laser tutorial 3 december 11, 2012
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solved problems in laser physicsTRANSCRIPT
Department of Physics – 3rd Class Laser Tutorial 1 December 3, 2012
Dr. Qahtan Al-zaidi – Physics Dept. – Mobile: +9647702981421 E-mail: [email protected] 1
1. In thermal equilibrium at T=300 K, the ratio of level populations N2/N1 for some
particular pair of levels is given by 1/e. Calculate the frequency for this transition. In
what region of the EM spectrum does this frequency fall? Homework
2. If levels 1 and 2 are separated by an energy E2 – E1 such that the corresponding
transition frequency falls in the middle of the visible range, calculate the ratio of the
populations of the two levels in thermal equilibrium at room temperature.
Solution:
At thermal equilibrium, the ratio N2/N1 is given as follows:
The middle of the visible range is taken at
(
( )
)
3. Consider a lower energy level situated 200 cm-1 from the ground state. There are no other energy levels nearby. Determine the fraction of the population found in this level compared to the ground state population at a temperature of 300 K.
Solution: Boltzmann's constant is equal to The conversion from cm-1 to joules is given by: , where h is Planck's constant ( ) and c is the speed of light in a vacuum ( ). Boltzmann's Law is used:
By considering the energy of the ground state to be zero and calling 0 the ground state and 1 the lower energy level:
(
) (
)
After converting cm-1 to joules:
Thus 38% of the population is in the lower energy level.
4. A helium-neon laser emitting at 633 nm makes a spot with a radius equal to 100 mm at 1/e2 at a distance of 500 m from the laser. What is the radius of the beam at the waist (considering the waist and the laser are in the same plane)?
Solution: The problem can be solved by using the formula that links the divergence of the beam and the waist size:
Department of Physics – 3rd Class Laser Tutorial 1 December 3, 2012
Dr. Qahtan Al-zaidi – Physics Dept. – Mobile: +9647702981421 E-mail: [email protected] 2
5. The amplifying medium of a helium-neon laser has an amplification spectral band
equal to at 633 nm. For simplicity, the spectral profile is assumed to be
rectangular. The linear cavity is 30 cm long. Calculate the number of longitudinal
modes that can oscillate in this cavity.
Solution:
The number of longitudinal modes is equal to the spectral band divided by the
interval between the two longitudinal modes:
Note: in general, 2 modes can successfully oscillate but, if one of the modes is
perfectly in the center of the transition, the number can rise to 3.
6. The spontaneous emission rate to the stimulated emission rate is given by:
(
)
Determine the spontaneous emission to the stimulated emission for tungsten of
temperature of 2000 K in the visible range.
Solution:
(
)
(
)
e., the spontaneous emission rate is about times the stimulated emission rate.
Department of Physics – 3rd Class Laser Tutorial 2 December 3, 2012
Dr. Qahtan Al-zaidi – Physics Dept. – Mobile: +9647702981421 E-mail: [email protected] 1
1. The brightness of the probably the brightest lamp so far available (PEK Labs type
107/109TM excited by 100 W of electrical power) is about 95 W/cm2 sr in its most
intense green line . Compare this brightness with that of a 1 – W argon
laser which can be assumed to be diffraction – limited.
Solution:
For diffraction limited beam, the divergence is given by:
[ ]
where and D are the wavelength and the diameter of the beam. The factor is a numerical coefficient of the order of unity whose value depends on the shape of the amplitude distribution. The brightness B of a given source of e.m. waves can be defined as the power P emitted per unit surface area ΔA per unit solid angle ΔΩ. Mathematically given as follows:
( )
( )
which is the maximum brightness that a beam of power P can have. Therefore, the brightness of the above 1 – W argon laser will be:
[ ]
For comparison purpose, the ratio
Or, the brightness
2. For laser light of wavelength =1.06*10-3 mm, D=3 mm, =1.1, calculate the beam divergence and compare it with convensional sources.
Solution: For diffraction limited beam, the divergence is given by:
[ ]
𝜃
R
ΔΩ
ΔA ΩTotal (sphere) = 4π
steradians
Ω Δ𝐴
𝑅 𝜋𝑟
𝑅 𝜋 𝜗𝑅
𝑅 𝜋𝜃
R
D
r
Department of Physics – 3rd Class Laser Tutorial 2 December 3, 2012
Dr. Qahtan Al-zaidi – Physics Dept. – Mobile: +9647702981421 E-mail: [email protected] 2
. Compare this value with a
normal flashlight, the divergence is about 250, a searchlight has a divergence angle of 100 , the high directionality of laser light is obvious.
