# Spectral Modification by Diffraction and Scattering

Authors: M. Taghi Tavassoly; Masoomeh Dashtdar; Mohammad Amiri
Journal: Advances in Optical Technologies
 (2010)

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## Abstract

Different effects induce spectral changes, for example, correlation of source
fluctuations, propagation of light, random changes in optical properties of
a medium, diffraction and scattering from objects, and rough interfaces. We review the spectral changes that occur as a result of light
diffraction from phase steps, and particularly we discuss the spectral changes
in the neighborhood of phase singularities. We also review the redshift and
blueshift in the spectra of the lights coherently and diffusely scattered from
rough interfaces. In addition, we study the effects of roughness and incident
angle on the spectral profiles of scattered lights in reflection and transmission
modes.


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## Body

## 1. Introduction

After being realized in 1986 [1] that the spectrum of radiation may change on propagation, a considerable volume of works has been carried out dealing with spectral changes induced by the correlated fluctuations in sources and changes in coherence properties of radiations [2, 3]. Since the spatial coherence width of a light beam increases with the distance from the source, any light beam is coherent to some extent that depends on the source size and the distance from the source. Thus, as a coherent or partially coherent polychromatic beam of light passes through a medium that imposes different phases on different parts of the beam, the beam scatters from these parts and interference of the scattered lights leads to spectral changes. In this paper we study the spectral changes that occur as a result of light diffraction from phase steps and light scattering from rough interfaces in reflection and transmission modes.


