OPTI 517 Image Quality Richard Juergens Senior Engineering Fellow Raytheon Missile Systems 520-794-0917 [email protected] Why is Image Quality Important? • Resolution of detail – Smaller blur sizes allow better reproduction of image details – Addition of noise can mask important image detail Original Blur added Noise added OPTI 517 Pixelated 2 Step One - What is Your Image Quality (IQ) Spec? • There are many metrics of image quality – Geometrical based (e.g., spot diagrams, RMS wavefront error) – Diffraction based (e.g., PSF, MTF) – Other (F-theta linearity, uniformity of illumination, etc.) • It is imperative that you have a specification for image quality when you are designing an optical system – Without it, you don't know when you are done designing! OPTI 517 3 You vs. the Customer • Different kinds of image quality metrics are useful to different people • Customers usually work with performance-based specifications – MTF, ensquared energy, distortion, etc. • Designers often use IQ metrics that mean little to the customer – E.g., ray aberration plots and field plots – These are useful in the design process, but they are not end-product specs • In general, you will be working to an end-product specification, but will probably use other IQ metrics during the design process – Often the end-product specification is difficult to optimize to or may be time consuming to compute • Some customers do not express their image quality requirements in terms such as MTF or ensquared energy – They know what they want the optical system to do • It is up to the optical engineer (in conjunction with the system engineer) to translate the customer's needs into a numerical specification suitable for optimization and image quality analysis OPTI 517 4 When to Use Which IQ Metric • The choice of appropriate IQ metric usually depends on the application of the optical system – Long-range targets where the object is essentially a point source • Example might be an astronomical telescope • Ensquared energy or RMS wavefront error might be appropriate – Ground-based targets where the details of the object are needed to determine image features • Example is any kind of image in which you need to see detail • MTF would be a more appropriate metric – Laser scanning systems • A different type of IQ metric such as the variation from F-theta distortion • The type of IQ metric may be part of the lens specification or may be a derived requirement flowed down to the optical engineer from systems engineering – Do not be afraid to question these requirements – Often the systems engineering group doesn't really understand the relationship between system performance and optical metrics OPTI 517 5 Wavefront Error • Aberrations occur when the converging wavefront is not perfectly spherical Real Aberrated Wavefront Reference sphere (centered on ideal image point) Rays normal to the reference sphere form a perfect image Ideal image point Real rays proceed in a direction normal to the aberrated wavefront Optical Path Difference (OPD) OPTI 517 Optical path difference (OPD) and wavefront error (WFE) are just two different names for the same error 6 Tilt and Focus of a Wavefront • When calculating the wavefront error, the reference sphere is centered on the "expected" image location (usually the image surface location of the chief ray) • It may be that a different reference sphere will fit the actual wavefront better – If the center of the better reference sphere is at a different axial location than the expected image location, there is a "focus error" in the wavefront – If the center of the better reference sphere is at a different lateral location than the expected image location, there is a "tilt error" in the wavefront Actual wavefront Better fitting reference sphere Center of better fitting reference sphere Tilt error Reference sphere Focus error Expected image location OPTI 517 7 Optical Path Difference Peak-to-valley OPD is the difference between the longest and the shortest paths leading to a selected focus Typical Wavefront RMS wavefront error is given by: Specific OPD W n [W ( x, y )] dxdy = òò òò dxdy n Wrms = W 2 - W Peak-To-Valley OPD 2 For n discrete rays across the pupil Reference Sphere RMS = S OPD2/n This wavefront has the same P-V wavefront error as the example at the left, but it has a lower RMS OPTI 517 8 Peak-to-Valley vs. RMS • The ratio of P-V to RMS is not a fixed quantity • Typical ratios of P-V to RMS (from Shannon's book) – Defocus 3.5 – 3rd order spherical 13.4 – 5th order spherical 57.1 – 3rd order coma 8.6 – 3rd order astigmatism 5.0 – Smooth random errors ~5 • In general, for a mixture of lower order aberrations, P-V/RMS ≈ 4.5 • When generating wavefront error budgets, RMS errors from different sources can be added in an RSS fashion – P-V errors cannot be so added • In general, Peak-to-Valley wavefront error is a poor choice to use for error budgeting – However, Peak-to-Valley surface error or wavefront error is still commonly used as a surface error specification for individual optical components and even for complete optical systems OPTI 517 9 Rayleigh Criterion • Lord Rayleigh observed that when the maximum wavefront error across a wavefront did not exceed l/4 peak-to-valley, the image quality was "not sensibly degraded" – This quarter-wave limit is now called the Rayleigh Criterion • This is approximately equivalent to the RMS wavefront error being about 0.07 wave or less (using the value for defocus) • The Strehl Ratio is a related measure of image quality – It can be expressed (for RMS wavefront error < 0.1 wave) as Strehl Ratio = e -(2pF ) » 1 - ( 2pF )2 2 where F is the RMS wavefront error (in waves) – For F = 0.07 wave, the Strehl Ratio » 0.8 • Requiring the Strehl Ratio to be 0.80 or greater for acceptable image quality is often called the Maréchal Criterion OPTI 517 10 Diffraction-limited Performance • Many systems have "diffraction-limited" performance as a specification – Taken literally, this might mean that no aberrations are allowed – As a practical matter, it means that diffraction dominates the image and that the geometric aberrations are small compared to the Airy disk • There is a distinction between the best possible performance, as limited by diffraction, and performance