ECEN 4616/5616 Optoelectronic Design Class website with past lectures, various files, and assignments: (The first assignment will be posted here by 1/27) To view video recordings of past lectures, go to: and select “course login” from the upper right corner of the page.

Radiometry Radiometry concerns the measurement of EM radiation (and light in particular). A closely related field, Photometry, is basically Radiometry scaled by the sensitivity of a particular detector. EM waves are radiated into space and (generally) spread out. A basic concept used in measuring the energy flow is the solid angle. A solid angle is a two-dimensional angle (e.g., a cone), whose unit is the steradian, abbreviated sr and usually represented by the Omega symbol: Ω. A one sr cone intercepts an area r 2 in a sphere of radius r, if the cone’s vertex is at the center of the sphere. Hence there are a total of 2 π sr in a complete sphere.

The solid angle of a circular cone with half angle θ is: Small angle approximation to Ω To get a small angle approximation, convert to sin : Using the small angle approximations, We will often be concerned with the brightness of relatively distant objects, where the small angle approximation is appropriate.

The subject of radiometry contains a fair number of defined technical terms, which you can find defined in the text:  Flux (Φ): Just another term for power; J/s, but referring to propagating light.  Intensity (I): Flux per unit solid angle; W/sr (sr = steriadian)  Irradiance (E): Flux per unit area, incident; W/m 2  Exitance (M): Flux per unit area, exiting; W/m 2  Radiance (L): Flux per unit projected area per unit solid angle; W/(A∙sr) Where, in the above definitions, “Flux” refers to “Power carried by EM waves”, into, out of, or through a surface. If the above quantities are scaled by the response curve of a detector (like the eye), then they are called “Luminous Flux”, “Luminous Intensity”, …, “Luminance (for “Luminance Radiance”), etc. These values depend on wavelength, where the strict power values don’t.

Lambertian Sources: (Why is the term “projected” in the definition of Radiance?) Lambert’s Law describes most diffuse, area sources of radiation. It says that the power emitted per unit area per steradian (i.e., the ‘radiance’) varies as the cosine of the angle between the radiation direction and the normal to the surface, as shown in the following figure:

Does this make such a source become dim, as the angle of incidence increases? Consider a telescope looking at a fraction of the surface of such a lambertian source, and consider what happens when the angle of the source changes: Field of view of one pixel. θ Entrance pupil Ω D Let the FOV of a single pixel in the telescope’s detector be a square of extent D at the source. Let the source have a radiance of: For case ‘A’: A B For case ‘B’: Emitting areaRadiance Power received X=XSolid Angle Hence the apparent brightness of the source is not a function of it’s angle.

This is the reason that the Sun, for example, does not appear dimmer at the edge than at the center – as you look toward the edge, the surface has less radiance per area than at the center (as per Lambert’s Law), but there is also more surface radiating into the observer’s FOV.

Energy transmitted through an optical system: u u’ h h’ Object (radiance = R) image The power radiated to the lens is: Assuming no losses in the optical system, this is the same as the power radiated to the image. The magnification is:From the Optical Invariant: Hence: Therefore and the radiance of the image is the same as the radiance of the object.

Exposure The exposure is defined as the energy deposited on the detector per unit area; therefore the exposure is the irradiance of the image (in w/m 2 ) times the exposure time in seconds. The irradiance of the image is simply the radiance, in w/(m 2 sr), times the solid angle, Ω’, over which the (incoming) radiance exists: (w/m 2 ) f D u’ Ω’Ω’ We will assume that the object is far from the camera compared to the focal length, so that the image is essentially one focal length away from the lens. Hence: and therefore Hence, the irradiance (w/m 2 ) of the image is: This is known as the Camera Equation.

The Camera Equation The Camera Equation shows how to set exposures for objects of given brightness (radiance), R: For an exposure of T seconds, the energy deposited on the detector is E’T Joules/meter 2. An interesting feature of this relation is that the proper exposure for an object of a given radiance (brightness) is independent of the distance to that object. The correct exposure (for a given camera and detector) of the surface of the sunlit Moon, for example, is the same whether you are on the surface of the Moon, or taking the picture from Earth. The camera equation is derived from geometric optics, and no longer holds when the geometrical image of an object is significantly smaller than the real (diffraction) image, e.g., the point spread function.

