How to Make a Fresnel Lens - Illinois Institute...

Transcript of How to Make a Fresnel Lens - Illinois Institute...

$Page 1: How to Make a Fresnel Lens - Illinois Institute ofcsrri.iit.edu/~segre/phys570/10F/lecture_15.pdf · How to Make a Fresnel Lens The ideal refracting lens has a parabolic shape but$

How to Make a Fresnel Lens

The ideal refracting lens has a parabolicshape but this is impractical to make.

h(x) = Λ

(x√

2λo f

)2

when h(x) = 100Λ ∼ 1000µm

x = 10√

2λo f ∼ 100µm

aspect ratio too large for a stable structureand absorption would be too large!

C. Segre (IIT) PHYS 570 - Fall 2010 October 12, 2010 1 / 8

h(x) = Λ

(x√

2λo f

)2

when h(x) = 100Λ ∼ 1000µm

x = 10√

2λo f ∼ 100µm

h(x) = Λ

(x√

2λo f

)2

when h(x) = 100Λ ∼ 1000µm

x = 10√

2λo f ∼ 100µm

h(x) = Λ

(x√

2λo f

)2

when h(x) = 100Λ ∼ 1000µm

x = 10√

2λo f ∼ 100µm

h(x) = Λ

(x√

2λo f

)2

when h(x) = 100Λ ∼ 1000µm

x = 10√

2λo f ∼ 100µm

Λ

2Λ

3Λ

4Λ

5Λ

6Λ

7ΛMark off the longitudinal zones (of thicknessΛ) where the waves inside and outside thematerial are in phase.

Each block of thickness Λ serves nopurpose for refraction but only attenuatesthe wave.

This material can be removed and theremaining material collapsed to producea Fresnel lens which has the same opticalproperties as the parabolic lens as long asf � NΛ where N is the number of zones.

1

2

3

4

5

6

7 Mark off the longitudinal zones (of thicknessΛ) where the waves inside and outside thematerial are in phase.

1

2

3

4

5

6

7 Mark off the longitudinal zones (of thicknessΛ) where the waves inside and outside thematerial are in phase.

Fresnel Lens Dimensions

1

2

3

4

5

6

7

The outermost zones become very small andthus limit the overall aperture of the zoneplate. The dimensions of outermost zone, Ncan be calculated by first defining a scaledheight and lateral dimension

ν =h(x)

Λξ =

x√2λo f

Since ν = ξ2, the position of the Nth zoneis ξN =

√N and the scaled width of the Nth

(outermost) zone is

∆ξN = ξN − ξN−1 =√N −

√N − 1

1

2

3

4

5

6

7

ν =h(x)

Λ

ξ =x√

2λo f

(outermost) zone is

∆ξN = ξN − ξN−1 =√N −

√N − 1

1

2

3

4

5

6

7

ν =h(x)

Λξ =

x√2λo f

(outermost) zone is

∆ξN = ξN − ξN−1 =√N −

√N − 1

1

2

3

4

5

6

7

ν =h(x)

Λξ =

x√2λo f

(outermost) zone is

∆ξN = ξN − ξN−1 =√N −

√N − 1

1

2

3

4

5

6

7

ν =h(x)

Λξ =

x√2λo f

(outermost) zone is

∆ξN = ξN − ξN−1 =√N −

√N − 1

1

2

3

4

5

6

7

∆ξN = ξN − ξN−1 =√N −

√N − 1

=√N

(1−

√1− 1

N

)

≈√N

(1−

[1− 1

2N

])∆ξN ≈

1

2√N

The diameter of the entire lens is thus

2ξN = 2√N =

1

∆ξN

1

2

3

4

5

6

7

∆ξN = ξN − ξN−1 =√N −

√N − 1

=√N

(1−

√1− 1

N

)

≈√N

(1−

[1− 1

2N

])∆ξN ≈

1

2√N

2ξN = 2√N =

1

∆ξN

1

2

3

4

5

6

7

∆ξN = ξN − ξN−1 =√N −

√N − 1

=√N

(1−

√1− 1

N

)

≈√N

(1−

[1− 1

2N

])

∆ξN ≈1

2√N

2ξN = 2√N =

1

∆ξN

1

2

3

4

5

6

7

∆ξN = ξN − ξN−1 =√N −

√N − 1

=√N

(1−

√1− 1

N

)

≈√N

(1−

[1− 1

2N

])∆ξN ≈

1

2√N

2ξN = 2√N =

1

∆ξN

1

2

3

4

5

6

7

∆ξN = ξN − ξN−1 =√N −

√N − 1

=√N

(1−

√1− 1

N

)

