Nonlinear Near-Field Tomography

George Y. Panasyuk

University of Pennsylvania

Philadelphia, PA

SPM (AFM, STM, NSOM, …)

NSOM (Near-Field Scanning Optical Microscopy)

Near-Field Optical Tomography

NSOM measurements 3D sample structure

Near-Field Scanning Optical Microscopy(NSOM)

E. Betzig and J. Trautman. Science 1992

- Illumination mode

- Collection mode

3D Effects in NSOM

collection

modez measured in units of wavelength

Topographic images

Topographic image

Near-Field images

Near-Field images

Local Spectra of Clusters of Silver Particles

V.Markel, V.Shalaev, M Moskovits at al, PRB 1999

Reconstruct the medium from measurements of

the scattered field

Incident wave

Scattered wave

Scattering medium

Inverse Scattering

Collection Mode Near-Field Tomography

Vary direction of incidence and scan probe

measure amplitude and phase

.

.

.

Near-Field Optical Tomography

z measured in units of wavelength

S.Carney, J.Schotland, App. Phys. Lett. 2000

Photon Scanning tunneling microscopy (PSTM) Experiment

Tip height 200 nm

FOV = 300 nm x 500 nm

= 633 nm

Sample: gold nanoparticle

DATA

PSTM Reconstructions

Reconstructed Image AFM Image

S.Carney, S.Bozhevolnyi, J.Schotland, Phys. Rev. Lett. 2004

Re((gold)) = -0.9

Collection Mode Near-Field Tomography

Vary direction of incidence and scan probe

measure amplitude and phase

.

.

.

Basic EquationsMaxwell Equation for Electric Field

2 2 20 0( ) ( ) ( ) 4 ( ) ( )E r n z k E r k r E r

Equivalent integral equation

20( ) ( , ')[ ( ') ( ')] ( ')scat inc scatE r k drG r r E r E r r

�

Linearization: 20( ) ( , ') ( ') ( ')scat incE r k drG r r E r r

�

r

'r'( , )G r r

�

Forward series

;G�

incE

1 2 2 2 2 1 1 1

0 0

( ; , ) ( ; ) ( ; , ) ( ; ) ( , )L L

incddz dz dpG q Q z z p q Q z G p z z q p z E q z

����

= + + + …

= = η; =

=

=

0

( , ) ( , ) ( , )L

incdzG q Q z Q z E q z K ���

scatE

Inverse Series

η = η(1) + η(2) + η(3) + η(4) …

η(1) = = K+Es,

η(2) =

η(3) = + +

η(4) = 7 diagrams, …

= K+

Linear = η(1)

η(1)+η(2)+η(3)+η(4)

η(1) + η(2)

η(1)+η(2)+η(3)

Equatorial crossection

recon

spher

x/λ

Conclusions:

- The main result is an approach developed for solving the non-linear inverse problem;

- Non-linear corrections improve significantly the accuracy and quality of a reconstructed image.

Future work :

- Developing of a non-linear approach for solving the inverse problem for the apertureless modality.