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optics

Optics 5th edition Eugene Hecht - Solutions

Write an expression for the transform A(Ï‰) of the harmonic pulse of Fig. P.7.54. Check that sinc u is 50% or greater for values of u roughly less than Ï€/2. With that in mind, show that Î”v Î”t ‰ˆ 1, where Î”v is the bandwidth of the
Examine Fig. P. 7.53, which shows three periodic functions and their corresponding Fourier frequency spectra. Discuss the graphs explaining what€™s happening in successive parts. What happens to the envelopes of the frequency spectra as the wavelength increases? Why are the same number of
Figure P.7.42 depicts an electric field in time and the Fourier components that compose it. The units are arbitrary. Given that(a) Explain why the series contains both sine and cosine terms.(b) Why does the series contain harmonic terms having arguments with odd and even multiples of
Change the upper limit of Eq. (7.59) from ˆž to Î± and evaluate the integral. Leave the answer in terms of the so-called sine integral:which is a function whose values are commonly tabulated. Si(z) sinc w dw 0. 00 f(x): E̟L sinc (kL/2) coskx dx (7.59) т
Show that the Fourier series representation of the function Æ’(Î¸) = |sin Î¸| is cos 2m0 f(0) : 4m2 – 1 TT TT т3D 1
Take the function Æ’(Î¸) = Î¸2in the interval 0 < Î¸ < 2Ï€ and assume it repeats itself with a period of 2Ï€ . Now show that the Fourier expansion of that function is Σ(- 4п? 4 cos mo 4т sin me f(x) 3 т т т3D1
Consider the periodic function defined over one wavelength byƒ(x) = (kx)2 where -π < kx < πwhich repeats over and over again with a period of 2π. Draw a diagram of ƒ(x) and determine the corresponding Fourier series representation.
Given the function ƒ(x) = A cos (πx/L), determine its Fourier series.
Determine the Fourier series for the periodic function depicted in Fig. P.7.45. Fig. P.7.45. f(x) -4 -2 4
Compute the Fourier series components for the periodic function shown in Fig. 7.35.Fig 7.35 (a) f(x) х -A/a 0 1/a л f(x) One cosine term -Two cosine terms Three cosine terms 1 DC term 2 -/a A/a (b)
Show thatwhere Î± ‰ 0, b ‰ 0, and Î± and b are positive integers. [7.44] sin akx cos bkx dx = 0 [7.45] 8 cos akx cos bkx dx = dab [7.46] dab sin akx sin bkx dx 2
Analytically determine the resultant when the two functions E1 = 2E0 cos ωt and E2 = 1/2 E0 sin 2ωt are superimposed. Draw E1, E2, and E = E1 + E2. Is the resultant periodic; if so, what is its period in terms of ω?
Using the dispersion equation,show that the group velocity is given byfor high-frequency electromagnetic waves (e.g., X-rays). Keep in mind that since Æ’j are the weighting factors, Î£jÆ’j = 1. What is the phase velocity? Show that vvg ‰ˆ c2. Nq.
An ionized gas or plasma is a dispersive medium for EM waves. Given that the dispersion equation iswhere Ï‰p is the constant plasma frequency, determine expressions for both the phase and group velocities and show that vvg = c2. o² = w, + c²k? @p
For a wave propagating in a periodic structure for which ω(k) = 2ω0 sin (kℓ/2), determine both the phase and group velocities. Write the former as a sinc function.
For light at a wavelength of λ1 = 656.3 nm water (at 20° C) has an index of n1 = 1.3311. At a wavelength of λ2 = 89.3 nm water has an index of n2 = 1.333 0. Determine the approximate value of the group velocity of light in water. Is v̅ > vg? [Hint: Reread Problem 7.36, approximate the
Show that the group velocity can be written as Лс dn Og п п? dл
Determine the group velocity of waves when the phase velocity varies inversely with wavelength.
