Questions and Answers of Optics

A beam of light from an argon laser (λ0 = 500 nm) traveling in a glass block (ng = 3/2) is totally internally reflected at the flat air–glass interface. If the beam strikes the interface at 60.0°
A large crystal of Fabulite is covered by a layer of carbon tetrachloride. A beam of light comes up through the crystal and impinges on the solid–liquid interface. At what incident angle (at
Figure P.4.91 shows a laserbeam incident on a wet piece of filter paper atop a sheet of glass whose index of refraction is to be measured€”the photograph shows the resulting light pattern.
Consider the common mirage associated with an in homogeneous distribution of air situated above a warm roadway. Envision the bending of the rays as if it were instead a problem in total internal
Figure P.4.94 shows a prism-coupler arrangement developed at the Bell Telephone Laboratories. Its function is to feed a laserbeam into a thin (0.000 01-inch) transparent film, which then serves as a
The shape of the interface pictured in Fig. P.5.1 is known as a Cartesian oval after RenÃ© Descartes, who studied it in the 1600s. It€™s the perfect configuration to carry any
Someone views a flag through a yellow filter. The flag has five colored horizontal bands, which are, starting from the top, blue, cyan, magenta, yellow, and white. What colors, if any, will she see
A wall is painted with stripes of red, cyan, white, yellow, green, and magenta. A person wearing yellow sunglasses views the wall through a piece of cyan-colored stained glass. What colors, if any,
The graphs in Fig. P.4.98 are the reflection spectra for several roses seen in white light. The flowers were white, yellow, light pink, dark pink, blue, orange, and red. Associate each graph with a
Figure P.4.99 depicts a ray being multiply reflected by a transparent dielectric plate (the amplitudes of the resulting fragments are indicated). As in Section 4.10, we use the primed coefficient
A wave, linearly polarized in the plane-of-incidence, impinges on the interface between two dielectric media. If ni> ntand Î¸i= Î¸'p, there is no reflected wave, that is,
Making use of the Fresnel Equations, show that t||(Î¸p)t'p||(Î¸'p) = 1, as in the previous problem.Data from Prob. 4.100A wave, linearly polarized in the plane-of-incidence,
A prism, ABC, is configured such that angle BCA = 90° and angle CBA = 45°. What is the minimum value of its index of refraction if, while immersed in air, a beam traversing face AC is to be totally
Using a block of a transparent, unknown material, it is found that a beam of light inside the material is totally internally reflected at the air–block interface at an angle of 48.0°. What is its
What is the critical angle for total internal reflection for diamond in air? What, if anything, does the critical angle have to do with the luster of a well-cut diamond?
Referring back to Problem 4.21, note that as θi increases θt increases. Prove that the maximum value θt may have is θc.Data from Prob. 4.21Make a plot of θi versus θt for an air– glass
Calculate the critical angle beyond which there is total internal reflection at an air–glass (ng = 1.5) interface.
Show that when θi > θc at a dielectric interface, r|| and r⊥ are complex and r⊥r*⊥ = r||r*|| = 1.
Establish that at near-normal incidence the equationis a good approximation. [Use the results of the previous problem, Eq. (4.43), and the power series expansions of the sine and cosine functions.]
Examine the three photos in Fig. P. 4.41. Part(a) Shows a single wide block of Plexiglas.(b) Shows two narrow blocks of Plexiglas, each half as wide as the first, pressed lightly against one
According to the mathematician Hermann Schwarz, there is one triangle that can be inscribed within an acute triangle such that it has a minimal perimeter. Using two planar mirrors, a laserbeam, and
Derive the expressions for rŠ¥and r|| given by Eqs. (4.70) and (4.71). cos 6; – (n – sin² 0;)'/² cos 0; + (n – sin² 0;)'/2 (4.70) ni cos 0; – (ni – sin² e,)'/2 n cos e; +
Show that at normal incidence on the boundary between two dielectrics, as nti S → 1, R → 0, and T → 1. Moreover, prove that as nti → 1, R → 0, and T → 1 Moreover, prove that as
Making use of the expressionI(y) = I0e-αy [4.78]for an absorbing medium, we define a quantity called the unit transmittance T1. At normal incidence,
Suppose that we look at a source perpendicularly through a stack of N microscope slides. The source seen through even a dozen slides will be noticeably darker. Assuming negligible absorption, show
Using the results of Problem 4.72, that is, Eqs. (4.98) and (4.99), show thatData from Prob. 4.72Show thatand sin 20; sin 20, т (4.98) sin?(0; + 0,) cos²(0; – 6,) sin 20; sin 20, T1 sin (0; + 0)
Make a sketch of R⊥ and R|| for ni = 1.5 and nt = 1 (i.e., internal reflection) versus the incident angle.
