Space Optics

Space Opcs (2) AstrOpt2016 Derive Etendu . AΩ J. B. Breckinridge Adjunct professor Caltech & College of Opcal Sciences Transmiance, throughput, & vigneng

• How bright is my image? • Can I record it? • Parameters that describe the ability of the opcal system to transmit power • What is the diameter of my opcal elements? • Can they be fabricated or just designed!

4/15/14 Etendu & radiometry 2 Kirchoff’s Laws If a body of mass is at thermal equilibrium with its surrounding environment, conservaon of energy requires that Φ = Φ + Φ + Φ incident absorbed reflected transmitted

By dividing both sides by , we write Φincident α + r + t = 1, where α is absorbtance, r is reflectance, and t is transmiance. For an opaque body where there is no transmiance (t = 0), the radiaon is either absorbed or reflected. Therefore,

watts absorbed = α ⋅ E ⋅area = ε ⋅ M ⋅area = watts radiated.

4/12/12 Etendu & radiometry 3 Minimize reflectance loss to maximize transmiance • Anreflecon coat has limitaons I λ R Reflected ( ) i (λ) = n n n I λ 1 2 3 Incident ( )

If n2 (λ) = n1(λ)n3(λ) Then reflectance is zero & transmiance is maximized

Oen a physical material with just the right n2 (λ) does not exist

4/15/14 Etendu & radiometry 4 Power at the focal plane

• Determined by • Transmiance • Etendu or “through-put” • Polarizaon (discussed later)

4/15/14 Etendu & radiometry 5 Étendú or throughput [pp103]

• Calculated using tools of 1st order opcs • Expresses the geometric ability of an opcal system to pass radiaon from object space to image space • Ray trace can indicate an excellent image but if no light gets through the system, there is no image – SNR=0.0! • Consider 2 general rays, pass them through the opcal system, then look at pupil and image planes

4/15/14 Etendu & radiometry 6 Étendú or throughput Consider 2 general rays

Plane 1 Plane 2 φ Plane 3 2 n n1 2 y1 y2 Red u2 u1 u1 y1 y u 2 From Ch 2: 2

n u n u y n u n u y 2 2 = 1 1 − 1φ2 2 2 = 1 1 − 1φ2

4/15/14 Etendu & radiometry 7 Étendú or throughput Consider 2 general rays

n u n u y n u n u y 2 2 = 1 1 − 1φ2 2 2 = 1 1 − 1φ2

(n1 − n2 ) Recall that: φ2 = (n1 − n2 )C2 = R The opcal power is the same for both rays2

(n2u2 − n1u1 ) (n2u2 − n1u1 ) = φ2 = y1 y1 Re-group the terms Then we discover that there is an invariant between any two planes in the opcal system Invariant on n1u1y1 − n1u1y1 = n2u2 y1 − n2u2 y1 = H

4/15/14 refracon Etendu & radiometry 8 Étendú, Helmholtz, LaGrange Invariant

n u y − u y = n u y − u y = H 1 ( 1 1 1 1 ) 2 ( 2 2 2 2 ) Rewrite this equaon with the object plane on the LHS y 0 1 = And and the pupil plane on the right hand side y 0 2 = Then nuy nuy H 1 = 2 =

H has units of angle × distance, e.g., radians × cenmeters. 4/15/14 Etendu & radiometry 9 Area solid angle product

Pupil plane +y +y +z Object plane chief AP

ΩO ΩP A Ω = A Ω AO O P P O

4/15/14 Etendu & radiometry 10 Confusion?

• Transmiance, transmission, transmissivity - dimensionless • Throughput, etendu, Opcal Invariant • Units of solid angle × Area, when calculang opcal system capacity to transmit radiave power

• Units of radians × length for opcal ray trace design

Etendu & radiometry 11 Useful relaonships π Ω = 4( f #)2

Etendu & radiometry 12 Vigneng: Étendú is not conserved at field points

1

2

For no vigneng, the radius R ≥ y + y . of the kth surface must be k k k Etendu & radiometry 13 Space Opcs (2) AstrOpt2016 Geometric Aberraon Theory