A well-known Optics book gives the equationCould this possibly be correct? Explain. [Check the units.] c dn = vl 1 1 dn do Vg n dk n? dk dk п
With the previous problem in mind show thatData from Prob. 32With the previous problem in mind prove that dn п — Л- ал Пg dn(v) n(v) + v dv ng
With the previous problem in mind prove that dn(v) n(v) + v dv ng
Show that the group velocity can be written as n + w(dn/dw)
Show that the group velocity can be written as dv = v – - di Og
The speed of propagation of a surface wave in a liquid of depth much greater than Î» is given bywhere g = acceleration of gravity, Î» = wavelength, Ï = density, Y = surface tension. Compute the group velocity of pulse in the long wavelength limit (these are called
In the case of lightwaves, show that v dn п Vg c dv
Using the relation 1/vg= dÎº/dv, prove that v dv v? dv Vg
At a wavelength of 1100 nm pure silica glass has an index of refraction of 1.449. Use Fig. 7.22 to (a) Determine its group index at that wavelength. (b) Find its group velocity. (c) Compare that to its phase velocity.Fig. 7.22 1.49 1.48 1.47 Пg 1.46 1.45 п 1.44 500 700 900 1100 1300 1500 $1.49 1.48 1.47 Пg 1.46 1.45 п 1.44 500 700 900 1100 1300 1500 1700 1900 Light Infrared Wavelength (nm) Index of refrac$
With the previous problem in mind show thatAnd then sinceprove thatCheck this expression by confirming that the units are correct. dn v|1 An² d(1/A) || d dv d (1/λ) d(1/λ) d
Show thatFirst prove that vg = dv/d(1/Î»). c d(1/n) л d(1/A) Og п
Beginning with vg= dÏ‰/dk prove that dv Vg = -X? ал
Given the dispersion relation ω = αk2, compute both the phase and group velocities.
Figure P.7.21 shows a carrier of frequency vc being amplitude modulated by a sine wave of frequency Ï‰m, that is,E = E0(1 + Î± cos Ï‰mt) cos Ï‰ctShow that this is equivalent to the superposition of three waves of frequencies Ï‰c, Ï‰c +
As we€™ve seen, Eq. (7.33) describes the beat pattern. Let€™s now derive a different version of that expression assuming that the two overlapping equal-amplitude cosine waves have angular spatial frequencies of kc+ Î”k and kc- Î”k, and angular temporal
Use the phasor method, described in conjunction with Fig. 7.17, to explain how two equal-amplitude waves of sightly different frequencies generate the beat pattern shown in Fig. 7.19 or Fig. P.7.19a. The curve in Fig. P.7.19b is a sketch of the phase of the resultant measured with respect to one of
Imagine that we strike two tuning forks, one with a frequency of 340 Hz, the other 342 Hz. What will we hear?
Show that a standing wave created by two unequal-amplitude wavesandhas the formHere Ï is the ratio of the amplitude reflected to the amplitude incident. Discuss the meaning of the two terms. What happens when Ï = 1? E1 = E, sin (kx + wt) ER = pE, sin (kx ± wt)
A standing wave is given byDetermine two waves that can be superimposed to generate it. E = 100 sin 7x TX Cos 5Tt
Microwaves of frequency 1010 Hz are beamed directly at a metal reflector. Neglecting the refractive index of air, determine the spacing between successive nodes in the resulting standing-wave pattern.
Considering Wiener€™s experiment (Fig. 7.14) in monochromatic light of wavelength 550 nm, if the film plane is angled at 1.0° to the reflecting surface, determine the number of bright bands per centimeter that will appear on it.Fig. 7.14 Litit (b) Antinodal planes Film 1/21 plate 1/2]
The electric field of a standing electromagnetic plane wave is given byDerive an expression for B(x, t). (You might want to take another look at Section 3.2.) Make a sketch of the standing wave. [7.30] E(x, t) = 2E0 sin kx cos wt
Using phasors, determine the amplitude and phase of the waveform given byIn other words, knowing that Î¨(t) = A cos (Ï‰t + Î±) find A and Î± using a ruler and protractor. = 16 cos wt + 8 cos (wt + T/2) (t) + 4 cos (wt + T) + 2 cos (wt + 3/2)
Using phasors, determine the amplitude and phase of the waveform given byDraw an appropriate diagram. In other words, knowing that Ïˆ(t) = A cos (Ï‰t + Î±) find A and Î± with a ruler and protractor. (t) = 6 cos wt + 4 cos (wt + T/2) + 3 cos (wt + 7)
Consider the functions E1= 3 cos Ï‰t and E2= 4 sin Ï‰t. First prove that E2= 4 cos (Ï‰t - Ï€/2). Then, using phasors and referring to Fig. P.7.10, show that E3= E1+ E2= 5 cos (Ï‰t - Ï†); determine Ï†. Discuss the values of
Use the complex representation to find the resultant E = E1 + E2, whereE1 = E0 cos (k x + ωt) and E2 = -E0 cos (k x - ωt)Describe the composite wave.
