Refraktiva element. Bländartal (f-tal)

Refraktiva element Bländartal (f-tal) f tal = f /# = f där f är systemets brännvidd och D inträdespupillens diameter tex D f /2 = 50 där f /2 skall tolkas som bländartalet 2 för fallet brännvidd 50 mm och 25 aperturdiameter 25 mm. Numerisk apertur NA = n sinθ där θ är maximala acceptansvinkeln från normalen (=halva konvinkeln). Härav ser vi att i extremfallet mikroskop där s = f blir numeriska ingångsaperturen för objektivet NA = n sin n D 2 f eller i luft (n = 1) NA = 1. Nedan en allmän bild. 2 f tal Lägg märke till att den vanliga laterala förstoringen ges av M = s' ' s = NA NA' ' Dvs. om man känner de numeriska in- och utgångsaperturerna för ett system så vet man också därmed förstoringsgraden. Begreppet numerisk apertur används flitigt i samband med optiska fibrer. T.ex. så är den numeriska ingånsaperturen för en optisk fiber given av material och diameter Antag att den numeriska aperturen anges till 0.25 för en viss 100 μm fiber. Om vi önskar koppla in ljuset från en ljuskälla med en utsträckning av 1 mm in i fibern kan vi beräkna den numeriska ingångsapertur, som det avbildande systemet bör ha. Förstoringen blir M = 100 μm/1 mm = 0.1 dvs. s"/s = 0.1. Den efterfrågade numeriska ingångsaperturen för det avbildande systemet framför fibern bör då vara NA = 0.1 0.25 = 0.025. 1

Gaussiska linsformeln Den vanliga linsformeln fungerar utmärkt också för tjocka linser eller linssystem om man mäter avstånd till och från huvudpunkterna dvs. huvudplanens skärningar med den optiska axeln. 1 f = 1 s 1 s ' ' Nedan visas ett exempel på ett sammansatt linssystem där huvudplanenen antytts och effektiva brännvidder markerats. Observera att huvudplanen ligger utanför linssystemet (och är krökta). 2

Vidgning av ljusstråle Ett vanligt förekommande önskan är att expandera eller motsatsen komprimera en ljusstråle i endast en ledd. Detta kan vi åstadkomma med t.ex. ett par cylinderlinser. En mera kompakt lösning bygger på prismor vilket tex används i korta laserkaviteter. Parallelliteten hos en ljusstråle Det är lätt att göra misstaget att föreställa sig att en oerhört lång laserkavitet i en kontinuerlig laser skulle kunna generera en divergensfri stråle att t.ex. sända t.o.r. månen. Emellertid kan vi inte undgå stråldivergens. En smal rund Gaussisk stråle har en minsta midjediameter D någonstans. Denna genererar en intensitetsprofil vars totala divergens ges av Φ = 4 λ π D = 1.27 λ på stort avstånd från midjan. (Detta kan jämföras med att D Airyfläcksdiametern, som man får från diffraktionen i ett cirkulärt hål, svarar mot vinkeln 2 1.22λ/D ) Av detta följer att man måste arbeta med stora stråldiametrar för att överföra elektromagnetisk strålning långa sträckor. Självfokusering Detta fenomen uppstår i material genom att brytningsindex varierar tvärs strålen pga strålintensitetens avtagande radiellt ut från centrum tex vid Gaussisk strålprofil. Strålen blir självinstängande och krymper till en minsta diameter av ca 1 μm. Ingen optik klarar sådana påkänningar. Minsta möjliga brännpunkt Besläktat med ovanstående problem är att besvara frågan hur liten kan vi göra fokalpunkten? Svaret ges av avbildningsgeometrin och ljuskällans storlek. Men om vi har en enastående parallell laserstråle (ljuskällan oändligt långt bort) så blir fläckens storlek ändå f Φ, där f är objektivets brännvidd och Φ som ovan den ursprungliga laserstrålens divergens. Ex. 10 cm lins och Φ = 0.5 mrad ger ca 0.1 0.5 10-3 = 50 μm. 3