Show thatand sin 20; sin 20, т (4.98) sin?(0; + 0,) cos²(0; – 6,) sin 20; sin 20, T1 sin (0; + 0) (4.99)
Making use of the definitions of the azimuthal angles in Problem 4.69, show that
It is often useful to work with the azimuthal angle Î³, which is defined as the angle between the plane-of-vibration and the plane-of-incidence. Thus for linearly polarized light,Figure
Show that the polarization angles for internal and external reflection at a given interface are complementary, that is, θp + θ'p = 90° (see Problem 4.66).Data from Prob. 4.66Show that tan θp =
Beginning with Eq. (4.38), show that for two dielectric media, in general tan Î¸p= [Ïµt(ÏµtÎ¼i-
Show that tan θp = nt/ni and calculate the polarization angle for external incidence on a plate of crown glass (ng = 1.52) in air.
Use the Fresnel Equations to prove that light incident at θp = 1/2π - θt results in a reflected beam that is indeed polarized.
Verify thatfor Î¸i = 30° at a crown glass€“air interface (nti = 1.52). [4.49] ti +(-r1) = 1
Prove thatfor all Î¸i, first from the boundary conditions and then from the Fresnel Equations. ti +(-r1) = 1 [4.49]
In Fig. 4.49 the curve of rŠ¥approaches -1.0 as the angle-of-incidence approaches 90°. Prove that if Î±Š¥is the angle the curve makes with the vertical at
Prove that for a vacuum-dielectric interface at glancing incidence rŠ¥†’-1, as in Fig. 4.49. Fig. 4.49. 1.0 0.5 -0.5 56.3° -1.0 30 90 0; (degrees) Amplitude coefficients
Use Eq. (4.42) and the power series expansion of the sine function to establish that at near-normal incidence we can obtain a better approximation than the one in Problem 4.45, which is
Compare the amplitude reflection coefficients for an air–water (nw = 4/3) interface with that of an air–crownglass (ng = 3/2) interface, both at near-normal incidence. What are the corresponding
Quasimonochromatic light having an irradiance of 400 W/m2 is incident normally on the cornea (nc = 1.376) of the human eye. If the person is swimming under the water (nw = 1.33), determine the
Show that energy is conserved in the previous problem.
A beam of unpolarized light carries 2000 W/m2 down onto an air–plastic interface. It is found that of the light reflected at the interface 300 W/m2 is polarized with its E-field perpendicular to
We know that 1000 W/m2 of unpolarized light is incident in air on an air–glass interface where nti = 3/2. If the transmittance for light with its E-field perpendicular to the plane of incidence is
Considering the previous problem calculate R⊥, R||, T⊥, T||, and the net transmittance T and reflectance R.
Unpolarized light is incident in air on the flat surface of a sheet of glass of index 1.60 at an angle of 30.0° to the normal. Determine both amplitude coefficients of reflection. What is the
Using the Fresnel Equations show thatand cos 0; – Vn – sin² 0; cos 0; + Vn – sin² 0; nị cos 0; - Vni – sin' 0; ni cos 0; + Vni – sin² 0;
A beam of quasimonochromatic light having an irradiance of 500 W/m2 is incident in air perpendicularly on the surface of a tank of water (nw = 1.333). Determine the transmitted irradiance.
Light is incident in air perpendicularly on a sheet of crown glass having an index of refraction of 1.522. Determine both the reflectance and the transmittance.
Considering the previous problem, compute the corresponding values of the amplitude coefficients of reflection for both the normal transits of light from air-to-glass and glass-to-air. Show that Eq.
