What Is Depth of Focus of Camera Lens?

Home , Aperture, Camera, Camera lens, Circle of confusion, Deep focus, Depth of field

What is depth of focus of camera lens?

• Depth of focus – Range of lens-to-cameraback distances for which one image is tolerably sharp (reasonably in focus) • Inﬁnite for pinhole camera • image is always in focus • ≈ one ray connects each object pt to image pt image depth of focus blurry here – For lens-camera depth of focus but depends on tolerable • lens diameter, • focal length and • object distance – Circle of confusion circles of • locus of rays that focus to pt elsewhere f confusion

• If diameter of circle of confusion is missing small enough blur is tolerable ray – Lens with smaller diameter has larger depth of focus smaller circle of 1 • More like a pinhole (image is dimmer) confusion f What is depth of ﬁeld of a lens?

• Depth of field – related to depth of focus – maximum separation distance (along axis) between two objects such that both objects are tolerably Depth of sharp (in reasonably good focus) field diameter of – Deep focus: term for large depth of circle of confusion field in movie-making (maximum tolerable blur) – Pinhole camera depth of field is ∞ • all objects in focus at all lens-to- camera-back distances. • Smaller lens opening (aperture) ⇒ – larger depth of field but – dimmer image • Wide angle lenses generally have 2 more depth of field than telephotos Circles of confusion

3 A camera is focused by changing the distance between the lens and the ﬁlm or CCD at the back of the camera

• xi in lens eqn is distance between • Given: f = 1 cm, xo = 100 cm lens and back of the camera (where • Find: image distance, xi image is recorded) • 1/f = 1/xo + 1/xi • 1/f = 1/xo + 1/xi • 1/1 = 1/100 + 1/xi • Suppose you take a picture of your • 1 - 0.01 = 1/xi friend using a camera with a lens of • 0.99 = 1/xi focal length f = 1 cm • xi = 1/0.99 = 1.01 cm • The friend is a distance xo = 100cm – Both (a) and (c) are both correct from the camera lens • When take picture of object a • For what value of xi is the image at distance xo away from camera which the back of the camera in focus? is much larger than the focal length, a) The value obtained by solving f, the image distance, xi, will always the lens eqn for xi be a number very close to f b) 100 cm • This is because rays from distant c) Roughly the same as the focal object are almost parallel and length, f therefore go through the focal point!! • Can also see from lens equation 4 What are apertures, f-numbers and stops?

• Diameter of lens aperture can be reduced by using diaphragm. 10 5 – Largest aperture is full diameter of lens – Smaller apertures are called stops 40 mm focal 40 mm focal length lens length lens • Aperture is measured by f-number: at full aperture with aperture focal length of lens divided by of diameter of diameter diameter of aperture: d = 10 mm d = 5 mm – f-number (or f-stop) = f/d – Large f-number means small lens diameter f f-number = - = 4 ( = f/4) • What is f-number (f-stop) of same 10 lens when diaphragm reduces diameter of lens from 10 mm to 5 mm? a) f/2, b) f/4, c) f/6, d) f/8, e) f/10 5 Camera exposure is related to Poynting ﬂux carried by light wave.

• Wave energy density and flux of wave energy density ∝ E2 – Flux of energy density = • Poynting flux [velocity×energy-density – (m/s)×J/m3] • Intensity [power/m2 - watts/m] rays converge here • When brighter (more intense) light enters your eye electric force fields add here – Electric force field of light is stronger (more photons) – More absorption in your retina (back of eye) 6 Lens with larger aperture lets more light energy reach image pts. at back of camera (where CCD records image) • Larger aperture lens brings more rays from each object pt. to corresponding image pt. • Pinhole only lets "one" ray from real nose converge at Pinhole image nose. – Image is dim (not intense) • Small aperture lens lets a few Small aperture lens more rays from nose converge to image nose. – Brighter image • Large aperture lens allows more rays from nose to converge to image nose. Large aperture lens – Image even brighter with same focal length 7 – Why do we squint in bright light? as smaller aperture lens Light is more intense when there is more energy per unit area. • Intensity of light (Poynting flux) = energy per second per unit area

• Power = energy per second (watts) Light r bulb • Intensity = power divided by area – Power from 60 watt light bulb same as move away but intensity decreases – Alex sees intensity (power reaching his eye = intensity times his eye area) Draw imaginary sphere whose radius, r = 2 • Intensity ∝ 1/r distance from your eye to center of light bulb – r = distance from center of light bulb to your eye • Intensity of light is same everywhere on – Intensity falls off as square of the sphere = power divided by area of sphere distance – Star seems dim! – Camera ﬂash sometimes can't supply ∝ 2 ∝ -2 enough light? Area of sphere r so intensity r 8 Light energy reaching ﬁlm each second is proportional to the AREA of lens

