Questions and Answers of Optics

Imagine that we have an opaque mask into which are punched an ordered array of circular holes, all of the same size, located as if at the corners of the boxes of a checkerboard. Now suppose our robot
A fine square wire mesh with 50 wires per cm is placed vertically in the object plane of the optical computer of Fig. 13.50. If the lenses each have 1.00-m focal lengths, what must be the
Replace the cosine grating in the previous problem with a €œsquare€ bar grating, that is, a series of many fine alternating opaque and transparent bands of equal width. We now
Suppose we insert a mask in the transform plane of the previous problem, which obscures everything but the m = +1 diffraction contribution. What will the reformed image look like on Σi? Explain your
Imagine that you have a cosine grating (i.e., a transparency whose amplitude transmission profile is cosinusoidal varying between 0 and 1) with a spatial period of 0.01 mm. The grating is illuminated
A diffraction grating having a mere 50 grooves per cm is the object in the optical computer shown in Fig. 13.41. If it is coherently illuminated by plane waves of green light (543.5 nm) from a He-Ne
With Fig. 13.36 in mind, show that the transverse magnification of the system is given by -Æ’i/Æ’tand draw the appropriate ray diagram. Draw a ray up through the center of the
Returning to Fig. 13.37, what kind of spatial filter would produce each of the patterns shown in Fig. P.13.43?Fig. 13.37P.13.43? (b) (a) (a) (b)
Repeat the previous problem using Fig. P.13.42 this time.Fig. P.13.42 (a) LOOK AT YOUR FUTURE ngsentatiwa iintervie at b) LOOK AT YOUR FUTURE
Repeat the previous problem using Fig. P.13.41 instead.previous problem Make a rough sketch of the Fraunhofer diffraction pattern that would arise if a transparency of Fig. P.13.40a served as the
Make a rough sketch of the Fraunhofer diffraction pattern that would arise if a transparency of Fig. P.13.40a served as the object. How would you filter it to get Fig. P.13.40b? Fig. P.13.40a
What would the pattern look like for a laserbeam diffracted by the three crossed gratings of Fig. P.13.39?Fig. P.13.39?
A He-Ne laser operating at 632.8 nm has an internal beam waist diameter of 0.60 mm. Calculate the full-angular width, or divergence, of the beam.
Show that the maximum electric-field intensity, Emax, that exists for a given irradiance I iswhere n is the refractive index of the medium. 1/2 in units of V/m Emax 27.4 max
Determine the threshold gain coefficient for a semiconductor laser where α ≈ 10 cm-1, the resonator is 0.03 cm long, and the “mirror” reflectances are both only 0.4.
A He-Ne c-w laser has a Doppler-broadened transition bandwidth of about 1.4 GHz at 632.8 nm. Assuming n = 1.0, determine the maximum cavity length for single-axial mode operation. Make a sketch of
A gas laser has a Fabry–Perot cavity of length 40 cm. The index 13.41 of refraction of the gas is 1.0. Operating at 600 nm, determine the mode number, that is, the number of half-cycles fitting
The 488.0-nm line from an argon ion laser is Doppler broadened to 2.7 × 109 Hz. Given that the laser’s mirrors are 1.0 m apart, determine the approximate number of longitudinal modes. Assume the
Determine the frequency difference between adjacent axial resonant cavity modes for a typical gas laser 25 cm long (n ≈ 1).
Given that a ruby laser operating at 694.3 nm has a frequency bandwidth of 50 MHz, what is the corresponding linewidth?
A solid-state laser has an active region consisting of a rod 10 mm in diameter and 0.20 m long that is operating with an efficiency of 2.0%. The rod contains 4.0 × 1019 participating ions per cubic
What is the transition rate for the neon atoms in a He-Ne laser if the energy drop for the 632.8 nm emission is 1.96 eV and the power output is 1.0 mW?