that is below this limit but produces acceptable image quality (e.g., Strehl Ratio > 80%) Diffraction spot size Geometrical spot size Spot Size Total spot size Rule of Thumb: Total 80% blur = [(Geo 80% blur)2 + (Airy diameter)2]1/2 Amount of Aberration OPTI 517 11 Image Quality Metrics • The most commonly used geometrical-based image quality metrics are – Ray aberration curves – Spot diagrams – Seidel aberrations – Encircled (or ensquared) energy – RMS wavefront error – Modulation transfer function (MTF) • The most commonly used diffraction-based image quality metrics are – Point spread function (PSF) – Encircled (or ensquared) energy – MTF – Strehl Ratio OPTI 517 12 Ray Aberration Curves • These are by far the image quality metric most commonly used by optical designers during the design process • Ray aberration curves trace fans of rays in two orthogonal directions – They then map the image positions of the rays in each fan relative to the chief ray vs. the entrance pupil position of the rays Sagittal rays Dy values for tangential rays Dx values for sagittal rays 0.1 Image position Tangential rays 1 -y +y -x +x -0.1 Pupil position OPTI 517 13 Graphical Description of Ray Aberration Curves • Ray aberration curves map the image positions of the rays in a fan – The plot is image plane differences from the chief ray vs. position in the fan Image plane differences from the chief ray Pupil position Image plane • Ray aberration curves are generally computed for a fan in the YZ plane and a fan in the XZ plane – This omits skew rays in the pupil, which is a failing of this IQ metric Transverse vs. Wavefront Ray Aberration Curves • Ray aberration curves can be transverse (linear) aberrations in the image vs. pupil position or can be OPD across the exit pupil vs. pupil position – The transverse ray errors are related to the slope of the wavefront curve ey(xp,yp) = -(R/rp) ¶W(xp,yp)/¶yp, ex(xp,yp) = -(R/rp) ¶W(xp,yp)/¶xp R/rp = -1/(n'u') » 2 f/# • Example curves for pure defocus: 0.001 inch Transverse 1.0 wave Wavefront error More on Ray Aberration Curves • The shape of the ray aberration curve can tell what type of aberration is present in the lens for that field point (transverse curves shown) Tangential fan Sagittal fan 0.05 0.05 1 1 -0.05 -0.05 Defocus Coma 0.05 0.05 1 1 -0.05 -0.05 Third-order spherical Astigmatism OPTI 517 16 The Spot Diagram • The spot diagram is readily understood by most engineers • It is a diagram of how spread out the rays are in the image – The smaller the spot diagram, the better the image – This is geometrical only; diffraction is ignored • It is useful to show the detector size (and/or the Airy disk diameter) superimposed on the spot diagram Different colors represent different wavelengths Detector outline • The shape of the spot diagram can often tell what type of aberrations are present in the image OPTI 517 17 Main Problem With Spot Diagrams • The main problem is that spots in the spot diagram don't convey intensity – A ray intersection point in the diagram does not tell the intensity at that point FIELD POSITION 0.00, 1.00 0.000,14.00 DG 0.00, 0.71 0.000,10.00 DG The on-axis image appears spread out in the spot diagram, but in reality it has a tight core with some surrounding lowintensity flare 0.00, 0.00 0.000,0.000 DG .163 DEFOCUSING MM 0.00000 Double Gauss - U.S. Patent 2,532,751 OPTI 517 18 Diffraction • Some optical systems give point images (or near point images) of a point object when ray traced geometrically (e.g., a parabola on-axis) • However, there is in reality a lower limit to the size of a point image • This lower limit is caused by diffraction – The diffraction pattern is usually referred to as the Airy disk Image intensity Diffraction pattern of a perfect image OPTI 517 19 Size of the Diffraction Image • The diffraction pattern of a perfect image has several rings – The center ring contains ~84% of the energy, and is usually considered to be the "size" of the diffraction image d d Very important !!!! • The diameter of the first ring is given by d » 2.44 l f/# – This is independent of the focal length; it is only a function of the wavelength and the f/number – The angular size of the first ring b = d/F » 2.44 l/D • When there are no aberrations and the image of a point object is given by the diffraction spread, the image is said to be diffraction-limited OPTI 517 20 Image of a Point Object and a Uniform Background D1 d D2 d Same f/# • For both systems, the Airy disk diameter is the same size – d = 2.44 l f/# • For both systems, the irradiance of the background at the image is the same – EB = LB(p/4f2) • The flux forming the image from the larger system is larger by (D2/D1)2 – We get more energy in the image, so the signal-to-noise ratio (SNR) is increased by (D2/D1)2 – This is important for astronomy and other forms of point imagery OPTI 517 21 Spot Size vs. the Airy Disk • Regime 1 – Diffraction-limited Airy disk diameter Image intensity Point image (geometrically) Strehl = 1.0 • Regime 2 – Near diffraction-limited Non-zero geometric blur, but smaller than the Airy disk Strehl ³ 0.8 • Regime 3 – Far from diffraction-limited Airy disk diameter Geometric blur significantly larger than the Airy disk OPTI 517 Strehl ~ 0 22 Point Spread Function (PSF) • This is the image of a point object including the effects of diffraction and all aberrations Intensity peak of the PSF relative to that of a perfect lens (no wavefront error) is the Strehl Ratio Image intensity Airy disk (diameter of the first zero) OPTI 517 23 Diffraction Pattern of Aberrated Images • When there is aberration present in the image, two effects occur – Depending on the aberration, the shape of the diffraction pattern may become skewed – There is less energy in the central ring and more in the outer rings Perfect PSF Strehl = 1.0 Strehl = 0.80 OPTI 517 25 25 0.002032 mm 0.002032 mm 24 PSF vs. Defocus OPTI 517 25 PSF vs. Third-order Spherical Aberration OPTI 517 26 PSF vs. Third-order Coma OPTI 517 27 PSF vs. Astigmatism OPTI 517 28 PSF for Strehl = 0.80 Defocus Balanced 3rd and 5th-order SA 3rd-order SA Coma Astigmatism OPTI 517 29 Encircled or Ensquared Energy • Encircled or ensquared energy is the ratio of the energy in the PSF that is collected by a single circular or