CX-0693
`
`36
`
`Illumination
`
`Form Factor and Average Projected Solid Angle
`
`
`Here the approximations of constant cosines cannot be
`used.
`
`
`
`The angles between
`the normal
`to the
`radiating
`surface
`and the directions
`to points on the
`illuminating—sur-
`face vary not only with the locations of the points on the
`illuminated surface, but also with the locations of points
`on the radiating surface. The concept of projected solid
`angle takes the former into account, but not the latter.
`What is needed is an average projected solid angle,
`Q toi, which is the projected solid angle subtended by the
`illuminated area and averaged over all points on the
`radiating area. Then the illuminating flux, @;, from a
`Lambertian radiator is
`
`0, =LA, Dyog = °. Q
`
`
`
`T
`
`rtot*
`
`In practice, the average projected solid angle is not used.
`However,
`its geometrical equivalent, called the form
`factor, Fatos,
`is used. The only difference between the
`form factor and the average projected solid angle is a
`multiplier of x:
`
`Fsob = Out®
`the
`The form factor measures in hemispheres what
`
`average projected solid angle measures in_projected
`steradians. The form factor also can be interpreted as the
`portion of the flux leaving a Lambertian radiator, a, that
`illuminates a surface, b:
`©,.=-0 Frtoi*
`Note that the form factor is directional, as are the solid
`and the projected solid angles. Fa w »
`is not in general
`equal to F» toa. However, the product of the area and the
`form factor is constant:
`A, FFatob =A, F, toa*
`
`
`
`PAGE 50 OF 154
`
`MASITC_01080430
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`36
`
`Illumination
`
`Form Factor and Average Projected Solid Angle
`
`
`Here the approximations of constant cosines cannot be
`used.
`
`
`
`The angles between
`the normal
`to the
`radiating
`surface
`and the directions
`to points on the
`illuminating—sur-
`face vary not only with the locations of the points on the
`illuminated surface, but also with the locations of points
`on the radiating surface. The concept of projected solid
`angle takes the former into account, but not the latter.
`What is needed is an average projected solid angle,
`Q toi, which is the projected solid angle subtended by the
`illuminated area and averaged over all points on the
`radiating area. Then the illuminating flux, @;, from a
`Lambertian radiator is
`
`0, =LA, Dyog = °. Q
`
`
`
`T
`
`rtot*
`
`In practice, the average projected solid angle is not used.
`However,
`its geometrical equivalent, called the form
`factor, Fatos,
`is used. The only difference between the
`form factor and the average projected solid angle is a
`multiplier of x:
`
`Fsob = Out®
`the
`The form factor measures in hemispheres what
`
`average projected solid angle measures in_projected
`steradians. The form factor also can be interpreted as the
`portion of the flux leaving a Lambertian radiator, a, that
`illuminates a surface, b:
`©,.=-0 Frtoi*
`Note that the form factor is directional, as are the solid
`and the projected solid angles. Fa w »
`is not in general
`equal to F» toa. However, the product of the area and the
`form factor is constant:
`A, FFatob =A, F, toa*
`
`
`
`PAGE 50 OF 154
`
`MASITC_01080430
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`Illumination Transfer
`
`37
`
`Configuration Factor
`
`
`The form factor and the average projected solid
`angle both link two extended areas. The form factor
`measures in hemispheres what
`the average projected
`solid angle measures in projected steradians. Another
`term, the configuration factor, C, is similarly related to
`the projected solid angle, linking a small area with an
`extended area. Like the form factor,
`the configuration
`factor measures in hemispheres what the projected solid
`angle measures in projected steradians:
`
`C=Q/n.
`
`for
`Tables of configuration factors and form factors
`myriad geometries
`can be
`found in handbooks on
`illumination, in books on radiative heat transfer (where
`the issues are identical to illumination by Lambertian
`radiators),
`and on the Internet. Three
`cases with
`applicability to many optical situations are listed here:
`
`Case 1: Small area to an extended circular area; both
`areas parallel and with axial symmetry.
`
`radius, r
`
`C=
`
`and
`
`
`
`go a 2
`
`= sin’ 0
`
`Zz
`Tr
`=—— = asin’ 6.
`r +d
`
`
`
`PAGE 51 OF 154
`
`MASITC_01080431
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`38
`
`Illumination
`
`Useful Configuration Factor
`
`
`Case 2: Small area to an extended circular area; both
`areas parallel, but without axial symmetry.
`
`
`
`or, equivalently:
`
`1
`
`1+tan® 6-—tan’ 0
`|tan‘ o+ (2tan’ 8)(1 —tan” 8) + sec! 6| =
`
`and
`
`O= nC,
`
`These expressions degenerate to the expressions for case 1
`above when x, or equivalently, 6, is equal to zero.