A nearly monochromatic laserbeam polarized with its electric field perpendicular to the plane of incidence impinges normally in air on glass (nt = 1.50). Determine the amplitude coefficient of
Prove that at normal incidence on the boundary between two dielectrics 2n; [4 le, = 0 = [t1 lo, = 0 ° n¡ + n;
A laserbeam is incident on the interface between air and some dielectric of index n. For small values of θi show that θt = θi/n. Use this and Eq. (4.42) to establish that at near-normal
A beam of light in air strikes the surface of a smooth piece of plastic having an index of refraction of 1.55 at an angle with the normal of 22.0°. The incident light has component E-field
Derive Eqs. (4.42) through (4.45) for rŠ¥, r||, tŠ¥, and t||. sin (0; – 0;) sin (0; + 0,) (4.42)
Suppose a lightwave that is linearly polarized in the plane-of incidence impinges at 30° on a crown-glass (ng= 1.52) plate in air. Compute the appropriate amplitude reflection and transmission
Discuss the results of Problem 4.38 in the light of Fermat€™s Principle; that is, how does the relative index n21affect things? To see the lateral displacement, look at a broad source
Show that the two rays that enter the system in Fig. P.4.39 parallel to each other emerge from it being parallel. Figure P.4.39 na П na П2 Па
Show analytically that a beam (in a medium of index n1) entering a planar transparent plate (of index n2and thickness d ), as in Fig. P.4.38, emerges parallel to its initial direction. Derive an
Derive the Law of Reflection, θi = θr, by using the calculus to minimize the transit time, as required by Fermat’s Principle.
In the case of reflection from a planar surface, use Fermat’s Principle to prove that the incident and reflected rays share a common plane with the normal Ûn, namely, the plane-of-incidence.
Derive a vector expression equivalent to the Law of Reflection. As before, let the normal go from the incident to the transmitting medium, even though it obviously doesn’t really matter.
Starting with Snell€™s Law, prove that the vector refraction equation has the form n, k, – n;k = (n,cos 0; – n;cos 0;) û, [4.7]
Making use of the ideas of equal transit times between corresponding points and the orthogonality of rays and wavefronts, derive the Law of Reflection and Snell€™s Law. The ray diagram of
With the previous problem in mind, return to Eq. (4.19) and take the origin of the coordinate system in the plane-of-incidence and on the interface (Fig. 4.47). Show that that equation is then
In Fig. P.4.30 the wavefronts in the incident medium match the fronts in the transmitting medium every where on the interface€”a concept known as wave front continuity. Write expressions
Figure P.4.8 shows what€™s called a corner mirror. Determine the direction of the exiting ray with respect to the incident ray. Figure P.4.8
A coin is resting on the bottom of a tank of water (nW = 1.33) 1.00 m deep. On top of the water floats a layer of benzene (nb = 1.50), which is 20.0 cm thick. Looking down nearly perpendicularly, how
Suppose that you focus a camera with a close-up bellows attachment directly own on a letter printed on this page. The letter is then covered with a 1.00-mm-thick microscope slide (n = 1.55). How high
Light is incident in the air on an air–glass interface. If the index of refraction of the glass is 1.70, find the incident angle such that the transmission angle is to equal 1/2θi.
A laserbeam impinges on the top surface of a 2.00-cm-thick parallel glass (n = 1.50) plate at an angle of 35°. How long is the actual path through the glass?
A block of glass of index 3/2 has a small flaw 3.0 cm below its flat horizontal top surface. A camera lens is 8.0 cm above the surface in air, looking straight down. How far will the flaw appear to
A bowl 10.0 cm deep is filled with olive oil. A coin on the bottom of the bowl is viewed directly from above. How far beneath the surface will the coin appear?
An exceedingly narrow beam of white light is incident at 60.0° on a sheet of glass 10.0 cm thick in air. The index of refraction for red light is 1.505 and for violet light it’s 1.545. Determine
A laserbeam having a diameter D in air strikes a piece of glass (ng) at an angle θi. What is the diameter of the beam in the glass?
Make a plot of θi versus θt for an air– glass boundary where ngα = 1.5. Discuss the shape of the curve.
