The High Contrast Performance Of An Optical Vortex Coronagraph

By

Dr. David M. Palacios

Jet Propulsion LaboratoryCalifornia Institute of Technology

Stuart Shaklan Jet Propulsion Laboratory

G.A. Swartzlander Jr.University of Arizona

Dimitri Mawet University of

Acknowledgements

1.) What is an Optical Vortex?

2.) Optical Vortex Mask Design

3.) Lyot Optimization

4.) Planet Light Throughput Efficiency

5.) Conclusions

Outline

E(r,,z;t)A(r,z)exp(im)exp[i(tkz)]

0

2ππ

The Complex Field

Amplitude Phase

What is an Optical Vortex?

Laser θ0

N =πθ02 2λ2

Optical Vortices in Speckle

HG01 LG01

1-D Gouy Phase Shift

Cylindrical lens

Astigmatic Mode Converter

z

x

E0 exp(imθ) E0 exπ(−ixΛ)

Λ

I( z 0)2E02[1+ cos(mθ +

xΛ)]

Optical Vortex Holograms

The Optical Vortex Mask

Mask Thickness

Coronagraph Architecture

Final Image PlaneIncident Light

Pupil

L1

OVM Lyot Stop

L2 L3

FP

0

2πm

The Optical Vortex Mask

Mask Thickness

d

d

n1

n0

dz

θt

Ray Trace Analysis of the Vortex Mask

€

tanφ =dzdφ

€

dz=mλdφ

2π n1 −n0( )

€

φ =tan−1 mλ

2π n1 −n0( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

d

d

n1

n0

dz

The Vortex Core

€

TIR =sin−1n0

n1

⎛

⎝ ⎜

⎞

⎠ ⎟

€

c =mλ

2π n1 −n0( ) tanTIR( )

When c E Transmitted = 0

Output Amplitude Profile

Transmitted amplitude for the E Polarization

€

Eφ

E0

=1−μ1μ2n2

2cos φ( )−n1 n2

2−n1

2sin2 φ( )

μ1μ2n2

2cos φ( ) +n1 n2

2−n1

2sin2 φ( )

€

Er

E0

=1−n1 cos φ( )−

μ1μ2

n2

2−n1

2sin2 φ( )

n1 cos φ( ) +μ1μ2

n2

2−n1

2sin2 φ( )

Transmitted amplitude for the Er Polarization

0

0.2

0.4

0.6

0.8

1

0 5 10 15

m=2m=4m=6

r(lambda)

0

0.2

0.4

0.6

0.8

1

0 5 10 15

m=2 m=4 m=6

r(lambda)

A Discrete Representation of an OVM

0 8π

Phase profile of an m=4 OVM

dz

d

dz

d

Coronagraph Leakage!

Ideal OVC6 Pupil Discretized OVC6 Pupil

OVC Discretization Leakage

Numerical Simulations

Array Size

Pupil Size

λ

f #

Mask Pixel Size

n1

μ1

4096 x 4096 pixels

100 pixels in diameter

600 nm

27

0.2 microns

1.5

1

m=2 m=4 m=6

Even charged OVMs theoretically cancel the entire pupil!

The Lyot Plane for Even Values of m

System Performance

€

C =I x,y( )∫∫ dxdy

Iopen x,y( )∫∫ dxdy ⋅ o x,y( )2dxdy∫∫

Contrast

I(x,y) = Intensity with the occulter in place

Iopen(x,y) = Intensity with the occulter removed

o(x,y) = Occulter transmission function

10-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

0 2 4 6 8 10

Average Radial Contrast

Average Contrast Between 2-3 λ/DC

ontr

ast

m=6

r (λ/D)

10-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

0 2 4 6 8 10

Con

tras

t

m=6

Average Contrast Between 2-8 λ/D

Average Radial Contrast

r (λ/D)

10-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

0 2 4 6 8 10

Con

tras

t

m=6

Average Radial Contrast

Average Contrast Between 4-5 λ/D

r (λ/D)

10-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

0 2 4 6 8 10

Con

tras

t

m=6

Average Radial Contrast

Average Contrast Between 4-10 λ/D

r (λ/D)

10-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

0.5 0.6 0.7 0.8 0.9 1

Con

tras

t

m=2

m=4

m=6

Lyot Size (r/Rp)

Contrast vs. Lyot Size

Average Contrast Between 2-3 λ/D

10-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

0.5 0.6 0.7 0.8 0.9 1

Con

tras

t

m=2

m=4

m=6

Lyot Size (r/Rp)

Contrast vs. Lyot Size

Average Contrast Between 2-8 λ/D

10-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

0.5 0.6 0.7 0.8 0.9 1

Con

tras

t

m=2

m=4

m=6

Lyot Size (r/Rp)

Contrast vs. Lyot Size

Average Contrast Between 4-5 λ/D

10-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

0.5 0.6 0.7 0.8 0.9 1

Con

tras

t

m=2

m=4

m=6

Lyot Size (r/Rp)