≈√N

(1−

[1− 1

2N

])∆ξN ≈

1

2√N

2ξN = 2√N =

1

∆ξN

Fresnel Lens Example

In terms of the unscaled variables

∆xN = ∆ξN√

2λo f

=

√λo f

2N

dN = 2ξN =

√2λo f

∆ξN= 2√N√

2λo f =√

2Nλo f

If we take

λo = 1A = 1× 10−10m

f = 50cm = 0.5m

N = 100

∆xN = 5× 10−7m dN = 2× 10−4m = 100µm

∆xN = ∆ξN√

2λo f =

√λo f

2N

dN = 2ξN =

√2λo f

∆ξN= 2√N√

2λo f =√

2Nλo f

If we take

λo = 1A = 1× 10−10m

f = 50cm = 0.5m

N = 100

∆xN = 5× 10−7m dN = 2× 10−4m = 100µm

∆xN = ∆ξN√

2λo f =

√λo f

2N

dN = 2ξN

=

√2λo f

∆ξN= 2√N√

2λo f =√

2Nλo f

If we take

λo = 1A = 1× 10−10m

f = 50cm = 0.5m

N = 100

∆xN = 5× 10−7m dN = 2× 10−4m = 100µm

∆xN = ∆ξN√

2λo f =

√λo f

2N

dN = 2ξN =

√2λo f

∆ξN

= 2√N√

2λo f =√

2Nλo f

If we take

λo = 1A = 1× 10−10m

f = 50cm = 0.5m

N = 100

∆xN = 5× 10−7m dN = 2× 10−4m = 100µm

∆xN = ∆ξN√

2λo f =

√λo f

2N

dN = 2ξN =

√2λo f

∆ξN= 2√N√

2λo f =√

2Nλo f

If we take

λo = 1A = 1× 10−10m

f = 50cm = 0.5m

N = 100

∆xN = 5× 10−7m dN = 2× 10−4m = 100µm

∆xN = ∆ξN√

2λo f =

√λo f

2N

dN = 2ξN =

√2λo f

∆ξN= 2√N√

2λo f =√

2Nλo f

If we take

λo = 1A = 1× 10−10m

f = 50cm = 0.5m

N = 100

∆xN = 5× 10−7m dN = 2× 10−4m = 100µm

∆xN = ∆ξN√

2λo f =

√λo f

2N

dN = 2ξN =

√2λo f

∆ξN= 2√N√

2λo f =√

2Nλo f

If we take

λo = 1A = 1× 10−10m

f = 50cm = 0.5m

N = 100

∆xN = 5× 10−7m

dN = 2× 10−4m = 100µm

∆xN = ∆ξN√

2λo f =

√λo f

2N

dN = 2ξN =

√2λo f

∆ξN= 2√N√

2λo f =√

2Nλo f

If we take

λo = 1A = 1× 10−10m

f = 50cm = 0.5m

N = 100

∆xN = 5× 10−7m dN = 2× 10−4m = 100µm

Making a Fresnel Zone Plate

Λ

Λ/2

The specific shape required for a zone plateis difficult to fabricate, consequently, itis convenient to approximate the nearlytriangular zones with a rectangular profile.

In practice, since the outermost zonesare very small, zone plates are generallyfabricated as alternating zones (rings for2D) of materials with a large Z-contrast,such as Au/Si or W/C.

This kind of zone plate is not as effi-cient as a true Fresnel lens would be in thex-ray regime. Nevertheless, efficiencies upto 35% have been achieved.

Λ

Λ/2

Λ

Λ/2

Tangential Focusing Mirror

The shape of an ideal mirror is anellipse, where any ray coming fromone focus will be projected to thesecond focus.

Consider a 1:1 focus-ing mirror. For an ellipse the sumof the distances from any point onthe ellipse to the foci is a constant.

F1P + F2P = 2a

F1B = F2B = a

sin θ =b

a

=b

2f

a

bF

1F

2

PB

θ

1

f=

1

o+

1

i=

2

a

f =a

2

The shape of an ideal mirror is anellipse, where any ray coming fromone focus will be projected to thesecond focus. Consider a 1:1 focus-ing mirror. For an ellipse the sumof the distances from any point onthe ellipse to the foci is a constant.

F1P + F2P = 2a

F1B = F2B = a

sin θ =b

a

=b

2f

a

bF

1F

2

PB

θ

1

f=

1

o+

1

i=

2

a

f =a

2

F1P + F2P = 2a

F1B = F2B = a

sin θ =b

a

=b

2f

a

bF

1F

2

PB

θ

1

f=

1

o+

1

i=

2

a

f =a

2

F1P + F2P = 2a

F1B = F2B = a

sin θ =b

a

=b

2f

a

bF

1F

2

PB

θ

1

f=

1

o+

1

i=

2

a

f =a

2

F1P + F2P = 2a

F1B = F2B = a

sin θ =b

a

=b

2f

a

bF

1F

2

PB

θ

1

f=

1

o+

1

i

=2

a

f =a

2

F1P + F2P = 2a

F1B = F2B = a

sin θ =b

a

=b

2f

a

bF

1F

2

PB

θ

1

f=

1

o+

1

i=

2

a

f =a

2

F1P + F2P = 2a

F1B = F2B = a

sin θ =b

a=

b

2f

a

bF

1F

2

PB

θ

1

f=

1

o+

1

i=

2

a

f =a

2

Saggital Focusing Mirror

Ellipses are hard figures to make,so usually, they are approximatedby circles. In the case of saggitalfocusing, an ellipsoid of revolutionwith diameter 2b, is used for focus-ing.

ρsaggital = b = 2f sin θ

The tangential focus is also usuallyapproximated by a circular cross-section with radius

ρtangential = a =2f

sin θ

F1

F2

2b