J. B. Breckinridge Adjunct professor Caltech & College of Opcal Sciences Class outline • The challenges of space opcs • Derive Etendu, throughput, transmiance – Power to the focal plane • Geometric aberraons: thermal, structural, metrology, tolerancing & A/O • Scalar wave image formaon • Vector-wave image formaon: polarizaon aberraons – paral coherence & correlated wave fields • Hubble trouble Image locaon, size, orientaon 16 Space opcs differ from ground opcs

• No atmospheric turbulence • High angular resoluon astronomy • Diffracon limited important • Aberraon really make a difference – Metering structure • Thermal (radiave, conducon, convecon), • Dynamics – Opcal surfaces – • Stac & dynamic wavefront control 1st order opcs • Two planes, normal to the axis are important in an opcal system – Pupil plane • Entrance pupil • Exit pupil – Image plane – A complicated opcal system can have several image planes, pupil planes, but ONLY one each: • Entrance pupil • Exit pupil

Aberraon Theory (1) 18 Two rays define locaon of image & pupil planes

marginal

Image plane chief

Pupil plane Y-Z Plane is called Object plane the Meridional plane • The volume required for an opcal system is defined by both the chief ray and the marginal ray radius of the clear aperture = r(z) = y + y

Aberraon Theory (1) 19 A fan of rays provides an esmate of image quality

marginal chief Image plane

Pupil plane Object plane

Spot diagram Aberraon Theory (1) 20 Tolerancing an opcal system

Sources of errors Mechanical & structural

21 Coordinate system for geometric aberraon analysis

4/12/12 Aberraon Theory (2) 22 Sign convenons

Aberraon Theory (1) 23 Sign convenon

4/12/12 Aberraon Theory (2) 24 Rays and Geometric Waves

reference ray path – ray path OPD ρx′, ρ′y W , ( ) , (ρx′ ρ′y ) = = λ λ

Independent of λ

Aberraon Theory (1) 25 Definion of Wavefront aberraon W The equaon for a spherical wave converging to a point at a distance f from the pupil (lens) is

⎡ k 2 2 ⎤ 2π u′(x, y) = u(x, y) ⋅ exp ⎢– j (x + y )⎥, k = ⎣ 2 f ⎦ λ And x and y are Cartesian coordinates across the pupil plane. An expression that includes the wave aberraon term W. k 2 2 E = – j (x + y )⎣⎡1 + W (x, y)⎦⎤ , 2 f Now what is W (x, y)? Aberraon Theory (1) 26 Geometrical wavefront error The geometrical wavefront error is independent of the wavelength. If one wanted to calculate how much material [as a funcon of posion] is needed to be removed From a mirror surface, then one need to mulply by wavelength.

Aberraon Theory (1) 27 Expand the wavefront error

• Zernike polynomials – Power series expansions on the unit circle • Seidel aberraons – Defocus, lt, spherical, coma, asgmasm, field curvature, Petzval Zernike polynomials

Zernike polynomials are a set of orthogonal polynomials defined on a unit circle. They are expressed in either Cartesian (x, y) or polar ( ) ρ,ψ coordinates. There are both even and odd Zernike polynomials:

m m Zn (ρ,ψ ) = Rn (ρ) cos(mψ )

−m m Zn (ρ,ψ ) = Rn ρ sin mψ ( ) ( )

4/12/12 Aberraon Theory (2) 29 Zernike polynomials where the radial polynomials are defined as

n – m k 2 –1 n – k ! Rm ( ) ( ) n – 2k n (ρ) = ∑ ρ k = 0 ⎡(n + m) ⎤ ⎡(n – m) ⎤ k!⎢ – k ⎥ ! ⋅ ⎢ – k ⎥! 2 2 ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ We will not use the Zernike polynomials here, but rather will analyze using the Seidel aberraons, which lend themselves to an intuive understanding of the physical origins of the aberraons.

4/12/12 Aberraon Theory (2) 30 Coordinate system for geometric aberraon analysis

4/12/12 Aberraon Theory (2) 31 The wavefront series expansion for the aberraons yields the expression m W a hk n cos , = ∑ knm ⋅ ρ ( ψ ) k, n, m By convenon, first-order aberraon terms are those for which k + n – 1 = 1, Third-order aberraon terms are those for which k + n + 1 = 3, and Fih-order aberraon terms are those for which k + n + 3 = 5 The third-order aberraon coefficients are: a040, a220, a400, a131, a311, and a222. The fih-order aberraon coefficients are: a060, a240, a420, a600, a151, a331, a511, a242, and a422. Aberraon Theory (1) 32 Aberraons independent of the opcs

Defocus a020

• The aberraon a020 is introduced by an axial change in the focus and is called defocus. For defocus, the aberraon is corrected by translang the image point along the axis unl the sphere corresponding to the aberrated image point is superposed onto the reference sphere. This

aberraon, like a111, is corrected by reposioning the focal plane; it is not necessary to correct by refiguring any opcal element.

