The role of coherence in near-field optics

Eugenio R. Mendez

Division de Fısica Aplicada, CICESE,Ensenada, Mexico

Outline

• Introduction

• A principle of equivalence

• Partial coherence in near field optics

• Conclusions.

In collaboration with:

• J-J. Greffet and R. Carminati, Ecole Centrale (Paris)

• A. A. Maradudin and T. A. Leskova, UC Irvine

Introduction

The Conventional Microscope

In the image plane:

Photographic film

CCD array, etc.

objective

object image

ocular eyeillumination

Mode of image formation determined by

the coherence of the illumination

Resolution determined by size of PSF

Conventional Optical Scanning Microscopy

objective

source

object

image

scan

detector

objective

source

detector

objectscan

Principle of equivalence (based on reciprocity) Linear optics!

Mode of image

formation determined by:

Size of source

Size of detector

Resolution determined by:

Size of PSF

Resolution

Periodic object: Period T Grating equation

-1

1

0

ObjectImage

-1

0

ObjectImage

Near-field optics

source

laser

sharpened

optical fiber

aperture << l

detector

object

scan

Scanning Near-field

Optical Microscopy

detector

detector

point

source

object

object

scan

scan

Synge (1929)

Ash (1982)

It is commonly believed that:

• The PSTM and the illumination mode SNOM are fundamentally dif-

ferent instruments

• The coherence of the illumination in the collection mode SNOM influ-

ences the resolution [OL 18, 2090 (1993)]

Detector

PSTM

Prism

optical

fiber

Lens

collimating

lens

spatial

filter

sample

Gaussian beam (laser)

Coupler

Lens

Detector

Illumination mode SNOM

scan

scan

Gaussian beam (laser)

sample

Reciprocity and equivalence

point

source

point

detector

point

detector

R

(a) (b)

point

detector

point

source

point

source

R

p1

p2

r 0

p1

r 0

p2

A1 · E(p1; r0) = A0 · E(r0;p1)

(a) E(p1; r0) electric field at p1 due to a dipole of amplitude A0 at r0.

(b) E(r0;p1) electric field at r0 due to a dipole of amplitude A1 at p1.

For randomly orientated dipoles of equal strengths,

I(p1; r0) = I(r0;p1)

optics

optical

fiber

point

source

sample

optics

point

detector

(a)

optics

optical

fiber

point

detector

sample

optics

point

source

(b)

r0

p1

r 0

p1

optics

optical

fiber

extended

detector

sample

optics

incoherent

source

(b)

optics

optical

fiber

incoherent

source

sample

optics

extended

detector

(a)

p

rr

p

Incoherent sources and “incoherent detectors”

Principle of equivalence (aas in conventional scanning microscopy).

detector

PSTM

focusing lens

prism

sample

optical

fiber

lens

incoherent

source

Equivalent illumination mode SNOM

collimating lens

prism

sample

optical

fiber

lens

(a) (b)

pinhole

pinhole

narrow band filter

Gaussian aperture

detector

polarized Gaussian

beam

polarizer

(a) (b)

sample

optical

fiber

polarized

Gaussian beam

coupler

lens

detector

lens

source

narrow

band filterdetector

Gaussian

aperture

sample

optical

fiber

pinhole

polarizer

Equivalent collection mode SNOMIllumination mode SNOM

The near-field intensity under partially coherent illumination

Assumptions:

• Perfectly conducting one-dimensional surface.

• S-polarized incoherent quasimonochromatic (∆ν << ν) source

• Passive probe.

x1

x3

Incoherentsource

β

cθ

I(k)

Objective:

Find a relation between ζ(x1) and the near-field intensity.

Partially coherent illumination

• Γ12(τ) = 〈ψ(P1, t+ τ)ψ(P2, t)〉 - mutual coherence function.

• J12 = Γ12(0) - mutual intensity.

These quantities involve averages over an ensemble of realizations.

Employing Hopkin’s formula, we can write the incident mutual intensity

Jp,s(x1, x3;x′1, x

′3|ω)inc =

∫σ

dkΨp,s(x1, x3|k|ω)incΨ∗p,s(x

′1, x

′3|k|ω)inc ,

where

Ψp,s(x1, x3|k|ω)inc =√I(k)eikx1−iα0(k,ω)x3 ,

For (x1, x3) = (x′1, x′3), we obtain the incident intensity

Ip,s(x1, x3)inc =

∫σ

dk |Ψp,s(x1, x3|k|ω)inc|2 =

∫σ

dk I(k) .