`
`
`
`MASITC_01080432 MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`PAGE 52 OF 154
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
` radius, /;
`
`area, A,
`
`area, A;
`
`CX-0693
`
`Illumination Transfer
`
`og
`
`Useful Form Factor
`
`
`Case 3: An extended circular area illuminating another
`extended circular area; both areas parallel and centered
`on the sameaxis.
`
`
`
` radiating__illuminated
`
`Pr.
`Pr.
`F
`z
`2
`1+(4)
`1+(4)
`1
`Fn [—_ Ss ||3| a5
`e)
`2
`r.
`
`1
`
`Some numerical values of F; oi for this case are shown in
`the table below for
`several
`sizes of
`radiating and
`illuminated disks (each expressed as a multiple of the
`distance between the two parallel circular areas that are
`centered on the sameaxis).
`
`Form Factor,F r toi
`
`rild
`
`0.03
`
`0.10
`
`0.30
`
`1.00
`
`38.00
`
`10.0
`
`0.03
`
`.001
`
`.010
`
`.083
`
`.500
`
`.900
`
`.990
`
`0.10
`
`.001
`
`.010
`
`.082
`
`.499
`
`.900
`
`.990
`
`0.30
`
`.001
`
`.009
`
`.077
`
`.489
`
`.899
`
`.990
`
`1.00
`
`.000
`
`.005
`
`.044
`
`.382
`
`.890
`
`.990
`
`rrid
`
`3.00
`
`.000
`
`.001
`
`.009
`
`.099
`
`.718
`
`.989
`
`10.0
`
`.000
`
`.000
`
`.001
`
`.010
`
`.089
`
`.905
`
`
`
`PAGE 53 OF 154
`
`MASITC_01080433
`
`MASIMO2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`40
`
`Illumination
`
`Irradiance from a Uniform Lambertian Disk
`
`
`as
`be modeled
`can
`illumination situations
`Many
`illumination by a uniform circular Lambertian disk,
`with the illuminated area parallel to the disk and at some
`distance from it.
`
`offset, x
`
`Radiating area
`
`Illuminated spot
`
`is equal to the
`The irradiance at the illuminated spot
`radiance of the radiating area times the projected solid
`angle of
`the radiating area when viewed from the
`illuminated spot:
`
`mH =0, De.
`If the illuminated spotis on axis (x = 0, 6= 0), then
`2
`E,=nL, sin’ 0=nrL, ——
`r'+d
`If the spot is offset from the axis, it is necessary to use the
`projected solid angle or the configuration factor discussed
`previously for case 2:
`
`1+tan* 5-—tan’* 6
`_t
`2 |tan’ 6+ (2tan’ 8)(1 —tan” 6) +sec* o\”
`
`form factor, and
`Note: The configuration factor,
`projected solid angle are useful mainly when the
`
`radiation pattern is Lambertian or nearly Lambertian.
`
`
`
`PAGE 54 OF 154
`
`MASITC_01080434
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`Illumination Transfer
`
`4]
`
`Cosine Fourth and Increase Factor
`SSSCg.za7)!
`
`CX-0693
`
`Consider the previous case of illumination by a uniform
`circular Lambertian disk, with the illuminated area
`parallel to the disk and at some distance from it. For
`many values of aperture size (8) and field angle (6), the
`irradiance falls off very nearly at cos‘d, a phenomenon
`often referred to as the cosine-fourth law.
`
`Two of the cosine terms in the cos‘ law are due to the
`fact
`that, off axis,
`the distance increases with the
`cosine of 6 and the inverse square law applies. The
`third cosine factor comes from the Lambertian source,
`and the fourth from the fact
`that the illuminated
`
`surface is inclined to the direction of propagation.
`
`In reality, the cost “law’ is not exactly true, and is far
`from true for large values of 8 and 6. The table below
`displays values of the increase factor, F’, which is the
`multiplier
`that must be
`applied to the irradiance
`calculated by using the axial irradiance and cos? falloff. F’
`compensates for
`the inaccuracy in the “cosine-fourth”
`assumption:
`
`E,=n L, sin’ @-cos* 5: F”
`
`Increase Factor, F"
`
`Q
`
`(deg)
`
`18
`
`3.6
`
`7.2
`
`14.5
`
`30.0
`
`45.0
`
`NA
`
`0.03
`
`0.06
`
`0.13
`
`0.25
`
`0.50
`
`0.71
`
`fi#
`
`0
`
`10
`
`16
`
`8
`
`4
`
`2
`
`1
`
`0.71
`
`1.00
`
`1.00
`
`1.00
`
`1.00
`
`1.00
`
`1.00
`
`1.00
`
`1.00
`
`1.00
`
`1.01
`
`1.08
`
`1.05
`
`1.20
`1.11
`1.038
`1.01
`1.00
`1.00
`20
`d(deg)
`30
`1.00
`1.00
`1.01
`105
`1.23
`1.49
`
`40
`
`60
`
`100
`
`1.00
`
`1.02
`
`1.08
`
`1.37
`
`1.94
`
`1.00
`
`1.01
`
`1.02
`
`1.09
`
`1.48
`
`2.69
`
`The cost approximation is valid within a few percent up to
`very large apertures andfield angles.
