Why do we not commonly see interference effects with visible light? With light from a source such as the Sun, an incandescent bulb, or a fluorescent bulb, we do not see regions of constructive and destructive interference; rather, the intensity at any point is the sum of the intensities due to the individual waves. Light from anyone of these sources is, at the atomic level, by electronic transitions from one energy level to another which can not be externally controlled. Hence two independent sources identical in all respects can not be coherent.
`I=I_1+I_2+2sqrt(I_1I_2) cosphi.......(1)`
Waves from independent sources are incoherent; they do not maintain a fixed phase relationship with each other(i.e. `phi` varies with time). We cannot accurately predict the phase (for instance, whether the wave is at a maximum or at a zero) at one point given the phase at another point. Incoherent waves have rapidly fluctuating phase relationships. It means average of third term of equation (i) is zero. Therefore, the result is an averaging out of interference effects, so that the total intensity (or power per unit area) is just the sum of the intensities of the individual waves. Only the superposition of coherent waves produces sustained interference. Coherent waves must be locked in with a fixed phase relationship. Coherent and incoherent waves are idealized extremes; all real waves fall somewhere between the extremes. The light emitted by a laser can be highly coherent two points in the beam can be coherent even if separated by as much as several kilometers.
`text(Important points about Coherent source :)`
The sources which produce sustained, i.e. observable interference are called coherent sources. In case of interference as `I=I_1+I_2+2sqrt(I_1I_2) cosphi`, interference will be sustained if the phase difference `phi` at a given point does not vary with time. If the interfering wave are :
`y_1=A_1 sin(omega_1t - k_1x_1 +phi_1)` and `y_2=A_2sin(omega_2t-k_2x_2+phi_2)`
`phi=(omega_1-omega_2)t+(k_2x_2-k_1x_1)+(phi_1+phi_2)`
So `phi` will not vary with time if:
(a) `(phi_1-phi_2)=phi_0` is constant, i.e., the initial phase difference between the wave does not val)' with time and
(b) `(omega_1-omega_2)t=0`, i.e., `f_1=f_2`. But for a wave as `v= flamda`, `f_1=f_2` will also means `lamda_1=lamda_2` i.e., `k_1=k_2` [as `k=(2pi)/lamda`]. i.e., the two wave are of same frequency and wavelength. So two sources will be coherent if and only if they produce wave of same frequency (and hence wavelength) and have a constant initial phase difference. So in case of two coherent sources.
`phi=(2pi)/lamda Deltax + phi_0` with `phi_0=(phi_1-phi_2)`
Now as in general emission of light from atoms is random, rapid and independent of each other, `phi_0` cannot remain constant with time and hence two independent light source identical in all respects cannot be coherent.
Why do we not commonly see interference effects with visible light? With light from a source such as the Sun, an incandescent bulb, or a fluorescent bulb, we do not see regions of constructive and destructive interference; rather, the intensity at any point is the sum of the intensities due to the individual waves. Light from anyone of these sources is, at the atomic level, by electronic transitions from one energy level to another which can not be externally controlled. Hence two independent sources identical in all respects can not be coherent.
`I=I_1+I_2+2sqrt(I_1I_2) cosphi.......(1)`
Waves from independent sources are incoherent; they do not maintain a fixed phase relationship with each other(i.e. `phi` varies with time). We cannot accurately predict the phase (for instance, whether the wave is at a maximum or at a zero) at one point given the phase at another point. Incoherent waves have rapidly fluctuating phase relationships. It means average of third term of equation (i) is zero. Therefore, the result is an averaging out of interference effects, so that the total intensity (or power per unit area) is just the sum of the intensities of the individual waves. Only the superposition of coherent waves produces sustained interference. Coherent waves must be locked in with a fixed phase relationship. Coherent and incoherent waves are idealized extremes; all real waves fall somewhere between the extremes. The light emitted by a laser can be highly coherent two points in the beam can be coherent even if separated by as much as several kilometers.
`text(Important points about Coherent source :)`
The sources which produce sustained, i.e. observable interference are called coherent sources. In case of interference as `I=I_1+I_2+2sqrt(I_1I_2) cosphi`, interference will be sustained if the phase difference `phi` at a given point does not vary with time. If the interfering wave are :
`y_1=A_1 sin(omega_1t - k_1x_1 +phi_1)` and `y_2=A_2sin(omega_2t-k_2x_2+phi_2)`
`phi=(omega_1-omega_2)t+(k_2x_2-k_1x_1)+(phi_1+phi_2)`
So `phi` will not vary with time if:
(a) `(phi_1-phi_2)=phi_0` is constant, i.e., the initial phase difference between the wave does not val)' with time and
(b) `(omega_1-omega_2)t=0`, i.e., `f_1=f_2`. But for a wave as `v= flamda`, `f_1=f_2` will also means `lamda_1=lamda_2` i.e., `k_1=k_2` [as `k=(2pi)/lamda`]. i.e., the two wave are of same frequency and wavelength. So two sources will be coherent if and only if they produce wave of same frequency (and hence wavelength) and have a constant initial phase difference. So in case of two coherent sources.
`phi=(2pi)/lamda Deltax + phi_0` with `phi_0=(phi_1-phi_2)`
Now as in general emission of light from atoms is random, rapid and independent of each other, `phi_0` cannot remain constant with time and hence two independent light source identical in all respects cannot be coherent.