## 2. Modified Spectrum of Diffracted Light

When a fully or partially coherent beam of light experiences a sharp change in amplitude or in phase, redistribution of intensity occurs in space that is called diffraction. A sharp change in amplitude occurs as the beam passage is partly obstructed by an opaque object. A sharp change in phase happens as a beam of light is reflected from a step, Figure1(a), or passes through a transparent plate immersed in a liquid or gas where a sharp change in refractive index is imposed at the plate edge, Figure 1(b). The lights diffracted from the neighborhood of the change-affected area interfere and lead to different spectra at different points in the diffraction field. The amount of spectral change varies from point to point, but it is remarkable and anomalous in the neighborhoods of phase singularities.(a) The geometry of the Fresnel diffraction of light from a 1D phase step. By displacing pointp0 on the step, the point p on the screen is shifted to satisfy equation θr=θ. Therefore, the complex amplitude varies as a function of r. (b) A phase step in transmission is formed at the edges of a transparent plate of refractive index N that is immersed in a transparent gas or liquid of refractive index N′.
(a)(b)Gbur et al. [4] used the Fresnel-Kirchhoff integral and obtained the following expression:(1)M(r,λ)=1λ|∫∫WeikRRd2r′|2
for spectral modifying function for a fully coherent spherical wave front passing through a lens, at points in the neighborhood of the geometrical focal point. In (1), r, λ, k, d2r′, and R stand for the distance between the focal point and observation point, wavelength, wavenumber, surface element, and the distance between a point on the aperture and observation point. The above function leads to drastic spectral changes in the neighborhoods of the phase singularities on the symmetry axis. But intensities at these parts are so low that challenges the experimental studies [5]. However, the diffracted intensities are quite appreciable at points corresponding to the neighborhoods of the phase singularities that are formed by the phase steps. In the following we describe these singularities with more details.In Figure1(a), the plane wave ∑ is incident to a 1D step of height h at incident angle θ. Using the Fresnel-Kirchhoff integral, the diffracted complex amplitude at point p, the intersection of the screen and the ray that is specularly reflected from the step, can be given as [6, 7](2)U(r,λ)=-iλA(λ)eikrr[∫-∞x0eikx2/2rdx+ei2khcosθ∫x0∞eikx2/2rdx],
where A(λ) and x0 stand for the amplitude of the incident light and the distance from p0 to the step edge, p0o, respectively. Using the abbreviations(3)kx2r=πυ2,2khcosθ=ϕ,eikrr=u0,
the complex amplitude at point p, for p0 on the left side of the step edge, can be given by(4)UL(r,λ)=K′A(λ)[∫-∞υLeiπυ2/2dυ+eiϕ∫υL∞eiπυ2/2dυ]
and for p0 on the right side of the step edge by(5)UR(r,λ)=K′A(λ)[∫-∞-υReiπυ2/2dυ+eiϕ∫-υR∞eiπυ2/2dυ],
where K′=-i/2u0; υR and υL are the Fresnel variables corresponding to the distance between the step edge and point p0, for p0 on the left and right sides of the step edge. The corresponding intensities are [6, 7](6)IL(r,λ)=I0(λ)r[cos2(ϕ2)+2(CL2+SL2)sin2(ϕ2)-(CL-SL)sinϕ(ϕ2)],IR(r,λ)=I0(λ)r[cos2(ϕ2)+2(CR2+SR2)sin2(ϕ2)+(CR-SR)sinϕ(ϕ2)],
where, CL, SL and CR, SR are the Fresnel cosine and sine integrals for the cases p0 lying on the left and right sides of the step edge. These equations show that the diffracted light spectrum is modified, and the modifying functions are(7)ML(r,λ)=[cos2(ϕ2)+2(CL2+SL2)sin2(ϕ2)-(CL-SL)sinϕ(ϕ2)],MR(r,λ)=[cos2(ϕ2)+2(CR2+SR2)sin2(ϕ2)+(CR-SR)sinϕ(ϕ2)].Immersing a transparent plate of refractive indexN in a transparent gas or liquid of refractive index N′, a phase step is formed in transmission. Calculating the amplitudes and intensities in the same manner that is done for a phase step in reflection leads to similar equations, but the phase change should be replaced by the following:(8)ϕ=kN′h[n2-sin2θ-cosθ],
where n=N/N′.At the exact edge, that is, forυR=υL=0 and(9)ϕ=(2m+1)π,m=0,±1,…,
we have phase singularity. Thus, by changing the step height or the incident angle θ, one can create a phase singularity with the desired depth. For a phase singularity at λ0, that is, for 4πhcosθ/λ0=(2m+1)π, the modifying function for wavelengths close to λ0 at points near the step edge can be approximated by(10)ML(r,λ)≈2CL(CL+α),MR(r,λ)≈2CR(CR-α),
where α denotes the phase change that is obtained for replacing λ0 by λ. For λ>λ0, we have α<0. That means that the spectra at points on the right side of the step edge are redshifted and at points on the left side are blueshifted, and the amount of modification increases by the distance from the step edge. However, the spectral modification becomes complicated by the further increase of the distance as a result of entering in Cornu spirals.Phase steps can be formed in different ways. For example, if in the Michelson or Mach-Zehnder interferometer we block the alternative halves of the beams incident to the mirrors by two opaque plates, a 1D phase step is formed [8], and by displacing one of the mirrors, the height of the step is varied. Another simple way for fabricating 1D step is to coat a film on a substrate in the form of a step. In this case the phase ϕ can vary by changing the incident angle.On the top of Figure2 the spectrum of the light beam is plotted that was used to illuminate a 1D step formed in the Michelson interferometer. The other plots are the spectral profiles of the diffracted lights from the step at different distances from the step edge. The plots on the left side are the corresponding spectral profiles obtained by simulation. There are remarkable differences between the spectra corresponding to the same distances on the sides of the step edge.(a) The spectral profile of the light incident to a phase step formed by the Michelson interferometer versusλ/λ0 (λ0=510 nm). The plots (b′), (c′), and (d′) on the right are the experimental spectral profiles of the incident lights (a) diffracted from the step at points corresponding to different distances from the step edge. The plots (b), (c) and (d) on the left are the corresponding spectral profiles obtained by simulation.In Figure3 the diffraction patterns of white lights are shown, that diffracted from a step formed by coating aluminum on a substrate for different incident angles. The film thickness was 440 nm.Diffraction patterns of spatially coherent white light diffracted from a 1D phase step of heighth=440 nm constructed by coating aluminum film on a glass substrate. The changes in colors occur by changing the light incident angle.
(a)(b)(c)(d)Another interesting phase singularity happens at Brewster angle for TM component of incident light beam illuminating the surface of a dielectric at angles close to the Brewster angle. For this component we have a step of height equivalent to a phase change ofπ at Brewster angle. Using the Fresnel-Kirchhoff integral and applying the Fresnel equations for the amplitude reflection coefficients, the following modifying function is obtained [9]:(11)M(θ,λ)=K2Rλ[C2(θ,λ)+S2(θ,λ)],
where K is the propagation factor, R is the distance between the light incident point and observation point, C(θ,λ), and S(θ,λ) are the Fresnel sine and cosine integrals at incident angle θ that is close to the Brewster angle. In Figure 4 the modified spectral profiles in the neighborhood of the Brewster angle versus λ are plotted for an incident light of Gaussian spectral profile. As the plots shown for angles less than Brewster angle θB, η<1 (η=θ/θB), the spectrum is blueshifted, and for η>1 it is redshifted.Figure 4
Modified spectral profiles for TM component of the incident light in the neighborhood of Brewster angleθB for five different incident angles, η=θ/θB.