• Light energy/s ∝ lens Camera area, A • A ∝ d2 • d ∝ √A

• If A is doubled by what factor does d increase? Lens a) √2 d b) 2 diameter of lens = d c) 4 Area of lens = A = π(d/2)2

9 More light energy reaches ccd when lens f-stop is lower number (lower number means larger area)

• f-stop = (focal length)/d – f-stop at right has diameter d = 5 so f- 5 5·√2 stop is 40/5 = 8 10

• Double diameter, d from 5 to 10 mm – New f-stop is 40/2d = 4 f = 40 mm lens new diameter 40 mm lens – New area of aperture, A, increases by with aperture = √2·d mm at full aperture factor 4 d = 5 mm = 7.1 mm 2d = 10 mm – Lets in 4 times amount of light energy/s (since energy ∝ Area) 40/(√2·d) = f/5.6 40/d = f/8 40/(2d) • To double energy reaching CCD: = f/4 – Multiply old diameter, d, by √2 – New area is doubled because √22 = 2 Area 2 Area = Area – New f-stop ∝1/(√2d) decreased by = π ·(d/2) π ·(√2·d/2)2 2 = 6.25 mm2 = π·(2d/2) factor of 1/√2 from f/8 to f/5.6 = 12.5 mm2 = 25 mm2 (2x energy (4x 10energy as at f/8) as at f/8) Sequence of f-stops which each let in twice the light energy per sec

• f/22 • Smaller f-stops mean larger • f/16 diameters, less depth of ﬁeld, • f/11 larger aperture areas and more • f/8 light energy gets to CCD • f-stop or f-number • f/5.6 = (focal length)/(lens diameter) • f/4 • Each f-stop down list Smaller • f/2.8 • has diameter, d, larger by f-stops factor √2 than the previous stop • f/2 • has aperture area (∝d2) larger • f/1.4 by factor 2 • Lets in twice the light energy (energy ∝ area)

11 A light wave continues to deliver energy as long as the wave keeps moving into an eye or a lens

• You get more of a sunburn if the suns rays hit your skin longer – Each second more light energy hits your skin • Less light energy goes into your your eye if you open and close it again quickly • More light arrives at CCD at back of a camera when shutter is left open longer – Image gets brighter if shutter is left open longer

12 Sequence of shutter speeds in which each faster speed lets in half the light energy

• 1/15 sec • 1/30 sec • 1/60 sec • 1/125 sec • 1/250 sec Faster speeds, better able to stop (freeze) fast motion • 1/500 sec but give darker image • 1/1000 sec • 1/2000 sec

13 What is meant by exposure of a picture?

• Exposure at CCD is • Intensity of light on CCD = proportional to total light energy delivered per second energy falling on CCD divided by AREA of CCD during time shutter is left • Exposure ∝ light intensity × open time interval shutter is open – Exposure depends on both f- stop (controls intensity) and – shutter speed (time shutter is left open)

14 Equivalent combinations of f/stops and shutter speeds (which deliver the same energy to the ﬁlm or CCD). All give same exposure!

• f/22 • 1/15 sec • Large depth of field • 1/30 sec because of small aperture, • f/16 • 1/60 sec • f/11 • 1/125 sec – but slow shutter speed near • 1/250 sec top of list may give motion • f/8 • 1/500 sec blur • f/5.6 • 1/1000 sec • 1/2000 sec • f/4 • 1/4000 sec • Fast shutter speed but • f/2.8 small depth of field near • f/2 bottom of list due to large apertures. • f/1.4 – but image may be out of Twice the Half the time for focus due to small depth of energy gets in energy to get in field 15 each second Different exposures correspond to different alignments of sequence of f-stops and sequence of shutter speeds

• 1/15 sec • 1/30 sec • f/22 • f/22 • 1/60 sec • f/16 • 1/125 sec • 1/15 sec • f/16 • f/11 • 1/250 sec • 1/30 sec • f/8 • f/11 • 1/500 sec • 1/60 sec • f/5.6 • 1/1000 sec • 1/125 sec • f/8 • f/4 • 1/2000 sec • 1/250 sec • f/2.8 • f/5.6 • 1/4000 sec • 1/500 sec • f/2 • 1/1000 sec • f/4 • f/1.4 • 1/2000 sec • f/2.8 • 1/4000 sec • f/2 • f/1.4