Add the two waves of Problem 7.7 directly to find Eq. (7.17).Data from Problem 7.7Using Eqs. (7.9), (7.10), and (7.11), show that the resultant of the two wavesandis k Δr Δx E = 2E01 cos (7.17) sin wt – k x + E1 = E01 sin [wt – k(x + Ax)] %3D
Using Eqs. (7.9), (7.10), and (7.11), show that the resultant of the two wavesandis = E01 sin [wt – k(x + Ax)] E1 %3D E2 = E01 sin (wt – kx)
Determine the optical path difference for the two waves A and B, both having vacuum wavelengths of 500 nm, depicted in Fig. P.7.6; the glass (n = 1.52) tank is filled with water (n = 1.33). If the waves start out in-phase and all the above numbers are exact, find their relative phase difference at
Answer the following:(a) How many wavelengths of λ0 = 500 nm light will span a 1-m gap in vacuum?(b) How many waves span the gap when a glass plate 5 cm thick (n = 1.5) is inserted in the path?(c) Determine ∧ the OPD between the two situations.(d) Verify that ∧/λ0 corresponds to the
Show that the optical path length, defined as the sum of the products of the various indices times the thicknesses of media traversed by a beam, that is, Σinixi, is equivalent to the length of the path in vacuum that would take the same time for that beam to negotiate.
Show that when the two waves of Eq. (7.5) are in-phase, the resulting amplitude squared is a maximum equal to (E01+ E02)2, and when they are out-of-phase it is a minimum equal to (E01- E02)2. (7.5a) E1 = E01 sin (wt + aj) (7.5b) E2 = E02 sin (ot + a2)
Considering Section 7.1, suppose we began the analysis to find E = E1+ E2with two cosine functions E1= E01cos (Ï‰t + Î±1) and E2= E02 cos (Ï‰t + Î±2). To make things a little less complicated, let E01= E02 and Î±1 = 0. Add
Determine the resultant of the superposition of the parallel waves E1 = E01 sin (ωt + ε1) and E2 = E02 sin (ωt + ε2) when ω = 120π, E01 = 6, E02 = 8, ε1 = 0, and ε2 = π/2. Plot each function and the resultant.
Given a fused silica fiber with an attenuation of 0.2 dB/km, how far can a signal travel along it before the power level drops by half?
Given a fiber with a core diameter of 50 μm and nc = 1.482 and nƒ = 1.500, determine the number of modes it sustains when the fiber is illuminated by an LED emitting at a central wavelength of 0.85 μm.
Figure P.6.34 shows the distribution of light corresponding to the image arising when a monochromatic point source illuminates two different optical systems each having only one type of aberration. Identify the aberration in each case and justify your answer.Figure P.6.34 (a) (b)
Referring back to Fig. 6.18a, show that when PÌ…'CÌ… = Rn2/n1and PÌ…CÌ… = Rn1/n2all rays originating at P appear to come from P'. Fig. 6.18a Пу П2 R- R
With the previous two problems in mind, compute the magnification that results when the image of a flower 4.0 m from the center of a solid, clear-plastic sphere with a 0.20-m diameter (and a refractive index of 1.4) is cast on a nearby wall. Describe the image in detail.