square detector to the total amount of energy that reaches the image plane from that object point – This is a popular metric for system engineers who, reasonably enough, want a certain amount of collected energy to fall on a single pixel – It is commonly used for systems with point images, especially systems which need high signal-to-noise ratios • For %EE specifications of 50-60% this is a reasonably linear criterion – However, the specification is more often 80%, or even worse 90%, energy within a near diffraction-limited diameter – At the 80% and 90% levels, this criterion is highly non-linear and highly dependent on the aberration content of the image, which makes it a poor criterion, especially for tolerancing Ensquared Energy Example Ensquared energy on a detector of same order of size as the Airy disk Perfect lens, f/2, 10 micron wavelength, 50 micron detector Airy disk (48.9 micron diameter) Detector Approximately 85% of the energy is collected by the detector Modulation Transfer Function (MTF) • MTF is the Modulation Transfer Function • Measures how well the optical system images objects of different sizes – Size is usually expressed as spatial frequency (1/size) • Consider a bar target imaged by a system with an optical blur – The image of the bar pattern is the geometrical image of the bar pattern convolved with the optical blur Convolved with = • MTF is normally computed for sine wave input, and not square bars to get the response for a pure spatial frequency • Note that MTF can be geometrical or diffraction-based OPTI 517 32 Computing MTF • The MTF is the amount of modulation in the image of a sine wave target – At the spatial frequency where the modulation goes to zero, you can no longer see details in the object of the size corresponding to that frequency • The MTF is plotted as a function of spatial frequency (1/sine wave period) MTF = OPTI 517 Max - Min Max + Min 33 MTF of a Perfect Image For an aberration-free image and a round pupil, the MTF is given by 2 MTF( f ) = [j - cos j sinj] p j = cos -1(f / fco ) = cos -1( lf ) 2NA DEFOCUSING 0.00000 1.0 This f is spatial frequency (lp/mm) and not f/number 0.9 0.8 0.7 MTF • 0.6 A 0.5 0.4 0.3 Cutoff frequency fco = 1/(lf/#) 0.2 0.1 50 150 250 350 450 550 650 750 850 950 Spatial frequency (lp/mm) OPTI 517 34 Abbe’s Construct for Image Formation • Abbe developed a useful framework from which to understand the diffractionlimiting spatial frequency and to explain image formation in microscopes • If the first-order diffraction angle from the grating exceeds the numerical aperture (NA = 1/(2f/#)), no light will enter the optical system for object features with that characteristic spatial period OPTI 517 35 Example MTF Curve Direction of field point FOV OPTI 517 36 MTF as an Autocorrelation of the Pupil • The MTF is usually computed by lens design programs as the autocorrelation of the OPD map across the exit pupil Relative spatial frequency = spacing between shifted pupils (cutoff frequency = pupil diameter) Perfect MTF = overlap area / pupil area Complex OPD computed for many points across the pupil Overlap area MTF is computed as the normalized integral over the overlap region of the difference between the OPD map and its shifted complex conjugate OPTI 517 37 Typical MTF Curves Introductory Seminar f/5.6 Tessar DIFFRACTION LIMIT AXIS DIFFRACTION MTF T R 0.7 FIELD ( 14.00 O ) T R 1.0 FIELD ( 20.00 O ) WAVELENGTH 650.0 NM 550.0 NM 480.0 NM WEIGHT 1 2 1 MTF is a function of the spectral weighting DEFOCUSING 0.00000 1.0 MTF curves are different for different points across the FOV 0.9 0.8 MTF is a function of the focus 0.7 Diffraction-limited MTF (as good as it can get) M O 0.6 D U L 0.5 A T I O 0.4 N 0.3 0.2 0.1 20 40 60 80 100 120 140 160 180 200 SPATIAL FREQUENCY (CYCLES/MM) OPTI 517 38 Phase Shift of the OTF • Since OPD relates to the phase of the ray relative to the reference sphere, the pupil autocorrelation actually gives the OTF (optical transfer function), which is a complex quantity – MTF is the real part (modulus) of the OTF OTF = Optical Transfer Function MTF = Modulus of the OTF PTF = Phase of the OTF When the OTF goes negative, the phase is p radians OPTI 517 39 What Does OTF < 0 Mean? • When the OTF goes negative, it is an example of contrast reversal OPTI 517 40 Example of Contrast Reversal 1.00 0.75 DEFOCUSING 0.00000 1.0 0.9 0.50 0.8 0.7 0.6 0.25 0.5 0.4 At best focus 0.3 0.00 0.2 -0.1234 -0.0925 -0.0617 -0.0308 0.0000 0.0308 0.0617 0.0925 0.1234 -0.0922 -0.0614 -0.0307 0.0000 0.0307 0.0614 0.0922 0.1229 0.1 1.0 6.0 11.0 16.0 21.0 26.0 31.0 36.0 41.0 46.0 51.0 DISPLACEMENT ON IMAGE SURFACE (MM) 56.0 1.00 0.75 DEFOCUSING 0.00000 1.0 0.9 0.50 0.8 0.7 0.6 Defocused 0.5 0.25 0.4 0.3 0.00 -0.1229 0.2 0.1 1.0 6.0 11.0 16.0 21.0 26.0 31.0 36.0 41.0 46.0 51.0 DISPLACEMENT ON IMAGE SURFACE (MM) 56.0 OPTI 517 41 More on Contrast Reversal Original Object OPTI 517 42 Effect of Strehl = 0.80 • When the Strehl Ratio = 0.80 or higher, the image is considered to be equivalent in image quality to a diffraction-limited image • The MTF in the mid-range spatial frequencies is reduced by the Strehl ratio DEFOCUSING 0.00000 1.0 0.9 0.8 0.7 M O 0.6 D U L 0.5 A T I O 0.4 N Diffraction-limited MTF 0.3 1.0 0.2 0.8 0.1 13 91 169 247 325 403 481 559 637 715 793 SPATIAL FREQUENCY (CYCLES/MM) OPTI 517 43 Aberration Transfer Function • Shannon has shown that the MTF can be approximated as a product of the diffraction-limited MTF (DTF) and an aberration transfer function (ATF) DTF(n ) = 2é -1 2ù cos n n 1 n úû p êë 2 n = f / fco ( æW ö ATF(n ) = 1 - ç rms ÷ 1 - 4(n - 0.5)2 è 0.18 ø ) 1.0 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.5 0.4 0.025 waves rms 0.050 waves rms 0.3 0.075 waves rms 0.2 0.100 waves rms 0.125 waves rms 0.1 0.150 waves rms 0.0 0.00 Diff. Limit 0.025 waves rms 0.050 waves rms 0.075 waves rms 0.100 waves rms 0.6 MTF ATF Bob Shannon 0.20 0.40 0.60 0.80 1.00 Normalized Spatial Frequency 0.125 waves rms 0.150 waves rms 0.5 0.4 0.3 0.2 0.1 0.0 0.00 0.20 0.40 0.60 0.80 1.00 Normalized Spatial Frequency OPTI 517 44 Demand Contrast Function • The eye requires more modulation for smaller objects to be able to resolve them – The amount of modulation required to resolve an object is called the demand contrast function – This and the MTF limits the highest spatial frequency that can be resolved The limiting resolution is where the Demand Contrast Function intersects the MTF System A will produce a superior image although it has the same limiting resolution as System B System A has a lower limiting resolution than System B even though it has higher MTF at lower frequencies OPTI 517 45 Example of Different MTFs on RIT Target OPTI 517 46 Central Obscurations • In on-axis telescope designs, the obscuration caused by the secondary mirror is typically 30-50% of the diameter – Any obscuration above 30% will have a noticeable effect on the Airy disk, both in terms of dark ring location and in percent energy in a given ring (energy shifts out of the central disk and into the rings) • Contrary perhaps to expectations, as the obscuration increases the diameter of the first Airy ring decreases (the peak is the same, and the loss of energy to the outer rings has to come from somewhere) OPTI 517 47 Central Obscurations • Central obscurations, such as in a Cassegrain telescope, have two deleterious effects on an optical system – The obscuration causes a loss in energy collected (loss of area) – The obscuration causes a loss of MTF A B C D OPTI 517 So/Sm = 0.00 So/Sm = 0.25 So/Sm = 0.50 So/Sm = 0.75 48 Coherent Illumination • Incoherent illumination fills the whole entrance pupil • Partially coherent illumination fills only part of the entrance pupil – Coherent illumination essentially only fills a point in the entrance pupil OPTI 517 49 MTF of Partially Coherent Illumination OPTI 517 50 Partial Coherent Image of a 3-Bar Target GEOMETRICAL SHADOW RNA (X,Y) FIELD SCAN INC ( 0.00, 0.00) R 1.50 ( 0.00, 0.00) R 1.00 ( 0.00, 0.00) R 0.50 ( 0.00, 0.00) R 0.00 ( 0.00, 0.00) R DIFFRACTION INTENSITY PROFILE PARTIALLY COHERENT ILLUMINATION WAVELENGTH 500.0 NM WEIGHT 1 13-Oct-02 1.25 DEFOCUSING 0.00000 RELATIVE INTENSITY 1.00 0.75 0.50 0.25 0.00 -5.0 -3.8 -2.5 -1.3 0.0 DISPLACEMENT ON IMAGE SURFACE OPTI 517 1.3 2.5 (MICRONS) 3.8 5.0 51 Example of Elbows Imaged in Partially Coherent Light 0.2145 0.00504 mm With 1 wave of spherical aberration 0.1168 0.00504 mm OPTI 517 52 The Main Aberrations in an Optical System • Defocus – the focal plane is not located exactly at the best focus position • Chromatic aberration – the axial and lateral shift of focus with wavelength • The Seidel aberrations – Spherical Aberration – Coma – Astigmatism – Distortion – Curvature of field OPTI 517 53 Defocus • Technically, defocus is not an aberration in that it can be corrected by simply refocusing the lens • However, defocus is an important effect in many optical systems Spherical reference sphere centered on defocused point Ideal focus point Defocused image point Actual wavefront When maximum OPD = l/4, you are at the Rayleigh depth of focus = 2 l f2 OPTI 517 54 Defocus Ray Aberration Curves Wavefront map Spot diagram 2.5 0.02 -2.5 -0.02 Wavefront error Transverse ray aberration OPTI 517 55 MTF of a Defocused Image • As the amount of defocus increases, the MTF drops accordingly A B C D E OPTI 517 OPD = 0 OPD = l/4 OPD = l /2 OPD = 3l /4 OPD = l 56 Sources of Defocus • One obvious source of defocus is the location of the object – For lenses focused at infinity, objects closer than infinity have defocused images – There's nothing we can do about this (unless we have a focus knob) • Changes in temperature – As the temperature changes, the elements and mounts change dimensions and the refractive indices change – This can cause the lens to go out of focus – This can be reduced by design (material selection) • Another source is the focus procedure – There are two possible sources of error here • Inaccuracy in the measurement of the desired focus position • Resolution in the positioning of the focus (e.g., shims in 0.001 inch increments) – The focus measurement procedure and focus position resolution must be designed to not cause focus errors which can degrade the image quality beyond the IQ specification OPTI 517 57 Chromatic Aberration • Chromatic aberration is caused by the lens's refractive index changing with wavelength 1.524 Blue Green Red Refractive Index 1.522 1.520 1.518 1.516 0.800 1.514 480. 520. 560. 600. 640. 680. Wavelength (nm) The shorter wavelengths focus closer to the lens because the refractive index is higher for the shorter wavelengths FOCUS SHIFT (in) 0.400 0.000 480. 520. 560. 600. 640. 680. -0.400 -0.800 -1.200 OPTI 517 WAVELENGTH (nm) 58 Computing Chromatic Aberration • The chromatic aberration of a lens is a function of the dispersion of the glass – Dispersion is a measure of the change in index with wavelength • It is commonly designated by the Abbe V-number for three wavelengths – For visible glasses, these are F (486.13), d (587.56), C (656.27) – For infrared glasses they are typically 3, 4, 5 or 8, 10, 12 microns – V = (nmiddle-1) / (nshort - nlong) • For optical glasses, V is typically in the range 35-80 • For infrared glasses they vary from 50 to 1000 • The axial (longitudinal) spread of the short wavelength focus to the long wavelength focus is F/V – Example 1: N-BK7 glass has a V-value of 64.4. What is the axial chromatic spread of an N-BK7 lens of 100 mm focal length? • Answer: 100/64.4 = 1.56 mm • Note that if the lens were f/2, the diffraction DOF = ±2lf2 = ±0.004 mm – Example 2: Germanium has a V-value of 942 (for 8 – 12 m). What is the axial chromatic spread of a germanium lens of 100 mm focal length? • Answer: 100/942 = 0.11 mm Note: DOF(f/2) = ±2lf2 = ±0.08 mm OPTI 517 59 Chromatic Aberration Example - Germanium Singlet • We want to use an f/2 germanium singlet over the 8 to 12 micron band • Question - What is the longest focal length we can have and not need to color correct? (assume an asphere to correct any spherical aberration) • Answer – Over the 8-12 micron band, for germanium V = 942 – The longitudinal defocus = F / V = F / 942 – The 1/4 wave depth of focus is ±2lf2 – Equating these and solving gives F = 4*942*l*f2 = 150 mm waves 0.25 FIELD HEIGHT ( 0.000 O) DEFOCUSING 0.00000 1.0 0.9 0.8 0.7 Strehl = 0.86 M O 0.6 D U L 0.5 A T I O 0.4 N 0.3 -0.25 0.2 0.1 1.0 6.0 11.0 16.0 21.0 26.0 31.0 36.0 41.0 46.0 51.0 56.0 61.0 SPATIAL FREQUENCY (CYCLES/MM) OPTI 517 60 Correcting Chromatic Aberration • Chromatic aberration is corrected by a combination of two glasses – The positive lens has low dispersion (high V number) and the negative lens has high dispersion (low V number) Red Blue Green Red and blue focus together – This will correct primary chromatic aberration • The red and blue wavelengths focus together • The green (or middle) wavelength still has a focus error – This residual chromatic spread is called secondary color OPTI 517 61 Secondary Color • Secondary color is the residual chromatic aberration left when the primary chromatic aberration is corrected These two wavelengths focus together 0.050 FOCUS SHIFT (mm) 0.040 0.030 0.020 Secondary color (~ F/2400) This wavelength has a focus error 0.010 0.000 480. 