Diffractive Optical Elements and

(1)

Diffractive Optical Elements and Micro-Optics

이 병 호

서울대 전기공학부 byoungho@snu.ac.kr

(2)

OEQELab

Seoul National University

Diffraction

Sommerfeld:

“any deviation of light rays from rectilinear paths which cannot be

interpreted as reflection or refraction”

(3)

OEQELab

Examples of Fraunhofer Diffraction

Fraunhofer diffraction from a

rectangular aperture. The central lobe of the pattern has half-angular widths

y y

x

x  /D and  / D

  

The Fraunhofer diffraction

pattern from a circular aperture produces the Airy pattern with the radius of the central disk

subtending an angle _ ₁_.₂₂_ _/ _D

(4)

OEQELab

Fresnel Diffraction by Square Aperture

(b) Diffraction pattern at four axial positions marked by the arrows in (a) and

corresponding to the Fresnel numbers N_F=10, 1, 0.5, and 0.1. The shaded area represents the geometrical shadow of the slit. The dashed lines at represent the width of the Fraunhofer pattern in the far field. Where the dashed lines coincide with the edges of the geometrical shadow, the Fresnel number N_F=

 D^d

x  /

. 5 0

2 / d .

a  

Fresnel Diffraction from a slit of width D = 2a. (a) Shaded area is the geometrical shadow of the aperture. The dashed line is the width of the Fraunhofer diffracted beam.

(5)

OEQELab

Diffractive Optics Technology

Provides new degrees of freedom for the design and optimization of optical systems

 Features

Large aperture, lightweight optical elements

Aberration correction and achromatization

Eliminates need for exotic materials

Increase in performance over conventional systems

System weight, complexity, and cost can be reduced significantly

 Applications

(6)

OEQELab

Fabrication of a Surface-Relief Master

 Lithography – Optical & E-Beam

Staircase blaze profiles

 Binary optics – multiple e-beam masks

Continuous blaze profiles

 Gray-scale masks

 Variable e-beam exposure

 Single-Point Diamond Turning

Linear and spherical blaze profiles

 Single-Point Laser Pattern Generation

Vary exposure to shape blaze profile

X-Y and R-theta scan geometries

(7)

OEQELab

Diffractive Optical Element (DOE) (I)











 



 

 



 



 

N q N

q q q

2 sinc 2

sinc 2 sinc 2

0 2

2 0 2











(8)

OEQELab

Diffractive Optical Element (DOE) (II)

(9)

OEQELab

Advances in Single-Point Diamond Turning

1989

Diffraction Efficiency

 : 85% - 90%

1989

Diffraction Efficiency

 : 85% - 90%

1999

Diffraction Efficiency

 : 97% - 99%

G. M. Morris

(10)

OEQELab

Laser Pattern Generator

G. M. Morris

(11)

OEQELab

Micro-Optics Manufacturing Technique

using Reactive Ion Etching

(12)

OEQELab

Microlens (I)

(13)

OEQELab

Microlens (II)

(14)

OEQELab

Microlens (III)

(15)

OEQELab

Wavefront Sensing

(16)

OEQELab

- Micro-sized convex lens array

- Optical switching device for optical communication system

- Focusing device for optical data storage - 3-D display component for super-multi view - Various application for photonic devices

[3D-image system]

Microlens Array

(17)

OEQELab

- Patterned electrode type - Surface-relief structure type

- FLC microlens using surface relief - Input polarization dependence

• In practical applications,

Independence of Input Polarization is needed !!

Liquid Crystal Microlenses

(18)

OEQELab

Lord Rayleigh

“…If it were possible to introduce at every part of the aperture of the grating an arbitrary retardation, all the light might be concentrated in any desired spectrum. By supposing the retardation to vary uniformly and continuously we fall upon the case of an ordinary prism; but there is then no diffraction spectrum in the usual sense. To obtain such it would be necessary that the retardation should gradually alter by a wave-length in passing over any element of the grating, and then fall back to its previous value, thus springing suddenly over a wave-length. It is not likely that such a result will ever be fully attained in practice; but the case is worth

stating, in order to show that there is no theoretical limit to the concentration of light of assigned wave-length in one spectrum…”

Encylopaedia Britannica, 9^thed., Vol. 24, “Wave Theory of Light” (New York, Charles Schribner’s Sons, 1888), p. 437

J. W. Strutt (1842-1919)

G. M. Morris

Never say ‘Never’!

