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Lab 10: Spatial Profile of a Laser Beam

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Lab 10: Spatial Profile of a Laser Beam Lab 10: Spatial Pro¯le of a Laser Beam 2 Background We will deal with a special class of Gaussian beams for which the transverse electric ¯eld can be written as, r2 1 Introduction ¡ 2 E(r) = Eoe w (1) Equation 1 is expressed in plane polar coordinates Refer to Appendix D for photos of the appara- where r is the radial distance from the center of the 2 2 2 tus beam (recall r = x +y ) and w is a parameter which is called the spot size of the Gaussian beam (it is also A laser beam can be characterized by measuring its referred to as the Gaussian beam radius). It is as- spatial intensity pro¯le at points perpendicular to sumed that the transverse direction is the x-y plane, its direction of propagation. The spatial intensity pro- perpendicular to the direction of propagation (z) of ¯le is the variation of intensity as a function of dis- the beam. tance from the center of the beam, in a plane per- pendicular to its direction of propagation. It is often Since we typically measure the intensity of a beam convenient to think of a light wave as being an in¯nite rather than the electric ¯eld, it is more useful to recast plane electromagnetic wave. Such a wave propagating equation 1. The intensity I of the beam is related to along (say) the z-axis will have its electric ¯eld uni- E by the following equation, formly distributed in the x-y plane. This implies that " cE2 I = o (2) the spatial intensity pro¯le of such a light source will 2 be uniform as well. However, in reality, a light beam 2 2 does not extend to in¯nity. Therefore, this picture Here, "o = 8:85x10¡12C =Nm is the permittivity of of a light beam as an in¯nite plane wave is just an free space and c = 3:00x108m=s is the speed of light. idealization. Substituting equation 1 into equation 2 we get, 2r2 ¡ 2 In this experiment, we will investigate the spatial pro- I(r) = Ioe w (3) ¯le of a Gaussian laser beam. The shape of a 2 Here, I = "ocEo is the (maximum) intensity on the laser cavity is responsible for the intensity pro¯le. A o 2 z-axis (r = 0). Equation 3 is similar to equation 1; it Gaussian pro¯le is characteristic of most lasers. We represents a Gaussian function centered at r = 0. will discuss some properties of such Gaussian beams. In particular, we will concentrate on the fact that a Figure 1 is a plot of the Gaussian beam intensity pro- Gaussian beam has a non-uniform intensity pro¯le. ¯le given by equation 3. Notice that the Gaussian is Furthermore, by studying the pro¯le of a laser beam, a radially symmetric distribution. we are able to extract a characteristic radius of the Gaussian beam, known as the spot size, at a partic- If we set r = w in equation 3, we obtain I = I =e2. ular location along the direction of propagation of the o This means that at a distance w from the z axis, the beam. intensity of light is reduced to 1=e2 of its maximum value. If we put a small screen perpendicular to the The experiment consists of two parts. In the ¯rst part, z-axis at that position, we would observe a spot of you will verify that the laser beam has a Gaussian radius » w. intensity pro¯le. You will use this pro¯le to extract a value for the spot size. In the second part of the The spot size is a function of position along the di- experiment, you will study an alternative technique rection of propagation z of the Gaussian beam. It is for measuring spot size. This technique is suitable for given by, laser beams that have large diameters. s ¸z 2 w(z) = wo 1 + ( 2 ) (4) EXERCISES 3-4 PERTAIN TO THE BACK- ¼wo GROUND CONCEPTS AND EXERCISES 1- where ¸ is the wavelength of the light. Notice that 2 AND 5 PERTAIN TO THE EXPERIMEN- wo is the minimum spot size which occurs at the par- TAL SECTIONS. ticular position z = 0. The size of the beam at the 10.1 Figure 2: Propagation of a fundamental Gaussian beam. Il- lustrates the intensity pro¯le, variation of the spot size with longitudinal position and changes in the curvature of the beam. Figure 1: Gaussian intensity pro¯le of a laser ditions a process known as stimulated emission can beam/determination of the spot size. be induced inside the cavity. This causes further am- pli¯cation of light. The ampli¯ed electric ¯eld has to satisfy the boundary conditions imposed by the z = 0 position is called the beam waist. In practice, cavity walls. This determines the number of trans- z = zo will often correspond to the position of