Optiska fibrer Användningsområdet är huvudsakligen signalöverföring - det är svårt att skicka in och ta ut höga effekter genom en liten yta, så då får man arbeta med fiberbuntar. Nedan visas två möjliga strålgångar en axiell mod och en högre ordnings mod. Man ser omedelbart att gångvägarna är olika långa. Vid informationsöverföring skickas ett pulståg genom fibern. Resultatet blir så småningom en utsmetning av från början väldefinierade pulser om öppningsvinkeln är stor. För att transportera strålen en längre sträcka krävs totalreflektion mellan fiberns kärna och den s.k "claddingen". Stor skillnad i brytningsindex medför reflektion vid stora infallsvinklar (= stor NA) dvs. stora skillnader i gångväg. Man använder ibland variabelt brytningsindex för att kompensera för den längre randstråletiden. Den numeriska aperturen för denna typ av fiber ges av 2 2 NA = n kärnan n claddingen. Enda sättet att undvika geometriska gångskillnader är att göra kärnan så tunn att endast axialmoden är möjlig dvs. singelmod fiber (< 10 μm). En noggrann analys visar att modtalet blir < 2 om d λ 2.4 π NA. Renheten hos glaset avgör sedan dämpningen i fibern. Förutom denna geometriska gångvägsaspekt måste också dispersionen i materialet beaktas. Olika färger hos ljuset har olika transporttid allteftersom brytningsindex varierar med våglängden. Dels har varje puls i sig alltid en viss frekvensbredd och dels uppstår sidoband med annan frekvens om vi frekvensmodulerar ljuset. Bäst är att använda singelmodlasrar som är frekvensstabiliserade såsom (distributed feedback) diodlasrar. Och i första rummet långa våglängder (1.5 μm) eftersom då Rayleighspridningen ( f 4 ) och därmed dämpningen blir liten. Å andra sidan beror den temporala spridningen per längdenhet på dispersionen och frekvensbredden enligt d τ L = λ c d 2 n d λ 2 λ = M λ, där τ är tiden och L sträckan, så att spridningen beror av materialet och våglängden - för kisel är dispersionsspridningen nära noll vid våglängden 1.25 μm. För att ytterliga öka informationsöverföringskapaciteten används flera färger och då vanligen från ett antal sammankopplade diodlasrar, som arbetar vid något olika våglängder. 4

Övertongenerering Det är fullt möjligt att från infrarött ljus skapa såväl synligt som ultraviolett ljus. I ett material med susceptibiliteten χ sker polarisationen P = 0 χ E. Om intensiteten ökas sker så småningom mättnad och responsen kan skrivas P = 0 χ E χ 2 E 2 χ 3 E 3... Om ingående fältet E = E 0 sinωt kan P uttryckas som P = 0 χ E 0 sin ωt χ 2 2 E 2 0 1 cos2 ωt χ 3 4 E 3 0 3sin ωt sin 3ωt... I ett isotropt material eller ett material med ett inversionscentrum följer P riktningen hos det inkommande elektriska fältet. Alltså finns inga jämna termer (dessa skulle inte tillåta att polarisationen, P, byter tecken när E byter tecken) i utvecklingen av P. Härav följer att cos(2ωt) inte heller existerar. Exempelvis gaser visar sådana egenskaper. I material som saknar ett symmetricentrum kan däremot andra harmoniska övertonen (SHG) uppstå. KDP (potassium dihydrogen phosphate) är ett exempel som dessutom visar piezoelektriska egenskaper. P i formeln bör egentligen ses som en vektorrespons i 3 dimensioner där χ är en tensoroperator. Den skapade dubbla (eller högre övertoner) frekvensen medför att en våg utbreder sig i materialet med annan hastighet än den inkommande elektromagnetiska vågen pga dispersion. Endast om sträckan är kort blir fasförskjutningen mellan dessa blygsam. Högsta intensiteten för dubbla frekvensen uppnås för sträckan l = 1 4 n ω n 2ω. KDP är ett enaxligt dubbelbrytande kristallmaterial med olika brytningsindex n eo och n o. När man arbetar med planpolariserat ljus kommer 2ω-planet att vara ω-planet Genom att välja infallsvinkeln så att n 2ω = n o = n ω (fasmatchning) kan koherenssträckan ökas avsevärt. En alternativ metod (kvasi-fasmatchning) går ut på att sammanfoga många korta polvända domäner så att man ideligen fasvänder den annars destruktivt genererade övertonen. Dylika material framställs av tex litiumniobat (PPLN = periodically poled Lithium Niobate). Ett annat användningsområde för detta material är att generera korta pulser i våglängdsområdet mellan mikrovågor och infrarött ljus, vilket motsvarar THz-frekvens, (potentiellt intressant t.ex. på elektroniksidan) genom att utgå från optiska femtosekundpulser. λ 0 5