Aberraon Theory (1) 33 Aberraons independent of the opcs

Tilt a111

• The term a111 represents a lateral shi. The aberrated wave is lted about the vertex of the reference wave. This aberraon can be corrected by reposioning the focal plane. The aberraon

term a111 represents a lt in the wavefront in the y direcon. This aberraon can be corrected by reposioning the focal plane; it is not necessary to correct by refiguring any opcal element.

Aberraon Theory (1) 34 Defocus and Tilt The actual surface is the dashed black sphere. Standing at the blue point looking le you have created the blue reference surface. To super-impose your reference surface to the actual surface you need to move to the right (+), or refocus the system to make: a020 = 0.

To super-impose your reference surface to the actual surface you need to lt your reference surface up, or lt your focal plane the system to make:

a111 = 0. Aberraon Theory (1) 35 The 3rd order (Seidel) terms

We will examine in detail the first and third order aberraon

m W a hk n cos = ∑ knm ⋅ ρ ( ψ ) k, n, m

Aberraon Theory (1) 36 A point in object space is imaged …

A point in object space is imaged into several points in image space by a geometric aberraons

Aberraon Theory (1) 37 Tool for analysis: the spot diagram

Aberraon Theory (1) 38 CAD ray-trace objecves

• For each point [in object space] • Use trigonometry to trace rays from a point in object space – through a grid (hexapolar) on the physical surface of the lens, – refract and – then translate to the next surface unl it – penetrates the focal plane at a point.

Aberraon Theory (1) 39 A point in object space is imaged …

A point in object space is imaged into several points in image space by a geometric aberraons

Aberraon Theory (1) 40 The ray-fan plot

Plot of ε yas a function of where on the pupil the ray intercepted it.

ε y = f (ρy )

Aberraon Theory (1) 41 CAD ray-trace objecves

• For each point [in object space] • Use trigonometry to trace rays from a point in object space – through a grid (hexapolar) on the physical surface of the lens, – refract and – then translate to the next surface unl it – penetrates the focal plane at a point.

Aberraon Theory (1) 42 Sign convenon

Aberraon Theory (1) 43 The third order monochromac terms

W = 4 3 2 a040ρ + a131hρ cosψ + a222h ρ2 cos2ψ + +a h2ρ 2 + a h3ρ cosψ 220 331

Spherical aberraon

Aberraon Theory (1) 44 4 Spherical a040ρ

Aberraon Theory (1) 45 Spherical in the presence of defocus

Aberraon Theory (1) 46 2 4 6 8 W = a020ρ + a040ρ + a060ρ + a080ρ

The only set of aberraons on axis for a properly aligned opcal system • spherical aberraons – defocus, – 3rd order, – 5th order order spherical aberraon rd – 7 th order 3 Aberraon Theory (1) 47 Spherical aberraon region of the causc

Aberraon Theory (1) 48 Change in the wavefront as a funcon of pupil posion ∂W 4a 3 2a sin . = ( 040 ρx + 020ρx ) ψ ∂ρx ∂W = 4a ρ3 + 2a ρ sinψ ∂ρ ( 040 y 020 y ) y The radius of the circle at the focal plane is 2R 2 2 a 2a 3 . R 2 εr = ε x + ε y = ( 040 ρ + 040 ρ ) = nr r F # Aberraon Theory (1) 49 At the paraxial focus a020=0 The diameter of the image is 8R d = a040 nr

At the marginal focus ρ = 1 and a020 = −2a040 4R R 2 3 a . εr = (ρ − ρ ) 040 = # nr r F

Aberraon Theory (1) 50 At what value of a minimum ρ is εr

d ε 4R ( r ) 1 2 a 0. = ( − 3ρ ) 040 = d ∂ nr 1 ρ 2 = . 3

Aberraon Theory (1) 51 Spherical aberraon spot size

4/12/12 Aberraon Theory (2) 52 Depth of focus The distance between the R2 marginal and chief rays is z 4a r 2 2a . Δ = 2 ( 040 + 020 ) λ nr