520. 560. 600. 640. 680. -0.010 WAVELENGTH (nm) • Secondary color can be reduced by selecting special glasses – These glasses cost more (naturally) OPTI 517 62 Lateral Color • Lateral color is a change in focal length (or magnification) with wavelength – This results in a different image size with wavelength – The effect is often seen as color fringes at the edge of the FOV – This reduces the MTF for off-axis images Red Green Blue OPTI 517 63 Higher-order Chromatic Aberrations • For broadband systems, the chromatic variation in the third-order aberrations are often the most challenging aberrations to correct (e.g., spherochromatism, chromatic variation of coma, chromatic variation of astigmatism) – These are best studied with ray aberration curves and field plots OPTI 517 64 The Seidel Aberrations • These are the classical aberrations in optical design – Spherical aberration – Coma – Astigmatism – Distortion – Curvature of field • These aberrations, along with defocus and chromatic aberrations, are the main aberrations in an optical system OPTI 517 65 The Importance of Third-order Aberrations • The ultimate performance of any unconstrained optical design is almost always limited by a specific aberration that is an intrinsic characteristic of the design form • A familiarity with aberrations and lens forms is an important ingredient in a successful optimization that makes optimal use of the time available to accomplish the design • A knowledge of the aberrations – Allows "spotting" lenses that are at the end of the road with respect to optimization – Gives guidance in what direction to "kick" a lens that has strayed from the optimal solution OPTI 517 66 Orders of Aberrations OPTI 517 67 Spherical Aberration • Spherical aberration is an on-axis aberration • Rays at the outer parts of the pupil focus closer to or further from the lens than the paraxial focus This is referred to as undercorrected spherical aberration (marginal rays focus closer to the lens than the paraxial focus) Ray aberration curve Paraxial focus • The magnitude of the (third-order) spherical aberration goes as the cube of the aperture (going from f/2 to f/1 increases the SA by a factor of 8) OPTI 517 68 Third-order SA Ray Aberration Curves Spot diagram Wavefront map Transverse ray aberration curve Wavefront error OPTI 517 69 Spherical Aberration Minimum spot size Marginal focus Minimum RMS WFE Paraxial focus ½L ¾L L Marginal focus Minimum spot size OPTI 517 Minimum RMS WFE Paraxial focus 70 Scaling Laws for Spherical Aberration µ q0 µ (f/#)-3 5.0 4.0 f/3 Field Angle (deg) 3.0 f/4 2.0 1.0 0.0 -1.0 -2.0 f/5 -3.0 -4.0 Spot size goes as the cube of the EPD (or inverse cube of the f/#) -5.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 Field Angle (deg) Spot size not dependent on field position OPTI 517 71 Spherical Aberration vs. Lens Shape • The spherical aberration is a function of the lens bending, or shape of the lens OPTI 517 72 Spherical Aberration vs. Refractive Index • Spherical aberration is reduced with higher index materials – Higher indices allows shallower radii, allowing less variation in incidence angle across the lens n = 1.50 Notice the bending for minimum SA is a function of the index n = 1.95 OPTI 517 73 Spherical Aberration vs. Index and Bending b at K min = r j 3 3 4n2 - n 16(n - 1) (n + 2) 2 n = 1.5 n = 2.0 n = 3.0 n = 4.0 OPTI 517 74 Example - Germanium Singlet • We want an f/2 germanium singlet to be used at 10 microns (0.01 mm) • Question - What is the longest focal length we can have and not need aspherics to correct spherical aberration? • Answer – Diffraction Airy disk angular size is b diff = 2.44 l/D – Spherical aberration angular blur is b sa = 0.00867 / f3 – Equating these gives D = 2.44 l f3 / 0.00867 = 22.5 mm – For f/2, this gives F = 45 mm DEFOCUSING 0.00000 1.0 0.9 0.8 Strehl = 0.91 0.7 waves 0.25 M O 0.6 D U L 0.5 A T I O 0.4 N FIELD HEIGHT ( 0.000 O) 0.3 0.2 0.1 1.0 5.0 9.0 13.0 17.0 21.0 25.0 29.0 33.0 37.0 41.0 SPATIAL FREQUENCY (CYCLES/MM) -0.25 OPTI 517 75 45.0 49.0 Focus Shift vs. Wavelength (Germanium singlet) 0.1000 FOCUS SHIFT (mm) 0.0500 0.0000 8000. 8500. 9000. 9500. 10000. 10500. 11000. 11500. 12000. -0.0500 Diffraction-limited depth of focus -0.1000 WAVELENGTH (nm) OPTI 517 76 Spherical Aberration vs. Number of Lenses • Spherical aberration can be reduced by splitting the lens into more than one lens SA = 1 (arbitrary units) SA = 1/4 (arbitrary units) SA = 1/9 (arbitrary units) OPTI 517 77 Spherical Aberration vs. Number of Lenses OPTI 517 78 Spherical Aberration and Aspherics • The spherical aberration can be reduced, or even effectively eliminated, by making one of the surfaces aspheric 1.0 mm spherical 0.0001 mm aspheric OPTI 517 79 Aspheric Surfaces • Aspheric surfaces technically are any surfaces which are not spherical, but usually refer to a polynomial deformation to a conic z(r ) = r2 / R 1 + 1 - (k + 1)(r / R)2 + A r 4 + B r 6 + C r 8 + D r10 + ... • The aspheric coefficients (A, B, C, D, …) can correct 3rd, 5th, 7th, 9th, … order spherical aberration • When used near a pupil, aspherics are used primarily to correct spherical aberration • When used far away (optically) from a pupil, they are primarily used to correct astigmatism by flattening the field • Before using aspherics, be sure that they are necessary and the increased performance justifies the increased cost – Never use a higher-order asphere than justified by the ray aberration curves OPTI 517 80 Optimizing Aspherics For an asphere far away (optically) from a pupil, the ray