Telescopes and Optics IIObservational Astronomy 2018 Part 4 Prof. S.C. Trager

Fermat’s principle

Optics using Fermat’s principle

Fermat’s principle

The path a (light) ray takes is such that the time of travel between two fixed points is stationary with respect to small changes in that path

In other words, the travel time of a ray is infinitesimally close to that of a neighboring path

This is a generalization of the “principle of least time” or “principle of least action”

To be precise, given a surface lying between two points P0 and P1, consider two paths from P0 to P1

If the travel time from P0 to P1 is τ, then

where y and z are the coordinates of the intersection of the path with the surface

P0

P1

(y,z)

d⌧/dy = d⌧/dz = 0

If we replace the words “time of travel” from Fermat’s principle above with the words “optical path length (OPL)”, c dt, we see that Fermat’s principle is recovered

The OPL is defined as

where dt is an infinitesimal travel time, v is the the speed of light in a medium of index of refraction n, and ds is the infinitesimal geometrical path length

P0

P1

(y,z)

d(OPL) = c dt = (c/v)v dt = nds

Then

Note that n can be a function of position here, so consider n(y,z) and

Letting y′=dy/dz, Fermat’s principle is δ(OPL)=0 or

P0

P1

(y,z)

�

Z P1

P0

n(y, z)q

(1 + y02)dz = 0

ds =p

dy2 + dz2

OPL = c

Zdt =

Znds

If we call then

where

P0

P1

(y,z)

F (y, y0, z) = n(y, z)q

1 + y02,

�

Z P1

P0

F (y, y0, z)dz =

Z P1

P0

�F (y, y0, z)dz = 0,

�F =@F

@y�y +

@F

@y0�y0 =

@F

@y�y +

@F

@y0d

dz(�y)

Substituting and integrating by parts, we have

where the second term vanishes because δy=0 at P0 and P1, and then

P0

P1

(y,z)

Z P1

P0

@F

@y�ydz +

@F

@y0�y

����P1

P0

�Z P1

P0

d

dz

✓@F

@y0

◆�ydz = 0

Z P1

P0

@F

@y� d

dz

✓@F

@y0

◆��ydz = 0

This must vanish for arbitrary δy, so

Substituting we have (after some algebra)

P0

P1

(y,z)

@F

@y� d

dz

✓@F

@y0

◆= 0

F (y, y0, z) = n(y, z)q

1 + y02,

q1 + y02

@n

@y� n

d

dz

y0p

1 + y02

!� y0p

1 + y02dn

dz= 0

This must vanish for arbitrary δy, so

Substituting we have (after some algebra)

P0

P1

(y,z)

@F

@y� d

dz

✓@F

@y0

◆= 0

F (y, y0, z) = n(y, z)q

1 + y02,

q1 + y02

@n

@y� n

d

dz

y0p

1 + y02

!� y0p

1 + y02dn

dz= 0

Now, this looks ugly (and it is), but the presence of terms like suggests that trigonometric solutions would be useful:

Here α is the angle made by the tangent of the path with the z-axis

P0

P1

(y,z)

q(1 + y02)

tan↵ = dy/dz = y0, sin↵ = dy/ds = y0/q

(1 + y02)

cos↵ = dz/ds = 1/q

(1 + y02), d(sin↵)/dz = cos↵(d↵/dz)

Then noting that

we can write

P0

P1

(y,z)

dn

dz=

@n

@z+ y0

@n

@y,

cos↵@n

@y� sin↵

@n

@z� n cos↵

d↵

dz= 0

Finally, note that we can write the curvature K of a path as

and therefore

P0

P1

(y,z)

K =d↵

ds=

d↵

dz

dz

ds= cos↵

d↵

dz

nK = n cos↵d↵

dz= cos↵

@n

@y� sin↵

@n

@z(15)

This is the equation for the local curvature of a light ray subject to Fermat’s principle in a medium in which n is a smoothly varying function of position (in the y,z-plane)