`
`
`PAGE 55 OF 154
`
`MASITC_01080435
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`42
`
`Illumination
`
`Known Irradiance
`
`
`
`If a surface is illuminated by a source of uniform intensity
`at a distance d and the irradiance on the surface is
`known, then the intensity of the source is
`
`Aj
`
`5
`
`E
`
`ee
`
`a.
`
`.
`2,
`
`
`i= sn ;
`cos&
`
`For any surface that is illuminated by
`uniform irradiance,
`the
`total
`flux
`illuminating the surface is
`
`@=E-A.
`
`The radiance of the surface, caused by the light reflecting
`from the surface, depends on the reflecting properties of
`the surface.
`
`If the surface is Lambertian over all angles of reflection
`(for this incident geometry), then
`
`-E
`p=P=,
`
`Tt
`
`wherepfis the reflectance of the surface for the relevant
`incident geometry.
`If the surface is not Lambertian over all angles but is
`Lambertian over the direction of concern, then
`i= ah
`,
`
`-E
`
`T
`
`where RF is the reflectance factor of the surface for the
`relevant
`incident geometry and for
`the direction of
`concern.
`
`
`PAGE 56 OF 154
`
`MASITC_01080436
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`Illumination Transfer
`
`43
`
`a, Q, NA, and f/# for a Circular Cone
`
`
`The case of a circular disk subtending a knownhalf-angle,
`8, shows up often in illumination situations.
`
`There are at least four common ways of describing the
`cone:
`solid angle (@), projected solid angle (Q),
`numerical aperture (NA), and f-number (//#):
`
`w = 2n(1— cos 0)
`NA =n-sin@
`
`Q = msin’ 0
`f/# =1/2sin8,
`
`where n is the index of refraction.
`
`Cone subtended by a circular disk
`O(deg)
`@
`Q
`NA/n
`1.8
`0.003
`0.003
`0.03
`
`fi#
`16.00
`
`3.6
`
`7.2
`
`12.7
`
`14.5
`
`20.0
`
`25.0
`
`30.0
`
`35.0
`
`40.0
`
`45.0
`
`50.0
`
`60.0
`
`70.0
`
`80.0
`
`90.0
`
`0.012
`
`0.049
`
`0.154
`
`0.200
`
`0.379
`
`0.5389
`
`0.842
`
`1.14
`
`1.47
`
`1.84
`
`2.24
`
`3.14
`
`4.13
`
`5.19
`
`6.28
`
`0.012
`
`0.049
`
`0.152
`
`0.196
`
`0.367
`
`0.561
`
`0.785
`
`1.03
`
`1.30
`
`1.57
`
`1.84
`
`2.36
`
`2.77
`
`3.05
`
`3.14
`
`0.06
`
`0.13
`
`0.22
`
`0.25
`
`0.34
`
`0.42
`
`0.50
`
`0.57
`
`0.64
`
`0.71
`
`0.77
`
`0.87
`
`0.94
`
`0.98
`
`1.00
`
`38.00
`
`4.00
`
`2.27
`
`2.00
`
`1.46
`
`1.18
`
`1.00
`
`0.87
`
`0.78
`
`0.71
`
`0.65
`
`0.58
`
`0.53
`
`0.51
`
`0.50
`
`
`
`MASITC_01080437 MASIMO2054
`Apple v. Masimo
`IPR2022-01300
`
`PAGE 57 OF 154
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`44
`
`Illumination
`
`Invariance of Radiance
`
`
`Unlike intensity, which is associated with a specific point,
`and irradiance, which is associated with a specific surface,
`radiance is associated with the propagating light rays
`themselves. This distinction is not trivial and implies that
`the radiance of a surface can be considered separate from
`the actual physical emitter or reflector that produces the
`radiance.
`
`Consider a uniform Lambertian radiating source, A,, with
`radiance, L,, illuminating an area, Ai, through a limiting
`aperture that limits the solid angle of the source to o:
`
`Ey
`
`A
`
`The physical location of the radiating source is irrelevant.