3. The beam from a ruby laser is sent to the moon after passing through
a telescope of 1 – m diameter. Calculate the approximate value of the beam diameter
on the moon assuming that the beam has perfect spatial coherence. (The distance
between earth and moon is approximately 384,000 km).
Solution:
The diameter of the laser spot formed on the moon surface is given by:
D
D
Therefore:
4. The distance from Earth surface to Moon surface is: . What beam
divergence is needed for a beam to expand to 1 km diameter on the surface of the
moon?
Solution:
The beam divergence of the laser beam is given by:
5. A laser cavity consists of two mirrors with reflectivities and , while the
internal loss per pass is . Calculate the logarithmic losses per pass. If the length
of the active material is and the transition cross section is
, calculate the threshold inversion.
Solution:
𝜃
R=384,000 km
Department of Physics – 3rd Class Laser Tutorial 2 December 3, 2012
Dr. Qahtan Al-zaidi – Physics Dept. – Mobile: +9647702981421 E-mail: [email protected] 3
The single pass loss of the laser cavity is given by the following equation:
Where is called the logarithmic internal loss of the cavity. While
represent the logarithmic losses of the two cavity mirrors, given by:
Therefore, the single pass loss of the laser cavity will be:
The critical inversion or the threshold population inversion, Nc can be determined as follows:
[ ] [ ]
6. A 1 – mW He – Ne laser beam with a divergence of 0.5 mrad enters the eye. Find the irradiance on the retina if the focal length of the eye, from cornea to retina, is equal to f = 1.7 cm.
Given: P = 1 mW , = 0.5 mrad , f = focal length of eye focusing system (1.7 cm) Solution:
The solution of this problem requires three steps:
Calculate focal spot diameter. (Note: Laser propagation theory shows that, when a laser beam of divergence is focused by a lens of focal length f to a spot of diameter d, the spot diameter d is given by d = f .)
Calculate the area of the spot.
Calculate the retinal irradiance.
7. The beam of a YAG laser with a power of 50 W, 2 – mrad beam divergence and 6 – mm beam diameter is focused with a lens of focal length of 5 cm. Calculate:
1. The power density before the lens. 2. The beam diameter at the focal plan. 3. The power density at the focal plan.
Solution The power density is the laser power divided by the cross section of the beam:
Department of Physics – 3rd Class Laser Tutorial 2 December 3, 2012
Dr. Qahtan Al-zaidi – Physics Dept. – Mobile: +9647702981421 E-mail: [email protected] 4
50 W/ = 177 W/cm2 The beam diameter at the focal plane:
The cross section of the beam at the focal point:
The power density at the focal plane:
8. The diameter of a beam emitted from He – Ne laser is 1.2 mm, and its divergence angle is 1 mrad. A Kepler beam expander is used made of 2 positive lenses with focal lengths of 1 cm and 6 cm. Calculate: i. The beam diameter at the output of the beam expander. ii. The beam divergence angle.
Solution The relation between the beam diameters and the beam divergence angles is:
where f1 = Focal length [m] of the input lens – ocular, f2 = Focal length [m] of the output lens – objective, d1 = Diameter of the input beam [m] and d2 = Diameter of the output beam [m]. = Divergence angle (Rad) of the beam at the input to the beam expander while = Divergence angle (Rad) of the beam at the output to the beam expander. i. The beam diameter at the output of the beam expander is:
(
)
ii. The divergence angle at the output of the beam expander
(
) (
)
The beam expander caused a reduction of 6 times of the beam divergence (the ratio of the focal lengths of the lenses).
Department of Physics – 3rd Class Laser Tutorial 3 December 10, 2012
Dr. Qahtan Al-zaidi – Physics Dept. – Mobile: +9647702981421 E-mail: [email protected] 1
1. Calculate the number of possible longitudinal modes in a He – Ne laser having a cavity of
length of one meter. What is the separation between two modes?