Radius of the marginal focus is

R2 With no spherical aberraon we find Δz = 2 2a020. nr 2 Δz = − 8 f / # a . ( ) 020

Aberraon Theory (1) 53 Depth of focus

2 ⎛ a ⎞ Δz = − 8 f / # 020 λ, ( ) ⎜ λ ⎟ ⎝ ⎠ The Rayleigh criterion is a quarter wave error a 1 020 = , λ 4 The depth of focus for this criterion is therefore

2 Δz = ± 2 f / # λ. At 0.5 microns (mid visible) ( )

2 Δz = f / # µm. ( )

Aberraon Theory (1) 54 The third order monochromac terms

We just finished looking at Spherical aberraon as a 4 2 W = a040ρ + a020ρ funcon of defocus 3 Coma +a131hρ cosψ 2 2 a222h ρ cos2ψ + Examine the behavior a h2 2 220 ρ of the term +a h3ρ cosψ 311 W a h 3 cos . = 131 ρ ψ

4/12/12 Aberraon Theory (2) 55 Coma is a field aberraon

• As you move off axis the exit pupil appears shorter in

the y direcon h than it does in the x direcon

Aberraon Theory (1) 56 Coma W a h 3 cos . = 131 ρ ψ Abbe To see the way in which this maps to the image plane, we calculate the slopes of the wavefronts- Remember the focal plane maps slopes (angles)

∂W a h 2 a h 2 2cos sin . = 131 ( ρx ρ y ) = 131 ρ ( ψ ψ ) ∂ρx ∂W a h 2 sin . = 131 ρ (2ψ ) ∂ρx ∂W a h 2 2 = a h 2 2 +cos . = 131 (ρx + 3ρ y ) 131 ρ ( 2ψ ) ∂ρ y 4/12/12 Aberraon Theory (2) 57 A point in object space is imaged into many points in image space

Transverse ray aberraon are

ε x′,ε y′

Aberraon Theory (1) 58 Coma W a h 3 cos . = 131 ρ ψ The transverse ray aberraons are then

R 2 a h sin2 , R 2 ε x = − ( 131 ρ ψ ) ⎡a h 2 +cos ⎤. nr ε y = − ⎣ 131 ρ ( 2ψ )⎦ nr

4/12/12 Aberraon Theory (2) 59 Coma

Now lets examine coma in the presence of defocus W a 2 a h 3 cos = 020ρ + 131 ρ ψ Then we find the ray intercept at the focal plane is R ⎡2a sin a h 2 sin 2 ⎤ ε x = − ⎣ 020 ρ ⋅ ψ + 131 ρ ( ψ )⎦ nr

4/12/12 Aberraon Theory (2) 60 Coma

R ⎡2a sin a h 2 sin 2 ⎤ ε x = − ⎣ 020 ρ ψ + 131 ρ ( ψ )⎦ nr R ⎡a h 2 2a a h 2 cos cos ⎤ ε y = − 131 ρ + ( 020ρ + 2 131 ρ ψ ) ψ . nr ⎣ ⎦

a hρ3 cosψ = a h ρ 2ρ + ρ3 cosψ 131 131 ( x y y ) This aberraon is clearly not rotaonally symmetric

4/12/12 Aberraon Theory (2) 61 Coma The sign on the coma term affects its appearance Aer calculang the the ray intercept, the displacements in the image plane are given by

R 2 R ε = −a hρ ρ a h 2 3 3 . x ( 131 y ) ε y = (− 131 ρx + ρ y ) nr nr R then If we let C = a131 h, nr

2 2 ε x = Cρ sin2ψ ε = Cρ 2 + cosψ . y ( )

4/12/12 Aberraon Theory (2) 62 Coma

The image plane appearance for (a) posive coma in the image plane on the le and (b) negave coma in the image plane on the right.