`Only the solid angle matters. In fact, the physical location
`(and shape) can be assumed to be anywhere (and any
`shape) as long as the solid angle is the same. All of the
`following descriptions of the radiating area, Ai, A2, and As,
`are equivalent to A, from an illumination point of view:
`
`
`
`
`
`PAGE 58 OF 154
`
`MASITC_01080438
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`Illumination in Imaging Systems
`
`45
`
`Image Radiance
`
`
`In an imaging system with no vignetting or significant
`aberrations,
`for Lambertian objects, point-by-point,
`the
`radiance of an imageis equalto the radiance of the object
`except
`for
`losses due to reflection, absorption, and
`scattering. These losses are usually combined into a
`single value of transmittance, t. This equivalence of
`radiance is true for virtual as well as real images, and for
`reflective or refractive imaging systems.
`
`Real Image
`
`Imaging
`System
`T
`
`Object
`L
`.
`Virtual Image
`
`.
`v
`
`eal
`Image
`L;= tLo
`
`
`
`Imaging
`System
`1
`
`Object
`L.
`
`A =
`
`;
`7
`Virtual
`Image
`L;= tLe
`
`Viewed from any point on a real image, the entire exit
`pupil of the optical system is also the radiance of the
`corresponding object point but reduced by T.
`
`Imaging
`System
`
`:
`v
`exit pupil = tLo
`exit pupil
`
`Object
`°
`
`
`
`MASITC_01080439 MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`PAGE 59 OF 154
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`46
`
`Illumination
`
`Limitations on Equivalent Radiance
`
`
`the image radiance only exists when the
`In all cases,
`image is viewed through the exit pupil of the imaging
`system. When viewed in a direction that doesn’t include
`the pupil, the radiance is zero.
`|
`:
`3
`
`|
`
`i
`
`ae
`
`ow\ . L;= zero
`
`
`
`exit pupil
`
`
`
`then the angular
`is not Lambertian,
`If the object
`image
`is
`also not
`radiance of
`the
`distribution of
`Lambertian. The
`relationship between the
`angular
`
`distributions of object and image is_notradiances
`
`straightforward and must be determined by ray tracing
`on the specific system. However, in many practical cases,
`the entrance pupil of the imaging system subtends a
`small angle from the object, and the source is essentially
`Lambertian over this small angle.
`
`If the object and the image are in media of different
`refractive indices, mo for the object and ni for the image,
`then the expression for equivalent radiance is
`
`
`
`The point-by-point equivalence of radiance from object to
`image is only valid for well-corrected optical systems. For
`systems that suffer from aberrations or are not in focus,
`each small point in the object is mapped to a “blur spot” in
`the image. Thus, the radiance of any small spot in the
`image is related to the average of the radiances of the
`corresponding spot in the object and its surrounding area.
`
`
`
`PAGE 60 OF 154
`
`MASITC_01080440
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`Illumination in Imaging Systems
`
`47
`
`Image Irradiance
`aSSSSSSS
`
`CX-0693
`
`Since the exit pupil, when viewed from the image, has the
`radiance of the object, then the irradiance at the imageis
`the sameas the irradiance from a source of the same size
`as the exit pupil and the same radiance as the object
`(reduced by t). In most imaging systems, the exit pupil is
`round and the irradiance is the same as the irradiance
`from a uniform Lambertian disk:
`
` 8
`
`Exit pupil, L =t-L,
`
`Image
`
`E, =n tL, sin’ 0-cos*8- F’.
`
`A table of values for the increase factor, F', is presented
`in the section on illumination transfer. F' is very close to
`1.0 except for a combination of large field angle (6) and
`large aperture (9), which is not a common combination in
`imaging systems.
`
`The cos‘d term contributes to substantial field darkening
`in wide-angle imaging systems—for example, cos!45 deg =
`0.25.
`
`If the physical aperture stop is not the limiting aperture
`for all the rays converging to an off-axis image point, the
`light is vignetted. The irradiance at image points where
`there is vignetting will be lower than predicted.
`
`On axis, cos‘6 = 1.0 and # = 1.0. The imageirradiance on
`axis, Kio, is
`
`_
`“aa
`E,, =u tL, sin’ 0.
`
`
`
`PAGE 61 OF 154
`
`MASITC_01080441
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`48
`
`Illumination
`
`fi#, Working f/#, T/#, NA, Q
`
`infinite conjugates (distant
`For a camera working at
`object, magnification,
`|m|<<1), the image irradiance can
`be expressed in terms of the lens’ f-number, //#:
`
`_ ai,
`ACA”
`
`is an “infinite
`This //#, usually associated with a lens,
`conjugates” quantity. When a lens is used at
`finite
`conjugates,
`the working f/-number, /f/#w, describes the
`cone angle illuminating the image:
`
`fit, = (Ft )-d—m),
`
`the image
`the lateral magnification of
`where m is
`(negative for real images), and the axial image irradiance
`is:
`
`— ath,
`A (fH)
`
`Note that fffw degenerates to the conventional “infinite
`conjugates”
`//# when the lens
`is used at
`infinite
`conjugates.