Make a rough estimate of the amount of energy that can be delivered by a ruby laser whose crystal is 5.0 mm in diameter and 0.050 m long. Assume the pulse of light lasts 5.0 Ã— 10-6s. The
The beam (λ = 632.8 nm) from a He-Ne laser, which is initially 3.0 mm in diameter, shines on a perpendicular wall 100 m away. Given that the system is aperture (diffraction) limited, how large is
The helium-neon laser is famous for its red-light emission at 632.8 nm. But electrons in that same high-energy level can jump down to nine other lower levels (each with appreciable probabilities),
Referring to Fig. 13.6, which shows two transitions for the He-Cd laser, determine the lifetime of the higher-energy d-state.Fig. 13.6 Cadmium+ A = 1.6 × 10 s-1 2d3/2 353.6 nm 2p3/2 325.0 nm A = 7.8
With the Example 13.7 in mind, determine the average power per cubic meter radiated by the Nd:YAG laser rod, given that the transition occurs with an upper-level lifetime of 230 μs.
Radiation at 21 cm pours down on the Earth from outer space. Its origin is great clouds of hydrogen gas. Taking the background temperature of space to be 3.0 K, determine the ratio of the transition
For a system of atoms (in equilibrium) having two energy levels, show that at high temperatures where kBT >> Ej - Ei, the number densities of the two states tend to become equal.
Determine the rate at which stimulated emission is happening in a 100-mW He-Cd laser emitting at 441.56 nm.
Given a two-level atomic system where level-2 is more energetic than the ground state level-1, what is the meaning of the expressionWhen in thermal equilibrium show thatA21N2 + B21uvN2 = B12uvN1 dN2
Redo the previous problem for a temperature of 30.0 × 103 Κ and compare the results of both calculations.
A system of atoms in thermal equilibrium is emitting and absorbing 2.0-eV light photons. Determine the ratio of the transition rates of stimulated emission to spontaneous emission at a temperature of
Show that for a system of atoms and photons in equilibrium at a temperature T the ratio of the transition rates of stimulated to spontaneous emission is given by hv - 1 eквт
A 50.0-cm3 chamber is filled with argon gas to a pressure of 20.3 Pa at a temperature of 0°C. All but a negligible number of these atoms are initially in their ground states. A flash tube
The solar constant is the radiant flux density at a spherical surface centered on the Sun having a radius equal to that of the Earth’s mean orbital radius; it has a value of 0.133–0.14 W/cm2. If
Suppose we have a 100-W yellow lightbulb (550 nm) 100 m away from a 3-cm-diameter shuttered aperture. Assuming the bulb to have a 2.5% conversion to radiant power, how many photons will pass through
Figure P.13.13 shows the spectral irradiance impinging on a horizontal surface, for a clear day, at sea level, with the Sun at the zenith. What is the most energetic photon we can expect to encounter
In the atomic domain, energy is often measured in electronvolts. Arrive at the following expression for the energy of a light-quantum in eV when the wavelength is in nanometers:What is the energy of
Start with Eq. (13.4) and show that it€™s equivalent towhere Î» is in meters, T is in kelvins, and the wavelength interval Î”Î» should be in nanometers.
The energy per unit area per unit time per unit wavelength interval emitted by a blackbody at a temperature T is given byAt a specific temperature, the total power radiated per unit area of the
An object resembling a blackbody emits a maximum amount of energy per unit wavelength in the red end of the visible spectrum (λ = 680 nm). What is its surface temperature?
When the Sun’s spectrum is photographed, using rockets to range above the Earth’s atmosphere, it is found to have a peak in its spectral exitance at roughly 465 nm. Compute the Sun’s surface
The surface temperature of a class O blue-white star is around 40 × 103 K. At what frequency will it radiate most of its energy?
What is the wavelength that carries away the most energy when an object resembling a blackbody radiates energy into a room-temperature (20°C) environment?
Your average skin temperature is about 33°C. Assuming you radiate as does a blackbody at that temperature, at what wavelength do you emit the most energy?
The temperature of an object resembling a blackbody is raised from 200 K to 2000 K. By how much does the amount of energy it radiates increase?
Suppose that we measure the emitted exitance from a small hole in a furnace to be 22.8 W/cm2, using an optical pyrometer of some sort. Compute the internal temperature of the furnace.
