Questions and Answers of Optics

Figure P.10.30 is a computer-generated Fraunhofer irradiance distribution. Describe the aperture that would give rise to such a pattern and give your reasoning in detail.Figure P.10.30
In Fig. P.10.29a and b are the electric field and irradiance distributions, respectively, in the far field for a configuration of elongated rectangular apertures. Describe the arrangement of holes
Figure P.10.28 is the irradiance distribution in the far field for a configuration of elongated rectangular apertures. Describe the arrangement of holes that would give rise to such a pattern and
With the results of Problem 10.25 in mind, discuss the symmetries that would be evident in the Fraunhofer diffraction pattern of an aperture that is itself symmetrical about a line (assuming normally
Show that Fraunhofer diffraction patterns have a center of symmetry [i.e., I(Y, Z) = I(-Y, -Z)], regardless of the configuration of the aperture, as long as there are no phase variations in the field
Consider the Fraunhofer diffraction pattern of a rectangular aperture 0.200 mm (in the y-direction) by 0.100 mm (in the z-direction). It is formed in 543-nm light from a helium–neon laser, on a
An opaque screen contains a rectangular hole 0.199 mm (along the z-axis) by 0.100 mm (along the y-axis). It is illuminated by light at 543 nm from a helium–neon laser. A big positive lens with a
Starting with the irradiance expression for a finite slit, shrink the slit down to a minuscule area element and show that it emits equally in all directions.
Consider the Fraunhofer diffraction pattern for eight very narrow parallel slits under monochromatic illumination.(a) Sketch the resulting irradiance distribution.(b) Explain why the first minimum
Suppose we have 15 parallel long narrow slits in an opaque screen. Furthermore, suppose each slit is separated from the next by a center-to-center distance that is equal to 4 slit widths. Given that
Imagine two aperture screens arranged to produce two Fraunhofer diffraction patterns. One contains 8 very narrow closely spaced parallel slits, the other 16 such slits. All else being equal, compare
Let E01 be the electric-field amplitude on a distant screen due to each one of three very narrow parallel slits illuminated by monochromatic plane waves. Compare the amplitude of the central
What is the relative irradiance of the subsidiary maxima in a three-slit Fraunhofer diffraction pattern? Draw a graph of the irradiance distribution, when α = 2b, for two and then three slits.
In a two-slit setup, each slit is 0.020 mm wide. These apertures are illuminated by plane waves of yellow sodium light (λ = 589.6 nm). The resulting Fraunhofer fringe pattern consists of 11 narrow
Two long slits 0.10 mm wide, separated by 0.20 mm, in an opaque screen are illuminated by light with a wavelength of 500 nm. If the plane of observation is 2.5 m away, will the pattern correspond to
Show that for a double-slit Fraunhofer pattern, if α = mb, the number of bright fringes (or parts thereof) within the central diffraction maximum will be equal to 2m.
A long narrow slit 0.20 mm wide is illuminated normally with collimated blue hydrogen light (λ = 486.1 nm). Immediately behind the slit is a large positive lens of focal length 60.0 cm. It produces
Plane waves of green light (λ = 546.1 nm) impinge normally on a long narrow slit (0.15 mm wide) in an opaque screen. A large lens with a focal length of +62.0 cm placed just behind the slit produces
Consider the single-slit Fraunhofer diffraction pattern formed on a screen by a lens of focal length Æ’. Show that the peak of the first subsidiary bright band is a distance Y (measured
Plane waves from a magnesium lamp (λ = 518.36 nm) arrive perpendicularly on an opaque screen containing a long 0.250-mm-wide slit. A large nearby positive lens forms a sharp image of the Fraunhofer
A collimated beam of microwaves impinges on a metal screen that contains a long horizontal slit that is 20 cm wide. A detector moving parallel to the screen in the far-field region locates the first
A narrow single slit (in air) in an opaque screen is illuminated by infrared from a He–Ne laser at 1152.2 nm, and it is found that the center of the tenth dark band in the Fraunhofer pattern lies
A single slit in an opaque screen 0.10 mm wide is illuminated (in air) by plane waves from a krypton ion laser (λ0 = 461.9 nm). If the observing screen is 1.0 m away, determine whether or not the
The angular distance between the center and the first minimum of a single-slit Fraunhofer diffraction pattern is called the half-angular breadth; write an expression for it. Find the corresponding
Consider the case of single-slit Fraunhofer diffraction. Calculate the ratio of the irradiance of the central maximum to the irradiance of the first secondary maximum on either side of it. Check your
Examine the setup of Fig. 10.3 in order to determine what is happening in the image space of the lenses; in other words, locate the exit pupil and relate it to the diffraction process. Show that the
Referring back to the multiple antenna system on p. 456, compute the angular separation between successive lobes or principal maxima and the width of the central maximum.