## 3. Spectral Modification of the Light Scattered from Rough Interfaces

When a spherical wave that is originated from a point object strikes a flat smooth surface, the reflected wave is coherent on a sphere centered at the image of the object, Figure5(a). However, if a spherical wave is incident to a flat rough surface, we face different sections of waves reflected from a large number of facets at different heights that are not correlated. Therefore, the image is not observed, Figure 5(b). But as the object gets closer to the rough interface, in other words, for large incident angles, the waves diffracted from different facets become correlated, and for very large incident angles the spherical wave is restored and the image reappears. Thus, at very large incident angles rough surfaces look smooth. In fact, it is easy to see from Figure 6 that the optical path difference between the lights reflected from two facets at height difference h, at two arbitrary locations, is the following:(12)Δ=2hcosθ.
Thus, we conclude that when a coherent parallel beam of light is incident to a rough interface, depending on the roughness, incident angle, and light wavelength, there may be coherently scattered light in specular direction. However, by the increase of incident angle the spatial coherence property of light improves, and beyond a threshold incident angle we have partially coherent light in specular direction. Also we have diffusely scattered light in all directions. The partially coherent scattered lights interfere and lead to spectral modification, including redshift and blueshift. There are reports on spectral modification of light scattered from rough interfaces [10, 11]. But recently rather detailed theoretical and experimental studies of the subject in reflection and transmission modes have been reported [12–15]. According to these works the amplitude of the coherently scattered monochromatic light from a randomly rough interface versus incident angle is the Fourier transform of the height distribution on the rough interface. In addition, for polychromatic incident light the coherently scattered light amplitude versus wave number is also the Fourier transform of the height distribution on the rough interface. Calculation of the coherently scattered light intensity leads to the following expression [13–15]:(13)S(k,Pn,f)=S0(k)β2|∫-∞+∞Pn(h)exp(-ikhf)dh|2,
where S0(k), β, and Pn(h) denote the illuminating light intensity at wavenumber k, a dimensionless constant, and the probability density function of height distribution on the rough interface. Also, we should substitute(14)f=2cosθi
in reflection mode and(15)f=N1(N2-sin2θi-cosθi)
in transmission mode, where θi is the incident angle and N=N2/N1 where N1 and N2 are the refractive indices of the media surrounding the rough interface. As shown in (13) the coherently scattered light intensity depends on k, f, and Pn(h). Thus, we have spectral modification with the following modifying function:(16)M(k,Pn,f)=β2|∫-∞+∞Pn(h)exp(-ikhf)dh|2.
For an incident light of Gaussian spectral profile(17)S0(k)=s0β2exp(-(k-k0)22σk2),
the modified spectral profile becomes(18)S(k,pn,f)=S0exp(-k02σ2f2)×exp[-(k-k0/(1+2σk2σ2f2))22σk2/(1+2σk2σ2f2)],
where σ is the RMS of height distribution.We see that when s0exp(-k02σ2f2) is appreciable, we have the following redshift and spectral width shrinkage:(19)δk0=k01+1/2σk2σ2f2,σM=σk1+2σk2σ2f2.
According to (18) and (19) when s0exp(-k02σ2f2) is appreciable, the coherently scattered light spectrum is always redshifted with a shrinkage in spectral width. This result implies that in this situation the diffusely scattered light is blueshifted. Theoretical predictions and experimental results show that blueshift reduces with the increase of scattering angle. In addition, the spectral modification is significant near the threshold angle, the angle at which the coherently scattered light intensity becomes nonzero. In Figure 7 the spectral profiles of the incident beam, (r), and the coherently scattered beams, (a) and (b), and the diffusely scattered lights at scattering angle θs=1∘, (c) and (d), are plotted for interfaces having roughness σ=0.42 μm and σ=0.08 μm, respectively.(a) The image of a point object is observed in a smooth surface because the reflected wave front is a continuous spherical surface. (b) The image is not observed in a very rough interface because the waves reflected from different parts of the rough interface do not form a continuous wave front.
(a)(b)Figure 6
The optical path difference between the rays reflected from two parallel facets at arbitrary locations,2hcosθ, reduces as incident angle increases.Figure 7
The measured spectral profiles of the incident light beam, (r), the coherently scattered lights, (a) and (b), and the diffusely scattered lights at scattering angleθs=1∘, (c) and (d), from interfaces of roughness σ=0.42 μm and σ=0.08μm, respectively.


## 4. Conclusions

(1)
Spectral modification by diffraction of light from phase steps is a rich subject and can provide adequate ground for deeper studies of phase singularities.(2)
Polychromatic light diffraction from phase steps has applications in metrology, optical switching, and in dealing with transparent nonlinear materials.(3)
The scattering of polychromatic lights from rough interfaces gives better understanding of the optical field near the rough interfaces and it can provide valuable information about the properties of rough interfaces.(4)
The results obtained can be applied to transparent materials that impose irregular or random phases on lights propagating through them.



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*Source: 613728-2010-05-23.xml*