(19)

OEQELab

Tracor Flat-Panel ANVIS E-HUD*

(20)

OEQELab

Application Example

G. M. Morris

(21)

OEQELab

Micropattering Using DOE and Femtosecond Laser

Y. Kuroiwa et al., Optics Express, vol. 12, no. 9, pp. 1908-1915, 2004.

(22)

OEQELab

Theoretical study on design and analysis of DOEs

DOE design problem

 Basic layout of paraxial beam shaping system

 ₂^, ₂  ₂^, ₂^, ₁^, ₁  ₁^, ₁ ₁ ₁^,

F x y h x y x y W x y dx dy

 

 



 



1, 1

  

1, 1

 

exp

 ^

1, 1

^ 

W x y  A  x y j x y

 Surface relief structure of DOE

Input field Output field



x y1, 1





 Diffraction image F x y



2, 2



(23)

OEQELab

Nonlinear optimization methods for DOE design

 Hybrid method

   

2

1 1 2 2

Find , for minimizing ,

L

x y E I F x y

  

 Minimization scheme

 Iterative methods

 Iterative Fourier transform algorithm (IFTA)

 Nonlinear conjugate gradient method (NCGM)

 Stochastic methods

 Simulated annealing method (SA)

 Genetic algorithm (GA)

 SA or GA + IFTA or NCGM

(24)

OEQELab

Iterative Fourier transform algorithm (IFTA) for DOE design

Soft constraint Amplitude and phase

freedoms Hard constraint

Functional relationship between phase and amplitude modulations

  

^exp



n n n

W  A  j

Input plane

Diffraction image Phase

hologram

Forward Fresnel transform

Inverse Fresnel transform

Output plane

IFTA

Existence of the analytic expression of the inverse Fresnel transform is the key of IFTA.

(25)

OEQELab

Regularization technique for obtaining the optimal trade-off between diffraction efficiency and uniformity

H. Kim, B. Yang, and B. Lee, Journal of the Optical Society of America A, vol. 21, p. 2353, 2004.

H. Kim and B. Lee, Japanese Journal of Applied Physics, vol. 43, no. 6A, p. L702, 2004.

76 78 80 82 84 86 88 90 92

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Uniformity

Diffraction efficiency (%)

Proposed Conventional IFTA

IFTA

76 78 80 82 84 86 88 90 92

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Uniformity

76 78 80 82 84 86 88 90 92

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Uniformity

Proposed Conventional IFTA

IFTA Proposed Conventional IFTA

IFTA

 IFTA scheme with Tikhonov regularization scheme (adaptive regularization parameter distribution)

Diffraction efficiency=84.7%

Uniformity=0.0365

(arb. units)(arb. units)

Uniformity=0.0365

Uniformity=0.0746

Proposed design Conventional design

(26)

OEQELab

Hybrid scheme of IFTA and GA for optimal IFTA convergence

Micro genetic algorithm + IFTA

 Micro genetic algorithm finds the optimal relaxation parameter vector of the imbedded IFTA scheme.

 IFTA shows non-monotonic optimal convergence.

Micro genetic algorithm

for ( , )x y S for ( , )x y S Fn

     

 

1 0 0

0 2

, ,

exp 1 2 tan

, exp

n n

n n n n

n D n n

F x y F x y

F j F

F x y

F i

   



  

    

 

    

 

Fn for ( , )x y S

for ( , )x y S Fn

     

 

1 0 0

0 2

, ,

exp 1 2 tan

, exp

n n

n n n n

n D n n

F x y F x y

F j F

F x y

F i

   



  

    

 

    

 

Fn

Fn

     

 

1 0 0

0 2

, ,

exp 1 2 tan

, exp

n n

n n n n

n D n n

F x y F x y

F j F

F x y

F i

   



  

    

 

    

 

Fn

     

 

1 0 0

0 2

, ,

exp 1 2 tan

, exp

n n

n n n n

n D n n

F x y F x y

F j F

F x y

F i

   



  

    

 

    

 

Fn

IFTA

H. Kim and B. Lee, Optics Letters, vol. 30, p. 296, 2005.

(27)

OEQELab

0 250 500 750 1000 1250 1500

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Uniformity

Iteration number

0 50 100 150 200

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

0.8

0.6

0.4

0.2

0 50 100 150 200

0 250 500 750 1000 1250 1500

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Uniformity

0 50 100 150 200

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

0.8

0.6

0.4

0.2

0 ⁰ 50 ⁵⁰ 100 ¹⁰⁰ ¹⁵⁰150 200²⁰⁰ 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