In Section 10.1.3 we talked about introducing an intrinsic phase shift Îµ between oscillators in a linear array. With this in mind, show that Eq. (10.18) becomeswhen the incident plane
To examine the conditions under which the approximations of Eq. (9.23) are valid:(a) Apply the law of cosines to triangle S1S2P in Fig. 9.11c to get(b) Expand this in a Maclaurin series yielding(c)
An optical filter can be described by a Jones matrix(a) Obtain the form of the emerging beam when the incident light is plane polarized at angle Î¸ to the horizontal (see Problem
Two light waves Ex = E0 cos (kz - ωt) and Ey = -E0 cos (kz - ωt) overlap in space. Show that the resultant is linear light and determine its amplitude and tilt angle θ.
Two waves Ez = 4 sin (ky - ωt) and Ex = 3 in (ky - ωt), both in SI units, overlap in space. Describe completely the state of polarization of the resultant.
Consider the following two waves expressed in SI units: Ex = 8 sin (ky - ωt + π/2) and Ez = 8 sin (ky - ωt). Which wave leads, and by how much? Describe the resultant wave. What is the value of
Describe completely the state of polarization of each of the following waves: (a) É = ÎEo cos (kz – wt) – jE, cos (kz – wt) |(b) E = ÎEo sin 2#(z/A – vt) – jEo sin 27(z/A – vt) (c) É
Consider the disturbance given by the expression vector E(z, t) = [iˆ cos ωt + jˆ cos (ωt - π/2)]E0 sin kz. What kind of wave is it? Draw a rough sketch showing its main features.
Analytically, show that the superposition of an R- and an L-state having different amplitudes will yield an E-state, as shown in Fig. 8.11. What must Îµ be to duplicate that
Write an expression for a P-state lightwave of angular frequency ω and amplitude E0 propagating along the x- xis with its plane-of-vibration at an angle of 25° to the xy-plane. The disturbance is
Write an expression for a P-state lightwave of angular frequency ω and amplitude E0 propagating along a line in the xy‑plane at 45° to the x-axis and having its plane-of-vibration corresponding
Write an expression for an R-state lightwave of frequency ω propagating in the positive x-direction such that at t = 0 and x = 0 the vector E-field points in the negative z-direction.
A beam of linearly polarized light with its electric field vertical impinges perpendicularly on an ideal linear polarizer with a vertical transmission axis. If the incoming beam has an irradiance of
Given that 300 W/m2 of light from an ordinary tungsten bulb arrives at an ideal linear polarizer, what is its radiant flux density on emerging?
A beam of vertically polarized linear light is perpendicularly incident on an ideal linear polarizer. Show that if its transmission axis makes an angle of 60° with the vertical only 25% of the
The transmittance of a real linear polarizer illuminated by linear light making an angle of Î¸ with its transmission axis is given by Tl = (T0 - T90) cos 2Î¸ + T90where
Suppose 1000 W/m2 of natural light is incident perpendicularly on a sheet of HN-22 polarizer. Describe the light leaving the filter. What is its irradiance?
If light that is initially natural and of flux density Ii passes through two sheets of HN-32 whose transmission axes are parallel, what will be the flux density of the emerging beam?
What will be the irradiance of the emerging beam if the analyzer of the previous problem is rotated 30°?