density need not be high, but there must be a sufficient number of overlapping fields to sample the surface accurately. This asphere primarily corrects field aberrations (e.g., astigmatism). For an asphere at (or near) a pupil, there need to be enough rays to sample the pupil sufficiently. This asphere primarily corrects spherical aberration. OPTI 517 81 Asphere Example • 2 inch diameter, f/2 plano-convex lens sphere 0.10 asphere 0.00001 Note: Airy disk diameter is ~ 0.0001 inch OPTI 517 82 Aspheric Orders Sag cont relative to base sphere (in) 0.000 -0.002 -0.004 -0.006 Aspheric Sum 4th order 6th order 8th order 10th order -0.008 -0.010 0.00 0.20 0.40 0.60 Radial position (in) 0.80 1.00 0.0025 Corresponds to ~114 waves of asphericity Delta Sag 0.0020 0.0015 0.0010 0.0005 0.0000 -1.000E+00 -5.000E-01 0.000E+00 Y Position OPTI 517 5.000E-01 1.000E+00 83 MTF vs. Aspheric Order DEFOCUSING 1.0 DEFOCUSING 1.0 0.00000 0.00000 0.9 0.9 0.7 0.7 M O 0.6 D U L 0.5 A T I O 0.4 N M O 0.6 D U L 0.5 A T I O 0.4 N 0.3 0.3 0.2 0.2 0.1 0.1 84 168 252 336 420 asphere A term only 0.8 sphere 0.8 504 588 672 756 840 78 SPATIAL FREQUENCY (CYCLES/MM) 156 234 312 390 468 546 624 702 780 SPATIAL FREQUENCY (CYCLES/MM) DEFOCUSING 1.0 0.00000 DEFOCUSING 1.0 0.00000 0.9 0.9 asphere A,B terms 0.8 0.7 M O 0.6 D U L 0.5 A T I O 0.4 N 0.7 M O 0.6 D U L 0.5 A T I O 0.4 N 0.3 0.3 0.2 0.2 0.1 0.1 78 156 234 312 390 468 546 asphere A,B,C terms 0.8 624 702 780 858 SPATIAL FREQUENCY (CYCLES/MM) 78 156 234 312 390 468 546 624 702 780 858 SPATIAL FREQUENCY (CYCLES/MM) OPTI 517 84 Coma • Coma is an off-axis aberration • It gets its name from the spot diagram which looks like a comet (coma is Latin for comet) • A comatic image results when the periphery of the lens has a higher or lower magnification than the portion of the lens containing the chief ray Chief ray Spot diagram • The magnitude of the (third-order) coma is proportional to the square of the aperture and the first power of the field angle OPTI 517 85 Transverse vs. Wavefront 3rd-order Coma Spot diagram Wavefront map 5.0 0.001 -0.001 -5.0 Wavefront error Transverse ray aberration OPTI 517 86 Scaling Laws for Coma µ (f/#)-2 f/5 f/4 µ q1 f/3 5.0 4.0 Full Field 0.5 Field Field Angle (deg) 3.0 2.0 1.0 0.0 -1.0 -2.0 -3.0 -4.0 -5.0 On-axis -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 Field Angle (deg) Spot size goes as the square of the EPD (or inverse square of the f/#) Spot size is linearly dependent on field height OPTI 517 87 Coma vs. Lens Bending • Both spherical aberration and coma are a function of the lens bending Coma Spherical aberration OPTI 517 88 Coma vs. Stop Position • The size of the coma is also a function of the stop location relative to the lens Aperture stop Coma is reduced due to increased lens symmetry around the stop OPTI 517 89 Coma is an Odd Aberration • Any completely symmetric optical system (including the stop location) is free of all orders of odd field symmetry aberrations (coma and distortion) OPTI 517 90 Astigmatism • Astigmatism is caused when the wavefront has a cylindrical component – The wavefront has different spherical power in one plane (e.g., tangential) vs. the other plane (e.g., sagittal) • The result is different focal positions for tangential and sagittal rays Rays in YZ plane focus here Rays in XZ plane focus here • The magnitude of the (third-order) astigmatism goes as the first power of the aperture and the square of the field angle OPTI 517 91 Cause of Astigmatism Radius = RSphere Radius = RCut Rcut< RSphere Non-rotationally symmetric through an off-center part of the surface Astigmatism No astigmatism Rotationally symmetric through a centered part of the surface OPTI 517 92 Image of a Wagon Wheel With Astigmatism Wagon Wheel Radial lines Tangential Focus Tangential lines In Focus Tangential lines OPTI 517 Sagittal or Radial Focus Radial lines In Focus 93 Astigmatism vs. Field FIELD POSITION ASTIGMATIC FIELD CURVES ANGLE(deg) T 0.00, 1.00 0.000,5.000 DG S 5.00 0.00, 0.75 0.000,3.750 DG 3.75 0.00, 0.50 0.000,2.500 DG 2.50 0.00, 0.25 0.000,1.250 DG 1.25 0.00, 0.00 0.000,0.000 DG -0.10 -0.05 0.0 0.05 0.10 FOCUS (MILLIMETERS) .715E-01 MM DEFOCUSING -0.100 -0.090 -0.080 -0.070 -0.060 -0.050 -0.040 -0.030 -0.020 -0.010 -0.000 OPTI 517 94 Scaling Laws for Astigmatism µ (f/#)-1 q2 5.0 5.0 4.0 4.0 3.0 3.0 Field Angle (deg) Field Angle (deg) µ (f/#)-1 q2 2.0 1.0 0.0 -1.0 2.0 1.0 0.0 -1.0 -2.0 -2.0 -3.0 -3.0 -4.0 -4.0 -5.0 -5.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 5.0 1.0 2.0 3.0 4.0 5.0 Field Angle (deg) Field Angle (deg) Tangential focus Sagittal focus OPTI 517 95 Astigmatism Ray Aberration Plots TANGENTIAL 0.01 1.00 RELATIVE SAGITTAL FIELD HEIGHT 0.01 O ( 5.000 ) -0.01 -0.01 TANGENTIAL 0.01 0.01 ( 2.500 O) -0.01 0.01 -0.01 SAGITTAL 0.01 -0.01 FIELD HEIGHT ( 2.500 O) -0.01 0.00 RELATIVE 0.01 ( 5.000 O) TANGENTIAL 0.01 0.01 ( 0.000 O) -0.01 Tangential focus 0.01 0.01 -0.01 FIELD HEIGHT ( 0.000 O) -0.01 ( 5.000 O) SAGITTAL 0.01 -0.01 0.01 FIELD HEIGHT ( 2.500 O) -0.01 0.01 -0.01 0.00 RELATIVE 0.01 0.01 -0.01 -0.01 Medial focus (best diffraction focus) Occurs halfway between sagittal and tangential foci OPTI 517 FIELD HEIGHT 0.50 RELATIVE 0.00 RELATIVE FIELD HEIGHT 1.00 RELATIVE -0.01 0.50 RELATIVE FIELD HEIGHT -0.01 FIELD HEIGHT -0.01 0.50 RELATIVE 0.01 1.00 RELATIVE FIELD HEIGHT ( 0.000 O) 0.01 -0.01 Sagittal focus Note: the sagittal focus does not always occur at the paraxial focus 96 Transverse vs. Wavefront Astigmatism At medial focus Spot diagram Wavefront map 0.02 2.0 -0.02 -2.0 Transverse ray aberration Wavefront error OPTI 517 97 PSF of Astigmatism vs. Focus Position Tangential focus Medial focus (best diffraction focus) OPTI 517 Sagittal focus 98 Astigmatism of a Tilted Flat Plate • Placing a tilted plane parallel plate into a diverging or converging beam will introduce astigmatism t q • The amount of the longitudinal astigmatism (focus shift between the tangential and sagittal foci) is given by é n2 cos2q ù Ast = 1 ê 2 ú 2 n2 - sin2q ë n - sin q û t ( ) - t q 2 n2 - 1 Ast = n3 Exact Third-order OPTI 517 99 Correcting the Astigmatism of a Tilted Flat Plate • The astigmatism introduced by a tilted flat plate can be corrected by – Adding cylindrical lenses – Adding tilted spherical lenses – Adding