CCD-sensors Optical sensors are quite often based on the CCD-technique. Charge-coupled devices (CCDs) are silicon-based integrated circuits consisting of a dense matrix of photodiodes that operate by converting light energy in the form of photons into an electronic charge. Electrons generated by the interaction of photons with silicon atoms are stored in a potential well and can subsequently be transferred across the chip through registers and output to an amplifier. The schematic diagram illustrated in Figure 1 shows various components that comprise the anatomy of a typical CCD. CCDs were invented in the late 1960's by research scientists at Bell Laboratories (Boyle and Smith), who initially conceived the idea as a new type of memory circuit for computers. Later studies indicated that the device, because of its ability to transfer charge and the photoelectric interaction with light, would also be useful for other applications such as signal processing and imaging. Early hopes of a new memory device have all but disappeared, but the CCD is emerging as one of the leading candidates for an all-purpose electronic imaging detector, capable of replacing film in the emerging field of digital photomicrography. Fabricated on silicon wafers much like integrated circuits, CCDs are processed in a series of complex photolithographic steps that involve etching, ion implantation, thin film deposition, metallization, and passivation to define various functions within the device. The silicon substrate is electrically doped to form p-type silicon, a material in which the main carriers are positively charged electron holes. Multiple dies, each capable of yielding a working device, are fabricated on each wafer before being cut with a diamond saw, tested, and packaged into a ceramic or polymer casing with a glass or quartz window through which light can pass to illuminate the photodiode array on the CCD surface. Explore the sequence of steps necessary to build a CCD using our interactive Java tutorial, which is linked from the dialog box. When a ultraviolet, visible, or infrared photon strikes a silicon atom resting in or near a CCD photodiode, it will usually produce a free electron and a "hole" created by the temporary absence of the electron in the silicon crystalline lattice. The free electron is then collected in a potential well (located deep within the silicon in an area known as the depletion layer), while 6

the hole is forced away from the well and eventually is displaced into the silicon substrate. Individual photodiodes are isolated electrically from their neighbors by a channel stop, which is formed by diffusing boron ions through a mask into the p-type silicon substrate. The principal architectural feature of a CCD is a vast array of serial shift registers constructed with a vertically stacked conductive layer of doped polysilicon separated from a silicon semiconductor substrate by an insulating thin film of silicon dioxide. After electrons have been collected within each photodiode of the array, a voltage potential is applied to the polysilicon electrode layers (termed gates) to change the electrostatic potential of the underlying silicon. The silicon substrate positioned directly beneath the gate electrode then becomes a potential well capable of collecting locally-generated electrons created by the incident light. Neighboring gates help to confine electrons within the potential well by forming zones of higher potentials, termed barriers, surrounding the well. By modulating the voltage applied to polysilicon gates, they can be biased to either form a potential well or a barrier to the integrated charge collected by the photodiode. The most common CCD designs have a series of gate elements that subdivide each pixel into thirds by three potential wells oriented in a horizontal row. Each photodiode potential well is capable of holding a number of electrons that determines the upper limit of the dynamic range of the CCD. After being illuminated by incoming photons during a period termed integration, potential wells in the CCD photodiode array become filled with electrons produced in the depletion layer of the silicon substrate. Measurement of this stored charge is accomplished by a combination of serial and parallel transfers of the accumulated charge to a single output node at the edge of the chip. The speed of parallel charge transfer is usually sufficient to be accomplished during the period of charge integration for the next image. 7

