Diffraction

$Page 1: Diffraction$
Optical Engineering 5 - 0

Optical Engineering - Outline

Geometrical Optics General Theory

Fermat’s Principle

Gaussian Optics and paraxial behaviour of Components and Surfaces

Optical Systems and Aberrations Matrix Ray Tracing

Stops and Pupils

Introduction to Monochromatic Aberrations

Monochromatic Aberrations Gauss-Seidel Aberrations

Behaviour of Lenses and Mirrors – Aplanatic Points

Impact of Pupil on Aberrations

Aspheric Surfaces and Chromatic Aberration Use of Symmetric Aspheric Surfaces

3rd January 2012 Optical Engineering – Diffraction and Image Quality Slide 1

y p

Dispersion and Chromatic Aberration

Use of Zernike Polynomials in defining OPD or WFE profiles

Diffraction and Image Quality Diffraction - ’Near Field’ and ‘Far Field’

Gaussian Beam Propagation

Definitions of Image Quality

Diffraction and Image QualitySummary

Theoretical basis for Huyghens Principle

Rayleigh Formulae

Far Field (Fraunhofer) approximation

Far field diffraction of gaussian beams

Airy Disc

Point Spread Function and Strehl Ratio

Gaussian beam propagation


Gaussian beam propagation

Image Quality

Geometric Spot

PSF

MTF

Given a known amplitude, Huyghen’s principle can be used to determine the amplitude downstream. Huyghen’s principle can be used to account for diffraction effects. Diffraction effects are particularly evident when the lateral extent of a wavefront is of the order of the wavelength of light. This is true of laser beams. In fact, Huyghen’s principle can be derived directly from Maxwell’s equation.

The wavelets diminish in amplitude further from their source and follow and theirThe wavelets diminish in amplitude further from their source and follow and their intensity is proportional to 1/r2 and the amplitude to 1/r. Therefore, the wavelet’s amplitude can be described by the following equation:

5-1)r

krtAA

)sin(0

In fact the above equation really only applies for kr>>1.

r


All the predictions of diffraction theory originate from Maxwell’s equation:

5-2)

Rayleigh derived two specific equations describing diffraction. The first covers the situation where the amplitude itself is known across a semi-infinite plane This is

2

2

2

2

2

2

002

2

2

2

2

2

t

E

c

n

t

E

z

E

y

E

x

E

situation where the amplitude itself is known across a semi-infinite plane. This is the so called Rayleigh Formula of the first kind:

5-3)

The second formula covers the situation where only the gradient of the amplitude is k thi f Thi i th ll d Ra leigh Form la of the second

'''

)',','(2

1),,(

0'

dydxs

e

zzyxAzyxA

iks

z

known across this surface. This is the so called Rayleigh Formula of the second kind:

5-4)

Details of this can be found in Born & Wolf, Principles of Optics.

'''

)',','(

2

1),,(

0'

dydxs

e

z

zyxAzyxA

iks

z


Dealing with one dimension in the far field (x) and ignoring y:

1 iks 5-5)

In the farfield, k>>1/s and

5-6)

''

)','(2

1),(

0'

dxs

e

zzxAzxA

iks

z

cos''

eik

seik

e iksiksiks

22 '' zzxxs

Applying to equation DA5)

5-7)

Since x’ is vanishingly small:

'' szssz

zzxxs

')','(2

cos),(

0'

dxezxAR

ikzxA

z

iks

22 zxR

Since x is vanishingly small:

5-8)

and:

sin''

'2 22 xRR

xxRzxxxs

ik ikR


5-9) ')','(2

cos),(

0'

sin' dxezxAR

ikezxA

z

ikxikR

For rigorous analysis, the cos term should be included. However, for low NA this term may be ignored.



The gaussian distribution is only an approximation to a real far field distribution. However, its use is convenient, as it is easy to analyse. The x0 figure in the slide represents the much quoted (1/e)2 intensity point. Another often quoted figure is the full width half maximum (FWHM) the relationship between the two is:

5-10a)

The far field is derived from the Fourier Transform of the amplitude:00 177.1)2ln(2 xxxFWHM

p

u=kx

The fourier transform of is

duuNAedxkxNAeNAA kx

u

x

x

)sin()sin()(

2

0

2

0

2

0

kx

u 20

2

NAkx

Therefore we can express the far field amplitude as:

5-10b) where

E 5 10b) i ll d d i fi ld di ib i f f fi ld

0 kxe 2 e

2

0

0

NA

NA

eAAo

vacuum

oo nxxkxNA

2

0


Eq 5-10b) is all we need to derive near field distributions from far field etc. 0 ooo

It is straightforward to express the FWHM (full width half maximum) of the near and far field distributions in terms of NA0 and x0:

5-10a)

Substituting in the near and far field expressions we get:

5 11)

0)2ln(2 xxFWHM 0)2ln(2 NANAFWHM

5-11)

Similarly substituting expressions for the RMS size of a circular (i.e. 2D) gaussian beam:

FWHM

vacuumFWHM x

NA

)2ln(2FWHM

vacuumFWHM NA

x

)2ln(2

0x NA

Therefore:

5-12)

20x

xRMS 2

0NANARMS

vacuumRMSNA

vacuum

RMSx


RMSRMS x

NA2 RMS

RMS NA2

Farfield of Uniform ApertureAiry Disc

In most instruments, nearfield or farfield defined by uniformly illuminated (circular) aperture

Find fourier transform of uniform aperture of NA = NAx

2

012

)(

xx

xx

J

xIx

vac

NAx

20

NAx


0 x

Instrument Exit Pupil

Airy Disc

J1 = Bessel Function of First Kind

Airy Disc (1)

Central Spot


Concentric Rings

Airy Disc (2)


Airy DiscSize

FWHM is well defined

3.23266 * x0

RMS size is not defined (infinite)

Optimal fit of Airy to Gaussian gives a Gaussian RMS of:

Xrms = 2.64*x0

x

vacFWHM NA

x

*61633.1


Wider than comparable gaussian – Airy has more pronounced ‘tail’

x

vacRMS NA

x

*32.1

RMS

vacRMS NA

x

07.1or


The Point Spread Function (PSF) gives a realistic description of the image formed by a point source through the optical system and takes into account the impact of p g p y pdiffraction. For a system that is relatively close to the diffraction limit, the geometrical spot size considerably exaggerates the optical performance. In this situation, the PSF is significantly larger than the corresponding spot size, and the PSF is a faithful representation of the real image. The Strehl ratio gives a measure of whether the image is ‘diffraction limited’. The higher the Strehl ratio, the better the image quality.


The above diagram is a 2 dimensional map of wavefront error. The X and Y coordinates represent the entrance pupil (e.g. the lens aperture) normalised to some physical or arbitrary aperture. The physical aperture could be the diameter of the lens as defined by the lens holder. Alternatively an arbitrary aperture could be defined by the manufacturer’s specification of the lens aperture, or alternatively it could reflect the diameter of the laser beam. The Z co-ordinate on the above surface map represents the wavefront error.

Note wavefront error is expressed in waves. For example, a peak to valley error of /4 (quarter wave) is often used to define ‘aberration free’ performance.

The wavefront map above can be output as a numerical file with real values of the wavefront distortion. This can enable detailed coupling calculations to be made as will later be revealed.


Point Spread FunctionDiffraction Analysis

Two dimensional fourier transform of exit pupil amplitude

Contains uniform pupil intensity plus phase disturbance caused by

yxikxNA dNAdNAeNAAxA )()(

)()( NAikeNAA

aberration


If (NA) = 0 we obtain Airy pattern

Huyghens PSF


For small aberrations we can make the following approximation:

If we assume that the average value of the WFE is zero across the aperture (i.e. no piston term), then:

2

)()(1~

22)( NAk

NAike NAik

Expressed in terms of wavelength:

)(12

)(1 22

222

NAkNAk

Strehl

5-13) )(2

1 22

NAStrehl


Strehl RatioRepresentation of WFE by Zernikes

Total RMS WFE is simply the RSS sum of the Zernike Polynomial Coefficients that describe the WFE

22 mC

mimn

mn

mn eRaC ),(


nC

Physical OpticsGaussian Beam Propagation

Models the propagation of a gaussian beam profile through system

Commonly applied to laser beam propagation

Simple analysis that provides insight into physical optics propagation through complex system

Models near field, far field and intermediate locations

Strictly applies to low numerical aperture beams


Based on solution to wave equation

5-14) (Faraday’s Law) or

The latter equation assumes a magnetically homogeneous medium. 0 is the magnetic permeability of free space and the relative permeability of the medium.

5-15) (Ampère’s Law) or

t

BxE

t

HxE

0

5-15) (Ampère s Law) or

The latter equation assumes a electrically homogeneous medium. 0 is the permittivity of free space and the relative permittivity (dielectric constant) of the medium. It also assumes that the current density, J, is zero.

t

DJxH

t

ExH

0

If (zero space charge)

5-16)

c = speed of light in vacuo; n is the refractive index of the medium

t

E

txH

txEx

000 0. E

2

2

2

2

2

2

002

2

2

2

2

2

t

E

c

n

t

E

z

E

y

E

x

E


p g ;

We start off with Maxwell’s equation:

5-17)

Although E is strictly a vector, we represent it as a scalar equation. We now look for solutions in the form of a beam propagating along the z axis:

2

2

2

2

2

2

2

2

2

2

t

E

c

n

z

E

y

E

x

E

)()()( kztjAE 5-18)

where k is the wavevector in the medium and equal to n/c.