16 Which exposure lets in more light? More detailed examples of equivalent exposures

Exposure time 1/250 sec 1/125 sec 1/60 sec 1/30 sec f-number f/5.6 f/8 f/11 f/16

Diameter of 28mm 5 mm 3.5mm 2.5mm 1.75mm lens Area of aperture of 20 mm2 10 mm2 5 mm2 2.5 mm2 28 mm lens Light intensity 500 250 120 60 reaching ﬁlm (arb units) Exposure (arb. units) 2 2 2 2

17 How we can understand the concept of intensity in terms of the images from pinhole cameras.

• Intensity of light is energy/second per unit area (e.g., watts per m2) – Light is more intense when there is more energy delivered to each unit area each second • How does the intensity change when the image is larger? Is the intensity of light on the ﬁlm of the telephoto camera a) higher, b) lower, c) the same • The same light energy is spread over a larger area so the intensity goes down • To compensate for this lower intensity cameras use a lens to let in more light than a pinhole can. 18 Qualitative argument why f-number is deﬁned as f/d. (focal length divided by lens aperture diameter)

• Consider two lenses with different focal length but same f-number, say, f/2 20 mm – f/2 lens with longer focal length, f, must have larger diameter, d, by deﬁnition f-number = f/d – Larger d brings more rays to each f = 40 mm image point f/2 lens with focal length 40 mm means • If d were not larger, image would be less intense (dimmer, as in pinhole 2= 40/d, so lens has diameter d = 20 mm camera) • Deﬁnition f-number = f/d f/2 lens with focal length 60 mm means – guarantees that every f/2 lens gives same 2= 60/d, so lens has diameter d = 30 mm exposure for the same shutter speed 30 mm – regardless of whether focal length, f, is small (wide-angle lens) or large (telephoto lens). • Quantitative proof on next slide

19 fnew = 60 mm Quantitative argument why two lenses with same f-numbers give image of Alex with same intensity

• If fnew= 60 mm lens did not have larger diameter than f = 40 mm lens, 20 mm

the intensity, Iimage of its image would 2 – be lower by 1/xi , where xi is distance from center of lens to Alex's image – Object is usually far from lens so rays enter f = 40 mm lens almost parallel and xi ≈ f f/2 lens with f = 40 mm 2 • For same d, Iimage decreases as 1/fnew must have d = 20 mm 2 – Doubling f decreases Iimage by factor 2 2 • For same f, Iimage increases as d 2 f/2 lens with f = 60 mm – Doubling d increases Iimage by factor 2 • Different focal length lenses only have must have d = 30 mm 2 2 30 mm same Iimage if d /f is same for both. • Ratio d2/f2 = 1/(f-number)2 – If both lenses have same f-number there is no change in image intensity 20 fnew = 60 mm Common focal lengths, f, of f/2.8 lenses and required diameters for each

Lens focal 35mm 50mm 80mm 105mm 135mm length, f Max f/stop f/2.8 f/2.8 f/2.8 f/2.8 f/2.8

d = lens 12.5mm 17.9mm 28.6mm 37.5mm 48.2mm diameter = f/(f/stop)

21 Wide angle and telephoto effects in the pinhole camera only depend on distance from pinhole to ﬁlm

• As distance between pinhole and camera back increases, image gets larger – First camera gives wide-angle effect • camera back is covered by image that includes more than Alex – Longer camera gives telephoto effect • Alex's image covers entire camera back – Angle between crossed yellow lines smaller in longer camera – Image alwayss in focus (sharp) smaller angle • Wide angle and telephoto lenses have some similarities to pinhole camera – Telephoto lenses are long and wide angle lenses are short • There are also differences – Lens lets in more light ⇒ brighter image – f must be different for different image 22 sizes to maintain focus Digital and 35 mm camera lenses compared (lens may be a zoom lens)

Focal Length Equivalent focal Image Typical Uses of digital length of 35mm appearance camera lens film camera lens 5.4 mm 35 mm Object looks Wide-angle shots, smaller and landscapes, large farther away. buildings, groups of people

7.7 mm 50 mm Object looks "Normal" shots of about same as people and objects what eye sees.

16.2 mm 105 mm Object is Telephoto shots, magnified and close-ups appears closer.

23 Meaning of ISO on digital cameras

• ISO measures sensitivity of CCD at • High ISO number means image plane • high sensitivity of CCD • take pictures with less light – Formerly a measure of film sensitivity • (film "speed") or at smaller f/numbers with greater depth of field • However pictures taken at high ISO have less resolution • They are pixelated or grainy

24 Comparison of images