CCD choices : full frame; interline transfer; or interline transfer with integrated colour filters (see figure above). A full-frame CCD is an unobstructed rectangular array of detectors. Electronic charge accumulates in each pixel when light strikes the CCD. Readout occurs by sequentially shifting each row of pixels down into a read-out register until the entire array is cleared. An external shutter prevents light from reaching the CCD during the read-out cycle to avoid image smearing. In an interline-transfer CCD, alternate columns are masked with an opaque layer. During read out, the accumulated charge in each exposed column is rapidly transferred laterally to a nonimaging column where the charge is shifted into the read-out register. This allows read out to occur while another exposure is being acquired. Both of these CCD types respond only to incident luminance and cannot measure colour directly. To measure colour, separate exposures must be made through red, green and blue colour filters. However, a variant of the interline-transfer architecture has each pixel overlaid with either a red, green or blue colour filter specifically for direct colour measurement. There are several important CCD-dependent performance characteristics to be aware of, starting with fill factor. When an image is formed on a full-frame CCD, virtually none of the information is lost. On the other hand, a significant percentage of the surface of an interlinetransfer CCD is opaque, so image features that fall on this blocked area will not be seen. In practice, interline-transfer CCDs often use a microlens array to focus some of the light that would normally be blocked into the active area. This can increase fill-factor efficiency by up to 70%. The fill-factor problem is further exacerbated by the presence of integrated colour filters. Next figure shows how missing data can cause erroneous results when the output of a single LED is focused onto a 5 x 5 pixel area of the CCD. 8

Another major difference between CCD types is their dynamic range. The definition of a CCD's dynamic range is the maximum capacity of each pixel in electrons (called the full-well capacity) divided by the RMS dark noise (the number of electrons read from the device with no input light). Full-well capacity increases with pixel size. Full-frame CCDs typically have larger pixels, with full-well capacities of between 200,000 and 700,000 electrons, making dynamic ranges of 14 (16384:1) to 16 bit (65536:1) possible. In contrast, the full-well capacity of most interline-transfer CCDs is in the 10,000-20,000 electrons range, resulting in a dynamic range of 12 bit (4096:1) or less. There are also several practical differences between CCD types. A full-frame CCD requires an external shutter and active cooling to minimize noise while an interline CCD does not. These features add to system size, weight, cost and complexity. An interline-transfer CCD can be read out faster than a full-frame CCD, which could be a consideration in high-speed production applications. Color Accuracy A CCD cannot distinguish between different wavelengths of light. Therefore, recording color images requires somehow separating the various wavelengths before they reach the CCD surface. There are several ways to accomplish this. On Detector Filters In this approach, a rectangular array (called a Bayer 1 pattern) of red, green and blue color filters is put directly onto the surface of an interline transfer CCD. The advantages of this method are low cost, small package size and mechanical ruggedness. 1 Bayer filter mosaic is a color filter array (CFA) for arranging RGB color filters on a square grid of photosensors. The term derives from the name of its inventor, Dr. Bryce E. Bayer of Eastman Kodak, and refers to a particular arrangement of color filters used in most single-chip digital image sensors used in digital cameras, camcorders, and scanners to create a color image. The filter pattern is 50% green, 25% red and 25% blue, hence is also called RGBG or GRGB. 9

A drawback of this approach is that it reduces the fill factor of the CCD for each color even further, making it possible to miss imaging small details. Also, the interline transfer CCDs typically used by device manufacturers have small pixels, which limits detector dynamic range. Color accuracy is also compromised because the filters used do not match well with CIE spectral responsivity curves. The lowered signal to noise ratio also compromises color accuracy. X-Cube Beamsplitter This method uses a dichroic cube beamsplitter designed to separate the red, green and blue wavelengths and send them to three different interline transfer CCDs. Because all colors are detected simultaneously, this technique offers good read speed and is mechanically rugged. However, it is more costly and less compact than the Bayer array filter approach. Because the commercially available product that uses this approach is based on interline transfer CCDs, it still suffers from limited fill factor and dynamic range. Most significantly, the thin film coatings used to separate wavelengths don t match well to CIE curves, making it difficult to achieve good color accuracy. Moving Filter Wheel This technique mates a single, scientific grade, cooled, full frame CCD with a series of color filters that are moved over the detector sequentially by a motorized filter wheel. An external shutter is necessary to block light from the detector between separate exposures. Depending upon the particular CCD chosen, this can deliver very high dynamic range, low noise, high spatial resolution and high fill factor. The filters can be chosen to deliver the best possible match to CIE responsivity curves. This approach also enables the use of other specialized wavelength filters The disadvantages of this method are slower measurement times, higher cost and increased system size and weight. 10