A somewhat typical person has a total naked area of about 1.4 m2 and an average skin temperature of 33°C. Determine the net power radiated per unit area, the irradiance or more precisely the
After a while, a cube of rough steel (10 cm on a side) reaches equilibrium inside a furnace at a temperature of 400°C. Knowing that its total emissivity is 0.97, determine the rate at which the cube
While studying the star Arcturus with a Michelson stellar interferometer the fringes vanished when the two mirrows were 24 ft apart. Assuming light of a mean wavelength of 500 nm, what angle did the
Consider the Michelson stellar interferometer. Under what conditions will the fringes vanish when the light comes from two equally bright stars? Compare this to the situation in which there is only
Earlier as an example we used dc‰ˆ Î»Ì…0/Î¸sto calculate the approximate lateral coherence distance for sunlight. Now find that same quantity, the
Imagine that we have a wide quasimonochromatic source (λ = 500 nm) consisting of a series of vertical, incoherent, infinitesimally narrow line sources, each separated by 500 μm. This is used to
Show that Eqs. (12.34) and (12.35) follow from Eqs. (12.32) and (12.33). (12.34) (A1(t + t)A½(t))t = |ř12(7)|? (Ah(t + t)Ab(t))T = (1)t(12)T|Ÿ12(7)|? (12.35)
Look at the Young€™s Experiment depicted in Fig. 9.10. Thermal quasimonochromatic light, filtered to a mean of 500 nm, impinges from the left on the 0.10-mm-diameter hole in the source
Suppose we set up Young’s double-pinhole experiment with a small circular hole of diameter 0.1 mm in front of a sodium lamp (λ̅0 = 589.3 nm) as the source. If the distance from the source to the
Under what circumstances will the irradiance on Î£oin Fig. P.12.15 be equal to 4I0, where I0is the irradiance due to either uncorrelated point source alone?Fig. P.12.15 O" S2. S' O' S"
Carry out the details leading to the expression for the visibility given by Eq. (12.22). 2Vī, Vī, |ĩ12(7)| (12.22) I + I½
Referring to the slit source and pinhole screen arrangement of Fig. P.12.13, show by integration over the source thatFig. P.12.13 sin (па/Al)b cos (2παΥ/As) I(Ү) с b + па/Al S2 S' S1 Σο
Imagine that we have the arrangement depicted in Fig. 12.8. If the separation between fringes (max. to max.) is 1 mm and if the projected width of the source slit on the screen is 0.5 mm, compute the
With the previous problem in mind, return to the autocorrelation of a sine function, shown in Fig. 11.51. Now suppose we have a signal composed of a great many sinusoidal components. Imagine that you
With the previous problem in mind, now consider things spread across space at a given moment in time. Each wave separately would result in an irradiance distribution I1 and I2. Plot both on the
We wish to examine the irradiance produced on the plane of observation in Young’s Experiment when the slits are illuminated simultaneously by two monochromatic plane waves of somewhat different
Suppose we set up a fringe pattern using a Michelson Interferometer with a mercury vapor lamp as the source. Switch on the lamp in your mind’s eye and discuss what will happen to the fringes as the
Even though the coherence area increases as Î£Î±moves away from Î£s, there is a quantity that doesn€™t change; that€™s the solid angle
The Sun’s disk subtends an angle of about 9.3 × 10-3 rad as seen from the Earth’s surface. If sunlight is filtered to a mean wavelength of 550 nm, roughly what is the area of coherence on an
Let Î©sbe the solid angle subtended by the source when viewed from the center of the aperture screen. Show thatrepresents the coherence area. This equation is useful when we
A small thermal source of quasimonochromatic light with a mean wavelength of 500 nm, and an area of 1.0 × 10-6 m2, is used to illuminate an opaque screen containing two pinholes, each 0.10 mm in
Show that Eq. (12.2)is reasonable. Then approximating As as d2s, show thatNotice that Ac gets larger as l gets larger. Ac = (Ao/0,)? (12.2) Ac
With Fig. 12.3 in mind, establish that when two incoherent cosine-squared fringe systems, each of the form I0 cos2 Î±, overlap so that peaks fall on troughs, the resultant is I
Two monochromatic point sources radiate in-phase. At the usual distant plane of observation (parallel to the line connecting the sources) the irradiance from one of them is 100 times the irradiance
Consider the function in Fig. 11.49 as a cosine carrier multiplied by an exponential envelope. Use the frequency convolution theorem to evaluate its Fourier transform.Fig. 11.49
Imagine two uniformly illuminated small circular holes in an opaque screen, as shown in Fig. P.11.48. Construct its autocorrelation. Discuss the irradiance distribution for each resulting individual
Figure P.11.47 shows a function Æ’(x) consisting of a periodic array of equally spaced delta functions. Construct its autocorrelation and discuss whether or not it is periodic.Figure
Given that F{ƒ(x)} = F(κ) and F{h(x)} = H(κ), if α and b are constants, determine F{αƒ(x) + bh(x)}.