0.8

0.6

0.4

0.2

0 50 100 150 200

0 250 500 750 1000 1250 1500

50 55 60 65 70 75 80 85 90

Diffraction efficiency

0 50 100 150 200

50 55 60 65 70 75 80 85

9090

80

70

60

50

0 ⁰ 50 ⁵⁰ 100 ¹⁰⁰ 150 200¹⁵⁰ ²⁰⁰

50 55 60 65 70 75 80 85

9090

80

70

60

50

0 50 100 150 200

0 250 500 750 1000 1250 1500

50 55 60 65 70 75 80 85 90

Diffraction efficiency

0 50 100 150 200

50 55 60 65 70 75 80 85

9090

80

70

60

50

0 ⁰ 50 ⁵⁰ 100 ¹⁰⁰ 150 200¹⁵⁰ ²⁰⁰

50 55 60 65 70 75 80 85

9090

80

70

60

50

0 50 100 150 200

x (sam pling in

dex) y (sampling index)

Intensity (a.u.)

Diffraction efficiency=85.7%

Uniformity=0.113 x (sam

pling in

Intensity (a.u.)

Uniformity=0.113

Proposed design

dex) y (sampling index)

Intensity (a.u.)

Uniformity=0.401 x (sam

pling in

Intensity (a.u.)

Uniformity=0.401

Conventional design

Non-monotonic optimal convergence of IFTA

(28)

OEQELab

Speckle reducing problem in DOE design

Phase profile of conventional DOE Obtained diffraction image

Optical vortices appear in the diffraction image generated by a conventional DOE

(29)

OEQELab

Cross-section of input beam

Obtained diffraction image Phase profile of the proposed DOE

H. Kim and B. Lee, Japanese Journal of Applied Physics, vol. 43, no. 12A, p. L1530, 2004.

Boundary modulated DOE design

Optical vortices are eliminated in the diffraction image generated by the proposed boundary modulated DOE

(30)

OEQELab

Accurate modeling of the transmittance functions of DOEs



t

Incident plane wave

Transmitted wave

Internal reflections

nI

nII

nI

Back scattered wave



t

Incident plane wave

Transmitted wave

Internal reflections

nI

nII

nI

Back scattered wave

H. Kim and B. Lee, Optical Engineering, vol. 43, no. 11, p. 2671, 2004.

The transmittance function of a multilevel DOE is calculated considering internal multiple reflections inside the DOE structure.

(31)

OEQELab

Phase error (deg.)

X direction position index Y dire

ction position

index

Phase error (deg.)

ction position

index

Phase error (deg.)

ction position

index Normalized thickness

ction position

index

Normalized thickness

ction pos ition

inde x Normalized thickness

ction pos ition

inde x Normalized thickness

ction pos ition

inde x

Normalized thickness

X direction pos

ition index Y dire

ction pos ition

X direction pos

ition index Y dire

ction pos ition

X direction pos

ition index Y dire

ction pos ition

index

Phase error (deg.)

X direction position index Y di

rec tion pos

ition inde

x Phase error (deg.)

rec tion pos

ition inde

x Phase error (deg.)

rec tion pos

ition inde

x

Conventional design

Normal incidence

Oblique incidence

Phase error

Proposed design without phase errors

Refractive index=3.4

Comparison of the proposed transmittance function with

the conventional transmittance function

(32)

OEQELab

 Laser display system and speckle problem

 The speckle patterns impair the image quality (strong deblurring of the color edges)

 Low transmission efficiency and deterioration of the laser beam quality

 Coherence degradation of the light source, diffuser, digital image processing of signals

R

B

G M

M

Laser M

Modulator Dichroic Mirror

Projection Scanning system

Screen

Resolution Spot

Observers Sync.

signals

Surface height fluctuation

 Uncorrelated speckle =c / 

d = 2.2mm f# = 3.5 ~ 5.6

1/ 2

det _

( _proj / ) 2 _pixel /( _pupil _size #_eye)

C    h d  f

Speckle reduction for laser display

(33)

OEQELab

Seoul National University IEEE LEOS ’99, pp.354-355.

35 97.2

0.009

All vibrated components & lens arrays

0 89.5

0.034 Vibrated projection screen

5 85.1

0.048 Vibrated diffractive diffuser

20 82.4

0.057 Vibrated fiber

20 18.9

0.262 Fiber, 2m

14 4.3

0.309 Refractive lens array(68)

5 3.4

0.312 Stationary diffractive diffuser

Insertion loss(%) Speckle reduction(%)

Speckle contrast Component

 Speckle configuration

- Speckle contrast ratio: “Severe” speckle is associated with contrast measurements of C > 0.30

 Comparison of speckle degradation methods





 

i i i

I I I

C

2 2 where, <I_i> : intensity of the i^thpixel of a CCD

 : standard deviation, : mean value

Speckle reduction for laser display

(34)