How many pixels does a 3 MP sensor have? There is a catch here in that pixel really means picture element. An output pixel contains information on colour and intensity level. The sensor manufacturer does of course count the number of photosites. If wavelength information is not required these numbers coincide at best. (The image is usually somewhat smaller than the sensor) So for a 3 MP camera one could argue that in reality it is at most a 1 MP! The number of photosites is of course 3 M unless the manufacturer is lying. Why are small image sensors noisy? The dynamic range can be understood from the definition full well capacity DR CCD = 20 log rms noise dark [db ] As usual you have to fight statistics (Poisson) If for a particular example a Sony ICX285 is being used the saturation number of electrons is 18000, noise is 6 so dynamic range is 69.5 [db]. A suitable A/D-converter should have 12 bit resolution (4095 steps) or in other words 4.4 electrons per step. There is no point in going further. 11

The following pages are are taken from an article by Robert M. Atkins. There's a lot of attention paid in the digital camera world to pixel count. Cameras are often categorized by the number of pixels they have in their image sensor. However not all pixels are equal and, as in many contexts, size matters! In this article I'm going to take a look at several digital cameras with different physical sensor sizes but all with a nominal 3 megapixel (3MP) pixel count. A standard 35mm frame is 36mm x 24mm, so lets call that "full frame". As you can see from row #1 in the table, all the cameras listed here have sensors smaller than a "full frame sensor". Just looking at the "short" side of the sensor compared to the "short" side of the 35mm frame we can see that the Canon D30 sensor is 0.63x full frame, the Nikon Coolpix 995 0.22x full frame and the Minolta Xi sensor is 0.167x full frame. You can see the difference in relative sensor size from the figure below. It's pretty dramatic! "So what" you might think, "if you have enough pixels, what does the sensor size matter?". Cameras with smaller sensors use shorter focal length lenses to get the same angular coverage as cameras with larger sensors do with longer focal length lenses. So if you have a 28-105mm zoom on a Canon D30, a 10-37 mm zoom on a Nikon 995 or a 7.4-28 mm zoom on a Minolta Xi, you get approximately the same shot. What's the big deal about the physical size of the sensor? Why does it matter? One reason why sensor size matters is shown in the table below which I'll go through line by line since it's a little complex! Please note that the numbers are intended only for illustrative purposes and in some cases may be approximations or upper limits. These numbers are not intended to be accurate predictions of the exact resolution you would see from these cameras in practice. However the trends which these numbers illustrate certainly can be seen! 12