Show that if Æ’(x) is real and even, its transform is real and even. Start with Eq. (11.5), use the Euler formula from Section 2.5, and assume that Æ’(x) has both a real and an
With the previous problem in mind show that the inverse transform of
Consider the functionand first check that the exponents are unitless. Then show that the Fourier transform of E(t) isYou might want to use the integral identity ,-ίωρ! E(t) = Ege-i@ote-7/27²
Determine the Fourier transform of the function ƒ(x) = A cos κ0x.
Show that F{1} = 2πδ(κ).
Determine the Fourier transform ofMake a sketch of F(Ï‰), then sketch its limiting form as T†’ ±ˆž. Scos w,t |1| T
Determine the Fourier transform ofMake a sketch of it. |x| < L |x| > L S sin-kpx f(x) =
Determine the Fourier transform of the functionMake a sketch of F{E(x)}. Discuss its relationship to Fig. 11.11.Fig. 11.11 |x| < L |x|> L SEo sinkp.x E(x) = (a) f(x) (a) F(k) = A(k) %3D 5т т п х
A long narrow horizontal opaque rectangular object of width 0.70 mm is illuminated by 600-nm light. Consider a point-P, at the level of the lower edge of the object, 1.0 m from it. Determine the
A long horizontal narrow slit of width 0.70 mm is illuminated with 600-nm light. A point-P, 1.0 m away from the aperture screen, is opposite the lower edge of the screen. If 100 W/m2 arrives at
A long narrow slit 0.10 mm wide is illuminated by light of wavelength 500 nm coming from a point source 0.90 m away. Determine the irradiance at a point 2.0 m beyond the screen when the slit is
Suppose the slit in Fig. 10.76 is made very wide. What will the Fresnel diffraction pattern look like?Fig. 10.76 P2 (b) (a)
Make a rough sketch of a possible Fresnel diffraction pattern arising from each of the indicated apertures (Fig. P.10.89).Fig. P.10.89
Plane waves from a collimated He–Ne laserbeam (λ0 = 632.8 nm) impinge on a steel rod with a 2.5-mm diameter. Draw a rough graphic representation of the diffraction pattern that would be seen on a
What would you expect to see on the plane of observation if the half-plane Î£ in Fig. 10.81 were semi€‘transparent?Fig. 10.81 (5) (4) (3)
The Fresnel integrals have the asymptotic forms (corresponding to large values of w) given byUsing this fact, show that the irradiance in the shadow of a semi-infinite opaque screen decreases in
No lens can focus light down to a perfect point because there will always be some diffraction. Estimate the size of the minimum spot of light that can be expected at the focus of a lens. Discuss the
For large values of uUse that relationship to show that the angular separation (Î”Î¸) between consecutive minima far from the center of an Airy pattern is given byWrite an
Verify that the peak irradiance I1of the first €œring€ in the Airy pattern for far-field diffraction at a circular aperture is such that I1/I(0) = 0.0175. You might want to use
Imagine that you are staring at a star. You have dilated pupils, each with a diameter of 6.00 mm. The retina is about 21.0 mm from the pupil in a typical eye. Considering that the index of refraction
We wish to use the 15-cm-diameter objective from an amateur telescope to form an image on a CCD of a distant star. Assuming a mean wavelength of 540 nm and a focal length of +140 cm, determine the
A 2.4-cm-diameter positive lens with a focal length of 100 cm forms an image of a small far-away red (656 nm) hydrogen lamp. Determine the linear size of the central circular spot appearing on the
In light of the five previous questions, identify Fig. P.10.32, explaining what it is and what aperture gave rise to it.Fig. P.10.32
Figure P.10.31 is the electric-field distribution in the far field for a hole of some sort in an opaque screen. Describe the aperture that would give rise to such a pattern and give your reasoning in

Showing 100 - 200 of 821