We now make the approximation that A(x,y) varies slowly. This is the slowly varying envelope approximation or the paraxial approximation. More specifically the approximation means:

)(),(),,( kztjeyxAzyxE

specifically the approximation means:

A’(x,y) << kA(x,y).

Substituting equation GB5) into Maxwell’s equation

5 19)22 AAA A


5-19)

(Paraxial Helmholtz Equation)

0222

z

Akj

y

A

x

A022

z

AkjAT

At the nearfield, there is a so called beam waist where the angular divergence is zero. The beam starts to spread and in the limit of infinite distance, the divergence equals that predicted by the more elementary diffraction theory.


If we propose a solution for a gaussian beam as:

5-20)

Most importantly, (z) includes imaginary terms. Then by applying the paraxial Helmholtz equation and by collecting terms in (x^2)*e-x^2, we get:

5 21)

2)(0

xzeAA

0)('2)(4 2 k5-21)

if we now substitute =1/, then we get the following:

5-22) or and

0)('2)(4 2 zkjz

0)(

)('2

)(

422

z

zkj

z

kjz

2)('

k

zjCz

2)(

C is an arbitrary constant of integration. It is real at z=0, since we assume here we are at the beam waist and there is no curvature and hence imaginary component. In fact since =1/w2(z), =w2(z). If w0 is the size of the beam waist at z = 0, then C = w0^2

2

212 z

jzjw


5-23)

240

2

20

20

2

240

0

20

41

11

421

)(1)(

kw

z

kwj

w

k

zw

kjw

k

zjw

zz

Referring to original description of Gaussian Beam

5-24)

The beam size and radius are described in terms of the real and imaginary components of (z):

22 )(2)(

1

0),(x

zR

kj

zweAyxA

k

From equation, we obtain

5-25) and or

)(Re

1)(2

zzw

)(Im2

)(z

kzR

24

2

0

41)(

zwzw z

kwkw

z

zR

4

41

)(24

024

0

2

5-26) and

240

0 1)(kw

wzw z

kw

z 44)(

240

2

2

0 1)(RZ

zwzw z

z

ZzR R

2

)(


ZR is the Rayleigh Distance where: 20

20

2

wkwZ R

For propagation of laser beams in a turbulent atmosphere, the situation is more complicated. This might be a situation that is encountered in LIDAR measurements of atmospheric properties etc. An empirical formula exists which describes the propagation of a beam where the atmospheric turbulence is described by the structure parameter, Cn2.

5-27) ZR is the Rayleigh Distance and ZT is h i i b l di

Z

z

Z

z11

2

2

0a characteristic turbulence distance

The turbulence distance is given by:

5-28)

TR ZZ 20

TZ2

370

For z < ZR, then the beam is in the near field and not spreading. For ZR<z<ZT, then spreading is dominated by diffraction. For z > ZT, spreading is dominated by turbulence. For 0 = 15 mm and = 600 nm, ZR is equal to about 1.2 km. At ground level, a typical value for the structure parameter might be 10-14 m-2/3. This gives ZT as about 4km. At about 20 km above the earth, where the structure

Rn

T ZCZ

258.3


parameter might be 10-18 m-2/3, then ZT would be about 40000 km!

Manipulation of Gaussian BeamsImpact of Optical System Propagation

Complex parameter, q, provides complete description of gaussian beam

q = z + jZR

Paraxial optical system represented by matrix

Uniquely determine new gaussian beam from following relation that bl l l ti f l l

DC

BAM


enables calculation of new complex q value:

DCq

BAqq

1

12

Physical Optics PropagationGaussian-Hermite Polynomials

Gaussian beam not the only solution to the Paraxial Helmholtz Equation

Gaussian-Hermite Polynomials represent complete solution set for the Paraxial Equation

Any solution can be represented by a superposition of terms

Useful for beam propagation modelling


G0G1 G2

Diffraction TheoryLimitations of ‘Scalar Theory’

Maxwell’s equations deal with vector quantities

Simple analysis has presented scalar amplitudes Simple analysis has presented scalar amplitudes

Ignores the impact of polarisation

Scalar theory depends on the assumption that the axial component of the electric field may be neglected

Breaks down for systems with very large NA

Analysis then requires rigorous solution of vector wave


y q gequations

Example – analysis of blazed diffraction gratings

Image QualitySome Measures

RMS spot size and distribution Related to geometrical optics

Distribution gives some indication of underlying aberrationsg y g

Not so useful where system is (nearly) diffraction limited

Point Spread Function (PSF) Related to physical optics

Gives the real image intensity distribution in the presence of diffraction

Peak intensity of the PSF gives the so called ‘Strehl Ratio’