An underwater swimmer shines a beam of light up toward the surface. It strikes the air–water interface at 35°. At what angle will it emerge into the air?
A laserbeam impinges on an air–liquid interface at an angle of 55°. The refracted ray is observed to be transmitted at 40°. What is the refractive index of the liquid?
Light of wavelength 600 nm in vacuum enters a block of glass where ng= 1.5. Compute its wavelength in the glass.What color would it appear to someone embedded in the glass (see Table 3.4)?Table 3.4 v
A beam of 12-cm planar microwaves strikes the surface of a dielectric at 45°. If nti = 4/3 compute(a) The wavelength in the transmitting medium.(b) The angle θt.
Given an interface between water (nw = 4/3 ) and glass (ng = 3/2), compute the transmission angle for a beam incident in the water at 45°. If the transmitted beam is reversed so that it impinges on
A ray of yellow light from a sodium discharge lamp falls on the surface of a diamond in air at 45°. If at that frequency nd = 2.42, compute the angular deviation suffered upon transmission.
Figure P.4.14 is a plot of the sine of the angle-of-incidence versus the sine of the transmission angle measured as light passed from air into a more optically dense medium. Discuss the curve. What
A laserbeam in air strikes the flat surface of a sheet of glass (ng = 1.50) at an angle of incidence of 30.0°. Rather than continuing straight into the glass the beam bends toward the normal through
The construction in Fig. P.4.12 corresponds to Descartes€™s erroneous derivation of the Law of Refraction. Light moves from S to O in the same time it travels from O to P. Moreover, its
Calculate the transmission angle for a ray incident in air at 30° on a block of crown glass (ng = 1.52).
Return to Fig. 4.33 and Huygens€™s refraction method and prove that it leads to Snell€™s Law.Figure 4.33 Incident Transmitted 1/n; 1/ni n, > n;
A beam of light strikes mirror-1 and then mirror-2 in Fig. P.4.9. Determine angles Î¸r1and Î¸r2.Figure P.4.9 30° Mirror-1 Mirror-2 45°
On entering the tomb of FRED the Hero of Nod, you find yourself in a dark closed chamber with a small hole in a wall 3.0 m up from the floor. Once a year, on FRED’s birthday, a beam of sunlight
A very narrow laserbeam is incident at an angle of 58° on a horizontal mirror. The reflected beam strikes a wall at a spot 5.0 m away from the point of incidence where the beam hit the mirror. How
Imagine that we have a nonabsorbing glass plate of index n and thickness Î”y, which stands between a source S and an observer P.(a) If the unobstructed wave (without the plate present) is
The equation for a driven damped oscillator is(a) Explain the significance of each term.(b) Let E = E0eiÏ‰t and x = x0ei(Ï‰t-Î±), where E0 and x0 are real quantities.
Figure P.4.3 depicts light emerging from a point source. It shows three different representations of radiant energy streaming outward. Identify each one and discuss its relationship to the
A white floodlight beam crosses a large volume containing a tenuous molecular gas mixture of mostly oxygen and nitrogen. Compare the relative amount of scattering occurring for the yellow (580 nm)
Work your way through an argument using dimensional analysis to establish the λ-4 dependence of the percentage of light scattered in Rayleigh Scattering. Let E0i and E0s be the incident and
If an ultraviolet photon is to dissociate the oxygen and carbon atoms in the carbon monoxide molecule, it must provide 11 eV of energy. What is the minimum frequency of the appropriate radiation?
In 1871 Sellmeier derived the equationwhere the Aj terms are constants and each Î»0j is the vacuum wavelength associated with a natural frequency v0j, such that
Crystal quartz has refractive indexes of 1.557 and 1.547 at wavelengths of 410.0 nm and 550.0 nm, respectively. Using only the first two terms in Cauchy’s Equation, calculate C1 and C2 and
Referring to the previous problem, realize that there is a region between each pair of absorption bands for which the Cauchy Equation (with a new set of constants) works fairly well. Examine Fig.
Augustin Louis Cauchy (1789€“1857) determined an empirical equation for n(Î») for substances that are transparent in the visible. His expression corresponded to the power series

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