Integrating Sphere Work Principle

Integrating Sphere Work Principle

If the total flux of light enter integrating sphere is Φ, irradiate on the inner wall area S3 of the integrating sphere. The light diffuse on the surface of the inner wall several times. Examine the illumination of the inner wall at any point M. Since the light entering the integrating sphere directly illuminates at S3, one part of the light diffused by each point on the S3 will irradiate on the observation M point directly, the illuminance of all these light illuminated to point M is called the direct illumination, which is denoted by E0. In addition, there are some light diffused from S3 to each point on the inner wall of the integrating sphere by multiple diffuse reflections and then reach to point M, the sum illuminance of this part light is called multiple diffuse reflection illuminance, which is denoted by E∑. Thus, the illuminance E of the observation point M is the sum of the two parts of the illuminance: E=E0+E∑         (1-1)

Direct Illuminance E0 www.LisunGroup.cn

In the S3 range, take a small element dSA at any point A, the total luminous flux on this surface is dΦ, then the illuminance EA of the position A: EA=dΦ/dSA, the inner wall of integrating sphere can be seen as ideal Reflector, so the brightness LA at A: LA=EAρ/π(where ρ is the diffuse reflection coefficient). If take a small element dSM at observation position A , the luminous flux from the element dSA  with brightness LA to the element dSM: dΦA=LAdSAcosi1dSMcosi1’/rA2.

dE0=dΦA/dSM=LAdSAcosi1cosi1’/rA2=LAdSA/4R2 

The physical quantities in the formulas are shown in the figure. The figure also can show: i1=i1’, rA=2Rcosi1(R is the radius of the inner wall of the integrating sphere). The illuminance of the light emitted from element dSA at the observation position M: dE0=dΦA/dSM=LAdSAcosi1cosi1’/rA2=LAdSA/4R2.

 

The physical quantities in the formulas are given and sorted and the direct illuminance of the entire S3 diffuse light at M is obtained as follows: E0=ρ∫S3dΦ/4πR2=ρΦ/4πR2                    (1-2)

Φ in the formula is the total luminous flux into the integrating sphere.

Multiple diffuse reflections illuminance E∑

Firstly analyze the light that the direct light from S3 at any position N on the inner wall and then diffuse again and directly reaches the observation position M. This part is called an additional illumination E1.

Since N also gets the direct illumination E0, the luminance L0 is L0=ρE0/π. Take surface element dSN at point N, An additional illuminance dE1 formed at position M from dSN: dE1=dΦ1/dSN=L0dSNcosi2dSMcosi2’/ 4R2cosi2cosi2’ dSM =L0dSN/4R2

Diffused by the entire interior of the integrating sphere, the total primary illuminance E1 formed at position M: E1=ρE0∫SdSN/4πR2=ρE0S/4πR2=ρE0(1-f)S1/4πR

S1 is the area of the inner wall of the whole sphere; f is the opening ratio (f=S2/S1, S2 is the spherical area at the opening)

Substitute S1=4πR2into the above equation：E1=ρ(1-f)E0                     (1-3)

Follow the same method, we could derive the second illuminance E2, the third illuminance E3 and so on formed at the position M from the first illumiance inside the inner wall:

E2=ρ(1-f)E1=[ρ(1-f)]2E0

E3=ρ(1-f)E2=[ρ(1-f)]3E0  www.lisungroup.cn

In this way, multiple diffuse reflections total illuminance is: E∑=E1+E2+E3+……=E0ρ(1-f)/[1-ρ(1-f)]                    (1-4)

Thus, the total illuminance E at the position M is: E=E0+E∑=E0/[1-ρ(1-f)]                        (1-5)

Feed the formula (1-2) above, then we can obtain:

E=ρΦ/4πR2[1-ρ(1-f)]                   (1-6)

We can see from the above formula, the illuminance at any position on the inner wall is proportional to the total luminous flux entering the integrating sphere. This is the basic formula for applying the luminous flux integrating sphere.