Two sheets of HN-38S linear polarizer are in series one behind the other with their transmission axes aligned. The first is illuminated by 1000 W/m2 of natural light. Determine the approximate
Discuss in detail what you see in Fig. P.8.33. The crystal in the photograph is calcite, and it has a blunt corner at the upper left. The two Polaroids have their transmission axes parallel to their
Show by direct calculation, using Mueller matrices, that a unitirradiance beam of natural light passing through a linear polarizer with its transmission axis at +45° is converted into a P-state at
Show by direct calculation, using Mueller matrices, that a beam of horizontal P-state light passing through a 1/4 λ-plate with its fast axis horizontal emerges unchanged.
With Lloyd’s mirror, X-ray fringes were observed, the spacing of which was found to be 0.002 5 cm. The wavelength used was 8.33 Å. If the source–screen distance was 3 m, how high above the
A thin film of ethyl alcohol (n = 1.36) spread on a flat glass plate and illuminated with white light shows a color pattern in reflection. If a region of the film reflects only green light (500 nm)
One of the mirrors of a Michelson Interferometer is moved, and 1000 fringe-pairs shift past the hairline in a viewing telescope during the process. If the device is illuminated with 500-nm light, how
The irradiance of a beam of natural light is 400 W/m2. It impinges on the first of two consecutive ideal linear polarizers whose transmission axes are 40.0° apart. How much light emerges from the
Imagine four HN-32 Polaroids one behind the other with their transmission axes all parallel. If the irradiance of natural light incident on the first filter is Ii, what is the transmitted irradiance
Natural light of irradiance Ii is incident normally on an HN-32 polarizer.(a) How much light emerges from it?(b) A second identical polarizer is placed parallel to and behind the first. How much
Natural light of irradiance Ii is incident normally on three identical sheet linear polarizers aligned with parallel transmission axes. If each has a principal transmittance of 64% and a high
As we saw in Section 8.10, substances such as sugar and insulin are optically active; they rotate the plane of polarization in proportion to both the path length and the concentration of the
The light from an ordinary flashlight is passed through a linear polarizer with its transmission axis vertical. The resulting beam, having an irradiance of 200 W/m2, is incident normally on a
Linearly polarized light (with an irradiance of 200 W/m2) aligned with its electric-field vector at +55° from the vertical impinges perpendicularly on an ideal sheet polarizer whose transmission
Two ideal linear sheet polarizers are arranged with respect to the vertical with their transmission axis at 10° and 60°, respectively. If a linearly polarized beam of light with its electric field
Imagine a pair of crossed polarizers with transmission axes vertical and horizontal. The beam emerging from the first polarizer has flux density I1, and of course no light passes through the analyzer
Imagine that you have two identical perfect linear polarizers and a source of natural light. Place them one behind the other and position their transmission axes at 0° and 50°, respectively. Now
Given that 200 W/m2 of randomly polarized light is incident normally on a tack of ideal linear polarizers that are positioned one behind the other with the transmission axis of the first vertical,
Two ideal HN-50 linear polarizers are positioned one behind the other. What angle should their transmission axes make if an incident unpolarized 100 W/m2 beam is to be reduced to 30.0 W/m2 on
An ideal polarizer is rotated at a rate Ï‰ between a similar pair of stationary crossed polarizers. Show that the emergent flux density will be modulated at four times the rotational
Figure P.8.31 shows a ray traversing a calcite crystal at nearly normal incidence, bouncing off a mirror, and then going through the crystal again. Will the observer see a double image of the spot on
A pencil mark on a sheet of paper is covered by a calcite crystal. With illumination from above, isn’t the light impinging on the paper already polarized, having passed through the crystal? Why
The calcite crystal in Fig. P.8.34 is shown in three different orientations. Its blunt corner is on the left in (a), the lower left in (b), and the bottom in (c). The Polaroid€™s
In discussing calcite, we pointed out that its large birefringence arises from the fact that the carbonate groups lie in parallel planes (normal to the optic axis). Show in a sketch and explain why
A beam of light enters a calcite prism from the left, as shown in Fig. P.8.36. There are three possible orientations of the optic axis of particular interest, and these correspond to the x-, y-, and
Compute the critical angle for the ordinary ray, that is, the angle for total internal reflection at the calcite–balsam layer of a Nicol prism.
