考前复习参考0711

Chapter 1

light waves

photons

modes & states coherence

temporal coherence spatial coherence coherence area coherence volume temporal coherence length transverse coherence length thermal radiation Planck’s Law absorption

induced emission spontaneous emission. transition probability the Einstein coefficient basic elements of a laser population inversion threshold condition…

The number of modes

1. In the visible part of the spectrum (λ = 500 nm; ν = 6 x 1014 s-1), the number of modes per unit

volume within the Doppler width of a spectral line (?ν=109 s-1) is .

2. In the microwave region (λ = 1 cm; ν = 3 x 1010 s-1), the number of modes per unit volume

within the typical Doppler width ?ν=105 s-1 is .

3. In the X-ray region (λ = 1 nm; ν = 3 x 1017 s-1), the number of modes per unit volume within

the typical Doppler width ?ν=1011 s-1 of an X-ray transition is .

Light Sources

a) Examples of real radiation sources with spectral energy distributions close to the Planck

distribution are the sun, the bright tungsten wire of a light bulb, flash lamps, and high-pressure discharge lamps.

b) Spectral lamps which emit discrete spectra are examples of nonthermal radiation sources. In

these gas-discharge lamps, the light-emitting atoms or molecules may be in thermal

equilibrium with respect to their translational energy, which means that their velocity

distribution is Maxwellian. However, the population of the different excited atomic levels may not necessarily follow a Boltzmann distribution and there is generally also no thermal equilibrium between the atoms and the radiation field. The radiation may nevertheless be isotropic.

c) Lasers are examples of nonthermal and anisotropic radiation sources. The radiation field is

concentrated in a few modes, and most of the radiation energy is emitted into a small solid angle. This means that the laser represents an extreme anisotropic radiation source. Induced Emission

a) In the thermal radiation field of a 100 w light bulb, 10 cm away from the tungsten wire, the

number of photons per mode at λ = 500 nm is about 10-8. If a molecular probe is placed in this field, the induced emission is therefore completely negligible.

b) In the center spot of a high-current mercury discharge lamp with very high pressure the

number of photons per mode is about 10-2 at the center frequency of the strongest emission line at λ = 253.6 mm. This shows that even in this very bright light source the induced

emission only plays a minor role.

c) Inside the cavity of a He-Ne laser (output power 1 mw with mirror transmittance T = 1%)

which oscillates in a single mode, the number of photons in this mode is about 107. In this example the spontaneous emission into this mode is completely negligible. Note, however,

that the total spontaneous emission power at λ= 632.8 nm, which is emitted into all directions, is much larger than the induced emission. This spontaneous emission is more or less uniformly distributed among all modes. Assuming a volume of 1 cm3for the gas discharge, the number of modes within the Doppler width of the neon transition is about 108, which means that the total spontaneous rate is about 10 times the induced rate.

Temporal Coherence

What is the temporal coherence length Lc of

(1) a mercury vapor lamp emitting in the green portion of the spectrum at a wavelength of

546.1nm with an emission bandwidth of ?ν = 6 x 108 Hz, and

(2) a helium-neon laser operating at a wavelength of 632.8nm with an emission width of ?ν

= 106 Hz?

Spatial Coherence

A laser-produced plasma consisting of a 100 μm diameter ball radiates very strongly at a

wavelength of 10 nm. At a distance of 0.5 m from source, what is the spatial coherence

resulting from light emitted from opposite sides of the plasma?

思考题：

1.1* 设一光子的波长λ = 5×10-1μm，单色性?λ/λ=10-7，试求光子位置的不确定量?x。若光子的波

长变为5×10-4μm（x射线）和5×10-8μm (γ射线), 则相应的?x又是多少？

1.2* 假定工作物质的折射率η=1.73,试问ν为多大时，A21/B21=1J·S/m3，这是什么光范围？

1.3* 如果工作物质的某一跃迁波长为100nm的远紫外光，自发跃迁几率A10等于106S-1，试问：

[1] 该跃迁的受激辐射爱因斯坦系数B10是多少？[2] 为使受激跃迁几率比自发跃迁几率大三

倍，腔内的单色能量密度ρ应为多少？

1.4* 如果受激辐射爱因斯坦系数B10=1019m3s-3w-1，试计算在 [1]λ=6μm (红外光)； [2]λ=600nm

(可见光)；⑶λ=60nm(远紫外光)；⑷λ=0.60nm(X射线时)，自发辐射跃迁几率A10和自发辐射寿命。又如果光强I=10W/mm2，试求受激跃迁几率W10。