OEQELab

Nd:YAG, DPSS (500mW, 532nm)

VC44 CCD (756 581)

Laser

Image Processor

DOE

L₁

L₂ L₃

L₄ L₅

Polygon scanner Screen

f₁ f₂

Rotation

2

0

 phase (rad)

0 1 2 3 4 5

100 200 300 400 500

50 100 150 200 250 300 350 400 450 500

2

0

 phase (rad) 2

0

 phase (rad)

0 1 2 3 4 5

100 200 300 400 500

50 100 150 200 250 300 350 400 450

500 0

1 2 3 4 5

100 200 300 400 500

50 100 150 200 250 300 350 400 450 500

Speckle reduction using DOEs

(35)

OEQELab

Speckle pattern and its histogram of Nd:YAG laser source, (static DOE, and rotating DOE (5rps).

CR_Laser = 0.5126 CRDOE, static = 0.2729 CRDOE, rotation = 0.2388

Speckle reduction using DOEs

(36)

OEQELab

Theoretical studies on design and analysis of DOEs

DOE design for non-invertible Fresnel diffraction integral

IFTA is not applicable for the design of DOEs for making diffraction images on an arbitrarily curved surface.

(37)

OEQELab

Nonlinear conjugate gradient method

 

²

 

²

   

2

exp

N N

mn n n m

m n

E  ^ m G A  j I ^

 

 

 

Φ

 Objective function

Im

 

^m



Target image

Weight distribution

 Flow chart of NCGM

0  E

 

0

d Φ

Bracketing algorithm

Golden section line search

Fletcher-Reeves formula

1

k_  k k k

Φ Φ d

k _for min E



Φ_k_1;_k





 

1 1 k  E k k_ k_

d Φ d

 

2

1 2

1 k k

k

E E

 _



 



Φ Φ

1 k  k

(38)

OEQELab

Design example



₂ ₂



for x y, 

 Target image

 Curved output surface (defocus surface)

 

⁰ ⁷

  

¹

 

²



²

 

3

2 ^h 7 / 2 2 ^h 2

h h h

A  c c   ^   ^  



 



  

Amplitude function as a function of phase

 ⁰  ²

A  A  ^A^

 

⁰ ^ ^A^

 

²^

y (sam

pling p

oint) x (sampling point)

Intensity (normalized)

y (sam

pling p

oint) x (sampling point)

Intensity (normalized)

y (sam

pling point) x (sampling point)



^,



s x y

y (sam



^,



s x y

(39)

OEQELab

Diffraction simulation

Field amplitude distribution on the curved output surface

Field amplitude distribution on the flat plane (z₂=0)

y (sam pling

point) x (sampling point)

Intensity (a.u.)

y (sam pling

point) x (sampling point)

Intensity (a.u.)

y (sam

Intensity (a.u.)

y (sam

Intensity (a.u.)

y (sam pling p

oint)

x (sampling point)

Intensity (a.u.)

y (sam pling p

oint)

x (sampling point)

Intensity (a.u.)

y (sam

Intensity (a.u.)

y (sam

Intensity (a.u.)

DOE for curved surface

1 2 3 4 5 6

DOE for flat surface

(40)

OEQELab

Electromagnetic analysis on subwavelength scale DOEs

Fourier expansion of binary subwavelength scale grating



^,



mn ^exp ²

m n x y

m n

x y j x y

T T

 ^ ^  

 

  



 

   





pq

fpq

x y

L  N

^{p q}^, 

 

^{ }

2 2

2 1

1 1

,

1 [ ,

sinc exp ]

M M

mn mn pq

p q x y

pq pq mn pq

k x y

D p q n n f T T

m n

kf j k f L H k

T T





 





 

   

 

      





Binary surface relief structure on a subwavelength scale can modulate both phase and amplitude of the incident optical field.

(41)

OEQELab

wavelength

wavelen gth

permittivity

wavelength

wavelen gth

wavelength

wavelen gth

permittivity Phase (rad)

wavelen

gth wavelen

gth

Phase (rad)Phase (rad)

wavelen

gth wavelen

gth

wavelen

gth

wavelength

permittivity

wavelen

gth

wavelength

permittivity

wavelen gth

Phase (rad)

wavelen gth

Phase (rad)

Rigorous coupled wave analysis (RCWA)

Ex

wavelength wavelen

gth

Ex

wavelength wavelen

gth

Ex_x

E

wavelength wavelen

gth

Ex

wavelength wavelen

gth

Ex

wavelength wavelen

gth

Ex_x

E

wavelength wavelen

gth

The binary subwavelength scale gratings can modulate both phase and amplitude of the incident optical field.