Figure P.5.104, which purports to show an erecting lens system, is taken from an old, out-of-print optics text. What€™s wrong with it? P.5.104
Figure P.5.102 shows an arrangement in which the beam is deviated through a constant angle s, equal to twice the angle b between the plane mirrors, regardless of the angle-of-incidence. Prove that this is indeed the case.Figure P.5.102
Figure P.6.33 shows the image irradiance distributions arising when a monochromatic point source illuminates three different optical systems, each having only one type of aberration. From the graphs identify that aberration in each case and justify your answer. Figure P.6.33a (ЕН) Figure P.6.33b
Starting with the exact expression given by Eq. (5.5), show that Eq. (6.46) results, rather than Eq. (5.8), when the approximations for ℓ0 and ℓi are improved a bit. 1 ( n2Si R l; N¡So п) (5.5) Пу li lo
Figure P.6.29 shows two identical concave spherical mirrors forming a so-called confocal cavity. Show, without first specifying the value of d, that after traversing the cavity two times the system matrix isThen for the specific case of d = r show that after four reflections the system is back
Considering the lens in Problem 6.29, determine its focal length and the location of the focal points with respect to its vertices V1and V2.Data from Prob. 6.29Figure P.6.29 shows two identical concave spherical mirrors forming a so-called confocal cavity. Show, without first specifying the value
A concave-planar glass (n = 1.50) lens in air has a radius of 10.0 cm and a thickness of 1.00 cm. Determine the system matrix and check that its determinant is 1. At what positive angle (in radians measured above the axis) should a ray strike the lens at a height of 2.0 cm, if it is to emerge from
Starting with Eq. (6.35) and Eq. (6.37), show that the 2 × 2 matrix resulting from the product of the three 2 × 2 matrices in Eq. (6.33) has the formSince this matrix is the product of matrices each of which has a unit determinant, it has a unit determinant. Accordingly, show that a12do) a12 |
The system matrix for a thick biconvex lens in air is given byKnowing that the first radius is 0.5 cm, that the thickness is 0.3 cm, and that the index of the lens is 1.5, find the other radius. 0.6 -2.6 0.8 0.2
Compute the system matrix for a thick biconvex lens of index 1.5 having radii of 0.5 and 0.25 and a thickness of 0.3 in any units you like). Check that |A| = 1.
Show that the planar surface of a concave-planar or convex-planar lens doesn’t contribute to the system matrix.
Establish that Eqs. (6.41) and (6.42) are equivalent to Eqs. (6.3) and (6.4), respectively. Пii (1 — ај) (6.41a) VH| -d12 aj1) VH1 (6.41b) -dj2
Prove that the determinant of the system matrix in Eq. (6.31) is equal to 1. Dzdy -D1 – D2 + п п a12 Did1 пу di 1 A22 a21 пj (6.31)
A positive meniscus lens with an index of refraction of 2.4 is immersed in a medium of index 1.9. The lens has an axial thickness of 9.6 mm and radii of curvature of 50.0 mm and 100 mm. Compute the system matrix when light is incident on the convex face and show that its determinant is equal to 1.
Show that Eq. (6.36), relating the object and image distances measured from the vertices of a lens, reduces to Gauss€™s Formula [Eq. (5.17)] for thin lenses. Remember that when s0> 0, d1O< 0 and when si> 0, dI2 > 0. —а21 + аг2do di a11 – aj2do (6.36) (5.17) Si So
Starting with Eq. (6.33) derive Eq. (6.34) when both the object and image are in air. Гs-ein e |прад 1 1 По@о a12 1||a21 a22]-do/no Уг Уo lu/'p (6.33) Ул — аolaz1 — az2do + (аj1 — а12dojdj] (6.34) + yolaz2 + a12d)
A thick biconvex lens in air has an index of 1.810 and a thickness of 3.00 cm. Its first radius of curvature is 11.0 cm and its second is 120 cm. Determine its system matrix A.
A convex-planar lens of index 3/2 has a thickness of 1.2 cm and a radius of curvature of 2.5 cm. Determine the system matrix when light is incident on the curved surface.
A compound lens is composed of two thin lenses separated by 10 cm. The first of these has a focal length of +20 cm, and the second a focal length of -20 cm. Determine the focal length of the combination and locate the corresponding principal points. Draw a diagram of the system.
Imagine two identical double-convex thick lenses separated by a distance of 20 cm between their adjacent vertices. Given that all the radii of curvature are 50, the refractive indices are 1.5, and the thickness of each lens is 5.0 cm, calculate the combined focal length.
A crown glass double-convex lens, 4.0 cm thick and operating at a wavelength of 900 nm, has an index of refraction of 3/2. Given that its radii are 4.0 cm and 15 cm, locate its principal points and compute its focal length. If a television screen is placed 1.0 m from the front of the lens, where
It is found that sunlight is focused to a spot 29.6 cm from the back face of a thick lens, which has its principal points H1 at +2.0 cm and H2 at -4.0 cm. Determine the location of the image of a candle that is placed 49.8 cm in front of the lens.
A thick glass lens of index 1.50 has radii of +23 cm and +20 cm, so that both vertices are to the left of the corresponding centers of curvature. Given that the thickness is 9.0 cm, find the focal length of the lens. Show that in general R1 - R2 = d/3 for such afocal zero-power lenses. Draw a
A spherical glass bottle 20 cm in diameter with walls that are negligibly thin is filled with water. The bottle is sitting on the back seat of a car on a nice sunny day. What’s its focal length?
