Notes Computer Assisted Optical System Design, Appunti di Ottica

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Notes CAOSD

Giacomo Fiorentini

A.Y. 2022/

Prof. D. Contini

Contents

Preface

This text arises from my need to study from a computer-taken text instead of a handmade one. The first four chapters are the transcriptions of the lectures held by the professor during the course with some personal notes that I found useful to include. In the last chapter I tried to sum up all the important things to remember about the software and I commented with the main characteristics the examples we made during the semester. Of course this text can’t replace the active participation at the lectures held by the professor, especially of the part regarding the use of Zemax OpticStudio since as any other software needs a lot of practice in order to acquire confidence with it.

1 Introduction to the optical design

In the design of an optical system we can play with some parameters to obtain the desired effect in terms of magnification, focal position, aberration and field of view. Those parameters are called degrees of freedom of the optical system which are:

• radii of curvature of optical surfaces
• thickness of optical elements
• air gap between optical elements
• materials
• diameters of the lenses
• dimensions and positions of apertures and stops

In the ideal case, once all the degrees of freedom have been chosen, the optical system has the property to conjugate one object point in only one image point, maintaining the aspect ratio of the object. The deviations from this ideal definition are called aberrations and are:

• spherical aberration
• coma aberration
• astigmatism
• distortion
• chromatic aberration

We will deal in detail with aberrations in the dedicated chapter, but just for example we can say that spherical aberration arises from the different angle between parallel incident rays with respect to the normal to the surface as in Fig. 1.1 that will result in a different focus position due to Snell’s law. A similar problem can arise for example with rays at different wavelengths that hit the surface in the same position but are focused in different spots because the refractive index of the lens depends on the wavelength n(λ) (dispersive medium).

Figure 1.2: The sag is the pink segment VM.

An analytical expression of this parameter is

SAG = R −

p R^2 − y^2 = R(1 −

p 1 − (y/R)^2 ) ≃ R − R(1 − y^2 / 2 R^2 ) =

y^2 2 R

where y is the height of the ray represented in Fig. 1.2 with the dashed line. A spherical surface with radius R and vertex in the origin has an analytical expression

z^2 − 2 zR + y^2 = 0

and the formula can be generalized to aspherical surfaces if it’s written

P z^2 − 2 zR + y^2 = 0

with P = 1 + k, k = −e^2 and e^2 =

a^2 − b^2 a^2

numerical eccentricity of the surface with

a and b semi-axis of the ellipse that represents the surface. Given a certain height y the sag is given by

R −

p R^2 − P y^2 P

y^2 2 R

P

y^4 R^3

P 3

y^6 R^6

and it contains a parabolic term and many non parabolic ones. If for example we consider a parabolic surface and a spherical one, the difference in the sag is given by the non parabolic terms and considering an height of 5 mm and a radius of 3. cm we obtain ∆z = 1. 8 μm that in the visible is in the order of 3λ. We now list the different types of surface and their relative values of P and k.

• circle P = 1 k = 0

• parabola P = 0 k = − 1
• hyperbole P < 0 k < − 1
• prolate ellipse 0 < P < 1 − 1 < k < 0
• oblate ellipse P > 1 k > 0

1.2 Anti-reflection coating

To reduce the power losses due to Fresnel reflection at surfaces, sometime it is re- quired to coat the surfaces with an anti reflection layer. To estimate the lost power we remember that the reflectivity of a surface for perpendicular incidence, where we consider perpendicular all those rays with an incident angle i which is smaller than 8 degrees with respect to the normal to the surface, is given by

R =

n 1 − n 2 n 1 + n 2

For glass in air where n 1 = 1, n 2 = 1.5 we obtain R = 4% so if many lenses are required this problem can’t be neglected and it’s possible to buy lenses with anti reflection coating or it’s even possible to cement adjacent optical surfaces with the same radius deleting in this way two interfaces air-glass / glass-air.