another plate tilted in the orthogonal plane To correct for this Do not do this (it will double the astigmatism) OPTI 517 Do this 100 Reducing the Astigmatism of a Tilted Flat Plate • Astigmatism of a flat plate can be reduced by adding a slight wedge to the plate Flat plate 0.2 -0.2 0.47° wedge 0.2 -0.2 Transverse ray aberration OPTI 517 101 Rectilinear Imaging • Most optical systems want to image rectilinear objects into rectilinear images h Object q Image s' q s h' • This requires that m = -s'/s = -h'/h = constant for the entire FOV • For infinite conjugate lenses, this requires that h' = F tanq for all field angles h' q F OPTI 517 102 Distortion • If rectilinear imaging is not met, then there is distortion in the lens • Effectively, distortion is a change in magnification or focal length over the field of view Paraxial image height Real image height less than paraxial height implies existence of distortion Plot of distorted FOV • Negative distortion (shown) is often called barrel distortion • Positive distortion (not shown) is often called pincushion distortion OPTI 517 103 More on Distortion • Distortion does not result in a blurred image and does not cause a reduction in any measure of image quality such as MTF • Distortion is a measure of the displacement of the image from its corresponding paraxial reference point • Distortion is independent of f/number • Linear distortion is proportional to the cube of the field angle • Percent distortion is proportional to the square of the field angle DISTORTION ANGLE(deg) 20.00 15.27 10.31 5.20 -2 -1 0 1 2 % DISTORTION OPTI 517 104 Implications of Distortion • Consider negative distortion – A rectilinear object is imaged inside the detector • This means a rectilinear detector sees a larger-than-rectilinear area in object space OPTI 517 105 Curvature of Field • In the absence of astigmatism, the focal surface is a curved surface called the Petzval surface Petzval Surface Lens Flat Object OPTI 517 106 Third-order Field Curvature µ (f/#)-1 q2 TANGENTIAL 0.15 1.00 RELATIVE FIELD HEIGHT ( 15.00 O) -0.15 SAGITTAL 0.15 -0.15 0.66 RELATIVE ASTIGMATIC FIELD CURVES 0.15 FIELD HEIGHT ( 10.00 O) 0.15 ANGLE(deg) T 15.00 S -0.15 -0.15 0.33 RELATIVE 0.15 11.36 FIELD HEIGHT ( 5.000 O) -0.15 0.15 -0.15 7.63 0.00 RELATIVE 0.15 FIELD HEIGHT ( 0.000 O) 0.15 3.83 -0.15 -0.15 Aberrations relative to a flat image surface -0.02 -0.01 0.0 0.01 0.02 FOCUS (MILLIMETERS) OPTI 517 107 The Petzval Surface • The radius of the Petzval surface is given by 1 RPetzval æ 1 ö ÷÷ = å çç i è ni Fi ø – For a singlet lens, the Petzval radius = n F • Obviously, if we have only positive lenses in an optical system, the Petzval radius will become very short – We need some negative lenses in the system to help make the Petzval radius longer (i.e., flatten the field) • This, and chromatic aberration correction, is why optical systems need some negative lenses in addition to all the positive lenses OPTI 517 108 Field Curvature and Astigmatism • As an aberration, field curvature is not very interesting • As a design obstacle, it is the basic reason that optical design is still a challenge • The astigmatic contribution starts from the Petzval surface – If the axial distance from the Petzval surface to the sagittal surface is 1 (arbitrary units), then the distance from the Petzval surface to the tangential surface is 3 Field curvature and astigmatism can be used together to help flatten the image plane and improve the image quality 3 1 OPTI 517 109 Flattening the Field • The contribution of a lens to the focal length is proportional to yF where F is lens power (1/F) • The contribution of a lens to the Petzval sum is proportional to F/n • Thus, if we include negative lenses in the system where y is small we can reduce the Petzval sum and flatten the field while holding the focal length Y Y Lens With Field Flattener (Petzval Lens) Cooke Triplet • Yet another reason why optical systems have so darn many lenses Flat-field lithographic lens Negative lenses in RED OPTI 517 110 Original Object OPTI 517 111 Spherical Aberration Image blur is constant over the field OPTI 517 112 Coma Image blur grows linearly over the field OPTI 517 113 Astigmatism Image blurs more in one direction over the field OPTI 517 114 Distortion No image degradation but image locations are shifted OPTI 517 115 Curvature of Field Image blur grows quadratically over the field OPTI 517 116 Combined Aberrations – Spot Diagrams OPTI 517 117 Balancing of Aberrations • Different aberrations can be combined to improve the overall image quality – Spherical aberration and defocus – Astigmatism and field curvature – Third-order and fifth-order spherical aberration – Longitudinal color and spherochromatism – Etc. • Lens design is the art (or science) of putting together a system so that the resulting image quality is acceptable over the field of view and range of wavelengths OPTI 517 118 Resolution • Resolution is an important aspect of image quality • Every image has some resolution associated with it, even if it is the Airy disk – In this case, the resolution is dependent on the aberrations of the system • Resolution is the smallest detail you can resolve in the image – It determines whether you can resolve two closely spaced objects 25 25 25 Well resolved 0.002016 mm 0.002016 mm 0.002016 mm Rayleigh criterion peak of 2nd at 1st zero of first OPTI 517 Sparrow criterion overlap at FWHM 119 Resolution vs. P-V Wavefront Error • The 1/4 wave rule was empirically developed by astronomers as the greatest amount of P-V wavefront error that a telescope could have and still resolve two stars separated by the Rayleigh spacing (peak of one at 1st zero of the other) Perfect 1/4 wave P-V 25 25 0.002016 mm 0.002016 mm 3/4 wave P-V 1/2 wave P-V 25 25 0.002016 mm 0.002016 mm OPTI 517 120 Resolution Examples • Angular resolution is given by b » 2.44 l/D – Limited only by the diameter, not by the focal length or f/number • U of A is building 8.4 meter diameter primary mirrors for astronomical telescopes • For visible light (~0.5 mm), the Rayleigh spacing corresponds to an angular separation of (2.44 * 0.5x10-6 / 8.4)/2 = 0.073x10-6 radian (~0.015 arc second) • Assume a binary star at a distance of 200 light years (~1.2x1015 miles) – This would