Intensity of Near-Infrared Spectroscopy Laser

[creative_wind]

Homework Statement

Light in the near-infrared (close to visible red) can penetrate surprisingly far through human tissue, a fact that is being used to "illuminate" the interior of the brain in a noninvasve technique known as near-infrared spectroscopy (NIRS). In this procedure an optical fiber carrying a beam of infrared laser light with a power of 1.5 mW and a cross-sectional diameter of 1.4 mm is placed against the skull. Some of the light enters the brain, where it scatters from hemoglobin in the blood. The scattered light is picked up by a detector and analyzed by a computer.

(a) According to the Beer-Lambert law, the intensity of light, I, decreases with penedtration distance, d, as I=I₀e^-µd, where I₀ is the initial intensity of the beam and µ = 4.7 cm-1 for a typical case. Find the intensity of the laser beam after it penetrates through 3.5 cm of tissue.

Answer should be in mW/m²

(b) Find the electric field of the initial light beam.

Answer should be in kV/m

Homework Equations

I₀=P/A
I=I₀e^-µd
I=c$$\epsilon$$₀E² (I think this is the equation I need to answer b)
$$\epsilon$$₀=8.85*10^-12

The Attempt at a Solution

a)
Step 1) Solve for I₀ (I'm assuming the cross-sectional area refers to a circle)
I₀=P/A = 1.5mW/(pi*(1.4*10^-4m/2)² = 9.744*10⁵mW/m²

Step 2) Solve for I
I=I₀e^-µd = (9.744*10⁵mW/m²)*e^{(-0.047m-1*0.035m)} = 9.728*10⁵mW/m²

This answer gets me "Your answer differs from the correct answer by orders of magnitude." Since the e term is about 0.998, my I₀ must be incorrect. The math is correct (as far as me quadruple-checking can affirm ) so is there a different equation I should be using to determine I₀?

 

[borgwal]

Homework Statement

Light in the near-infrared (close to visible red) can penetrate surprisingly far through human tissue, a fact that is being used to "illuminate" the interior of the brain in a noninvasve technique known as near-infrared spectroscopy (NIRS). In this procedure an optical fiber carrying a beam of infrared laser light with a power of 1.5 mW and a cross-sectional diameter of 1.4 mm is placed against the skull. Some of the light enters the brain, where it scatters from hemoglobin in the blood. The scattered light is picked up by a detector and analyzed by a computer.

(a) According to the Beer-Lambert law, the intensity of light, I, decreases with penedtration distance, d, as I=I₀e^-µd, where I₀ is the initial intensity of the beam and µ = 4.7 cm-1 for a typical case. Find the intensity of the laser beam after it penetrates through 3.5 cm of tissue.

Answer should be in mW/m²

(b) Find the electric field of the initial light beam.

Answer should be in kV/m

Homework Equations

I₀=P/A
I=I₀e^-µd
I=c$$\epsilon$$₀E² (I think this is the equation I need to answer b)
$$\epsilon$$₀=8.85*10^-12

The Attempt at a Solution

a)
Step 1) Solve for I₀ (I'm assuming the cross-sectional area refers to a circle)
I₀=P/A = 1.5mW/(pi*(1.4*10^-4m/2)² = 9.744*10⁵mW/m²

Step 2) Solve for I
I=I₀e^-µd = (9.744*10⁵mW/m²)*e^{(-0.047m-1*0.035m)} = 9.728*10⁵mW/m²

This answer gets me "Your answer differs from the correct answer by orders of magnitude." Since the e term is about 0.998, my I₀ must be incorrect. The math is correct (as far as me quadruple-checking can affirm ) so is there a different equation I should be using to determine I₀?

You made two mistakes, one small, one big:

First, you have been given the diameter of a circle, not the radius. That's a factor of 4 you miss.

Second, 1cm^-1 is certainly not equal to 0.01m^-1: this is where you're orders of magnitude off.

 

[creative_wind]

The 1.4e-4 was a typing error on my part, it should have been -3, which still yields the initial value for I that I wrote. I did completely err on the m^-1 though. Thank you so much for that!

As for part b, I solved for E using the correct I value of 9.744e11 W/m^2.
E=(I/c*constant)^1/2
E=(9.744e11/3e8*8.85e-12)^1/2=1.92e7 V/m=1.92e4 kV/m which is off by orders of magnitude again. BAH! Your help is greatly appreciated.

 

[Redbelly98]

Where are you getting 9.744e11 from? Your intensity is orders of magnitude off, which is the problem here.

 

[creative_wind]

It's from my calculation in the first post for I₀. It's in megawatts, I converted it to watts for the second step so I wouldn't have unit problems.

(9.744*10⁵mW/m²)*(1*10⁶/1mW)=9.744*10¹¹.

I used this number to calculate the answer for a) which was correct. Please tell me how it's orders of magnitude off for part b).

 

[Redbelly98]

"mW" is milliwatts. Megawatts would be "MW".

 

[creative_wind]

Forgive my lack of shift-usage. Can you help me answer the problem at all?

 

[Redbelly98]

It's supposed to be milliwatts (mW), as written originally. Nobody is going to shoot a megawatt laser through somebody's head in a noninvasive technique, since a megawatt would be quite invasive, to put it mildly.

Just do the conversion from mW to W.