1.3 Optical materials

The optical elements can be made of glass or of special plastics which have both pros and cons with respect to glasses. The pros are a lower cost also thank to the possibility of molding the lenses, the optical and the mechanical elements can be produced already together, they are lighter, it’s easier to produce aspherical surfaces and filters/dyes can be added directly to the optical elements. The cons of plastic are that there are less materials available, it’s softer so it’s more delicate to scratches, the thermal expansion is higher and so it’s the thermal dependence of the refractive index which can be also 100 times higher than the one of glass. With plastic is more difficult to obtain parallel surfaces and it’s more difficult to cement singlets together, lastly plastic accumulate electrostatic charge which attracts dust. In the choice of the material an important thing to keep in mind is the dependence of

One last important thing to keep in mind when selecting a material is the transmit- tance of the material with respect to the desired wavelength since it may change a lot as it’s shown in Fig. 1.6.

Figure 1.4: Abbe diagram from Schott catalog. Note that the increasing of the Abbe coefficient is in the left verse.

Figure 1.5: Data sheet of N-BK7 from Schott catalog.

Figure 1.6: Transmittance of two different glasses. The big absorption in the UV region is due to electronic interaction, instead the one in the IR is due to the interaction between light and molecular phonons.

2 Ray tracing

2.1 Definitions and conventions

Object and image

Defined the object space as the one on the left hand side of the lens and the image space as the one on the right hand side, we consider the object real if the rays are diverging in the object space and virtual if the rays are converging in object space, instead the image is real if the rays are converging in the image space and it’s virtual if the rays are diverging as summarized in Fig. 2.1. In the object space all the quantities are identified with capital letters, while in the image space with a capital letter plus a prime ’.

Figure 2.1: Real / virtual object / image.

Ray types

There exist three types of rays:

• meridional rays that lay on the meridional plane;
• paraxial rays that are meridional rays close to the optical axis and with a small angle u;
• skew rays that don’t lay on the meridional plane.

2.2 Trigonometrical ray tracing at a spherical surface

Considering the notation in Fig. 2.3 which is built tracing the pink line as the line parallel to the incident ray passing from C, where Q is the distance between the vertex of the surface and the incident ray and P T //Q.

Figure 2.3: Notation for the trigonometrical ray tracing.

We can write AF¯ = R sin(U ), P T¯ = R sin(I), Q = R [sin(I) − sin(U )] so we can derive the first equation of the ray tracing which is

sin(I) =

Q

R

• sin(U )

And applying the Snell’s law n sin(I) = n′^ sin(I′) we obtain the second equation

sin(I′) =

n n′^

sin(I)

Considering now the angle P CAˆ = I − U , P CVˆ = π − I′^ − |U ′| we obtain that P CAˆ = π − P CVˆ = I′^ + |U ′| = I′^ − U ′^ = I − U so the third equation is

U ′^ = U + I′^ − I

Considering now for reference the Fig. 2.4 with the pink line parallel to the refracted ray and passing through the vertex of the surface we can write the last equation of ray tracing.

Figure 2.4: Notation for the trigonometrical ray tracing.

It’s valid that BC¯ = R sin(I′) and that CD¯ = R sin(|U ′|) so

Q′^ = R [sin(I′) − sin(U ′)]

To allow the calculations with more surfaces we can write two transfer equations that give the parameters entering the following surface once we know the ones exiting the previous surface, so the first one is

U 2 = U 1 ′

since the slope of the ray doesn’t change in the propagation. The second equation describes the distance from the o.a. of the impinging ray on the second surface with respect to the distance from the o.a. of the ray exiting from the first surface and to the distance d between the two surfaces:

Q 2 = Q′ 1 + d sin(U 1 ′)

I 1 II 2 III 3 IV 4 V 5 VI 6

R 8.372 -7.2688 Inf 5.736 -3.807 -16. d 2.4 0.4 7.738 1.8 0. n 1 1.5224 1.6164 1 1.51625 1. Q 3. Q’ I I’ U 0 U’

Where the Arabic numbers indicate the surfaces and the Roman ones the portions of space between surfaces.