Canon D30 Nikon Coolpix 995 Minolta Xi 0 "Sensor size" - 1/1.8" 1/2.7" 1 Physical size (mm) 22.7 x 15.1 7.2 x 5.3 5.3 x 4.0 2 Size (pixels) [all nominal "3MP"] 2160 x 1440 2048 x 1536 2048 x 1536 3 Print size for 8x10 crop 8 x 12 8 x 10.7 8 x 10.6 4 Magnification for 8x10 crop 13.46x 38.3x 50.8x 5 Sensor pixels/mm 95.4 290 384 6 Sensor resolution limit 47.7 lp/mm 145 lp/mm 192 lp/mm 7 Max resolution 8x10 print 3.54 lp/mm 3.78 lp/mm 3.78 lp/mm 8 Sensor Resolution needed for 3 lp/mm in an 8x10 print 40.4 lp/mm 115 lp/mm 152.4 lp/mm 9 Corresponding MTF @ f8 0.75 0.31 0.14 1 0 1 1 1 2 Corresponding MFT @ f4 0.87 0.64 0.53 Corresponding MTF @ f2 0.94 0.81 0.76 Corresponding MTF @ f16 0.50 0.00 0.00 Line 0 is the industry name for the sensor size. Quite misleading and confusing! Line 1 shows the actual physical sensor size (in mm) Line 2 shows the size of the sensor in pixels Line 3 shows the minimum print size needed for an 8x10 cropped image Line 4 shows the magnification of the sensor needed to make an 8x10 image Line 5 shows the number of pixels per mm in the sensor Line 6 shows the theoretical resolution limit of the sensor ("Nyquist limit") Line 7 shows the maximum resolution you can get in an 8x10 cropped image OK, so those are the basic facts. Now lets take an example of an 8x10 print and let's say we want a fairly sharp print, so we are going to need a resolution of at least 3 lp/mm in the print. First, by looking at line 7 we can see that that's possible with all three cameras. Now for 3 lp/mm in the print, what resolution do we need from the sensor? Well, that's given in row #8 of the table. Since the smaller the sensor the more the image needs to be enlarged, to get the same resolution in the same sized print we need more resolution from smaller sensors. The table shows that for the D30 we need to record the image at up to 40.4 lp/mm on the sensor. For the Coolpix 995 we need to record 115 lp/mm on the sensor and for the Xi we need to record a whopping 152.4 lp/mm on the sensor. Now we get to MTF (Modulation Transfer Function). This is a measure of lens performance and shows how well a lens reproduces object detail in the image it produces. I'll deal with MTF and exactly what it is in a future article, but for now it's enough to know that MTF can range 13

from 1 to 0, and that high numbers mean high contrast, resolution and image fidelity, while lower numbers mean lower contrast, resolution and image fidelity. You want high numbers! Line 9 of the table shows the MTF of a perfect lens operating at f8 when recording detail on the sensor at the resolution given in line 8 of the table. So for the D30, we need 40.4 lp/mm on the sensor, and at 40.4 lp/mm a lens operating at f8 will have an MTF of 0.75. For the Coolpix 995 we need 115 lp/mm on the sensor, and at 115 lp/mm a lens operating at f8 will have an MTF of 0.31. Finally for the Xi we need 152.4 lp/mm on the sensor to get 3 lp/mm in an 8x10 print, and at 152.4 lp/mm an f8 lens has an MTF of 0.14. Lines 10 and 11 show the corresponding numbers for a perfect lens at f4 and f2. The larger sensor is still better, but the difference is less pronounced. Note however that the data here is the best possible case. In practice lenses are not perfect and the faster a lens is, the less perfect it is. Real MTF drops off much faster than that predicted for a perfect lens, so the differences in MTF between sensor sizes will actually be greater than the analysis here shows, especially at wide apertures, giving an even greater advantage to the larger sensors. So its clear that the larger the sensor, the higher up on the MTF curve of the lens it's operating at when delivering a particular resolution and the final output is a print of a given size. This is shown graphically in the figure below. This graph shows where on the lens' MTF curve each of the 3 cameras operate when at f8 and used to make an 8x10 print with detail up to 3 lp/mm in the print. As you can see the D30 uses the high part of the MTF curve and so yields an image of good contrast and resolution all the way up the the required 3 lp/mm. The Coolpix 995 with a significantly smaller sensor has to use part of the MTF curve which is lower than that used by the D30, and the Xi with the smallest sensor of all has to use almost the whole MTF curve, even the low part. The range of the curve used depends only on the size of the sensor, so we could replace the "D30" label with 22.7 mm x 15.1 mm, the "995" label with 7.2mm x 5.3 mm and the "Xi" label with 5.3 mm x 4mm. 14