Contrast vs. Lyot Size

Average Contrast Between 4-10 λ/D

Optimized Contrast

Lyot Stop Radius = 0.8Pr

Average Contrast

2-3 λ/D 2-8 λ/D 4-5 λ/D 4-10 λ/D

5.3x10-11 2.9x10-11 2.5x10-11 2.0x10-11

m = 2

m = 4

m = 6

1.2x10-10 5.1x10-11 4.1x10-11 2.5x10-11

2.8x10-10 9.3x10-11 6.4x10-11 3.5x10-11

0

0.2

0.4

0.6

0.8

1

0.5 0.6 0.7 0.8 0.9 1

Thr

ough

put

Lyot Size (r/Rp)

m=2

m=4

m=6

m=0

Throughput Efficiency vs. Lyot Size

Planet Located at 2λ/D

0

0.2

0.4

0.6

0.8

1

0.5 0.6 0.7 0.8 0.9 1

Thr

ough

put

Lyot Size (r/Rp)

m=2

m=4

m=6

m=0

Throughput Efficiency vs. Lyot Size

Planet Located at 4λ/D

m=6m=4m=2m=0

2λ/D

4λ/D

0.64

0.64 0.64 0.62 0.58

0.62 0.53 0.43

Optimized Planet Light Throughput

Lyot Stop Radius = 0.8Pr

Is an Achromatic OVC Possible?

10-13

10-12

10-11

10-10

10-9

10-8

C

m

6 6.0015.999

m must be maintained to ~5x10-4 across the bandpass!

f/30 beam

Holographic Vortex

Direction-compensating Grating

Zero-order blocker

Lyot Stop

Achromatic Holographic Vortex Coronagraph

System advantages

•Small inner working angle ~ 2λ/D

•High throughput (theoretically 100%)

• Same WFC architecture as other Lyot type coronagraphs

•Small polarization effects (dependent on creation method)

•Low aberration sensitivity to low-order Zernikes

•Large search area (radially symmetric)

•System can be chained in series

System Disadvantages

•Broadband operation requires further research on new OV creation techniques

• Issues with mask Fabrication or hologram fabrication are just beginning to be explored.

•The Useful throughput decreases with stellar size making operation at 2λ/D difficult on 0.1λ/D sized stars.

Conclusions

•An m=6 vortex coronagraph meets TPF contrast requirements

•Simulated 10-11 contrast at 2λ/D with a discretized OVM

• OVM discretized with 0.2 micron pixels

•Even charged OVMs theoretically cancel over the entire pupil

•With discretization errors the Lyot stop radius = 0.8Pr

•53% throughput efficiency at 2λ/D

•62% throughput efficiency at 4λ/D near optimal of 64%

Aberration Sensitivity

€

C = αΔγ

is the order of the aberration sensitivity

4th order linear sinc2 masks best demonstrated contrast

8th order masks presently being explored

Vortex masks possess a 2mth order aberration sensitivity

The Aberration Sensitivity

€

M(ρ ) = tanhm ρ wv( )

Mask Amplitude Transmission Function

€

E r ,θ( ) = P(r ) 1+i l

l!Φ l

l =1, 2, 3...

∑ r ,θ( ) ⎛

⎝ ⎜

⎞

⎠ ⎟

The Entrance Pupil

Assuming (r,θ) <<1,

€

M ρ( ) ≈ akρk

k =1, 2, 3...

∑

€

ak =1

wvk!

∂ k

∂ρ kM(ρ = 0)

More Math…

The Exit Pupil

€

Pexit r,θ( ) = E r,θ( )∗Hm M ρ( ){ }

€

H m ρ k f (ρ ){ } =−1

2π

⎛

⎝ ⎜

⎞

⎠ ⎟k

r m −k d k

dr k

1

r m −kH m −k f ρ( ){ }

⎡

⎣ ⎢ ⎤

⎦ ⎥

€

Pexit r ,θ( ) = P(r ) 1+i l

l!Φ l

l =1, 2, 3...

∑ r ,θ( ) ⎛

⎝ ⎜

⎞

⎠ ⎟∗ ak

−1

2π

⎛

⎝ ⎜

⎞

⎠ ⎟k

r m −k d k

dr k

1

r m −kδ r( )

⎡

⎣ ⎢ ⎤

⎦ ⎥k =m, m+1, m+ 2...

∑

Using the identity:

The Approximate Exit Pupil

The Approximate Solution

€

Pexit r ,θ( ) ≈ am

−1

2π

⎛

⎝ ⎜

⎞

⎠ ⎟m

dm

dr mP(r ) 1+

i l

l!Φ l

l =1, 2, 3...

∑ r ,θ( ) ⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

The first term in the expansion k=m

All terms with less than an rm dependence vanish!

The Intensity has a 2mth aberration sensitivity!

For the m=5 case:

10th order sensitivity predicted!

Low Order Zernike Modes

Z=4 Z=5 Z=6 Z=7

Z=8 Z=9 Z=10 Z=11

10-16

10-15

10-14

10-13

10-12

0.1 1

z=2z=4z=5z=7z=9z=11z=12

Numerical Simulations

Aberration size (waves peak to valley)

C

Coronagraph Comparisons

m=5 vortex8th OrderZernike #

23456789101112

88444444422

99--66445555

Improvement

Pupil Vortex Mask Lyot Stop

Lyot Plane Focal Plane

λ/D

The Lyot and Focal Plane Profiles

Amplitude Occulting Spots

E(x,y) = A(x,y)exp[i(x,y)]

Sinc2(r)

Hard Stop

The Lyot Stop

Hard Stop

Cat’s Eye Stop

The Final Image

Before After

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

m=4m=3

m=2

m=1

m=0

R/Rdiff

An Optical Limiting TechniqueA

mpl

itude

Contrast Simulations

Contrast Image

Compute the Radial Average Contrast