2.3 Paraxial ray tracing (first order optics)

Remembering that a paraxial ray is a ray which is close to the o.a. and with a small slope, both Q and U can be considered infinitesimal and all the parallel rays are focused in the same point. In paraxial approximation all the quantities are indicated with lowercase letters and it’s valid that

• Q and Q′^ → y
• sin(I′) = i′
• sin(U ) = u
• cos(I) = 1

So the previous equations for ray tracing can be rewritten introducing the curvature

c =

R

as      

    

i = cy + u

i′^ =

n n′^

i

u′^ = i′^ − cy

y 2 = y + du′

2.3.1 The (y-nu) method

Rewriting the first and third equation of the previous system and multiplying the first by the index of refraction n and the third by n′^ we obtain  ni = ncy + nu n′i′^ = n′cy + n′u′

The Snell’s law in paraxial approximation is n′i′^ = ni so we obtain n′u′^ = nu + c(n − n′)y and defining the power of the surface ϕ = c(n′^ − n) we obtain

ϕ =

nu − n′u′ y

We can also rewrite accordingly the transfer equation y 2 = y 1 + du′. This method was introduced mainly for two reasons: highlighting the power of the surface and find an easy formula for the inversion of the propagation of the ray.

2.3.2 The (l-l’) method

In this section we will refer to Fig. 2.6.

Figure 2.6: Notation for (l-l’) method.

Considering n′u′^ = nu + c(n − n′)y and dividing it by y noting that l =

y −u

and

l′^ =

y −u′^

we obtain

ϕ =

n′ l′^

n l and solving for l′

l′^ =

n′ ϕ −

n |l|

In this method the transfer equation becomes l 2 = l′^ − d

3 Calculation of optical systems

3.1 Cardinal points

To describe an optical system in paraxial approximation we just need four points, two principal points which identify also the two principal planes and two focal points. Each of these points is called frontal or rear/back if it’s referred to backward or forward propagating rays as in Fig. 3.1.

Figure 3.1: Cardinal points and principal planes. [From Kingslake and Johnson]

The distances F 1 P 1 = f and F 2 P 2 = f ′^ are the frontal and rear effective focal distances, instead the distance of F 1 from the vertex of the first surface and of F 2 from the vertex of the second surface are simply called frontal and rear focal distances.

3.1.1 Principal planes

Still referring to Fig. 3.1 consider the paths BRF 1 and AQF 2. They can be reversed without changing the optical path. This means that R and Q are conjugated one with the other, so the two principal planes are the image one of the other with magnification equal to one. To find the location of the principal points inside a lens we refer to Fig. 3.

Figure 3.2: Position of principal points inside a lens.

It’s possible to write −u′ 2 =

y 1 f ′^

y 2 f ′^ + δ′^

so δ′^ =

y 2 y 1

f ′^ < 0. From the

transfer equation we know that y 2 = y 1 + du 2 so δ′^ =

du 2 y 1

f ′, recalling the equation

of ray tracing at the first surface n′u′^ = (^) nu 1 − y 1 ϕ 1 and the transfer function for the angle u′ 1 = u 2

δ′^ = −

dϕ 1 n′^

f ′

and with similar considerations

V 1 ¯P 1 = δ = dϕ^2 n′^

f

3.1.2 Lens power expression

For a single surface we already found ϕ = (n′^ − n)c = (n′^ − n)

R

, now we want to

derive the power of a couple of surfaces called singlet. Referring to Fig. 3.2 with refractive indexes in the three spaces respectively no, n′, ni we can write at the first

surface n′u′ 1 = (^) nou 1 − y 1 ϕ 1 so u 2 = u′ 1 = −

y 1 n′^

ϕ 1. At the second surface

niu′ 2 = n′u 2 −y 2 ϕ 2 = n′

y 1 n′^

ϕ 1

y1 + d

y 1 n′^

ϕ 1

ϕ 2 = −y 1

ϕ 1 + ϕ 2 −

d n′^

ϕ 1 ϕ 2

so

ϕsinglet = ϕ 1 + ϕ 2 −

d n′^

ϕ 1 ϕ 2

The rear effective focal length f ′^ = −

y 1 u′ 2

so that

f ′^ =

y 1 ni y 1 ϕsinglet

ni ϕsinglet

We can introduce at this point also the power of a reflective surface as

ϕ = (n′^ − n)c = − 2 nc

since to maintain the sign convention we consider as negative the refractive index for the propagation after the reflection.