Star Pictures from the Moon One occasionally hears the argument that, because no stars are visible in the pictures taken by astronauts on the Moon, that therefore they must have been faked. Is this a reasonable argument? With a few facts and the camera equation we can test it:  The Moon, during Lunar day, is illuminated by bright sunlight – an irradiance of ~ 1.3 Kw/m 2.  The surface of the Moon is about 17% reflective, on average (about the same as a freshly paved asphalt parking lot). o Hence, the proper camera exposure for taking pictures on the sunlit Moon is similar to the proper exposure for taking pictures in a parking lot at noon on a cloudless day.  Solar facts: o Total light output of the Sun ~ 3.83 x W. o Radius of the Sun: 696,000 km  The camera used on the Moon (Hasselblad), 80 mm focal length, 28mm aperture. o The cameras were stopped down to F/16, so the astronauts wouldn’t have to focus. o Dynamic range of film (Ektachrome 400) ~ 1,000:1 o Dynamic range of camera ~ 100:1 (due to scattered light)

Moon-Star Imaging Calculations Radiance of the Moon’s Surface: Where 0.17 is the reflectivity of the surface, and the last factor of 2 is because the radiation into a hemisphere,, is not uniform, but Lambertian, so has a peak value of twice the average. We get the Irradiance of the Moon’s image in the camera from the camera equation: Assume a star with the same size and brightness as the Sun, but 4 light years (3.8 x m) away: Surface Area of Star: Radiance of Star:

Moon-Star Imaging Calculations (continued) We get the irradiance (w/m 2 ) of the star image from the Camera Equation: The size (radius) of the (geometric) star image is the radius of the star times the magnification: Total power in the star image is the geometric area of the image times the geometric irradiance: But, the actual diffraction image of the star is the radius of the Airy Pattern: and the area: Where

Moon-Star Imaging Calculations (continued) Finally, the actual irradiance of the star image is the power in the star image divided by the area of the diffraction image: Which is only 1/3203 of the Moon image’s irradiance on the film. This is beyond the film’s 1000:1 dynamic range, and undetectable in the camera, even with a perfect detector due to the 1% scattered light from the lenses. So, our conclusion has to be that if there were star images in the moon pictures then that would prove that they were faked!

‘Size’ of Diffraction Limited PSF d A calculation of the diffraction pattern of a circular aperture of radius a results in the ‘Airy Pattern’: The first null of which occurs at Hence, for a lens of focal length f and diameter D, the diameter of the Airy Pattern (to the first null) is:

Geometric image > Airy Pattern Diffraction image > Airy Pattern Strehl ratio=0.161 Geometric image = Airy Pattern Diffraction image ≈ Airy Pattern Strehl ratio=0.829 Geometric image < Airy Pattern Diffraction image = Airy Pattern Strehl ratio=0.955 The radius of the Airy Pattern (to the first null ring) is:

Graphical Ray Tracing For Mirrors Spherical mirrors have only a single focal point, located halfway between the center of curvature and the vertex of the surface. This focal point is traditionally labeled f’, the secondary focal point. For graphical ray tracing, we will only cover spherical mirrors. Other conic sections have special rules that are covered in the textbook. Cf’ R S Z Z=0 Surface S has radius of curvature R, center of curvature is at C (note C<0) and curvature c=1/R. The focal point, f’, is halfway between C and the vertex of the surface (which is also the local coordinate origin).

Cf’ Z Graphical Ray Tracing with Mirrors Rules for graphical rays with mirrors: 1.A ray parallel to the axis crosses the axis at f’ after reflection. 2.A ray crossing the axis at f’ will emerge parallel to the axis after reflection. 3.A ray crossing the axis at C will be reflected back on itself Rays for negative curvature (positive power) mirrors ‘concave mirror’

f’C Z Rays for positive curvature (negative power) mirrors Rules for graphical rays with mirrors: 1.A ray parallel to the axis crosses the axis at f’ after reflection. 2.A ray crossing the axis at f’ will emerge parallel to the axis after reflection. 3.A ray crossing the axis at C will be reflected back on itself. ‘convex mirror’

Cf’ Real Object and Image (concave mirror) The object and image positions are interchangeable, since the ray paths are reversible.

C f’ Virtual Object and Image (convex mirror) Again, the object and image positions are interchangeable, since the ray paths are reversible. See Chapter 2 of the text for more examples.