Standard deviation when estimating a Poisson regression using R

I'm interested in plotting the estimator of the standard deviation in a Poisson regression. The variance is $Var(y)=\phi⋅V(\mu)$ where $\phi=1$ and $V(\mu)=\mu$. So the variance should be $Var(y)=V(\mu)=\mu$. (I'm just interested in how the variance should be, so if overdispersion occurs $(\hat{\phi} \ne 1)$, I don't care about it). Thus an estimator of the variance should be $Var(\widehat{y})=V(\widehat{μ})=\widehat{\mu}$ and an estimator of the standard deviation should be $\sqrt{Var(\widehat{y})}=\sqrt{V(\widehat{\mu})}=\sqrt{\widehat{\mu}}=\sqrt{exp(x\widehat{\beta})}=exp(x\widehat{\beta}/2)$ when using the canonical link. Is this correct? I haven't found a discussion about standard deviation in the context with poisson regression yet, that's why I'm asking.

So here is an easy example (which makes no sense btw) of what I'm talking about.

>data1<-function(x){x^(2)}

>numberofdrugs<-data(1:84)

>data2<-function(x){x}

>healthvalue<-data2(1:84)
>
>plot(healthvalue,numberofdrugs)
>
>test<-glm(numberofdrugs~healthvalue, family=poisson)

>summary(test) #beta0=5.5 beta1=0.042
>
>mu<-function(x){exp(5.5+0.042*x)}
>
>plot(healthvalue,numberofdrugs)