have a resolution of 90 million miles – Perhaps enough resolution to "split the binary" • A typical cell phone camera has an aperture of about 0.070 inch – This gives a Rayleigh spacing of about 0.34 mrad (for reference, the human eye has an angular resolution of about 0.3 mrad) – For an object 10 feet away, this is an object resolution of about 1 mm OPTI 517 121 Film Resolution • Due to the grain size of film, there is an MTF associated with films • A reasonable guide for MTF of a camera lens is the 30-50 rule: 50% at 30 lp/mm and 30% at 50 lp/mm • For excellent performance of a camera lens, use 50% at 50 lp/mm • Another criterion for 35 mm camera lenses is 20% at 30 lp/mm over 90% of the field (at full aperture) • As a rough guide for the resolution required in a negative, use 200 lines divided by the square root of the long dimension in mm OPTI 517 122 Detectors • All optical systems have some sort of detector – The most common is the human eye – Many optical systems use a 2D detector array (e.g., CCD) • No matter what the detector is, there is always some small element of the detector which defines the detector resolution – This is referred to as a picture element (pixel) • The size of the pixel divided by the focal length is called the Instantaneous FOV (IFOV) – The IFOV defines the angular limit of resolution in object space – IFOV is always expressed as a full angle FOV Detector array IFOV OPTI 517 123 Implications of IFOV • If the target angular size is smaller than an IFOV, it is not resolved – It is essentially a point target – Example is a star • If the target annular size is larger than an IFOV it may be resolved – This does not mean that you can always tell what the object is OPTI 517 124 Practical Resolution Considerations • Resolution required to photograph written or printed copy: – Excellent reproduction (serifs, etc.) requires 8 line pairs per lower case e – Legible reproduction requires 5 line pairs per letter height – Decipherable (e, c, o partially closed) requires 3 line pairs per height • The correlation between resolution in cycles/minimum dimension and certain functions (often referred to as the Johnson Criteria) is – Detect 1.0 line pairs per dimension – Orient 1.4 line pairs per dimension – Aim 2.5 line pairs per dimension – Recognize 4.0 line pairs per dimension – Identify 6-8 line pairs per dimension – Recognize with 50% accuracy 7.5 line pairs per height – Recognize with 90% accuracy 12 line pairs per height OPTI 517 125 Johnson Resolution Criteria OPTI 517 126 Examples of the Johnson Criteria Detect 1 bar pair Maybe something of military interest Recognize 4 bar pairs Tank Identify 7 bar pairs Abrams Tank OPTI 517 127 MTF of a Pixel • Consider a fixed size pixel scanning across different sized bar targets When the pixel size equals the width of a bar pair (light and dark) there is no more modulation OPTI 517 128 MTF of a Pixel • If the pixel is of linear width D, the MTF of the pixel is given by MTF( f ) = The cutoff frequency (where the MTF goes to zero) is at a spatial frequency 1/D 1.0 0.8 0.6 0.4 MTF • sin( pfD ) pfD 0.2 0.0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 -0.2 -0.4 Normalized Spatial Frequency OPTI 517 129 Optical MTF and Pixel MTF • The total MTF is the product of the optical MTF and the pixel MTF 1.0 1.0 detector 0.9 1.0 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.6 0.6 optics 0.5 0.6 0.5 0.5 0.4 0.4 0.4 product 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.00 0.0 0.0 0.00 500.00 1000.00 1500.00 2000.00 Case 1 - Optics limited Best for high resolution over-sampling • 2500.00 Case 2 - Optics and detector are matched Best for most FLIR-like mapping systems 500.00 1000.00 1500.00 2000.00 Case 3 - Detector limited Best for detecting dim point targets Of course, there are other MTF contributors to total system MTF – Electronics, display, jitter, smear, eye, turbulence, etc. OPTI 517 2500.00 130 Effects of Signal/CCD Alignment on MTF A sampled imaging system is not shift-invariant OPTI 517 131 MTF of Alignment • When performing MTF testing, the user can align the image with respect to the imager to produce the best image – In this case, a sampling MTF might not apply • A natural scene, however, has no net alignment with respect to the sampling sites • To account for the average alignment of unaligned objects a sampling MTF must be added – MTFsampling = sin(pfDx)/(pfDx) where Dx is the sampling interval – This MTF an ensemble average of individual alignments and hence is statistical in nature OPTI 517 132 Aliasing • Aliasing is a very common effect but is not well understood by most people • Aliasing is an image artifact that occurs when we insufficiently sample a waveform – It is evidenced as the imaging of high frequency objects as low frequency objects Array of detectors OPTI 517 133 Sampling of a Sine Wave 1.0 1.0 0.5 0.5 0.0 0.0 11 -0.5 -0.5 -1.0 -1.0 OPTI 517 134 Nyquist Condition • If we choose a sampling interval sufficiently fine to locate the peaks and valleys of a sine wave, then we can reconstruct that frequency from its sample values • The Nyquist condition says we need at least two samples per cycle to reproduce a sine wave – For a sine wave period x, we need a sampling interval Dx < x/2 OPTI 517 135 MTF Fold Over • The effect of sampling is to replicate the MTF back from the sampling frequency – This will cause higher frequencies to appear as lower frequencies Nyquist frequency Sampling frequency Prefiltered MTF • The solution to this is to prefilter the MTF so it goes to zero at the Nyquist frequency – This is often done by blurring OPTI 517 136 Final Comments • Image quality is essentially a measure of how well an optical system is suited for the expected application of the system • Different image quality metrics are needed for different systems • The better the needed image quality, the more complex the optical system will be (and the harder it will be to design and the higher the cost will be to make it) • The measures of image quality used by the optical designer during the design process are not necessarily the same as the final performance metrics – It's up to the optical designer to convert the needed system performance into appropriate image quality metrics as needed OPTI 517 137