The total and scattered intensities

The total field in the region x3 > ζ(x1)max, is

Ψp,s(x1, x3|k|ω)tot =√I(k)eikx1−iα0(k,ω)x3 + Ψp,s(x1, x3|k|ω)sc ,

The total intensity can be written in the form

Ip,s(x1, x3)tot = Ip,s(x1, x3)inc + Ip,s(x1, x3)sc +

+

∫σ

dk 2<e{Ψp,s(x1, x3|k|ω)incΨ∗p,s(x1, x3|k|ω)sc} ,

where we have defined

Ip,s(x1, x3)sc =

∫σ

dk |Ψp,s(x1, x3|k|ω)sc|2 ,

and

Ip,s(x1, x3)tot =

∫σ

dk |Ψp,s(x1, x3|k|ω)tot|2 .

First order perturbation

The scattered field due to a plane wave component of the incident field is

given by

Ψp,s(x1, x3|k|ω)sc =√I(k)

∫ ∞

−∞

dq

2πRp,s(q|k)eiqx1+iα0(q,ω)x3.

To first order in ζ(x1), the scattering amplitude has the form

Rp,s(q|k) = −2πδ(q − k) + 2iα0(k, ω)ζ(q − k).

where

• ζ(Q) =∫ ∞−∞ dx1e−iQx1ζ(x1)

The Scattered Intensity

The integrated scattered intensity at the point (x1, x3) is

Is(x1, x3)sc = Is +

∫ ∞

−∞duζ(x1 − u)Fs(u, x3)sc,

where the functions Is and Fs(u, x3)sc are given by

Is = I0∆k = 2I0(ωc

)cos θc sinβ ,

Fs(u, x3)sc = 4I0Im

∫ kc+∆k/2

kc−∆k/2dkα0(k, ω)e−iku−iα0(k,ω)x3

∫ ∞

−∞

dq

2πeiqu+iα0(q,ω)x3.

F s(x

1,x 3

) sc / I

s

-30

-20

-10

0

10

F s(x

1,x 3

) sc / I

s

-30

-20

-10

0

10

x1 [wavelengths]

β=0.01o

β=30o

Impulse Response Function - Scattered Intensity

x3=λ/20

x3=λ/20

(a)

(b)

θ =0o0

θ =0o0

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

x3=λ/20

β=0.01o

β=90o

x3=λ/20

(c)

(d)

θ =30o0

θ =0o0

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

x1 [wavelengths]

Dependence of the impulse response function of the scattered field on the

kind of illumination.

-30

-20

-10

0

10

-30

-20

-10

0

10

(b)

(a)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

x1 [wavelengths]

x3=λ/20

x3=λ/40

θ =0o0

β=0.01o

θ =0o0

β=0.01o

(c)

(d)

θ =0o0

β=0.01o

x3=λ/10

x3=λ/5

θ =0o0

β=0.01o

F s(x

1,x 3

) sc / I

sF s

(x1,

x 3) s

c / I

s

Impulse Response Function - Scattered Intensity

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

x1 [wavelengths]

Dependence of the impulse response function of the scattered field on the

distance from the surface.

Fs(u, x3)sc is strongly peaked at u = 0 for small values of x3.

Thus,

Is(x1, x3)sc ' Is + ζ(x1)

∫ ∞

−∞duFs(u, x3)sc,

It can be shown, however, that the integral vanishes.

Returning to the convolution integral, we then expand ζ(x1 − u) in powers

of u, and integrate term-by-term, the first nonzero term yields

Is(x1, x3)sc ' Is +1

2ζ ′′(x1)

∫ ∞

−∞duu2F (u, x3)sc.

So, the scattered intensity at constant height will resemble more closely

the second derivative of the surface profile, rather than the profile itself.

The claim that Is(x1, x3)sc as a function of x1 for a fixed value of x3 follows

the surface profile function ζ(x1) is not generally valid.

The total intensity

To first order in ζ(x1), the total intensity at the point (x1, x3) is

Is(x1, x3)tot = Is(x3)tot +

∫ ∞

−∞duζ(x1 − u)Fs(u, x3)tot,

where

Is(x3)tot = 4I0

∫ kc+∆k/2

kc−∆k/2dk sin2 α0(k, ω)x3,

and

Fs(u, x3)tot = −8I0Re

∫ kc+∆k/2

kc−∆k/2dke−ikuα0(k, ω) sinα0(k, ω)x3 ×

×∫ ∞

−∞

dq

2πeiqu+iα0(q,ω)x3.