a focal verse (spatial) modes that the cavity can support. A spot of a lens introduced in the laser beam. good Laser cavity allows only the fundamental (or TEM00) mode to be ampli¯ed. This mode Figure 2 illustrates the variation of w as the beam has a Gaussian spatial pro¯le. The laser that propagates along the z-axis. It also shows that the is being used in this lab produces a number radius of curvature of the wavefronts varies along the of spatial modes. Equation 3 is therefore only axis of propagation. an approximation of the spatial pro¯le of the laser. Keep in mind that although the spot size and the radius of curvature vary along the z-axis, the inten- Finally, notice that Gaussian beams exhibit di®rac- sity pro¯le remains Gaussian at every position along tion (even without a lens!). As you can see in Figure the direction of propagation. The underlying reason 2, the beam is diverging, as if it is di®racting from is that Gaussian beams have a unique property that the minimum spot size wo. In fact, this imposes a they maintain their Gaussian form along their prop- limitation on our ability to focus a laser beam. It is agation direction. This is an important characteris- not possible to focus laser light to a single point be- tic of Gaussian beams, in contrast to most ordinary cause it is di®raction-limited by the very fact that it sources of light which cannot maintain a speci¯c pro- is Gaussian in nature. ¯le along the direction of propagation. For instance, if you shine a flashlight through a circular aperture, its spatial pro¯le is uniform close to the aperture. At distances far away from the aperture, the beam pro- 3 Suggested Reading ¯le changes because of di®raction. Another property of Gaussian beams is that they remain Gaussian as they propagate through any combination of optical Refer to the chapters 14-17, elements such as lenses and mirrors. A.E. Siegman, Lasers (University Science Books, The unique properties of Gaussian beams that we have 1986) been discussing so far provide the motivation to inves- tigate them. A light source which naturally produces Gaussian beams is a Laser. It is instructive to briefly 4 Apparatus mention how the laser operates, to gain insight as to how it produces Gaussian beams. A Laser consists ² Laser: N-type semiconductor diode laser (635nm of a \Lasing medium" which is located between two wavelength) spherical mirrors, one of which can partially transmit the light produced in the cavity. Under special con- ² Translation stage and pinhole 10.2 ² Photodiode ² Box for covering the photodiode ² Digital multimeter ² Two mirrors and mirror holders ² Three convex lenses with focal lengths » 8 cm, » 15 cm and » 30 cm ² Three lens holders ² Large iris Figure 3: Schematic diagram of the setup for experiment 1. ² Cylindrical rod machined with variable diameters MAKE SURE ALL MOUNTS ARE SE- ² 90 degrees clamp CURELY FASTENED ON THE OPTICAL TABLE. ² Five 3" postholders and six 2" postholders ² Ten posts and eleven bases ² Screws, bolts, 3/16" Allen key, cardboards, trans- 5 Experiment I: Gaussian pro¯le of parent lens tissue a laser beam - measurement of the ² Optical table spot size WARNING!!: KEEP TRACK OF YOUR The purpose of this part of the experiment is to ver- LASER BEAM AT ALL TIMES. NEVER ify that the transverse beam intensity for the laser POINT THE BEAM AT PEOPLE, OR LOOK has a Gaussian pro¯le. Once you measure the inten- IN THE APERTURE OF THE LASER OR BE sity pro¯le at a particular position, you will be able AT EYELEVEL WITH THE BEAM. to determine the spot size of the laser beam at that position. KEEP EYES AWAY FROM DIRECT OR REFLECTED LASER BEAMS. OTHERWISE The experimental setup is shown in Figure 3. The SERIOUS EYE DAMAGE WILL OCCUR. laser beam is reflected from two mirrors so that you can conveniently manipulate the beam by adjusting YOU SHOULD BE AWARE OF WHERE the mirrors. The beam then passes through a trans- THE LASER BEAMS STRIKE OPTICAL lation stage. In order to measure the pro¯le of the COMPONENTS. REFLECTIONS FROM beam, we need to measure the intensity at various OPTICAL COMPONENTS SHOULD BE points on the beam's cross-section. This can be done BLOCKED BY USING PIECES OF CARD- by scanning a pinhole attached to a translation stage BOARD THAT ARE PROVIDED. BE PAR- across the beam's cross-sectional area, and measur- TICULARLY CAREFUL WHEN YOU IN- ing the light transmitted through the pinhole. Ide- SERT OR REMOVE LENSES INTO A ally, one must measure the intensity at every point in LASER BEAM.
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