4/12/12 Aberraon Theory (2) 63 Coma related aberraons

• Coma

– Tilt a11 rd – 3 order a31 th – 5 order a51 th – 7 order a71

4/12/12 Aberraon Theory (2) 64 The third order monochromac terms

4 W = a040ρ 2 3 a020ρ + a131hρ cosψ + 2 2 2 Field curvature a222h ρ cos ψ + 2 2 a220h ρ Asgmasm +a h3ρ cosψ 311

Layout the next three VG

4/12/12 Aberraon Theory (2) 65 Invesgate field curvature and asgmasm in the presence of defocus W a 2 a h2 2 cos2 a h2 2. = 020ρ + 222 ρ ψ + 220 ρ In the meridional plane ψ = 0 => cosψ =1

From the first two terms in this equaon, we see that if we add a lile field curvature,

that is, a222 ≠ 0 such that a220 = –a222, and with a020 = 0, we have a cylinder that is curved in y direcon and flat in the x direcon:

W a h2 2. = − 020 ρx 4/12/12 Aberraon Theory (2) 66 Source of Asgmasm

4/12/12 Aberraon Theory (2) 67 Image plane in the presence of asgmasm • The image plane falls on three surfaces: • Tangenal • Medial • Sagial

4/12/12 Aberraon Theory (2) 68 Petzval (field) Curvature

4/12/12 Aberraon Theory (2) 69 Petzval (field) Curvature Field Curvature W a 2 a h2 2 cos2 a h2 2. = 020ρ + 222 ρ ψ + 220 ρ The best focus as a funcon of field point will, in general, lie on a paraboloidal surface If the asgmasm is zero, the image, in general, falls on a curved surface • Petzval developed the design methods to eliminate field curvature; thus • Photography with flat emulsion-coated glass plates was made praccal

4/12/12 Aberraon Theory (2) 70 Petzval (field) Curvature

It is possible to eliminate field curvature by using two separated elements, posioning the stop properly, and sasfying the Petzval condion: n f n f 0, 1 1 + 2 2 = φ φ φ φ φ 1 + 2 + 3 + 4 + 5 + ..... = 0 n n n n n 4/12/12 Aberraon Theory (2) 71 Petzval (field) Curvature

To obtain an esmate for the geometric spot size, we now calculate the ray intercept plot for defocus, asgmasm, and field curvature.

W a 2 a h2 2 cos2 a h2 2. = 020ρ + 222 ρ ψ + 220 ρ ∂W 2R a a h2 sin . ε x = − = − ( 020 ρ + 220 ) ρ ψ ∂x nr ∂W 2R ⎡a a a h2 ⎤ cos ε y = − = − ⎣ 020 + ( 220 + 222 ) ⎦ ρ ψ ∂y nr

4/12/12 Aberraon Theory (2) 72 Field Curvature

Recognizing that y = cos ψ and using sin2 ψ + cos2 ψ = 1, we find that the zonal image is an ellipse centered on the chief ray, given by

2 2 ε x ε y 2 + 2 = 1. ⎡ 2R nr a + a h2 r⎤ 2R nr ⎡a + a + a h2 ⎤r ⎣( c )( 020 220 ) ⎦ {( c )⎣ 020 ( 220 222 ) ⎦ }

4/12/12 Aberraon Theory (2) 73 Asgmasm

Sagial focus

Medial focus y

x

Tangenal focus line

4/12/12 Aberraon Theory (2) 74 Field Curvature and Asgmasm Tangenal Focus W a 2 a h2 2 cos2 a h2 2. = 020ρ + 222 ρ ψ + 220 ρ By definion, the Tangenal focus condion is when a = − a + a h2. 020 ( 220 222 )

In this case 2R 2 ε x = − a222h ρ sinψ , ε y = 0 nr The full length, d of the tangenal image is 4R R 2 2 = d = 2ε y = a222h . # nr r F 4/12/12 Aberraon Theory (2) 75 Field Curvature and Asgmasm Medial Focus W a 2 a h2 2 cos2 a h2 2. = 020ρ + 222 ρ ψ + 220 ρ By definion, the medial focus condion is when ⎛ 1 ⎞ a a a h2 020 = − ⎜ 220 + 222 ⎟ ⎝ 2 ⎠

2R 2 ε x = + a222h ρ sinψ . nr

2R 2 ε y = + a222h ρ cosψ nr

4/12/12 Aberraon Theory (2) 76 Field Curvature and Asgmasm Sagial Focus W a 2 a h2 2 cos2 a h2 2. = 020ρ + 222 ρ ψ + 220 ρ By definion, the Sagial focus condion is when a a h2. 020 = − 220 In this case