F s(x

1,x 3

) tot /

I s(x

3)to

t

-500-400-300-200-100

0

-200

-150

-100

-50

0

Impulse Response Function - Total Intensity

(b)

(a)

x3=λ/20

β=0.01o

θ =0o0

β=0.01o

θ =0o0

x3=λ/40

F s(x

1,x 3

) tot /

I s(x

3)to

t

-80

-60

-40

-20

0

-25-20-15-10

-505

10

(c)

(d)

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

x1 [wavelengths]

β=0.01o

θ =0o0

β=0.01o

θ =0o0

x3=λ/10

x3=λ/5

F s(x

1,x 3

) tot /

I s(x

3)to

tF s

(x1,

x 3) to

t / I s

(x3)

tot

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

x1 [wavelengths]

Dependence of the impulse response function of the total field on the

distance from the surface.

-200

-150

-100

-50

0

-200

-150

-100

-50

0

Impulse Response Function - Total Intensity

(a)

(b)

β=0.01o

β=30o

x3=λ/20

x3=λ/20

θ =0o0

θ =0o0

F s(x

1,x 3

) tot /

I s(x

3)to

tF s

(x1,

x 3) to

t / I s

(x3)

tot

-200

-150

-100

-50

0

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

-200

-150

-100

-50

0

(c)

(d)

x3=λ/20

β=0.01o

β=90o

x3=λ/20

θ =30o0

θ =0o0

x1 [wavelengths]

F s(x

1,x 3

) tot /

I s(x

3)to

tF s

(x1,

x 3) to

t / I s

(x3)

tot

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

x1 [wavelengths]

Dependence of the impulse response function of the total field on the kind

of illumination.

I s(x 1

,x3)

tot/

I s(x 3

) tot

0.0

0.5

1.0

I s(x 1

,x3)

tot/

I s(x 3

) tot

0.0

0.5

1.0

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

I s(x 1

,x3)

tot/

I s(x 3

) tot

0.0

0.5

1.0

(b)

(c)

(d)

β=0.01o

θ =0o0

x3=λ/20

x3=λ/10

x3=λ/5

x1 [wavelengths]

β=0.01o

θ =0o0

β=0.01o

θ =0o0

0.000

0.005

0.010

0.015

0.020

0.025

(a)

ζ (x 1

) [w

avel

engt

hs]

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

x1 [wavelengths]

The surface profile and the total intensity for a surface profile function

of the form ζ(x1) = A exp(−(x1 − B)2/R2) + A exp(−(x1 + B)2/R2) with,

A = 0.02λ, R = 0.1λ, and B = 0.2λ.

0.0

0.5

1.0

0.0

0.5

1.0

(b)

(a)I s(x 1

,x3)

tot/

I s(x 3

) tot

I s(x 1

,x3)

tot/

I s(x 3

) tot

β=0.01o

β=30o

x3=λ/20

x3=λ/20

θ =0o0

θ =0o0

0.0

0.5

1.0

0.0

0.5

1.0

(c)

(d)

I s(x 1

,x3)

tot/

I s(x 3

) tot

I s(x 1

,x3)

tot/

I s(x 3

) tot

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

x1 [wavelengths]

x3=λ/20

β=0.01o

β=90o

x3=λ/20

θ =30o0

θ =0o0

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

x1 [wavelengths]

The total near field intensity for the double Gaussian groove as a function

of the kind of illumination.

For small values of x3, the impulse response function is a sharply peaked

function of x1 and we can write

I(x1, x3)tot ' I(x3)tot + ζ(x1)

∫ ∞

−∞duF (u, x3)tot − ζ ′(x1)

∫ ∞

−∞duuF (u, x3)tot +

+1

2ζ ′′(x1)

∫ ∞

−∞duu2F (u, x3)tot.

It can be easily verified that:

• the term proportional to ζ ′(x1) vanishes for symmetrical illumination

modes.

• the term proportional to ζ ′′(x1) provides a small correction.

It follows that we can estimate the surface profile from the equation

ζ(x1) 'I(x1, x3)tot − I(x3)tot∫ ∞

−∞ duF (u, x3)tot.

Conclusions

• Equivalence between PSTM and SNOM.

• The resolution is not affected by the coherence of the illumination.

• The image (and resolution) deteriorates as one moves away from the

surface.

• The scattered intensity does not resembles the profile [ζ ′′(x1)].

• The total intensity does resemble the profile.