`Occasionally, a lens will be designated with a T-number,
`T/#, which combines the //# and the transmittance into a
`single quantity,
`T# =—— with axial image irradiance: E,, = th,
`#4
`Vt
`4 (T/#)
`
`Another descriptor of the image illumination cone angle is
`the numerical aperture, NA,
`
`NA = sin 0 with axial image irradiance: E,, = tL, NA’.
`
`In all cases, even without circular symmetry, on or off
`axis, the cone illuminating the image can be described by
`its projected solid angle, ©, with image irradiance:
`
`BE, =, Q.
`
`
`
`PAGE 62 OF 154
`
`MASITC_01080442
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`Illumination in Imaging Systems
`
`49
`
`Flux and Etendue
`
`
`The total flux reaching the image is the product of the
`image irradiance and the area of the image. The image
`irradianceis proportional to the projected solid angle of
`the exit pupil when viewed from the image:
`
`®, = th, a,Q.,,
`
`is the projected solid angle of the exit pupil
`where Qi;
`viewed from the image, ai is the area of the image, Lo is
`the [assumed uniform] radiance of the object, and tL. is
`the radiance of the exit pupil.
`
`The flux reaching the image also can be expressed in
`terms of the radiance of the exit pupil, tlo, the area of the
`exit pupil, ap, and the projected solid angle of the image
`when viewed from the exit pupil, Q,:
`
`®, = th, a,Q.,.
`
`The quantity aQ, representing the area of a plane in the
`optical system times the projected solid angle of another
`plane when viewed from it, appears equivalently in both
`expressions.
`This
`area-solid-angle-product
`is
`a
`fundamental property of
`the
`optical
`system that
`determines the amount of light that can get through the
`system. It is called the throughputor étendue.
`
`The radiance of an object is invariant and cannot be
`increased by an optical system, and the étendue is a
`fundamental property of an optical system. These two
`concepts mean that, for a source of given radiance and
`a given optical system, the maximum flux that can be
`
`transmitted through the system is predetermined.
`
`And, without “throwing away” light, the étendue cannot
`be decreased, but area and solid angle can be traded off.
`
`
`PAGE 63 OF 154
`
`MASITC_01080443
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`50
`
`Illumination
`
`Generalized Etendue
`
`
`The terminology for illumination in nonimaging systems
`is the same as that for imaging systems; however,
`the
`range of validity is extended to include all angular space,
`while that of imaging systems is limited to paraxial
`systems. With this taken into account, étendue is often
`called generalized étendue. In this domain the étendue
`cannot be regarded as the simple product of the area and
`solid angle;
`it must be integrated per
`the following
`equation andfigure:
`
`6&=n"
`
`cos 6dA,da,
`
`ae
`
`dA.
`
`Aperture
`
`differential solid angle.
`
`where nis the refractive
`index,
`9 is
`the angle
`from the normal, dAs is
`the differential
`source
`area,
`and dow is
`the
`
`The total flux through
`
`the aperture is found by
`integrating the radiance over the aperture:
`
`|
`
`@=
`
`|| L(r,A)cosedA,do,
`
`aperture
`
`where r and a denote the positional and directional
`aspects of source emission. Assuming that the source is
`Lambertian so radiance is independentof angle, then
`
`® = L,
`
`(|
`
`aperture
`
`cos 0dA,da =
`
`L,é
`
`ae
`nv
`
`Note that total flux is the product of the radiance and the
`geometrical
`étendue
`factor. This
`also
`shows’
`the
`conservation of étendue that
`follows
`from the
`conservations of radiance and energy.
`
`
`PAGE 64 OF 154
`
`MASITC_01080444
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`Illumination in Nonimaging Systems
`
`ol
`
`Concentration
`
`
`a term associated with the
`is
`Concentration (C)
`generalized étendue. It represents the ability to transfer
`more light into a desired area by using the conservation of
`étendue to alter the angle at the output of an optical
`system. It is defined as the ratio of the input area (A) to
`the output aperture
`area (A’)
`that
`transmits
`the
`prescribed flux from area A. For this reason it is called
`the concentration ratio:
`
`C=A/A'.
`
`of
`laws
`the
`of
`factor
`limit
`a
`expression,
`This
`is a forbearer of the invariance of
`thermodynamics,
`radiance and étendue.