A Wollaston prism is made of two 45° quartz prisms much like Fig. 8.34. Given that Î»0= 589.3 nm, determine the angle separating the two emerging rays. As compared to a calcite
The prism shown in Fig. P.8.40 is known as a Rochon polarizer. Sketch all the pertinent rays, assuming(a) Why might such a device be more useful than a dichroic polarizer when functioning with
Imagine that we have a transmitter of microwaves that radiates a linearly polarized wave whose vector E-field is known to be parallel to the dipole direction. We wish to reflect as much energy as
At what angle will the reflection of the sky coming off the surface of a pond (n = 1.33) completely vanish when seen through a Polaroid filter?
What is Brewster’s angle for reflection of light from the surface of a piece of glass (ng = 1.65) immersed in water (nw = 1.33)?
Given that the critical angle for some transparent material in air is 41.0°, determine its polarization angle.
A beam of light is reflected off the surface of some unknown liquid, and the light is examined with a linear sheet polarizer. It is found that when the central axis of the polarizer (that is, the
Light reflected from a glass (ng = 1.65) plate immersed in ethyl alcohol (ne = 1.36) is found to be completely linearly polarized. At what angle will the partially polarized beam be transmitted into
A beam of natural light is incident on an air–glass interface (nti = 1.5) at 40°. Compute the degree of polarization of the reflected light.
Prove that the degree of polarization (Vr) of reflected light can be expressed as[Hint: For unpolarized reflected light Ir|| = IrŠ¥, whereas for polarized reflected light Ip = IrŠ¥
A beam of natural light incident in air on a glass (n = 1.5) interface at 70° is partially reflected. Compute the overall reflectance. How would this compare with the case of incidence at, say,
A narrow beam of natural light is incident at 56.0° on a glass plate (n = 1.50) in air. The reflected light is partially polarized. Determine the degree of polarization.
A narrow beam of light strikes the surface of a block of clear material and it is determined that the reflected light is totally polarized. If the total reflectance is 10% find the transmittance at
A ray of yellow light is incident on a calcite plate at 50°. The plate is cut so that the optic axis is parallel to the front face and perpendicular to the plane-of-incidence. Find the angular
A beam of light is incident normally on a quartz plate whose optic axis is perpendicular to the beam. If λ0 = 589.3 nm, compute the wavelengths of both the ordinary and extraordinary waves. What are
The electric-field vector of an incident P-state makes an angle of +30° with the horizontal fast axis of a quarter-wave plate. Describe, in detail, the state of polarization of the emergent wave.
Take two ideal Polaroids (the first with its axis vertical and the second, horizontal) and insert between them a stack of 10 half-wave plates, the first with its fast axis rotated π/40 rad from the
Suppose you were given a linear polarizer and a quarter-wave plate. How could you determine which was which, assuming you also had a source of natural light?
Linear light at 135° to the horizontal, oscillating in the second and fourth quadrants, passes through a π/2 retarder having its fast axis vertical. Describe the polarization state of the emerging
Right-circular light passes through a Î»/4 retarder whose fast axis is vertical. Describe the emerging polarization state. Did the polarization state shift one quarter of the way around
Right-circular light passes through a quarter-wave plate with a horizontal fast axis. Explain why you can expect the light to emerge linearly polarized at 45° in the first and third quadrants.
Linear light oscillating at 135° in the second and fourth quadrants passes through a half-wave plate whose fast axis is vertical. Explain why you can expect the emerging light to be linear in the
Right-circular light passes through a half-wave plate whose fast axis is vertical. Describe the emerging polarization state.
Linear light oscillating at 60° above the horizontal x-axis in the first and third quadrants passes through a quarter-wave plate with its fast axis horizontal. Explain why the light emerges as left
Left-circular light of wavelength 590 nm traveling in the z-direction is to be converted into right-circular light by passing perpendicularly through a plate of quartz. The quartz has been cut and
An L-state traverses an eighth-wave plate having a horizontal fast axis. What is its polarization state on emerging?
Figure P.8.67 shows two Polaroid linear polarizers and between them a microscope slide to which is attached a piece of cellophane tape. Explain what you see.Figure P.8.67
Imagine that we have randomly polarized room light incident almost normally on the glass surface of a radar screen. A portion of it would be specularly reflected back toward the viewer and would thus

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