(II) Pulsed lasers used for science and medicine produce very ...

Question

(II) Pulsed lasers used for science and medicine produce very brief bursts of electromagnetic energy. If the laser light wavelength is 1062 mm (Neodymium–YAG laser), and the pulse lasts for 32 picoseconds, how many wavelengths are found within the laser pulse? How brief would the pulse need to be to fit only one wavelength?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem, light pulses with a wavelength of 1550 nanometers, which is standard for telecommunications transmit data in a vacuum. If a pulse lasts for 10 PICO secons determine how many wavelengths are contained within this pulse and the duration of a pulse containing exactly one wavelength. So it appears for this particular problem we're asked to solve for two separate answers. Our first answer is we're trying to figure out how many wavelengths are contained within this 10 PICO PICO pulse. And our second answer, we're trying to figure out what the duration of a pulse containing exactly one wavelength is. So now that we know that we're solving for two separate answers, let's read off our multiple choice answers to see what our final answer pair might be noting that our first answer is for the number of wavelengths contained in a 10 PCO pulse. And our second answer is for the pul duration for exactly one wavelength. And let us note that for our first answer, it is unit list. So A is 1.9 multiplied by 10 to the power of three and 5.2 femtosecond. And our answer for B is 1.9 multiplied by 10 to the power of three and 6.2. PICO secons C is 2.2 multiplied by 10 to the power of five and 5.2 femtosecond. And finally D is 2.2 multiplied by 10 to the power of five and 6.2 PICO secons. Awesome. So first off, we need to convert our wavelength value from nanometers to meters. So let us note that we are given our, our, our wavelength as 1550. So 1550 nanometers. So I'm gonna go ahead and give you this conversion on my own and you can use dimensional analysis to solve for this on your, on your own or you can just quickly look it up. But in an effort to save time, I'll just give you the conversion. So 1550 multiplied by 10 to the power of negative 9 m. Awesome. So now our second step is we need to convert the pul duration in two seconds. So let us note that the pul duration is given to us as 10 PICO secons. Well, we need to convert Picos to seconds. So when we do that we'll get 10, multiplied by 10 to the power of negative 12 seconds. Awesome. So now we need to recall and use the speed of light inside of a vacuum constant, which the speed of light in a vacuum is equal to 3.0 multiplied by 10 to the power of eight. And of course, its units of measurement is meters per second. Ok. So now moving right along here, our next step is we need to calculate the distance traveled by light during the pulse. So in order to do that, we need to recall and find the distance that light travels during the pulse. So as we should recall, we should recall that the distance D is equal to the speed of light multiplied by the pulse duration, which I'm gonna denote as let's call that P subscript D. Yes, P subscript D will denote the pulse duration. So now we need to substitute in our known value. So D is equal to 3.0 multiplied by 10 to the power of 8 m per second. And then our pulse duration is equal to 10, multiplied by 10 to the power of negative 12 seconds. Awesome. So when we plug that into our calculator, we will find that our distance D is equal to three multiplied by 10 to the power of negative 3 m. Fantastic. So now we need to calculate the number of wavelengths within the pulse. So in order to determine how many wavelengths will fit within the pulse duration. We need to divide the distance traveled by the wavelength. So we'll call the number of wavelengths capital N subscript lambda. So the number of wavelengths is equal to the distance D divided by lambda. The wavelength which playing in our known variables will take three multiplied by 10 to the power of negative 3 m and divide it by our wavelength which is 1550 meter. So if 1 550 multiplied by 10 to the power of negative 9 m, which will mean that the number of wavelengths is approximately equal to when we round to one decimal place or to two significant figures, 1.9 multiplied by 10 to the power of three. Awesome. That is our first answer. So let's put a box around it. So we don't forget. So now our next step is we need to calculate the pulse duration for one wavelength. So let us recall and note that the pulse duration that corresponds to one wavelength can be found using the following relationship which states that lambda divided by C is equal to the pul duration for one wave. So let's call it PD one. So this is a pul duration for one and then we'll just say one lambda. So plugging in our known variables. So we know that the wavelength is 1550 multiplied by 10 to the power of negative 9 m divided by RC value, which once again is our speed of light. So 3.0 multiplied by 10 to the power of 8 m per second. So let me plug that into our calculator. We will find that it's approximately equal to 5.2 when we round to two decimal places multiplied by 10 to the power of negative 15 seconds. But we have a bit of a problem here right now. Our final answer is in units of seconds. But as you can see all of our multiple choice answers are either in femtosecond or pepto seconds. So let's quickly take a leap of there and convert our final answer from seconds to femtosecond. And when we do that, I'm just gonna go ahead and do the conversion for you. But you will find that your final answer in femtosecond is equal to 5.2 femtosecond. And that is our final answer. Hooray. We did it. So looking at our multiple choice answers, the correct answer has to be the letter A which once again is the number of wavelengths contained in a 10 PICO pulse is 1.9 multiplied by 10 to the power three and the pul duration for exactly one wavelength is 5.2 femtosecond. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.

Key Concepts

Wavelength

Pulse Duration

Speed of Light

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