The number of possible modes can be estimated using the following equation:
where L is the length of the cavity, N is number of all possible modes, n is the refractive index of
the lasing medium and λ is the wavelength. Assuming refractive index of the medium as one, the
number of modes can be calculated as:
However, all these modes will not be supported. These are limited by the fluorescence curve and
only the modes for which the gain of laser of the laser medium G(λ) > 1 would be supported.
The frequency separation, Δf, between two successive modes, N and N+1, can be estimated
using simple relation:
[
] [
]
OR, one can make use of the following relation and find out the separation in terms of
wavelength :
This gives:
2. In the above example, if the laser gain profile bandwidth is 500 MHz, how many longitudinal
modes are possible?
As mentioned above that though the number of possible modes exceed three millions but the
number is restricted by fluorescence curve and only the modes for which the gain of laser of the
laser medium G (λ) > 1 would be supported. In the present case, this bandwidth is 500 MHz, and
the frequency spacing between two modes is 150 MHz. Thus,
Maximum number of modes = (bandwidth / frequency spacing) = 500 MHz / 150 MHz = 3.33.
In a practical case only three modes will be excited.
3. A He – Ne laser with a 20 cm long cavity, lasing at 632.8 nm. What is the frequency (and
wavelength) gap between two consecutive longitudinal modes?
[
] [
]
Department of Physics – 3rd Class Laser Tutorial 3 December 10, 2012
Dr. Qahtan Al-zaidi – Physics Dept. – Mobile: +9647702981421 E-mail: [email protected] 2
4. We want to use a spectral filter with a bandwidth of 1 nm. What value should we use for the
cavity length to select one (and only one!) longitudinal mode with this filter?
We want that which can be written as:
5. A certain He-Ne laser has a Doppler broadened linewidth of 1.5 GHz and its central
operating wavelength of 632.8 nm. The radii of curvature of both mirrors is 1 m and the
length of the cavity is 25 cm [assume n =1].
a) Is this cavity stable? (Show your calculation).
b) What is the frequency difference between the longitudinal modes of the cavity?
c) How many longitudinal modes of the laser are active?
d) When this laser is mode-locked, what is the temporal separation between the output
pulses as would be seen by a detector placed in the output beam?.
e) What is the maximum resonator length you would have chosen if single-
longitudinal-mode operation was desired?
6. A filter is used to obtain approximately monochromatic light from a white source. If the pass
band of the filter is 10 nm, what is the coherence length and coherence time of the filtered
light? The mean wavelength is 600 nm.
Relevant equations:
7. A resonator is composed of two mirrors, one concave and one convex, with radii of
curvature 1.5 m and -1 m, respectively. The distance between them is L. In what range
could we choose L so that the cavity remains stable?
The stability condition for a two mirror resonator is:
which could also be written as:
Department of Physics – 3rd Class Laser Tutorial 3 December 10, 2012
Dr. Qahtan Al-zaidi – Physics Dept. – Mobile: +9647702981421 E-mail: [email protected] 3
and finally:
8. Consider a confocal resonator of length L= 1 m used for an Ar+ laser at wavelength
. Calculate: (a) the spot size at the resonator center and on the mirrors; (b) the
frequency difference between consecutive longitudinal modes; (c) the number of non-
degenerate modes falling within the Doppler-broadened width of the Ar+ line
.
a) The spot sizes for symmetric resonators: a confocal resonator, for which one has
L=R, i.e., g=0. The spot size at each mirror and at the beam waist are
given as follows:
(
)
(
)
√
where the suffix c stands for confocal. The equation shows that the spot size at the beam
waist is, in this case,√ smaller than that at the mirrors . Therefore:
(
)
(
)
√
b) The frequency difference (or separation) between consecutive longitudinal modes
can be calculated as given below:
[
] [
]
c) the number of non-degenerate modes falling within the Doppler-broadened width
of the Ar+ line
is found as follows:
[ ]
L=R
W0=Wc/√ Wc= 𝐿𝜆 𝜋
Department of Physics – 3rd Class Laser Tutorial 3 December 10, 2012
Dr. Qahtan Al-zaidi – Physics Dept. – Mobile: +9647702981421 E-mail: [email protected] 4
where [
]is the frequency spacing between two consecutive non-degenerate modes of a confocal
resonator.