As a special case, consider n=constant. Then K=0, and the path of a light ray in a homogeneous medium is a straight line — as expected

nK = n cos↵d↵

dz= cos↵

@n

@y� sin↵

@n

@z(15)

A useful example: atmospheric refraction Assume the atmosphere is a flat, layered medium with n=n(z) only, where the z-axis points to the center of the Earth, and hence the curvature of the atmosphere is negligible Then Eq. (15) becomes

ground

top of the atmosphere

z

n=1.00000

n=1.00029

nK = n cos↵d↵

dz= � sin↵

@n

@z

Because the change in n from the top of the atmosphere to the surface is very small, the path of a ray from a star will not be significantly deviated if α is not close to 90º Then we can integrate the expression on the last slide to find where α0 is the zenith angle at the top of the atmosphere

�↵ = � tan↵0 �n

ground

top of the atmosphere

z

n=1.00000

n=1.00029

α0

For a ray passing down through the atmosphere, δn>0 and so δα<0 Thus the ray is bent towards the z-axis If α0=45º, then δα=0.00029 rad≈60″ Note that n=n(λ), so different wavelengths get bent by different amounts

Very important for wide-band spectroscopy!

ground

top of the atmosphere

z

n=1.00000

n=1.00029

α0

Another useful example is a dispersing prism with n=n(λ) in air (n=1) There is some wavelength whose rays follow paths parallel to the prism base

For these rays, the diagram is symmetric about the prism’s vertical bisector, so that

α1 α2A

n(λ)

θ(λ)

L L

t

s1 s2

a1 a2

s1 = s2 = s, ↵1 = ↵2 = ↵, and a1 = a2 = a

Another useful example is a dispersing prism with n=n(λ) in air (n=1) There is some wavelength whose rays follow paths parallel to the prism base

For these rays, the diagram is symmetric about the prism’s vertical bisector, so that

α1 α2A

n(λ)

θ(λ)

L L

t

s1 s2

a1 a2

s1 = s2 = s, ↵1 = ↵2 = ↵, and a1 = a2 = a

α αA

n(λ)

θ(λ)

L L

t

s s

a a

The OPL of the bottom ray (at the prism’s base) is just nt The OPL of the top ray (at the prism’s vertex) is Fermat’s principle says that these two OPLs must be the same, so

α1 α2A

n(λ)

θ(λ)

L L

t

s1 s2

a1 a2

2L cos↵

nt = 2L cos↵

α αA

n(λ)

θ(λ)

L L

t

s s

a a

We’re interested in the change of θ with wavelength, so let’s differentiate the OPL equation above:

From the figure, we see that

Thus

α1 α2A

n(λ)

θ(λ)

L L

t

s1 s2

a1 a2

tdn

d�= �2L sin↵

d↵

d�= �2L sin↵

d↵

d✓

d✓

d�

L sin↵ = a and ✓ = ⇡ �A� 2↵, so d↵/d✓ = �1/2

d✓

d�=

t

a

dn

d�

α αA

n(λ)

θ(λ)

L L

t

s s

a a

For most optical glasses, we can write (approximately)

where C0 and C1 are constants Therefore

The negative sign means that θ decreases as λ increases, so that blue light is deviated more than red light

The angular dispersion dθ/dλ is larger for shorter wavelengths

α1 α2A

n(λ)

θ(λ)

L L

t

s1 s2

a1 a2

d✓

d�= �2t

a

C1

�3

n(�) = C0 + C1/�2

α αA

n(λ)

θ(λ)

L L

t

s s

a a

Let’s return to the case of reflecting mirrors What shape must a mirror have to satisfy Fermat’s principle? Let’s consider three cases:

A concave mirror with one conjugate at ∞ A concave mirror with both conjugates finite A convex mirror with both conjugates finite

C OB′ z

yP

f

lΔ

y

Case 1: concave mirror, one conjugate at infinity For convenience, let f, l, and Δ be positive (note that this violates our sign convention!) Applying Fermat’s principle to a ray on the optical axis and one at height y, we see that