1.5* 有一支输出波长为6328A，线宽为103H z,输出功率为1mW的氦氖激光器，发散角为1mrad。

问：⑴每秒发出的光子数目是多少？⑵如果输出光束的直径是1毫米，那么对于一个黑体来说，要求它从相等的面积上以及整个相同的频率间隔内，发射出与激光器发射相同数目的光子，所需温度应多高？

Answer to the Questions of C hapter 1

1. ?λ/λ=λ/L c=6 ? 10–10

λ = 10μm);

2. ?n21/?t = P/hν= Pλ/hc =λ = 500nm);

ν = 3000MHz).

3. (a) n 2/n 1 = e –

4.8x10–4 ≈ 1; (b) n 2/n 1 = e –48 ≈ 0;

(c) T = – h ν/[k ln (n 2/n 1 )] = – hc/[k λ ln (n 2/n 1 )] = 6.26 ? 103 k 4. 三能级系统，在理想情况下有N/2的粒子产生激光(λ=694.3nm ) E max = h ν N/2 = (π/8) d 2l n hc/λ = 16.9 J

P = E max /τ = 1.69 ? 109 W = 1.69 GW

6. (a) τ 4 = 1/(A 41+ A 42 + A 43) = (1/9 ) ? 10–7 s ； (b)稳态时各能级上的粒子数保持不

变，忽略受激辐射以及E 3和 E 2能级的自发辐射对其他能级布居数的影响, 则有： n 3/τ 3 = n 4 A 43

n 3/n 4 = A 43τ 3=n 2/τ 2 = n 4 A 42 n 2/n 4 = A 42τ 2=n 2/τ 2 = n 4 A 42 n 1/n 4 = A 41τ 1= 15

8. (1) I/I 0 = e -αz = 1/e = 36.8%

(2) I/I 0 = e g z = 2, ? g = (ln 2)/z = 0.693m -1

Example

Nd 3+ ions are doped into YAG at a 1% atomic concentration, which corresponds to a density of Nd atoms of 1.38 x 1026 per cubic meter in the laser rod. If all of these atoms were instantly pumped to the upper laser level and then began to radiate, what would be the radiated energy per cubic meter and the average power per cubic meter radiated from this material at the emission wavelength of 1.06 μm? If the power radiated from one cubic centimeter could be concentrated into a spot 1mm in diameter, what would be the intensity (power/m 2)?

Each photon radiated at 1.06 μm would have an energy of

E = h ν = hc/λ = 1.87 x 10-19 J = 1.17 ev.

Thus, the total energy radiated per cubic meter would be

E R = ( 1.38 x 1026 photons/m 3 )( 1.87 x 10-19 J/photon ) = 2.58 x 107 J/m 3

P R, the total average power radiated per cubic meter, would thus equal the energy radiated per cubic meter multiplied by the transition rate of the level. If the level has radiative lifetime of 230 μs and we assume that the dominant decay occurs at 1.06 μm (which is only approximate, since there are other low-lying levels to which the upper laser level can decay), then according to A 21 = 1/τs , we have

A 21 = 1 /(2.3 x 10-4 s) = 4.35 x 10-4 s -1

P R = ( 2.58 x 107 J/m 3 ) (4.35 x 10-4 s -1 ) = 1.12 x 1011 W/m 3

This is a very high power, but we must remember that it lasts only for an average of 230 μs. The amount of power radiated from one cubic centimeter would thus be

(1.12 x 1011 W/m 3 ) (1 m 3/106 cm 3 ) (1 cm 3 ) = 1.12 x 105 W.