9. Consider a hemiconfocal resonator (plane-spherical resonator with L= R/2 of length L= 2 m
used for a CO2 laser at a wavelength of . Calculate: (a) the location of the beam
waist; (b) the spot size on each mirror; (c) the frequency difference between two consecutive
TEM00 modes; (d) the number of TEM00 modes falling within the laser linewidth (consider a
typical low pressure CO2 laser and thus take ).
(a) A common cavity arrangement used to obtain the smallest output-beam diameter from the
laser is the semi-confocal cavity as shown in the Figure herein. It consists of one curved and
one flat mirror, with the flat mirror used as the output mirror. This is the equivalent of taking
a stable cavity having two equal radii-of-curvature mirrors and replacing one of the curved
mirrors with a flat mirror located at the halfway point between the two curved mirrors. For
this arrangement the minimum beam waist w0 occurs at the flat mirror. This, in effect, is
one-half of the confocal resonator because it folds half of the resonator back on itself by
using a flat mirror at the halfway location. This is a common resonator to obtain a parallel
beam at the output of the laser, if the output mirror is the flat mirror.
(b) (
)
[
]
(
)
[
]
(
)
[
]
(
)
[
]
And: ( ( )
)
[
]
(c) The frequency difference between two consecutive modes can be calculated as follows:
(d) The number of TEM00 modes falling within the laser linewidth is equal to:
L=R2/2
W0
Wc= 𝐿𝜆 𝜋
[ ]
𝑅 ∞
Department of Physics – 3rd Class Laser Tutorial 3 December 10, 2012
Dr. Qahtan Al-zaidi – Physics Dept. – Mobile: +9647702981421 E-mail: [email protected] 5
10. Consider a resonator consisting of two concave spherical mirrors both with radius of
curvature 4 m and separated by a distance of 1 m. Calculate the spot size of the TEM00 mode
at the resonator center and on the mirrors when the laser oscillation is at the Ar+ laser
wavelength .
Solution:
For a symmetric resonator one has R1=R2=R and g1=g2=1-(L/R)
Therefore, the equations giving the spot radius at each mirror which as giving below:
(
)
[
]
(
)
[
]
These two equations will be reduced to:
(
)
[
]
(
)
[
]
On the other hand, the spot size, , at the beam waist is giving by the equation:
(
)
[
]
This equation is also reduces to the following:
(
)
[
]
Thus, the spot size of the TEM00 mode at the resonator center will be calculated as follows:
(
)
[
]
11. How is the spot size modified at each mirror if one of the mirrors of the above problem is
replaced by a plane mirror?
12. A given He-Ne laser, oscillating in a pure Gaussian TEM00 mode at λ= 632.8 nm with an
output power of P= 5 mW is advertised as having a far-field divergence angle of 1 mrad.
Calculate the spot size, the peak intensity and the peak electric field at the waist position.
Department of Physics – 3rd Class Laser Tutorial 3 December 10, 2012
Dr. Qahtan Al-zaidi – Physics Dept. – Mobile: +9647702981421 E-mail: [email protected] 6
The peak intensity will occur at the beam waist, and will be equal to:
The intensity of an e.m. wave is seen to be related to the field amplitude E0 by:
√
13. The beam of an Ar laser, oscillating in a pure Gaussian TEM00 mode at λ= 514.5 nm with an
output power of 1W, is sent to a target at a distance of 100 m from the beam waist. If the spot
size at the beam waist is , calculate, at the target position, the spot size, the
radius of curvature of the phase front and the peak intensity.
Solution:
[ ( )
]
[ (
)
]
The peak intensity:
11. A crystal laser has the design shown below. The refractive index of the crystal is n and its
length is l. The mirror separation is d. The output spectrum of the laser is shown in the figure
below where each frequency division is 200 MHz wide.
What is the length, d, of the laser resonator?
If one of the laser mirrors has a radius of curvature R1 and the other is flat, as shown:
What is the shortest possible length, ds, between the mirrors for stable operation?
What is the largest possible length, dl, between the mirrors for stable operation?
Department of Physics – 3rd Class Laser Tutorial 3 December 10, 2012
Dr. Qahtan Al-zaidi – Physics Dept. – Mobile: +9647702981421 E-mail: [email protected] 7
Solution: from the figure above, the number of longitudinal oscillating modes is 10 within the
frequency bandwidth of:
Therefore, the frequency spacing between two consecutive longitudinal modes will be:
[
] [
]