Strehl Ratio A measure of how close the system is to diffraction limited


A measure of how close the system is to diffraction limited

Strehl ratio of unity is perfect

Strehl ratio of anything greater than 0.85 is considered diffraction limited

Modulation Transfer Function (MTF) Measures the contrast ratio of an image of a sinusoidally varying pattern as a

function of spatial frequency

Quantitatively, the RMS spot size is usually given. Also, the ‘geometric’ spot size can be given, which is effectively the maximum radial size of the distribution. The RMS g yspot size can either be related to the centroid of the spot, or the chief ray. For off axis images, in an aberrated system, these two definitions will not be identical.


The Point Spread Function (PSF) gives a realistic description of the image formed by a point source through the optical system and takes into account the impact of p g p y pdiffraction. As the rms spot size of an Airy distribution is infinite, the PSF is usually denominated by its full width half maximum (FWHM). This is the displacement between those two points where the intensity falls to half the peak value.

For a system that is relatively close to the diffraction limit, the geometrical spot size considerably exaggerates the optical performance. In this situation, the PSF is significantly larger than the corresponding spot size, and the PSF is a faithful representation of the real image The Strehl ratio gives a measure of whether therepresentation of the real image. The Strehl ratio gives a measure of whether the image is ‘diffraction limited’. The higher the Strehl ratio, the better the image quality.


Image QualityMeasures of PSF

Full width half maximum (FWHM)

The displacement between the two points at which the intensity drops to half the peak

Encircled energy (EE)

Usually 50% or 80% encircled energy

— The radius of a circle that entirely encloses 50% (or 80%) of the PSF’s total flux

Ensquared energy

S i l d b t d i t f th i f


Same as encircled energy but expressed in terms of the size of a square that encloses some proportion of the PSF’s flux

Enslitted energy

Same as ensquared energy, but for one dimension only

Image QualityStrehl Ratio

Strehl ratio is the ratio of the peak intensity of the aberrated PSF to the peak intensity of the unaberratedPSFPSF

Forms a useful arbitrary definition of ‘diffraction limited’ imaging

Strehl Ratio > 0.8 is considered diffraction limited

Maréchal’s Criterion


22

21

Strehl

Equivalent to RMS WFE of ~ λ/14

For small WFE:

In the example shown, increasing amounts of spherical aberration are added to a nominally perfect image. Where the RMS spot size is significantly smaller than the y p g p g ydiffraction limit, then spot size has little influence over the PSF. The practical implication of this is that where optical systems are close to the diffraction limit, optimising the geometrical spot size will have little effect on system performance. For this reason, in system design, the average (RMS) OPD is used as an optimisation metric, where the system is close to diffraction limited. Otherwise, it is perfectly acceptable to use the geometric spot size as an optimisation metric.


Image QualityMTF

MTF is the contrast ratio of the image of a perfect sinusoidal object of some spatial frequency f

The MTF is given as a function of spatial frequencyg p q y

0

1

Object Intensity vs Displacement

1

Imax


minmax

minmax

II

IIMTF

0

Imin Image Intensity vs Displacement

Image QualityMTF (2)

Performance of an optical system is determined by the MTF at high spatial frequency (high lines per mm)


MTF is very commonly used in analysing the image quality of camera lenses. The reason for this is that resolution, as expressed by spatial frequency, provides a very p y p q y p ystraightforward means of capturing detector resolution. For example, for photographic film, a typical resolution for colour media might be 50 cycles per mm. For black and white film (depending on the speed etc.), this would be higher, 100 cycles per mm might be typical. In the case of digital cameras, from Nyquistsampling theory, the effective spatial frequency might be half the pixel spatial frequency. That is to say, a 5 μm pixel spacing would be equivalent to 100 cycles per mm.


Problem 5

A gaussian beam with a wavelength of 0.55 m is described in the near field and far field by the (1/e)2 intensity points, w0 and NA0, in the following way:

2

2

x

2

2

NA

NA

The near field radius, w0, is known and is 3.5 m. What is the Rayleigh Distance of this beam? What is the radius of the wavefront, R, at a point 50 m from the beam waist? What is the radius, w, of the beam at this position, taken at the (1/e)2

intensity point? A 50 mm focal length lens is placed 85 mm from the beam waist. Calculate the position of the new beam waist after this lens. Calculate the size,

0

0)(

weIxI 0

0)(

NAeIxI

w0‘, of the new beam, together with its Rayleigh Distance.


Diffraction

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known amplitude

wavelets amplitude

diffraction effects

image quality slide

roptical engineering

far field x

wfe profiles diffraction