C OB′ z

yP

f

lΔ

y

2f = l + (f ��) and so l = f +�

We also know that (from Pythagoras) that

Eliminating l from these two equations we see that

This is the equation of a parabola with a vertex at (0,0)

C OB′ z

yP

f

lΔ

y

l2 = y2 + (f ��)2

y2 = 4f� = �4fz

Using Eq. (7) from the last set of slides and applying the sign convention we have

where R is the radius of curvature and both R and z are negative In three-space, we replace y2 with x2+y2, and we find that a paraboloid satisfies Fermat’s principle in this case

C OB′ z

yP

f

lΔ

y

y2 = 2Rz

Case 2: concave mirror, both conjugates finite In this case we find (using the same analysis)

where This is the equation for an ellipse with center (0,a)

C OB′Bz

y

P

Rs′

s

lΔ

y

y2 � 2zb2

a+ z2

b2

a2= 0

2a = s+ s0 and b2 = ss0

(16)

Case 3: convex mirror, both conjugates finite

In this case we find (using the same analysis)

where

This is the equation for a hyperbola with center (0,0)

y2 + 2zb2

a� z2

b2

a2= 0

2a = s+ s0 and b2 = �ss0

All of these forms can be written in a simple way:

First, recall that Eq. (7) from the last set of slides says

(in the paraxial approximation)

Next, consider an ellipse with eccentricity e=c/a where c is the distance from one focus to the center of the ellipse: c2 = a2 � b2

1

s+

1

s0=

2

Ror

ss0

s+ s0=

R

2=

b2

2a

For our elliptical mirror above, then, we can write

We can then write Eq. (16) as

Although we’ve derived this equation for an ellipse with –1<e<1, it’s actually the correct formula for any conic section

y2 � 2Rz + (1� e2)z2 = 0 (17)

e2 =

✓s� s0

s+ s0

◆2

and 1� e2 =4ss0

(s+ s0)2=

b2

a2

We can write this equation for a surface of revolution as

where K is Schwarzschild’s conic constant and ρ2=x2+y2. Then

conic section e2 Kprolate ellipsoid <0 >0

sphere 0 0oblate ellipsoid 0—1 –1—0

paraboloid 1 –1hyperboloid >1 <–1

⇢2 � 2Rz + (1 +K)z2 = 0

Note that the above calculation suggests that for both “classic” Cassegrain and Gregorian telescopes, the primary should be a paraboloid, while the secondary should be

an ellipsoid for a Gregorian

a hyperboloid for a Cassegrain

Consider a perfect optical system — one that satisfies Fermat’s principle

Light from two neighboring sources — say, stars A and B — separated by angle θ fill the aperture D

Resolutionto A

to B

D

L

Δ

θθ

θA′B′

z★★

The wavelength theory of light says that two image points cannot be separated — resolved — if the difference in light travel time to them from opposite sides of an aperture is less than ~one period of the wave

Resolutionto A

to B

D

L

Δ

θθ

θA′B′

z★★

In other words, the points can’t be resolved if the OPL difference between the rays is less than ~one wavelength

Resolutionto A

to B

D

L

Δ

θθ

θA′B′

z★★

The OPL difference in the figure is Δ, so we require Δ⪆λ

This then implies that

The exact result from diffraction theory is

Resolution

✓min ⇠ �/D

✓min = 1.22�/D

to A

to B

D

L

Δ

θθ

θA′B′

z★★

Aberrations

Departures from Gaussian optics are called aberrations

They occur in imperfect optical systems

Aberrations come in two main types:

chromatic aberrations due to wavelength variation in the index of refraction

monochromatic aberrations, which are independent of wavelength

Aberrations

Monochromatic aberrations come in two types:

those that deteriorate the image

those that deform the image

These are inherent to each optical element and can be corrected in multiple-element systems (hopefully)