`
`In a 2D system, which is analogous to an extruded trough,
`and a 3D system, which is analogous to a well, we find
`that the respective concentrations are given by
`
`a
`Cay =
`a
`
`—
`
`.
`n’sind’
`
`‘
`nsin9
`
`and
`
`CW
`3D
`
`.
`A
`n’sin 0’
`
`—
`:
`nsin@
`
`A’
`
`2
`
`5
`
`where a and a’ are the aperture widths, A and A’ are the
`aperture areas, n is the input index, n’
`is the output
`index, 9’ is the output angle, and 0 is the input angle.
`Optimal concentration is realized when the output
`angle is n/2, giving
`
`Cossit =
`
`’
`
`nh
`
`nsing,
`
`and
`
`Carat -|
`
`’
`
`fT
`
`nsin0,
`
`2
`
`?
`
`where 0a, the acceptance angle, is the prescribed upper
`input angle over which conservation of étendue is
`maintained.
`
`
`
`PAGE 65 OF 154
`
`MASITC_01080445
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`52
`
`Illumination
`
`Skew Invariant
`eeeeeee
`
`CX-0693
`
`in
`limiting factor
`is another
`skew invariant
`The
`nonimaging system design.
`Its definition is
`rather
`esoteric:
`
`
`
`where § = fmink;, and rmin is the
`rays closest approach to the
`optical axis (z, as shown), and
`kt
`is
`the
`tangential
`com-
`ponent of
`the
`ray’s_ prop-
`agation direction.
`
` Z
`
`A simpler way to think about
`the
`skew invariant
`is
`to
`
`recognize in a_rota-that
`
`tionally symmetric system (e.g., a lens), loss is introduced
`
`from the input two_spatialto the output if the
`
`
`
`distributions are not the same shape. For example, if the
`object shape is a uniform square but a uniform round
`outputis desired, then transfer losses will be produced.
`
`different
`efficiency with
`transfer’
`To maximize
`distributions, the symmetry of the optical system must be
`broken; or, in other words, there must be a “twist” in the
`optical components to force rays out of their respective
`sagittal planes. Many nonimaging optical systems take
`advantage of this property by including faceted reflectors
`(e.g.,
`segmented headlights),
`segmented lenses
`(e.g.,
`pillow optics for projection displays), or 3D edge-ray
`concentrators that employ V-wedges near the source(i.e.,
`solar concentrators).
`
`For a rotationally symmetric system, the rotational
`skewness of each ray is conserved or invariant.
`This skew invariant is given bythe first derivative
`
`of the étendue.
`
`PAGE 66 OF 154
`
`MASITC_01080446
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`Fibers, Lightpipes, and Lightguides
`
`53
`
`Fibers—Basic Description
`
`
`lightpipes, and lightguides are all
`Optical fibers,
`variations on the same theme. They each contain a
`central transparent core, usually circular in cross-section,
`surrounded by an annular cladding. The cladding has a
`lower index of refraction than thecore.
`
`Core
`
`Cladding
`
`The core can transmit light for long distances with low
`loss because of total internal reflection at the interface
`between the core and the cladding. The primary purpose
`of the cladding is
`to maintain the integrity of this
`interface. Without it, total internal reflection would occur
`at a core-air interface, but dust, nicks, abrasions, oils, and
`other contamination on the interface would reduce the
`transmission to unacceptably low levels.
`
`Sometimes layers of buffering and/or jacketing are
`
`placed outside the cladding for additional protection.
`
`The core diameter can range from very small, on the order
`of the wavelength of light, to a centimeter or more. The
`very thin cores are essentially waveguides and not used
`for illumination. Flexible glass and quartz fibers have
`core diameters ranging from approximately 50 microns to
`about 1 millimeter. If they are thicker than that, they are
`rigid and called rods or light pipes. Plastic fibers are
`flexible at thicker core diameters. Sometimes liquid cores
`and plastic cladding are used to make flexible, high-
`transmittance lightguides that are over a centimeter in
`core diameter.
`
`
`
`PAGE 67 OF 154
`
`MASITC_01080447
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`54
`
`Illumination
`
`Numerical Aperture and Etendue
`
`
`The maximum angle that a fiber can accept and transmit
`depends on the indices of refraction of the core and
`cladding (as well as the index of the surrounding medium,
`usually air, no= 1).
`
`
`
`Index = ne
`______.____ Sladding, index=n»
`
`
`_-.---.-_-._ Core, index = m1
`
`.
`1
`2
`2
`siné... = 5vm — Ne
`
`and the NA is
`
`_
`.
`_ ae:
`NA =n,8in0,,,. = 7,
`—Ns-
`
`The fiber has a maximum acceptance projected solid
`angle, Q = msin20max, and an acceptance area, the cross-
`sectional area of
`the core. Together,
`they define a
`throughput or étenduefor thefiber in air:
`
`,
`r
`Etendue = qoNA’,
`
`where d is the core diameter.