Since all three of these cameras are nominal "3MP" cameras, we can predict that results from the D30 will be better than those from the Coolpix 995, which will in turn be better than those from the Xi, assuming we use a lens of the same optical quality on each camera. One final interesting point is shown by row 12 of the table which shows the MTF at f16. The D30 sensor is OK but using the 995 sensor or the Xi sensor the MTF is zero! What this means is that there would be a significant drop in image quality operating at f16 with either the Coolpix 995 or the Xi. In fact we could not obtain the desired 3 lp/mm in an 8x10 print if we were able to stop the lens down to f16 on those camera. That's why the smallest aperture available on these and most other small sensor cameras is f8. So now you know why "bigger is better" when it comes to image quality and digital sensors. Of course bigger is also more expensive, and bigger means bigger (hence heavier and more expensive) lenses, so you can see why many digital cameras stick with small sensors. It's cost, not quality that keeps sensors small.all text and images (C) Copyright 2003 Robert M. Atkins. Pixel count (8 Megapixels, wow!) grabs our attention in digital camera specifications. But it doesn't tell everything. Pixel and sensor size matters. It turns out there is an optimum range for pixel size and an advantage to large (i.e., costly) sensors. For reference, sensor diagonal measurements are 43.3 mm for full frame 35mm film; up to 11 mm for compact digital cameras, and 22 mm and over for digital SLRs. Small pixels have excellent resolution but suffer from increased noise (hence poorer signal-to-noise ratio, SNR), reduced exposure range (fewer f-stops), and reduced sensitivity (lower ISO speed). The reason is simple: they respond to fewer photons and can hold fewer electrons. These effects are most noticeable in compact digital cameras, which have pixels smaller than 4 µm. The exact relationship between noise and pixel size is difficult to quantify since there are several noise mechanisms, each of which scales differently. Large pixels have good SNR, ISO speed and exposure range, but suffer from aliasing-- low spatial frequency artifacts that appear when the lens has significant response above the Nyquist frequency: 1/(2*pixel spacing). Aliasing typically manifests as Moiré patterns on images with high frequency repetitive patterns, such as window screens and fabrics. It can be reduced by anti-aliasing (low pass) filters, which are expensive and unavoidably reduce resolution. In Optics for digital photography from Schneider Optics, the author states that aliasing will be adequately controlled if the MTF of the lens + sensor at Nyquist is no more than about 10%. Compact digital cameras, which have pixels smaller than 4 µm, don't need anti-aliasing filters: the lens is sufficient. This helps control cost. Small sensors run into problems with lens diffraction, which limits image resolution at small apertures-- starting around f/16 for the 35mm format. At large apertures-- f/4 and above-- resolution is limited by aberrations. There is a resolution "sweet spot" between the two limits, typically between f/5.6 and f/11 for good 35mm lenses. The aperture at which a lens becomes diffraction-limited is proportional to the format size: 22 mm diagonal sensors become diffraction-limited at f/8 and 11 mm diagonal sensors become diffraction-limited at f/4-- the same aperture where it becomes aberration-limited. There is little "sweet spot;" the total image resolution at optimum aperture is less than for larger formats. Of course cameras with small sensors can be made very compact, which is attractive to consumers. 15

Large sensors cost more. No getting around it. That's the major reason compact digital cameras are so popular. 11 mm diagonal sensors have 1/16 the area of a 35mm frame. The problem with large sensors is manufacturing yield-- the percentage of sensors that work properly. Suppose an 11 mm sensor has a 90% yield (pretty good). A 44 mm sensor (35mm format; 16x the area) with the same process would have a yield of 0.90 16 = 18% (not so hot). Larger sensors tend to have larger pixels, which helps the yield. Compact digital cameras have sensors with diagonal dimensions between 5 and 11 mm, and pixel pitches 3.4 µm or less. These cameras have acceptably low noise at low ISO speeds and the best of them-- the "prosumer" models-- can make excellent 8½x11 inch or larger prints, depending on pixel count. Thanks to noise and diffraction, overall image quality decreases for pixels smaller than 2 µm. The optimum pixel size for high quality imaging seems to be in the 6-9 µm range. Larger pixels have problems with aliasing and can't take advantage of high quality lenses. Smaller pixels have more noise and less dynamic range and sensitivity. Digital SLRs will stick with 6-9 µm pixels and evolve towards larger sensors with more pixels. 24x36 mm sensors with 16+ megapixels (7.4 µm or less pixel spacing) have performance approaching medium format (see The future of digital cameras), but they won't come cheap for quite some time. Så hur betydelsefull är sensorn för upplösningen och skärpan? Följande bild försöker illustrera MTF = Modulation Transfer Function 16