Aberrations

Since Gaussian optics are based on the paraxial approximation, where sin θ≈θ, this will clearly break down when considering rays that are either

at a large distance y from the optical axis

like the marginal rays

or at a large angle θ to the optical axis

Aberrations

So let’s expand sin θ in a Taylor series:

Taking the first (paraxial) and second terms, we have third-order (Seidel) aberration theory

Higher-order (Zernike) terms can of course still be present

sin ✓ = ✓ � ✓3

3!+

✓5

5!� · · ·

Aberrations

In third-order theory, there are five primary aberrations, which scale as

where m+n=3

ym✓n

aberration type scaling for a ray

scaling for a telescope

spherical aberration deterioration (y/R)3 F–3

coma deterioration θ(y/R)2 θF–2

astigmatism deterioration θ2(y/R) θ2F–1

field curvature deformation θ2(y/R) θ2F–1

distortion deformation θ3 θ3

For a telescope of radius R∝yF, we call the aberrations

Note that the last four aberrations are dependent on θ

these are off-axis aberrations

Spherical aberration is independent of θ

occurs even for on-axis sources!

This aberration occurs when rays do not come to a focus at the same point on the optical axis

a point source makes a blurred disk

Spherical aberration

This is the spherical aberration of the Hubble Space Telescope before (left) and after (right) correction

image on left covers ~2″; image on right covers ≈0.05″ This was due to a 1.3mm washer inserted during polishing of the primary mirror, which caused the margins of the primary to be too flat by ~λ/2!

Rays from an off-axis source converge at different points on the focal plane

This occurs off-axis, and is asymmetric because rays from the opposite side of the optical axis land on the same side of the axis

Coma

Coma from a parabolic mirror

This is a picture of a far off-axis part of a field with significant coma

note the “head-tail” structure like a comet Coma also appears when optical elements are misaligned

Astigmatism is caused by rays in the horizontal plane and the vertical plane coming to different foci

The best focus is then the circle of least confusion

Astigmatism

Astigmatism from a parabolic mirror

This is the image of a star in an astigmatic system

Field curvatureOutside of the paraxial region the image plane is curved

the mapping from object to image follows spherical surfaces

You can correct this by adding lenses or mirrors to flatten the field

If you can’t remove the curvature, you need to have a curved detector (or curved slit in a spectrograph)

Distortion

This is a radial change in plate scale with field angle

Does not change the focus but deforms the images

Can be calibrated out, but it’s annoying!

Chromatic aberrations

Caused by rays of light with different wavelengths coming to different foci

Note the large blue halos and smaller yellow and red halos

this is chromatic aberration

You can correct chromatic aberration by creating achromatic doublets (or triplets) by combining positive and negative lenses

Doublets bring light from two wavelengths to a common focus Triplets bring light from three wavelengths to a common focus — generally sufficient for broad-band sources without causing undue aberrations

Achromatic doublets and triplets demonstrate a general rule for compensating aberrations:

“one can generally correct n primary aberrations with n reasonably separated powered optical elements”

for example, we correct spherical aberration by using a single mirror shaped as a paraboloid instead of a sphere

Aberration compensation

But with a single mirror, the other aberrations — coma, astigmatism, field curvature, distortion — remain

We correct these by added more (and more) elements

For example, by choosing the correct shapes (the conic constants Ki) for the primary and secondary, we can correct two aberrations

Aberration compensation

Schwarzschild (1905) and Ritchey & Chrétien (1910) showed that two hyperboloidic mirrors — with the focus of the hyperboloid of the primary shared with the focus of the hyperboloid of the secondary — form an aplanatic telescope

an aplanat is an optical system free of both spherical aberration and coma

A Ritchey-Chrétien (RC) telescope is then limited by astigmatism, not coma like a “classical” Cassegrain

HST is an RC telescope

Aberration compensation

Non-spherical surfaces are difficult to manufacture!

But paraboloids can be made by spinning the liquid glass — so many large telescopes are now constructed as “classical” Cassegrain or Gregorian telescopes

The WEAVE PFCPFC=prime focus corrector 6 lenses!