the correlation structure brought

about by the perturbation.

The far-field pattern is displayed in Fig. 5 for the

perturbed and the unperturbed gratings. The main

effect of the perturbation at low randomness is to al-

ter the value of the diffraction peaks. In this particu-

lar case, the phase shift produced by the unperturbed

grating is Δ ¼ π. Then the zeroth order disappears.

With the perturbation, the zeroth order presents a

certain value and a small halo.

2CEOð1Þ. The subscript E stands for even. The ma-

C.

Degree of Randomness

It is important to analyze how the diffraction orders

of the perturbed grating varies in terms of random-

ness. In Fig. 6 we show the intensity of the diffraction

peaks 0, ?1, and ?3 in terms of σ=λ for two different

unperturbed gratings, Δ ¼ 0 and Δ ¼ π. Indepen-

dently from the initial parameters of the grating,

when a high-randomness perturbation is present,

the perturbed structure behaves as an amplitude

grating and a “scattering halo,” which is the far-field

diffraction pattern of a strip with width p=2. This ef-

fect is noticeable in Fig. 6, where the orders do not

change above σ ≃ λ.

Intensity of the Diffraction Orders in Terms of the

4.

To confirm the validity of the theoretical results, we

have carried out a numerical analysis based on the

Numerical Simulations

Fig. 4.

normalized to p=2.

Γ corresponding to σ ≪ λ. The xm and x0n axes are

Fig. 5.

Δ ¼ π. Although the pattern has been normalized, its maximum is higher than the maximum at high randomness.

Far-field pattern of (a) the perturbed grating at low randomness, σ ¼ 0:25λ, Δ ¼ π versus (b) the unperturbed structure, σ ¼ 0,

3066 APPLIED OPTICS / Vol. 48, No. 16 / 1 June 2009

Page 6

diffraction pattern estimation of gratings with

random heights, using a discrete version of the

Fraunhoffer approximation based on the fast Fourier

transform. For the simulation of the grating, we have

modified the height of the fluctuating strips accord-

ing to a Gaussian distribution. An example of a simu-

lated grating and its diffraction pattern is shown in

Fig. 7. We can see that the diffraction pattern of

a single grating is not completely symmetrical,

although, even for a low number of periods, the dif-

fraction orders clearly appear.

Since the results given in Eq. (14) are obtained

after an averaging process, we need to perform sev-

eral simulations, as shown in Fig. 7, and take the

average of the intensity of the different diffraction

patterns. In Fig. 8 we can see these averaging pro-

cesses for several parameters of randomness σ and

phase shift of the unperturbed grating Δ, according

to Eq. (14), and for the simulations obtained using

the fast Fourier transform. The results are very simi-

lar in all the cases.

5.

We investigated the far-field intensity distribution

for binary phase gratings with random perturbations

in their strips. Because of the stochastic properties of

the perturbations, we performed a statistical analy-

sis based on the mutual coherence function. The far

field was obtained after propagating the latter via

the Fraunhoffer approximation. Thus, the average

intensity distribution was computed. The equations

involved are similar to those of a perfect grating, but

for a new term that accounts for the interactions

among the diffraction orders owing to the random

Conclusions

Fig. 6.Intensity of the diffraction orders 0, ?1, and ?3 when the randomness of the strips σ is varied. (a) Δ ¼ 0 and (b) Δ ¼ π.

Fig. 7.

The phase shift of the unperturbed grating is Δ ¼ π=2 and the randomness parameter is σ ¼ 5λ.

(a) Diffraction grating with random heights and (b) intensity distribution for this diffraction grating (realization) numerically.

1 June 2009 / Vol. 48, No. 16 / APPLIED OPTICS3067

Page 7

perturbation. Low- and high-randomness limits have

been analyzed in detail. As randomness loses its

importance, we recover the usual characteristics in

the far-field zone. In this limit, the most remarkable

effect is a redistribution of the diffraction order in-

tensities. As the fluctuations grow, the existence of

a halo is more noticeable. Even more, in the limit

of high randomness, the correlation matrix of the

structure, Γ, is the sum of two separate contribu-

tions. Each one of them manages distinct features ob-

served in the average intensity. The diagonal term

rules the halo whereas the other term controls the

diffraction peak positions and their intensity. In

other words, the perturbation makes a phase grating

behave as an amplitude grating plus a nonperiodic

scatterer. In addition, the intensity of the diffraction

orders does not depend on the characteristics of the

unperturbed grating. The orders correspond to those

of an amplitude grating. To check the expression for

the average intensity, a Monte-Carlo-like simulation

was done. We found complete agreement between si-

mulation and theory. The structures between the two

limits can be easily analyzed within the formalism

described above. Moreover, the effects caused by

the strength of the randomness can be readily distin-

guished. To conclude, this approach has served as a

comprehensive tool to understand the performance

of this kind of grating. Its extension to multilevel

gratings would be direct, not only in the far field,

but also in the Fresnel region, thus paving the

way to study the effect of randomness in diffractive

optical elementswith

this paper.

The authors thank Eusebio Bernabéu, Alfredo

Luis, and Francisco José Torcal Milla for their inter-

estandhelp.Thisworkhasbeensupported byproject

CCG08-UCM/DPI-3952 of Dirección General de

thetools employed in

Fig. 8.

and (d) Δ ¼ π, σ ¼ 5λ. The number of experiments of the averaging process is N ¼ 100. There is a coincidence of the peaks corresponding to

the diffraction orders between the theoretical results and the simulation.

Average intensity distribution for several gratings with randomness. (a) Δ ¼ 0, σ ¼ 0:25λ, (b) Δ ¼ 0, σ ¼ 5λ, (c) Δ ¼ π, σ ¼ 0:25λ,

3068 APPLIED OPTICS / Vol. 48, No. 16 / 1 June 2009

Page 8

Universidades e Investigación de la Consejería de

EducacióndelaComunidaddeMadridyUniversidad

Complutensede Madrid

“Tecnologías avanzadas para los equipos y procesos

de fabricaciónde 2015.

e-máquina (eEe)” of the Ministerio de Industria,

Turismo y Comercio.

and CENIT project

e-eficiente, e-cológica,

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