If this power is collected and focused on a 1-mm (diameter) spot, or on an area of π (5 x 10-4)2 m 2 = 7.85 x 10-7 m 2, then the intensity would be

I = (1.12 x 105 W) (7.85 x 10-7 m 2) = 1.43 x 1011 W/m 2.

an extremely high intensity. This indicated the kind of intensity that would be obtained in a laser beam from a crystal of this size if all of the radiated energy could be concentrated in a single direction.

n 3 > n 2

Chapter 4

widths and profiles of spectral lines FWHM (full width at half maximum of line)

natural line width

collision broadening

Doppler width

Nonradiative decay Dephasing collision Amorphous crystal broadening Isotopes shifts homogeneous line broadening inhomogeneous line broadening Lorentzian line profile Gaussian line profile.

Example 1

a)The natural linewidth of the sodium D1 line at λ = 589.1 nm (ν0 = 5 x 1014 s-1) which corresponds to a transition from 3P3/2 level ( τ = 16 ns ) to the 3S1/2 ground state is 10 MHz (?νn = 109 / (16 x 2π) = 107 s-1).

b)The natural linewidth of a molecular transition between two vibrational levels of the electronic ground state with a wavelength in the infrared region is very small because of the long spontaneous lifetimes. For a typical life time of τ = 3 ms the natural linewidth becomes ?νn = 160 Hz.

c) Even in the visible or ultraviolet range, atomic or molecular electronic transitions with very small transition probabilities exist. In a dipole approximation these are “forbidden” transitions. One example is the 2S → 1S transition for the hydrogen atom. The upper level 2S cannot decay by electric dipole transition but a two-photon transition to the 1S ground state is possible. The natural life time τ is about 1S and the natural linewidth of such a two-photon line is therefore ?νn = 0.15 s-1 !

Example 2

a)Vacuum ultraviolet: For the Lymann a line (2P → 1S transition in the H atom) in a discharge with temperature T = 1000 K, M = 1, λ = 121.6nm, ν0 = 2.47 x 1015 s-1??νD =

5.6 x 109 s-1 = 5.6 GHz, ?λD = 2.8x 10-2 nm.

b)Visible spectral region: For the sodium D line (3P → 3S transition of the Na atom) in a

sodium-vapor cell at T = 500 K, λ = 589.1 nm, ν0 = 5.1 x 1014 s-1??νD = 1.7GHz, ?λD = 10-3nm.

c)Infrared region: For a vibrational transition between two rovibronic levels (quantum

numbers J, V) (J i, V i) ? (J k, V k) of the CO2 molecule in a CO2 cell at room temperature (T = 300 K), λ = 10 μm, ν0 = 3 x 1013 s-1, M = 44 ??νD = 5.6 x 107 s-1 = 56 MHz, ?λD =

1.9 x 10-2 nm.

Chapter 2

optical resonator/cavity passive resonator active resonator

plane-parallel (Fabry-Perot) resonator concentric (or spherical) resonator confocal resonator/cavity hemiconfocal resonator/cavity hemispherical resonator/cavity longitudinal modes

transverse modes

transverse mode spatial distributions, transverse mode frequencies; fractional power loss per transit

Fresnel number (confocal parameter) N = a 2/λL Mode volume

propagation factor M 2, Rayleigh range z R

模式测量思考题：

已知一He-Ne 激光管的腔长为L ，用共焦扫描干涉仪测出该管的两个纵模（如图所示）。并测得 ?x q = 0.5ms, ?x c =0.1ms ，试求这些纵模的线宽?νc .

One of the ways of describing the regions of stability of a two-mirror cavity

? IF BOTH MIRRORS ARE CONCAVE The cavity is stable if either the center of curvature of

each mirror lies beyond the other mirror or the center of curvature of each mirror lies between the other mirror and the center of curvature of the other mirror. ? IF ONE MIRROR IS CONVEX AND ONE MIRROR IS CONCAVE The cavity is stable if the

center of curvature of the concave mirror lies between the convex mirror and the center of curvature of the convex mirror.

课后练习题：

1. 设月球与地球表面距离为d , 用波长为λ, 发散角为θ0的激光束光束照射月球表面，试估算

月球表面被照亮的面积为

? 在一个腔镜口径为a 的对称稳定腔中，使用标准的高斯光束表达式，粗略计算从腔

x c