`
`the maximum flux-carrying
`defines
`étendue
`This
`capability of the fiber when presented with a source of
`radiance.
`
`Note: A fiber illuminated at less than its maximum
`acceptance angle will,
`theoretically, preserve the
`maximum illumination angle at its output. However,
`bending and scattering at the core-cladding interface
`broadens this angle toward the maximum allowable.
`This effect is not important in illumination systems in
`which it is desirable to utilize the maximum étendue of
`low-throughput components such asfibers and fill the
`full input NA.
`
`
`
`PAGE 68 OF 154
`
`MASITC_01080448
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`Fibers, Lightpipes, and Lightguides
`
`po
`
`Fiber Bundles
`
`
`To achieve high throughput with flexible glass or quartz
`fibers, multiple fibers are often arranged in a bundle,
`such as the 19-fiber tightly packed bundle shownbelow:
`
`space
`
`core
`
`cladding
`
`_
`
`interfiber
`
`The ratio of the light-carrying core area to the area of the
`entire bundle is called the packing fraction (pf), and
`can be as high as 85%. This packing fraction reduces the
`effective area of the bundle and, correspondingly,
`its
`étendue.
`
`In addition to flexibility, fiber bundles have other possible
`advantages in illumination systems:
`
`e
`
`In some situations, such as
`Shape Conversion:
`when illuminating a spectrometer, it can be useful to
`convert a circular cross-section of fibers to a line cross-
`section to align with, or actually become, the entrance
`slit to the spectrometer.
`
`e Splitting the Bundle: By feeding a large fiber
`bundle with a single light source and splitting the
`bundle into two or more branches,
`it is possible to
`illuminate multiple locations, from multiple angles,
`with one source.
`
`e Mixed Bundle: When illuminating with light over a
`wide spectral band, such as the full solar spectrum
`(~250 to 2500 nm), a mixed bundle of high OHsilica
`fibers for good UV transmission and low OH silica
`fibers for good IR transmission can compensate for the
`lack of an adequate single-fiber type.
`
`
`MASITC_01080449 MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`PAGE 69 OF 154
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`56
`
`Illumination
`
`Tapered Fibers and Bundles
`
`
`is possible to trade off
`it
`By tapering a single fiber,
`between area and solid angle while keeping the product
`(étendue) approximately constant.
`
`eeinput
`
`area, a;
`
`output
`area, a,
`
`input
`NA. NA,
`
`output NA
`“
`
`ai NA#? ~ do ‘NAo?
`
`On the other hand, when a bundle of straight fibers is
`tapered,
`the tradeoff is between the area and packing
`fraction.
`
`input
`area, a; ——
`
`input
`packing
`fraction,
`pf;
`
`input NA,
`NA;
`
`output
`area, a
`
`output
`packing
`fraction,
`pf,
`
`output
`NA, NA,
`
`NAo = NA; = NAfiber
`
`Ao Pf, = a;pf
`
`
`
`MASITC_01080450 MASIMO2054
`Apple v. Masimo
`IPR2022-01300
`
`PAGE 70 OF 154
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`Classical Illumination Designs
`
`57
`
`Spherical Reflector
`
`
`The light emitted from a source in the direction away
`from the optical system can be redirected toward the
`optical system by using a spherical mirror with the
`source located at the center of curvature.
`
`source
`
`source
`
`and
`image
`of
`
`--
`;
`
`If the source is solid, it is necessary to place the source
`slightly away from the center of curvature and the image
`just above, below, or alongside the physical source.
`
`— Ignoring
`losses
`on
`a reflection, the image has
`ow
`source, but the effective
`
`1
`
`the same radiance as the
`
`source area (source plus
`image)is doubled.
`
`Sometimes this technique is used to place the image of
`a source in a location where the physical source itself
`could not fit because of an obstruction such as a lamp
`
`envelope or socket.
`
`If the source is not solid, such as a coiled wire tungsten
`filament,
`imaging the source almost directly onto itself
`can help fill in the area between thecoils.
`
`source
`
`image
`
`In this case, the effective area of
`the source is not appreciably
`increased,
`but
`the
`apparent
`:
`radiance is nearly doubled.
`
`
`
`PAGE 71 OF 154
`
`MASITC_01080451
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`58
`
`Illumination
`
`Abbe Illumination
`a)
`
`Abbe illumination is characterized by imaging the
`source (or imaging an image of the source) directly onto
`the illuminated area. Since the uniformity of illumination
`is directly related to the uniformity of source radiance,
`Abbe illumination requires an extended source of uniform
`radiance such as a well-controlled arc, a ribbon filament
`lamp,
`the output of a clad rod, a frosted bulb, an
`illuminated diffuser, or
`the output of an integrating
`sphere.