Using Eq. (6.2), derive an expression for the focal length of a homogeneous transparent sphere of radius R. Locate its principal points. (n¡ – 1)d (n – 1) R1 (6.2) R2 n¡R¡R2
Prove that if the principal points of a biconvex lens of thickness dl overlap midway between the vertices, the lens is a sphere. Assume the lens is in air.
Suppose we have a positive meniscus lens of radii 6 and 10 and a thickness of 3 (any units, as long as you€™re consistent), with an index of 1.5. Determine its focal length and the locations of its principal points (compare with Fig. 6.3).Fig. 6.3
The radii of curvature of a thick lens are +10.0 cm and +9.0 cm. The thickness of the lens along its optical axis is 1.0 cm, it has an index of 1.50, and it is immersed in air. Find the focal length of the lens and explain the significance of its sign.
Write an expression for the thickness dl of a double-convex lens such that its focal length is infinite.
According to the military handbook MIL-HDBK-141 (23.3.5.3), the Ramsden eyepiece (Fig. 5.105) is made up of two planar-convex lenses of equal focal length Æ’' separated by a distance 2Æ’'/3. Determine the overall focal length f of the thin-lens combination and locate the
Work out the details leading to Eq. (6.8). 1 (6.8) f fi f2 fif2
Construct a Cartesian oval such that the conjugate points will be separated by 11 cm when the object is 5 cm from the vertex. If n1 = 1 and n2 = 3/2, draw several points on the required surface
Use Fig. P.5.3 to show that if a point source is placed at the focus F1of the ellipsoid, plane waves will emerge from the far side. Remember that the defining requirement for an ellipse is that the net distance from one focus to the curve and back to the other focus is constant. Figure P.5.3 A D F2
Diagrammatically construct an ellipto-spheric negative lens, showing rays and wavefronts as they pass through the lens. Do the same for an oval-spheric positive lens.
Making use of Fig. P.5.5, Snell€™s Law, and the fact that in the paraxial region Î± = h/so, Ï† ‰ˆ h/R, and Î² ‰ˆ h/si, derive Eq. (5.8). Figure P.5.5 ө, R- 02 П2 П1
Show that, in the paraxial domain, the magnification produced by a single spherical interface between two continuous media, as shown in Fig. P.5.6, is given byUse the small-angle approximation for Snell€™s Law and approximate the angles by their tangents. Пis; Mт Пzso Figure P.5.6 п1
Imagine a hemispherical interface, with a radius of curvature of radius 5.00 cm, separating two media: air on the left, water on the right. A 3.00-cm-tall toad is on the central axis, in air, facing the convex interface and 30.0 cm from its vertex. Where in the water will it be imaged? How big will
Locate the image of an object placed 1.2 m from the vertex of a gypsy’s crystal ball, which has a 20-cm diameter (n = 1.5). Make a sketch of the rays.
Return to Problem 5.7 and suppose we cut off the medium on the right, forming a thick water biconvex lens, with each surface having a radius of curvature of 5.00 cm. If the lens is 10.0 cm thick, determine the total magnification and everything you can about the toad’s image.Data from Prob
A biconvex glass (n1 = 1.5) thin lens is to have a +10.0-cm focal length. If the radius of curvature of each surface is measured to be the same, what must it be? Show that a spider standing 1.0 cm from the lens will be imaged at -1.1 cm. Describe that image and draw a ray diagram.
Going back to Section 5.2.3, prove that for a thin lens immersed in a medium of index nmThat done, imagine a double-concave air lens surrounded by water; determine if it€™s converging or diverging. (п — Пт) ( 1 R2. R1 Пт
A meniscus concave glass (nl= 1.5) thin lens (see Fig. 5.12) has radii of curvature of +20.0 cm and +10.0 cm. If an object is placed 20.0 cm in front of the lens, show that the image distance will be -13.3 cm. Describe that image and draw a ray diagram. CONCAVE CONVEX R 0 R >0 R, 0 R2>0 R >0 R2 >0
A biconcave lens (nl = 1.5) has radii of 20 cm and 10 cm and an axial thickness of 5 cm. Describe the image of an object 1-inch tall placed 8 cm from the first vertex. Use the thin-lens equation to see how far off it is in determining the final-image location.

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