2R 2 ε y = − a222h ρ cosψ , and ε x = 0 nr

4/12/12 Aberraon Theory (2) 77 Asgmasm aberraons

• Asgmasm

rd – 3 order a22 th – 5 order a42

th – 7 order a62

4/12/12 Aberraon Theory (2) 78 Field Curvature (review) An in-focus “image” lies on a curved surface

The designer, by changing the power on opcal surfaces can control asgmasm, but not completely depending on what the other requirements are. 4/12/12 Aberraon Theory (2) 79 The third order monochromac terms

Examine the behavior 4 W = a020ρ of the term a ρ 2 + a hρ3 cosψ 020 131 W a h3 cos . 2 2 = 311 ρ ψ a222h ρ cos2ψ + 2 2 a220h ρ +a h3ρ cosψ Distoron 311

4/12/12 Aberraon Theory (2) 80 W a h3 cos . Distoron = 311 ρ ψ Distoron in the presence of lt is W = a + a ⋅ h3 ρ ⋅ cosψ ( 111 311 ) The wavefront error W equals zero provided that

a a h3 111 = − 311 ⋅

R 3 ε y = − a311 ⋅ h , ε x = 0 nr

4/12/12 Aberraon Theory (2) 81 Distoron rescales object space as a funcon of field point Barrel

4/12/12 Aberraon Theory (2) 82 Distoron rescales object space as a funcon of field point

Pincushion shis the image posion a distance

This is not a change in magnificaon R 3 ε y = − a311 y because the shi is dependent on h the cube of the field posion

4/12/12 Aberraon Theory (2) 83 Example of wide angle lens with barrel distoron

4/12/12 Aberraon Theory (2) 84 Stop shi – controls aberraons Aperture Image Marginal Stop Ray y plane u O

Upper rim ray Entrance u Pupil y Chief Ray

O Lower rim ray

Aberraon Theory (1) 85 Telecentric opcal system

B A ′

B Stop located A′ The front focus Object Image plane of the lens Lens plane Desensize the image quality to shis in the BFD (thermal, flexure, creep, etc.) Changing focus does not not change image plane scale or the image centroid 86 The role of stop shi in distoron

R 3 ε y = − a311 y h

Barrel

Distoron is controlled by a stop shi Pincushion Simple zero distoron lens 1840

4/12/12 Aberraon Theory (2) 88 Petzval lens

• Front doublet is well corrected for spherical but introduces coma. • The second doublet corrects coma. • The posion of the stop corrects for asgmasm, but introduces addional field curvature and vigneng. Total FOV is restricted to 30 degrees. • F# as low a 3.7 were achieved. • Ideal portrait lens for the new invenon of photography.

4/12/12 Aberraon Theory (2) 89 2014 State of the art Commercial Canon lens layout is derived from the Petzval lens of 1840

Lens design resources: Arthur Cox book & Patent literature

4/12/12 90 3rd order aberraon summary • Spherical aberraon depends on 4th power of the aperture height ρ 4

• Coma depends on the square of the aperture height and is linear with field. ρ 2h • Asgmasm is caused by the difference in curvature between sagial and tangenal ray fans. It varies linear with aperture 2 and the square of the field. ρh N ⎛ ⎞ • Petzval curvature φ j ∝ ∑⎜ ⎟ j=1 ⎝ n j ⎠ • Distoron shis image locaon and is dependent on the cube of the field. h3 4/12/12 Aberraon Theory (2) 91 Spherical aberraon shown on Spoke Target Image Object

4/12/12 Aberraon Theory (2) 92 Coma through focus

4/12/12 Aberraon Theory (2) 93 Applicaons of aberraon theory • In this secon we describe the way in which the theory of aberraons is applied to a few opcal systems to show the reader the ulity of this analysis approach. • Opcal ray-trace design computer programs require input of the first-order properes on an opcal system. • The closer these first-order properes are to an opmized design, the faster and more accurately the CAD program will converge to an opmized design

4/12/12 Etendu & radiometry 94 Applicaons of aberraon theory

• The structural aberraon coefficients were developed to simplify the design process. • We will show how the structural coefficients are used to determine system aberraons. • Here we discuss aberraons introduced by a plane-parallel plate, derive the aberraon

coefficients a040, a131, a222, a220, and a311, analyze the effects of lens bending, and describe the Schmidt camera design.

4/12/12 Etendu & radiometry 95