`
`The paraxial layout below shows Abbe illumination used
`in a projection system. The source is
`imaged by a
`condenser onto the film. The projection objective images
`the film and the image of the source onto the screen. The
`purple dotted lines show the marginal and chief rays from
`the source. The black dotted lines show the marginal and
`chief rays from the film (and the image of the source). The
`marginal rays go through the on-axis points on the object
`and image and on the edges of the pupils (which are the
`lenses in this case). The chief rays go through the edges of
`the object and image andthe on-axis points of the pupils.
`
`screen
`
`source
`
`SOUrCE
`image
`ge,
`
`film
`
`projection
`lens
`
`J
`
`
`
`
`
`=. “a. - =
`
`condenser
`
`
`
`PAGE 72 OF 154
`
`MASITC_01080452
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`Classical Illumination Designs
`
`5g
`
`Kohler Illumination
`
`
`Kohler illumination is used when the source is not
`uniform, such as a coiled tungsten filament. Kohler
`illumination is characterized by imaging the source
`through the film onto the projection lens. The film is
`placed adjacent to the condenser, where the illumination
`is quite uniform, provided the source has a relative
`uniform angular distribution of intensity.
`
`The paraxial layout below shows Kohler illumination used
`in a projection system. The source is
`imaged by a
`condenser onto the projection lens. The projection
`objective images the film onto the screen. The purple
`dotted lines show the marginal and chief rays from the
`source. The black dotted lines show the marginal and
`chief rays from thefilm.
`
`Sou rce
`
`projection
`
`screen
`
`image mes ae mee
`generally depends upon the type of source available.
`
`~~. ~ ~
`
`~— ~
`
`de, ~
`
`condenser,
`film
`
`condenser NAs,
`similar
`sources,
`With similar
`source/condenser étendue as limiting étendue, and
`similar screen sizes, the average screen irradiance
`levels are the same for both Abbe and Kohler
`illumination systems. The choice between the two
`
`PAGE 73 OF 154
`
`MASITC_01080453
`
`MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`60
`
`Illumination
`
`Ellipsoidal and Paraboloidal Mirrors
`
`
`Very efficient collection of light from a source can be
`achieved using an ellipsoidal mirror, placing the source
`at one of the foci. The source is imaged at the other focus,
`with light collected over more than a hemisphere.
`
`ellipsoid
`
`source ~*~.
`
`oe
`
`-
`
`-
`
`to use a paraboloidal mirror to
`An alternative is
`collimate the light from a source andalens to reimageit.
`Again, the light from the source is collected over more
`than a hemisphere.
`
`paraboloid
`
`
`
`_--~ - - --------
`
`source
`
`TTS
`
`-
`
`lens
`
`> Binh
`
`we
`
`image
`
`The forward light is usually ignored in both of these types
`of designs.
`
`In both cases, the image of the source may not be good
`quality, but
`image quality may not be important
`in
`illumination systems. Also, obstructions like lamp bases,
`sockets, and mounting hardware can produce directional
`anomalies in the radiance of the image.
`
`If the quality of illumination is important, devices
`such as lenslet arrays or faceted reflectors may be
`
`used.
`
`MAS ITC_01 080454 MASIMO 2054
`
`Apple v. Masimo
`IPR2022-01300
`
`PAGE 74 OF 154
`
`MASIMO 2054
`Apple v. Masimo
`IPR2022-01300
`
`
`
`CX-0693
`
`Classical Illumination Designs
`
`61
`
`Spectral Control and Heat Management
`
`
`illumination systems often contain
`for
`Specifications
`spectral requirements. Some of these requirements can be
`partially met by the selection of lamp type, but usually
`some sort of filtering is needed. Also, for visual systems,
`especially those using tungsten lamps, unwanted heat
`from infrared light may need to be removed. Again,
`filtering is needed.
`
`The simplest type of filter is the absorbing filter placed
`in front of the light source. Filter glasses with a wide
`range of spectral characteristics are available from glass
`manufacturers. The primary concern with absorbing glass
`filters is cracking from excessive absorbed heat.
`
`
`
`
`
`
`Often a cracked filter will continue to workjustfine.
`
`Interference filters use multilayer thin-film coatings
`that
`either
`transmit
`or
`reflect
`light
`at
`specific
`wavelengths. Cracking is generally not a concern unless
`the filter is made of an absorbing substrate. These filters
`are available with a much wider variety of spectral
`properties
`than absorbing filters,
`including narrow
`bandwidth and sharp cut-off, and can be designed and
`manufactured to ac
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